The Degree of Fine-Tuning in our Universe -- and Others

TL;DR

This paper reviews the constraints on fundamental constants and cosmological parameters necessary for universe development and habitability, highlighting the range of viable fine-tuned conditions and exploring alternative astrophysical processes.

Contribution

It provides a comprehensive analysis of the parameter ranges allowing universe formation and habitability, including new insights into fine-tuning and alternative astrophysical processes.

Findings

Viable universes exist over a range of parameter space.

Constraints on constants ensure galaxy and star formation.

Alternative processes can support habitability in different universes.

Abstract

(abridged) Both fundamental constants that describe the laws of physics and cosmological parameters that determine the cosmic properties must fall within a range of values in order for the universe to develop astrophysical structures and ultimately support life. This paper reviews current constraints on these quantities. The standard model of particle physics contains both coupling constants and particle masses, and the allowed ranges of these parameters are discussed first. We then consider cosmological parameters, including the total energy density, the vacuum energy density, the baryon-to-photon ratio, the dark matter contribution, and the amplitude of primordial density fluctuations. These quantities are constrained by the requirements that the universe lives for a long time, emerges from the BBN epoch with an acceptable chemical composition, and can successfully produce galaxies. On smaller scales, stars and planets must be able to form and function. The stars must have sufficiently long lifetimes and hot surface temperatures. The planets must be massive enough to maintain an atmosphere, small enough to remain non-degenerate, and contain enough particles to support a complex biosphere. These requirements place constraints on the gravitational constant, the fine structure constant, and composite parameters that specify nuclear reaction rates. We consider specific instances of possible fine-tuning in stars, including the triple alpha reaction that produces carbon, as well as the effects of unstable deuterium and stable diprotons. For all of these issues, viable universes exist over a range of parameter space, which is delineated herein. Finally, for universes with significantly different parameters, new types of astrophysical processes can generate energy and support habitability.

Figures

Equations

$${\alpha }_G=\frac{G{m}_{\mathrm{p}}^2}{\hslash c}\approx 5.9\times 10^{-39}.$$

$$\sum _k{m}_{\nu k}\text{lower}2.15277pt\frac{\text{buildrel}<}{\sim }1\mathrm{eV}.$$

$${\Gamma }_q\sim C{V}_{qp}^2{G}_F^2{m}_q^5,$$

$${\mathcal{S}}_{\min }=\left\{m_{\mathrm{e}},m_{\mathrm{u}},m_{\mathrm{d}},\alpha ,{\alpha }_{\mathrm{w}},{\alpha }_{\mathrm{s}},{\alpha }_G\right\}.$$

$$\left\{\alpha (m_Z),{\alpha }_{\mathrm{w}}(m_Z),{\alpha }_{\mathrm{s}}(m_Z)\right\}\approx \left\{0.007818,0.03383,0.1186\right\}.$$

$$U_{\mathrm{s}}=-\frac{g_N^2}{4\pi r}\exp \left[-m_{\pi }r\right]\mathrm{and}{U}_{\mathrm{w}}=\frac{g_F^2}{4\pi r}\exp \left[-m_{\mathrm{w}}r\right].$$

$$\beta \equiv \frac{m_{\mathrm{e}}}{m_{\mathrm{p}}}\approx \frac{1}{1836},$$

$$m_{\mathrm{d}}<m_{\mathrm{u}}+m_{\mathrm{e}}+E_3,$$

$$m_{\mathrm{u}}<m_{\mathrm{d}}+m_{\mathrm{e}}+E_3.$$

$$(A,Z)\to (A-1,Z)+p+e^{+}+{\bar{\nu }}_e.$$

$$m_{\mathrm{d}}<m_{\mathrm{u}}+m_{\mathrm{e}}+{\Delta }_{\mathrm{em}}+B,$$

$$(A,Z)\to (A-1,Z-1)+n+e^{+}+{\nu }_e$$

$$m_{\mathrm{u}}<m_{\mathrm{d}}+m_{\mathrm{e}}-{\Delta }_{\mathrm{em}}+B.$$

$$m_{\mathrm{d}}>m_{\mathrm{u}}-m_{\mathrm{e}}+{\Delta }_{\mathrm{em}}.$$

$$m_{\mathrm{d}}>m_{\mathrm{u}}+m_{\mathrm{e}}+{\Delta }_{\mathrm{em}}.$$

$$m_{\mathrm{u}}<m_{\mathrm{d}}+m_{\mathrm{e}}-{\Delta }_{\mathrm{em}}+B_D.$$

$${\tilde{B}}_D=B_D-b\left(\frac{m_{\mathrm{u}}+m_{\mathrm{d}}}{(m_{\mathrm{u}}+m_{\mathrm{d}}{)}_0}-1\right),$$

$$\frac{m_{\mathrm{u}}+m_{\mathrm{d}}}{(m_{\mathrm{u}}+m_{\mathrm{d}}{)}_0}<1+\frac{B_D}{b},$$

$$\frac{m_{\mathrm{u}}+m_{\mathrm{d}}}{(m_{\mathrm{u}}+m_{\mathrm{d}}{)}_0}\text{lower}2.15277pt\frac{\text{buildrel}>}{\sim }0.4.$$

$$\frac{m_{\mathrm{u}}+{\Delta }_{\mathrm{em}}}{(1-f)m_{d0}}\le \frac{m_{\mathrm{d}}}{m_{d0}}\le \frac{m_{\mathrm{u}}+{\Delta }_{\mathrm{em}}+B}{(1-f)m_{d0}}.$$

$$\frac{{\Delta }_{\mathrm{em}}}{(1-f)m_{d0}}\le \frac{m_{\mathrm{d}}}{m_{d0}}\le \frac{{\Delta }_{\mathrm{em}}+B}{(1-f)m_{d0}}\Rightarrow 0.40\text{lower}2.15277pt\frac{\text{buildrel}<}{\sim }\frac{m_{\mathrm{d}}}{m_{d0}}\text{lower}2.15277pt\frac{\text{buildrel}<}{\sim }2.7.$$

$$\frac{m_{\mathrm{u}}}{(1-f)m_{d0}}\le \frac{m_{\mathrm{d}}}{m_{d0}}\le \frac{m_{\mathrm{u}}}{(1-f)m_{d0}}\Rightarrow \frac{m_{\mathrm{d}}}{m_{d0}}\approx 0.54\frac{m_{\mathrm{u}}}{m_{u0}}.$$

$${\beta }^{-3/4}{\alpha }^{3/2}\text{lower}2.15277pt\frac{\text{buildrel}<}{\sim }100\mathrm{or}\beta \text{lower}2.15277pt\frac{\text{buildrel}>}{\sim }10^{-8/3}{\alpha }^2.$$

$${\alpha }^2\beta \text{lower}2.15277pt\frac{\text{buildrel}>}{\sim }constant\approx 2.62\times 10^{-12}\approx 9.03\times 10^{-5}{\alpha }_0^2{\beta }_0,$$

$${\alpha }_G^{-1/2}{\alpha }^5{\beta }^{-1/2}\text{lower}2.15277pt\frac{\text{buildrel}>}{\sim }\frac{1}{2}{\beta }^{-3/4}{\alpha }^{3/2}\mathrm{or}\beta \text{lower}2.15277pt\frac{\text{buildrel}>}{\sim }\frac{1}{16}{\alpha }_G^2{\alpha }^{-14}.$$

$$\frac{{\alpha }_{\mathrm{s}}}{{\alpha }_{\mathrm{s}0}}\text{lower}2.15277pt\frac{\text{buildrel}>}{\sim }\frac{1}{4}{\left(\frac{\alpha }{{\alpha }_0}\right)}^{1/2},$$

$$\alpha \beta \ll {\alpha }_{\mathrm{s}},$$

$${\alpha }^2\beta \ll {\alpha }_{\mathrm{s}}^2.$$

$${\alpha }_{\mathrm{s}}\text{lower}2.15277pt\frac{\text{buildrel}<}{\sim }\frac{m_{\mathrm{p}}}{E_{56}}{\alpha }_{\mathrm{s}0}\approx 100{\alpha }_{\mathrm{s}0}.$$

$$Q(e^{-})+Q({\nu }_e)+3Q(\bar{d})=0.$$

Full text

††journal: Physics Reports

Table of Contents

1. Introduction 5

1.1. Types of Fine-Tuning Arguments

1.2. Scales of the Universe

1.3. Formulation of the Fine-Tuning Problem

1.4. Scope of this Review

2. Particle Physics Parameters 13

2.1. The Standard Model of Particle Physics

2.2. Constraints on Light Quark Masses

2.2.1. Stability of Quarks within Hadrons

2.2.2. Stability of Protons and Neutrons within Nuclei

2.2.3. Stability of Free Protons and Hydrogen

2.2.4. Unbound Deuterium and Bound Diprotons

2.2.5. Plane of Allowed Quark Masses

2.2.6. Summary of Quark Constraints

2.2.7. Mass Difference between the Neutron and Proton

2.2.8. Constraints on the Higgs Parameters

2.3. Constraints on the $\alpha$-$\beta$ Plane

2.4. Constraints on the Strong Coupling Constant

2.5. Additional Considerations

2.5.1. Charge Quantization

2.5.2. Constraint from Proton Decay

3. Cosmological Parameters and the Cosmic Inventory 33

3.1. Review of Parameters

3.2. Constraints on the Cosmic Inventory

3.3. The Flatness Problem

3.4. Quantum Fluctuations and Inflationary Dynamics

3.5. Eternal Inflation

4. The Cosmological Constant and/or Dark Energy 49

4.1. The Cosmological Constant Problem

4.2. Bounds on the Vacuum Energy Density from Structure Formation

5. Big Bang Nucleosynthesis 55

5.1. BBN Parameters and Processes

5.2. BBN Abundances with Parameter Variations

5.2.1. Variations in the Baryon to Photon Ratio

5.2.2. Variations in the Gravitational Constant

5.2.3. Variations in the Neutron Lifetime

5.2.4. Variations in the Fine Structure Constant

5.2.5. Variations in both $G$ and $\eta$

5.3. BBN without the Weak Interaction

6. Galaxy Formation and Large Scale Structure 68

6.1. Mass and Density Scales of Galaxy Formation

6.2. Structure of Dark Matter Halos

6.3. Bounds on the Amplitude of Primordial Fluctuations from Planet Scattering

6.4. Constraints from the Galactic Background Radiation

6.5. Variations in the Abundances of Dark Matter and Baryons

6.6. Gravitational Potential of Galaxies

6.7. Cooling Considerations

7. Stars and Stellar Evolution 83

7.1. Analytic Model for Stellar Structure

7.2. Minimum Stellar Temperatures

7.3. Stellar Lifetime Constraints

7.4. The Triple-Alpha Reaction for Carbon Production

7.5. Effects of Unstable Deuterium and Bound Diprotons on Stars

7.5.1. Universes with Stable Diprotons

7.5.2. Universes with Unstable Deuterium

7.6. Stellar Constraints on Nuclear Forces

7.6.1. Stellar Evolution without the Weak Interaction

7.6.2. Stellar Constraint on the Weak Interaction

7.6.3. Supernova Constraint on the Weak Interaction

7.6.3. Supernova Constraint on the Nucleon Potential

8. Planets 110

8.1. Mass Scale for Non-Degenerate Planets

8.2. Mass Scale for Atmospheric Retention

8.3. Allowed Range of Parameter Space for Planets

8.4. Planet Formation

8.5. Planets and Stellar Convection

9. Exotic Astrophysical Scenarios 118

9.1. Dark Matter Halos as Astrophysical Objects

9.1.1. Power from Dark Matter Annihilation

9.1.2. Time Evolution of Dark Matter Halos

9.2. Dark Matter Capture and Annihilation in White Dwarfs

9.3. Black Holes as Stellar Power Sources

9.4. Degenerate Dark Matter Stars

9.5. Nuclear-Free Universe

10. Conclusion 134

10.1. Summary of Fine-Tuning Constraints

10.2. General Trends

10.3. Anthropic Arguments

*10.4. Is our Universe Maximally Habitable? *

10.5. Open Issues

10.6. Insights and Perspective

Appendix A. Mass Scales in terms of Fundamental Constants 150

Appendix B. Number of Space-Time Dimensions 155

B.1. Stability of Classical Orbits

B.2. Stability of Atoms: Bound Quantum States

Appendix C. Chemistry and Biological Molecules 162

Appendix D. Global Bounds on the Structure Constants 166

Appendix E. Probability Considerations 169

Appendix F: Nuclei and the Semi-Empirical Mass Formula 173

References 178

1 Introduction

The laws of physics in our universe support the development and operations of biology — and hence observers — which in turn require the existence of a range of astrophysical structures. The cosmos synthesizes light nuclei during its early history and later produces a wide variety of stars, which forge the remaining entries of the periodic table. On larger scales, galaxies condense out of the expanding universe and provide deep gravitational potential wells that collect and organize the necessary ingredients. On smaller scales, planets form alongside their host stars and provide suitable environments for the genesis and maintenance of life. Within our universe, the laws of physics have the proper form to support all of these building blocks that are needed for observers to arise. However, a large and growing body of research has argued that relatively small changes in the laws of physics could render the universe incapable of supporting life. In other words, the universe could be fine-tuned for the development of complexity. The overarching goal of this contribution is to review the current arguments concerning the possible fine-tuning of the universe and make a quantitative assessment of its severity.

Current cosmological theories argue that our universe may be only one component of a vast collection of universes that make up a much larger region of space-time, often called the “multiverse” or the “megaverse” [119, 165, 172, 186, 196, 227, 261, 357, 445]. This ensemble is depicted schematically in Figure 1. Parallel developments in string theory and its generalizations indicate that the vacuum structure of the universe could be sampled from an enormous number of possible states [92, 264, 281, 309, 467, 501]. The potential energy function for this configuration space is depicted schematically in Figure 2, where each minimum corresponds to a different low-energy universe. If each individual universe within the multiverse (represented by a particular bubble in Figure 1) samples the underlying distribution of possible vacuum states (by choosing a particular local minimum represented in Figure 2), the laws of physics could vary from region to region within the ensemble. In this scenario, our universe represents one small subdomain of the entire space-time with one particular implementation of the possible versions of physical law. Other domains could have elementary particles with different properties and/or different cosmological parameters. A fundamental question thus arises: What versions of the laws of physics are necessary for the development of astrophysical structures, which are in turn necessary for the development of life?

1.1 Types of Fine-Tuning Arguments

Fine-tuning arguments have a long history [174, 177, 178, 222]. Although many previous treatments have concluded that the universe is fine-tuned for the development of life [49, 57, 61, 89, 122, 131, 132, 166, 179, 184, 280, 345, 446, 467, 528, 529], it should be emphasized that different authors make this claim with widely varying degrees of conviction (see also [94, 125, 165, 233, 281, 358, 364, 447]). We also note that this topic has been addressed through the lens of philosophy (see [158, 195, 217, 343, 484] and references therein), although this present discussion will focus on results from physics and astronomy. In any case, the concept of fine-tuning is not precisely defined. Here we start the discussion by making the distinction between two types of tuning issues:

The usual meaning of “fine-tuning” is that small changes in the value of a parameter can lead to significant changes in the system as a whole. For example, if the strong nuclear force were somewhat weaker, then the deuterium nucleus would no longer be bound. If the strong force were somewhat stronger, then diprotons would be bound. In both of these examples, relatively small changes (here, several percent variations in the strong force) lead to different nuclear inventories. A second type of tuning arises when a parameter of interest has a vastly different value from that expected (usually on theoretical grounds). The cosmological constant provides an example of this issue: The observed value of the cosmological constant is smaller than some expectations of its value by $\sim 120$ orders of magnitude. This second type of tuning is thus hierarchical. In the first example, the strong nuclear force can apparently vary by only $\sim 10$ percent without rendering deuterium unstable or diprotons stable. Nuclear structure thus represents a possible instance of Sensitive Fine-Tuning. In the second example, the value of the cosmological constant could be a million times smaller or larger (if the fluctuation amplitude $Q$ is also allowed to vary) and nothing catastrophic would happen, but the values would still be much smaller than the Planck scale (by $\sim 100$ orders of magnitude or more). The cosmological constant is thus an example of Hierarchical Fine-Tuning.

In addition, when an unexpected hierarchy arises due to some quantity being much smaller than its natural scale, one way to get such an ordering is for two large numbers to almost-but-not-quite cancel. This near cancellation of large quantities can be extremely sensitive to their exact values and could thus require some type of tuning. This state of affairs arises, for example, in the cosmological constant problem [110, 546] (see Section 4). This general concept is known as Naturalness. Although many definitions exist in the literature, the basic idea is that a quantity in particle physics is considered unnatural if the quantum corrections are larger than the observed value (for recent discussions of this issue, see [181, 551] and references therein; for a more critical point of view, see [283]). In such a situation, the quantum corrections must (mostly) cancel in order to allow for the observed small value to emerge. This cancellation is not automatic, so that it requires some measure of fine-tuning. One way to codify this concept, due to ’t Hooft, is to state a Principle of Naturalness: A physical quantity should be small if and only if the underlying theory becomes more symmetric in the limit where that quantity approaches zero [515].

1.2 Scales of the Universe

The physical constituents of our universe display a hierarchy of scales that allows it to function [122, 364, 443]. Before considering the details of fine-tuning, it is useful to assess the scope of our particular universe. Figure 3 depicts the range of length scales and mass scales that allow our universe to operate. The masses and lengths are given in units of the proton mass and the proton size, respectively. The triangular symbol at the origin $(0,0)$ thus marks the location of the proton. At the other end of the diagram, the mass and size of the observable universe is marked by the triangle at $(80,41)$. Objects that are smaller than their event horizons $(r<2GM/c^{2}$) fall below the red line, and lie in the black hole regime. Objects that are smaller than their Compton wavelengths ($r<h/cm_{\rm p}$) fall below the blue line and lie in the quantum regime. These two regions meet at the location of a Planck mass black hole, marked by the lower triangle at $(19,-19)$. Contours of constant density are shown by the dashed lines in the figure. A number of macroscopic bodies lie near the line of atomic density (middle dashed curve), which extends from the Hydrogen atom on the left to the black hole boundary on the right. In between, the green line segment shows the regime of known life forms, ranging from bacteria to whales. Planets are depicted by the square symbols and stars are depicted by the circles. The range of known black holes in shown by the heavy black line segment. Note that this segment is much shorter than the total possible range of black holes, which could span the entire red line. Finally, the region sampled by galactic structures is shown as the shaded region in the upper right portion of the diagram.

Figure 3 illustrates both the challenges and limitations posed by the scales of the universe. The full mass range spans approximately 80 decades. The range in radial scale, while large, is more constrained. The lower dashed curve shows the contour of nuclear density. At large mass scales, where gravity can crush material to higher density, objects become black holes. For lower masses, the nuclear forces dominate, so that our universe does not generally produce entities with sizes below the line of nuclear density. The upper dashed curve corresponds to the density of the universe as a whole. Objects above this curve would have densities lower than that of background space, and would be subject to tidal destruction. As result, our universe does not generally produce entities that fall above this line. The range of possible sizes for a given mass thus spans ‘only’ about 15 decades (with a smaller range at high masses because of the black hole limit). The universe, with its myriad structure, supports a parameter space that is about $80\times 15$ decades in extent. This large range of length and mass scales is enabled by the large hierarchy between the strength of gravity and the electromagnetic force. As emphasized previously [122, 364], if gravity were stronger, this range of scales would be correspondingly smaller: The red line would move upward in Figure 3 and the real estate available for astrophysical structures would shrink accordingly.

Another feature of the universe illustrated by Figure 3 is that the regions occupied by particular types of terrestrial and astrophysical objects are relatively small. The diagram shows the locations in parameter space populated by life forms, planets, stars, black holes, and galaxies. Moreover, the regions populated by atoms are tightly clustered near the point shown for the Hydrogen atom. Similarly, nuclei are clustered near the location of the proton. All of these regions are small compared to the total available parameter space and are widely separated from each other.

1.3 Formulation of the Fine-Tuning Problem

The overarching question under review is whether the parameters of physics in our universe are fine-tuned for the development of life. This question, which can be stated simply, is fraught with complications. This section outlines the basic components of the fine-tuning problem.

The first step is to specify what parameters of physics and astrophysics are allowed to vary from universe to universe. It is well known that the Standard Model of Particle Physics has at least 26 parameters, but the theory must be extended to account for additional physics, including gravity, neutrino oscillations, dark matter, and dark energy (Section 2). Specification of such extensions requires additional parameters. On the other hand, not all of the parameters are necessarily vital to the functioning of the low-energy universe (which does not depend on the exact masses of the heavy quarks). One hope — not yet realized — is that a more fundamental theory would have fewer parameters, and that the large number of Standard Model parameters could be derived or calculated from the smaller set. As a result, the number of parameters could be larger or smaller than the well-known 26. In addition to the parameters of particle physics, the Standard Cosmological Model has its own set of quantities that are necessary to specify the properties of the universe (Section 3). These parameters include the baryon-to-photon ratio $\eta$, the analogous ratio for dark matter $\delta$, the amplitude $Q$ of primordial density fluctuations, the energy density $\rho_{\scriptscriptstyle\Lambda}$ of background space, and so on. In principle, some or all of these quantities could be calculated from a fundamental theory, but this program cannot be carried out at present. Even if the cosmological parameters are calcuable, their values could depend on the expansion history of the particular universe in question, so that these values depend on the initial conditions (presumably set at the Planck epoch).

Once the adjustable parameters of physics and cosmology have been identified, a full description of the problem must consider their probability distributions. In the case of a single parameter, we need to know the underlying probability distribution for a universe to realize a given value of that parameter. For example, if the underlying probability distribution is a delta function, which would be centered on the value measured in our universe, then all universes must be the same in this regard. In the more general case of interest for fine-tuning arguments, the probability distributions are assumed to be sufficiently wide that large departures from our universe are possible. In particular, the range of possible parameters values (the minima and maxima of the distributions) must be specified. A full assessment of fine-tuning requires knowledge of these fundamental probability distributions, one for each parameter of interest (although they are not necessarily independent). Unfortunately, these probability distributions are not available at the present time.

The probability distributions described above are priors, i.e., theoretically predicted distributions that apply to a random point in space-time at the end of the inflationary epoch (or more generally whatever epoch of the ultra-early universe sets up its initial conditions). As emphasized by Ref. [511], one must also consider selection effects in order to test the theoretical predictions through experiment. For example, if a parameter affects the formation of planets, then the probability distribution for that parameter will be different when evaluated at a random point in space-time or at a random planet.

The crucial next step is to determine what range of the parameters allow for observers to develop. The question of what constitutes an observer represents yet another complication. For the sake of definiteness, this review considers a universe to be successful (equivalently, viable or habitable) if it can support the full range of astrophysical structures necessary for life or some type of complexity to arise. We then implicitly assume that observers will arise if the requisite structures are in place, and we won’t worry whether the resulting observers are mice or dolphins or androids. The list of required structures includes complex nuclei, planets, stars, galaxies, and the universe itself. In addition to their existence, these structures must have the right properties to support observers. Stable nuclei must populate an adequate fraction of the periodic table. Stars must be sufficiently hot and live for a long time. The galaxies must have gravitational potential wells that are deep enough to retain heavy elements produced by stars and not overly dense so that planets can remain in orbit. The universe itself must allow galaxies to form and live long enough for complexity to arise. And so on. The bulk of this review describes the constraints on the parameters of physics and astrophysics enforced by these requirements.

To summarize this discussion: In order to make a full assessment of the degree of fine-tuning of the universe, one must address the following components of the problem:

[$\mathbb{I}$] Specification of the relevant parameters of physics and astrophysics that can vary from universe to universe.

[$\mathbb{II}$] Determination of the allowed ranges of parameters that allow for the development of complexity and hence observers.

[$\mathbb{III}$] Identification of the underlying probability distributions from which the fundamental parameters are drawn, including the full possible range that the parameters can take.

[$\mathbb{IV}$] Consideration of selection effects that allow the interpretation of observed properties in the context of the a priori probability distributions.

[$\mathbb{V}$] Synthesis of the preceding ingredients to determine the overall likelihood for universes to become habitable.

This treatment focuses primarily on first two of these considerations. For both the Standard Model of Particle Physics and the current Consensus Model of Cosmology, we review the full set of parameters and identify those that have the most influence in determining the potential habitability of the universe. Most of the manuscript then reviews the constraints enforced on the allowed ranges of the relevant parameters by requiring that the universe can produce and maintain complex structures. Unfortunately, the underlying probability distributions are not known for either the fundamental parameters of physics or the cosmological parameters. As a result, these distributions and how they influence selection effects are considered only briefly. Similarly, selection effects depend on the probability distributions for the fundamental parameters and cannot be properly addressed at this time.

1.4 Scope of this Review

The consideration of possible alternate universes, here with different incarnations of the laws of physics, is by definition a counterfactual enterprise. This review considers the ranges of physical parameters that allow such a universe to be viable. Since alternate universes are not observable, this endeavor necessarily lies near the boundary of science [119, 196]. Nonetheless, this discussion is useful on several fronts: First, one can take the existence of the multiverse seriously, so that other universes are considered to actually exist, and the question of their possible habitability is relevant [445]. Moreover, if multiverse theory becomes sufficiently developed, then one could in principle predict the probability for a universe to have a particular realization of the laws of physics, and hence estimate the probability of a universe becoming habitable. Second, anthropic arguments [122, 61] are currently being used as an explanation for why the universe has its observed version of the laws of physics. In order to understand both of these issues, the first step is to determine the ranges of parameters that allow a universe to develop structure and complexity. Finally, and perhaps most importantly, studying the degree of tuning necessary for the universe to operate provides us with a greater understanding of how it works.

In this review, the term multiverse refers to the ensemble of other possible universes represented schematically in Figure 1 — other regions of space-time that are far away and largely disconnected from our own universe. For completeness, we note that the Many Worlds Interpretation of quantum mechanics [173, 207] describes physical reality as bifurcating into multiple copies of itself and this collection of possibilities is sometimes also called a multiverse [172]. Here we consider the multiverse only in the first, cosmological sense. The philosophy of quantum mechanics, and hence the second type of multiverse, is beyond the scope of this present treatment.

This review is organized as follows: We first consider the Standard Model of Particle Physics in Section 2. After discussing the full range of parameters, we focus on the subset of quantities that have the greatest influence in determining the properties of complex structures and then discuss constraints on those parameters resulting primarily from considerations of particle physics. Additional constraints resulting from astrophysical requirements are discussed in subsequent sections. The standard model of cosmology is presented in Section 3. The full range of cosmological parameters is reviewed, along with an assessment of the most important quantities for producing structure and some basic constraints on the cosmic inventory. The case of the cosmological constant (dark energy) is of particular interest and is considered separately in Section 4. The epoch of Big Bang Nucleosynthesis (BBN) is also considered separately in Section 5, which assesses how the abundances of the light elements change with varying values for the input cosmological parameters. Galaxy formation and galactic structure are considered in Section 6, which provides constraints on both fundamental and cosmological parameters due to required galactic properties. Section 7 considers the constraints due to the necessity of working stars, which are required to have stable nuclear burning configurations, sufficiently long lifetimes, and hot photospheres. This section also revisits the classic issues of the triple alpha resonance for carbon production, the effects of unstable deuterium, and the effects of stable diprotons. The required properties of planets are considered in Section 8, where the parameter constraints are found to be similar to — but less limiting than – those from stellar considerations. More exotic scenarios are introduced in Section 9, including alternate sources of energy generation such as dark matter annihilation and black hole radiation. The paper concludes in Section 10 with a summary of the fine-tuning constraints and a discussion of their implications. A series of Appendices provides more in-depth discussion, and presents some ancillary issues, including a summary of astrophysical mass scales in terms of the fundamental constants (A), the number of space-time dimensions (B), molecular bio-chemistry (C), global bounds on the structure constants (D), a brief discussion of the underlying probability distributions for the tunable parameters (E), and the range of possible nuclei (F).

A note on notation: The particle physics literature generally uses natural units where $\hbar=1$, $c=1$, and $G=M_{\rm pl}^{-2}$. Most of our discussion of particle physics topics follows this convention. On the other hand most of the astrophysical literature uses cgs units, so the discussion of stars and planets includes the relevant factors of $\hbar$ and $c$.

2 Particle Physics Parameters

A full assessment of the parameters of particle physics — along with an analysis of their degree of possible fine-tuning — is complicated by the current state of development of the field. On one hand, the Standard Model of Particle Physics provides a remarkably successful description of most experimental results to date. In addition to its myriad successes, the theory is elegant and well motivated. On the other hand, this theory is incomplete. We already know that extensions to the minimal version of the Standard Model are required to include neutrino oscillations, non-baryonic dark matter, dark energy, and quantum gravity. Additional extensions are likely to be necessary to account for cosmic inflation, or whatever alternate construction explains the relevant cosmological problems, along with baryon number violating processes that lead to the observed cosmic asymmetry. Against this background, this section reviews the parameters of particle physics that are known to be relevant, along with the sensitivity of the universe to their possible variations. Allowed ranges of parameter space are discussed for the fine structure constant, light quark masses, the electron to proton mass ratio, and the strong coupling constant. We also briefly consider constraints arising from physics beyond the Standard Model, including charge quantization and nucleon decay.

2.1 The Standard Model of Particle Physics

Specification of the the Standard Model itself requires a large number of parameters [220, 313]. In the absence of neutrino masses, the Lagrangian of the minimal Standard Model contains 19 parameters [280], and the inclusion of neutrinos raises the number to 26 [511]. Fortunately, however, only a subset of these parameters appear to require critical values for the successful functioning of the universe. Here we first review the full set of parameters (see [419] for current values) and then discuss the minimal subset necessary to consider variations in alternate universes.

In this treatment, we assume that the entire set of parameters can vary independently from universe to universe. Keep in mind, however, that if the current Standard Model of particle physics is the low-energy manifestation of a more fundamental theory, then the number of parameters could be smaller — or larger — than that considered here. Moreover, their variations could be correlated or even fully coupled. In any case, following previous treatments [280, 511], the Standard Model parameters can be organized and enumerated as follows:

$\bullet$ The masses of the six quarks and three leptons are specified by Yukawa coupling coefficients. Although the coefficients appear in the Standard Model Lagrangian, the masses appear in most phenomenological discussions of fine-tuning. In either case, this subset of parameters can be denoted as ${\cal S}_{\rm mass}$ = $\left\{u,d,s,c,t,b,e,\mu,\tau\right\}$. Here we denote the coupling coefficients as $G_{k}$, where the subscript $k=u,d,s,\dots$ refers to the type of particle. The corresponding particle masses are given by $m_{k}={\cal V}G_{k}/\sqrt{2}$, where ${\cal V}$ is the Higgs vacuum expectation value.

$\bullet$ The Higgs mechanism allows for non-zero particle masses. The Higgs parameters can be taken to be the Higgs mass and vacuum expectation value, ${\cal S}_{\rm higgs}=\left\{m_{\scriptscriptstyle H},{\cal V}\right\}$, or, equivalently, the quadratic and quartic coefficients of the Higgs potential ${\cal S}_{\rm higgs}=\left\{\mu^{2},\lambda\right\}$. The two choices of parameters are related by $m_{\scriptscriptstyle H}$ = $(-\mu^{2}/2)^{1/2}$ and ${\cal V}=(-\mu^{2}/\lambda)^{1/2}$. One should keep in mind that more complicated versions of the Higgs potential are possible [96].

$\bullet$ The quark mixing matrix (generally called the CKM matrix [108, 325]) specifies the strength of flavor-changing interactions among the quarks. The matrix is determined by three angles and one phase, and thus requires the specification of four parameters, which can be written in the form ${\cal S}_{\rm ckm}$ = $\left\{\sin\theta_{12},\sin\theta_{23},\sin\theta_{31},\delta_{q}\right\}$.

$\bullet$ The remaining parameters include a phase angle for the QCD vacuum and coupling constants for the gauge group $U(1)\times\mathit{SU}(2)\times\mathit{SU}(3)$. The latter three parameters are often specified by the strong and weak coupling constants (evaluated at a particular energy scale) and the Weinberg angle, so that the remaining subset of parameters can be written in the form ${\cal S}_{\rm g}$ = $\left\{\theta_{\rm qcd},g_{\rm s},g_{\rm w},\theta_{W}\right\}$. The latter three parameters, in conjunction with the Higgs expectation value, define more familiar entities. The mass of the $W^{\pm}$ particles are given by $m_{W}={\cal V}g_{\rm w}/2$ and the mass of the $Z$ particle is $m_{Z}={\cal V}g_{\rm w}/(2\cos\theta_{W})$. The electromagnetic coupling constant, evaluated at $m_{Z}$, is given by $e=g_{\rm w}\sin\theta_{W}$. The corresponding electromagnetic interaction strength then becomes $\alpha(m_{Z})=e^{2}/4\pi$ = $g_{\rm w}^{2}\sin^{2}\theta_{W}/4\pi\approx 1/128$; when scaled to zero energy one obtains $\alpha\approx 1/137$. The weak interaction strength can be written in the form $\alpha_{\rm w}$ = $g_{\rm w}^{2}/4\pi$. The strong interaction strength is given by $\alpha_{\rm s}=g_{\rm s}^{2}/4\pi$. Finally, we have the Fermi constant $G_{F}=1/(\sqrt{2}{\cal V}^{2})\approx(293{\rm GeV})^{-2}$.

$\bullet$ The neutrino sector includes Yukawa coupling constants to specify the mass of each neutrino, three mixing angles for the neutrino matrix, and an additional phase. Neutrino physics can thus be characterized by seven parameters, which can be written in the form ${\cal S}_{\nu}$ = $\left\{\nu_{e},\nu_{\mu},\nu_{\tau},\sin\theta_{\nu 12},\sin\theta_{\nu 23},\sin\theta_{\nu 31},\delta_{\nu}\right\}$.

$\bullet$ Although gravity is not part of the Standard Model of particle physics, a full accounting of the forces of nature requires a specification of its strength. Most approaches either use the gravitational constant $G$ or, equivalently, the gravitational fine-structure constant

[TABLE]

The incredibly small value of this dimensionless parameter is the source of many instances of Hierarchical Fine-Tuning.

In addition to the parameters of particle physics, a number of cosmological parameters are required to specify the properties of the universe (see Section 3). These properties include the inventory of baryons and dark matter in the universe [208, 307], as well as the amplitude of the primordial spectrum of density fluctuations. With a more comprehensive theory, these abundances — or perhaps their distribution of allowed values — could in principle be calculated from the parameters of particle physics. In the absence of such an overarching theory, however, current approaches consider the particle physics parameters and the cosmological parameters as separate and allow them to vary independently (e.g., see the discussions in [61, 280, 511], as well as references therein).

The successful operation of a universe does not depend on the specific values for all of the particle physics parameters found in the Standard Model. For example, the mass of the top quark plays little role in everyday life or in any astrophysical processes operating at the present epoch. As a result, when considering the possible fine-tuning of the universe, we can substantially reduce the set of 26 Standard Model parameters. Although not all existing treatments of this issue are identical (compare [109], [280], [446], [511], and others), the following reduction of parameters is representative:

The Higgs parameters and the Yukawa coupling constants determine the masses of quarks and leptons. Since only the first generation survives to form astrophysical structures (including nuclei), the reduced set of parameters must include masses for the up quark, the down quark, and the electron. All of the neutrino sector can be ignored, provided that the neutrino masses are small enough to not be cosmologically interesting [342, 431, 512]. In practice, this constraint requires

[TABLE]

The four parameters of the CKM matrix determine how rapidly the heavier quarks decay into the lighter ones. As long as the decay mechanisms operate, so that we only need to consider the first generation of particles, the particular values of the mixing matrix need not be fine-tuned. The decay width $\Gamma_{q}$ for a heavy quark of mass $m_{q}$ can be written in the general form

[TABLE]

where $C$ is a dimensionless factor and $V_{qp}$ is the matrix element corresponding to the decay $q\to p$. The CKM matrix represents the inverse of a fine-tuning problem. Unless the matrix elements were exactly zero, the heavier quarks would decay into lighter ones. We are also implicitly assuming that the masses of the heavy quarks are large compared to $m_{\rm u}$ and $m_{\rm d}$. With these reductions, the minimal set of parameters can be written in the form

[TABLE]

The value of the gravitational coupling constant is given by equation (1). The remaining coupling constants depend on energy. One common reference scale is the mass of the $Z$ particle, where current experimental measurements provide the values

[TABLE]

On the other hand, the coupling constants are sometimes given by their effective values at zero energy. In this limit, the fine structure constant approaches it usual value, $\alpha\rightsquigarrow 1/137$. For the strong and weak forces, particle interactions in the low energy limit can be described by potential energy functions of the forms

[TABLE]

In this treatment, the pion mass $m_{\pi}$ and the $W$ or $Z$ masses (represented here as a single value $m_{\rm w}$) determine the effective range of the forces. The coefficient for the weak force is related to the Fermi constant according to $g_{F}^{2}=4\pi G_{F}m_{\rm p}^{2}$ so that the weak coupling constant in this limit is given by $\alpha_{\rm w}$ = $G_{F}m_{\rm p}^{2}\approx 10^{-5}$. Similarly, the constant $g_{N}$ is the effective charge of the nucleon-nucleon interaction, and the corresponding coupling constant $\alpha_{\rm s}\approx 15$. As a result, the values for the coupling constants are sometimes quoted in the form $\{\alpha,\alpha_{\rm w},\alpha_{\rm s}\}\approx\{10^{-2},10^{-5},10\}$.

Another derived parameter that plays a role in many fine-tuning discussions is the ratio of the electron mass to the proton mass. This quantity,

[TABLE]

is a function of the more fundamental parameters given in equation (4). Note that some authors define $\beta$ as the inverse of that given in equation (7) and the ratio is sometimes denoted by the symbol $\mu$.

2.2 Constraints on Light Quark Masses

A large body of previous work has placed constraints on the allowed range of particle masses, including quark masses [54, 69, 73, 162, 184, 280, 296], the Higgs mass [185], the proton mass [415], and the Standard Model in general [261, 262]. This section reviews and reconsiders the conventional arguments for the allowed range of light quark masses. Constraints are imposed by the requirements that protons and neutrons do not decay within nuclei, and that both free protons and hydrogen atoms are stable. Previous work often invokes the additional requirements that deuterium nuclei are bound, and that diprotons must remain unstable [61, 280, 446]. However, recent studies of stellar evolution in other universes indicate that stars continue to operate with both stable diprotons [50] and unstable deuterium [15], so that the corresponding constraints on quark masses should be removed (see Section 7). With this generalized treatment, the allowed region in parameter space for the light quark masses is larger and thus exhibits less evidence for fine-tuning.

As discussed above, the Standard Model does not specify the values of the quark masses (or, equivalently, the values of the coupling constants that determine the quark masses). Moreover, the distribution of possible quark masses is also unknown. As a starting point, only the masses of the lightest two quarks (up and down) are allowed to vary in the discussion below.

Although we do not need to specify the possible distribution of quark masses to determine the range of possible values, it is useful to plot the allowed parameter space in logarithmic units. If the allowed quark masses were distributed in a log-random manner, then the allowed areas in such diagrams would reflect the probability of successful realizations of the parameters. The only direct input we have on this issue is the experimentally determined masses for the six known quarks. The distribution of these masses is shown in Figure 4, which indicates that the logarithmic quantities $\log{m_{q}}$ are relatively evenly spaced. This apparent trend holds up under more rigorous statistical tests [183]. As a result, as stated in [296], “there is reason to assume that the logarithms of the quark masses are smoothly distributed over a range of masses small compared to the Planck scale” (see their Figure 2). The measured lepton masses $(e,\mu,\tau)$ are also distributed in a manner that is consistent with log-uniform [183], but definitive conclusions are difficult with only three values. Notice also that the masses of all the quarks and leptons are small compared to the Planck scale (by 17 to 22 orders of magnitude), so that some degree of hierarchical fine-tuning is present.

2.2.1 Stability of Quarks within Hadrons

If the mass difference between up quarks and down quarks is too large, then the heavier quark can decay into the lighter one within a hadron (such as a proton or neutron). In order to prevent such decays, and allow for long-lived particles of interest, there exists an upper limit to the mass difference between the quarks, as outlined below [280, 54].

Down quarks can beta decay into up quarks inside of hadrons so that protons and neutrons could not exist. In this limit, the only stable particles would consist of only up quarks, so the universe would be composed of $\Delta^{++}$ = $(uuu)$. This condition places an upper limit on the down quark mass, which can be written in the from

[TABLE]

where $E_{3}$ is the energy required to produce an anti-symmetric state of three quarks. In our universe, we have $E_{3}\approx 300$ MeV.

In the opposite limit where the mass of the up quark is much larger than that of the down quark, the opposite decay can happen. This condition thus places an analogous upper limit on the up quark mass,

[TABLE]

2.2.2 Stability of Protons and Neutrons within Nuclei

Another constraint arises by requiring that both protons and neutrons are stable within atomic nuclei. If the mass differences between the up and down quark are too large, then beta decay can take place within nuclei.

First consider the usual beta decay of a neutron inside a nucleus:

[TABLE]

The requirement that this decay is not allowed on energetic grounds can be written in the form

[TABLE]

where $\Delta_{\rm em}$ is the contribution to the mass difference between the proton and neutron due to the electromagnetic force, and where $B$ is the binding energy of the proton within the nucleus. In our universe $\Delta_{\rm em}\approx 1.7$ MeV. The binding energy varies with the nuclear species in question, but has a typical value of $B\approx 10$ MeV.

Similarly, protons should also be stable within atomic nuclei, so that the process

[TABLE]

should be suppressed. This requirement, in turn, places an upper limit on the mass of the up quark,

[TABLE]

2.2.3 Stability of Free Protons and Hydrogen

In order for Hydrogen to exist, protons cannot spontaneously decay into neutrons via $p\to n+e^{+}+\nu_{e}$. Preventing this reaction from occurring implies the limit

[TABLE]

We get a similar but slightly stronger constraint by requiring that hydrogen atoms cannot convert themselves into neutrons through the reaction $p+e^{-}\to n+\nu_{e}$. This requirement implies the constraint

[TABLE]

2.2.4 Unbound Deuterium and Bound Diprotons

Many previous treatments consider a universe to be uninhabitable if deuterium becomes unbound or if the diproton becomes bound. Although these constraints are not necessary for a universe to be habitable, it is nonetheless instructive to consider the conditions required for unbound deuterium or bound diprotons.

The customary argument for the first case is that deuterium is a necessary stepping stone for nuclear reactions. The universe starts with only protons and neutrons, although the latter decay through the weak interaction. The reaction $p+p\to d+e^{+}+\nu$ is the first step of the reaction chain in the Sun, whereas $p+n\to d+\gamma$ is the first step in BBN. If deuterium is unstable, then – the argument goes – no complex nuclei can be made. However, recent work shows that stars can continue to make complex nuclei even if deuterium is unstable (see Section 7 and Refs. [15, 52]). Nonetheless, it is instructive to review the constraints that would be met if deuterium is required to be stable (see [67] for a more detailed treatment). The stability of deuterium to beta decay is essentially the same as equation (13) for the case where the binding energy is that of deuterium in our universe, so that $B$ = $B_{D}\approx 2.2$ MeV and the constraint becomes

[TABLE]

A weaker but more convincing constraint arises from the requirement that deuterium nuclei are stable to decay from the strong interaction where $d\to p+n$. This constraint requires that the binding energy of deuterium is positive. One model [54] writes the modified binding energy ${\widetilde{B}}_{D}$ in the form

[TABLE]

where $B_{D}=2.2$ MeV is the binding energy for deuterium in our universe. The parameter $b$ is not well-determined, but lies in the range $b=1.3-5.5$ MeV. The constraint thus has the form

[TABLE]

and requires the sum of the quark masses to be less about about twice their measured values.

Going in the other direction, if the quark masses are lighter, then the pion mass is smaller, and the strong force has a greater range. For sufficiently small quark masses, diprotons are stable, so one obtains the constraint

[TABLE]

The value appearing on the right hand side of this inequality varies with the author (compare [54] and [280]). As discussed below (Section 7), stable diprotons are not problematic for habitability, so the constraint of equation (19) is not required to be enforced (Section 7).

2.2.5 Constraints on Quark Masses

The treatment thus far allows for a three dimensional parameter space $(m_{\rm u},m_{\rm d},m_{\rm e})$. However, symmetry considerations [54] suggest that the electron mass could be a fixed fraction of the mass of the down quark, so that the ratio $f=m_{\rm e}/m_{\rm d}$ is constant under variations of the quark masses. Under this assumption we can evaluate the above constraints. Using the value $f\approx 0.107$ appropriate for our universe, the resulting parameter space is shown in Figure 5. The black dot marks the location of our universe in the diagram. (Keep in mind that other choices for $f=m_{\rm e}/m_{\rm d}$ are possible, and would lead to corresponding changes in the diagram.)

In the figure, the blue curves delimit the region for which quarks cannot decay within hadrons, where protons and neutrons are of primary interest (Section 2.2.1). The allowed region falls between the two curves. These constraints are not as confining as the others under consideration here due to the large value of the energy $E_{3}$ required to produce a bound state of three quarks. The green curves in the figure show the region for which nuclei are stable (Section 2.2.2). The allowed region again falls between the two curves. In the region above the upper curve, neutrons are unstable within nuclei, whereas in the region below the lower curve, protons are unstable. The most stringent constraints result from the requirement that protons cannot decay (Section 2.2.3). In the region below the lower red curve, free protons can decay into neutrons and positrons. In the region below the upper red curves, protons in hydrogen atoms can combine with the bound electron to form neutrons. This latter constraint is the most confining. Significantly, our universe lies close to this limit. If the down quark (and hence the neutron) were lighter by $\sim 1$ MeV, hydrogen atoms would decay via this channel.

Note that the two most important constraints are the upper limit on the down quark mass necessary to keep neutrons from decaying within nuclei (equation [11]) and the lower limit necessary to keep atomic hydrogen from combining into a neutron (equation [15]). We can thus write a combined constraint on the down quark mass

[TABLE]

In the limit of small mass for the up quark (left side of Figure 5), the allowed range for the down quark mass can be written in the form

[TABLE]

In the opposite limit of large up quark mass, we obtain

[TABLE]

These asymptotic forms show that in the limit of small $m_{\rm u}$, the mass $m_{\rm d}$ of the down quark can vary by a factor $1+B/\Delta_{\rm em}\sim 6.9$. In the limit of large $m_{\rm u}$, the allowed range of values narrows to (essentially) a line in the plane of parameters. Significantly, the up quark mass can vary (to lower values) by several orders of magnitude while the down quark mass has a range of $\sim 7$. The allowed parameter space is not overly restrictive.

Notice also that the two most restrictive bounds (from equations [20–22]) provide a bound on the composite parameter $(1-f)m_{\rm d}/m_{\rm d0}$. In this treatment, we specified the electron mass to be a fixed ratio ($f$) of the down quark mass, where $f=m_{\rm e0}/m_{\rm d0}$. For other choices of the ratio $f$, the ranges of allowed quark masses are similar, with the allowed region in the plane of Figure 5 moving up or down accordingly.

2.2.6 Summary of Quark Constraints

The allowed ranges for the light quark masses, shown in Figure 5, are significantly larger than reported in some earlier assessments. The region of allowed quark masses spans a factor of $\sim 7$ for the down quark over a range of several orders of magnitude for the up quark. One reason for this expanded range, compared with previous treatments, is that this work removes the unnecessary restrictions that deuterium must be stable and that diprotons must be unstable. Although these two constraints would reduce the allowed range of parameter space [280, 54], recent work shows that stars – and hence universes – can operate with either stable diprotons [50] or unstable deuterium [15] (see also [52]).

Although stars can operate with unstable deuterium, which requires $m_{\rm u}+m_{\rm d}$ to increase by a factor of $\sim 2$ (e.g., see Figure 11 of [200]), the sum of the quark masses cannot be made arbitrarily large. The quark masses determine the pion mass, which in turn sets the range of the strong force. If the quarks become too heavy, then the range of the strong force could become short enough to render all nuclei unstable. Although the required increase in quark masses has not been unambiguously determined, Figure 5 shows the contours where the sum of the light quark masses increases by factors of 2 and 4 (given by the lower and upper black curves in the diagram). This additional constraint cuts off only the tail of parameter space at large quark masses, and leaves most of the range viable.

The discussion thus far has considered only the two lightest quarks. For the case of three light quarks, the range of viable universes is even greater [296]. The band of congeniality found in that work is about 29 MeV wide in terms of the mass difference between the lightest two quarks (see [296] for further discussion; see also [31] for a less optimistic viewpoint). Note that an even wider range of possible universes may be viable if one considers more light quark masses, but such models have not been worked out.

Finally, notice that most of the constraints summarized in Figure 5 result from some type of beta decay, where neutrons and protons are transformed into each other. Universes can remain viable in the absence of the weak force [266], and such universes would not be subject to beta decay. As a result, for scenarios that are somewhat removed from our expectations, these constraints on the light quark masses could be significantly weaker (see also [245]).

2.2.7 Mass Difference between the Neutron and Proton

Recent work has provided an ab initio calculation of the mass difference $\Delta m$ between the neutron and proton using lattice QCD and QED calculations [83]. Historically, the calculation of $\Delta m$ has been notoriously difficult. Even this state-of-the-art treatment provides a mass splitting estimate of 1.5 MeV, which is somewhat larger than the measured value of $\Delta m$ = 1.29 MeV. In addition to the successful calculation of this quantity, these results provide estimates for the separate contributions to the mass differences from QCD effects (setting $\alpha=0$ and $m_{\rm d}-m_{\rm u}\neq 0$) and electromagnetic effects (setting $\alpha\neq 0$ and $m_{\rm d}-m_{\rm u}=0$). The result is that $(\Delta m)_{qcd}\approx-2.5(\Delta m)_{em}$.

One can use the results outlined above to determine how the mass difference between the neutron and proton depend on the mass difference between their constituent quarks (i.e., $\delta{m}$ = $m_{\rm d}-m_{\rm u}$) and the electromagnetic coupling ($\alpha$). The result is shown in Figure 6 (analogous to Figure 3 of Ref. [83]). The figure shows the contours of constant neutron-proton mass difference in the plane of parameters. For the value of $\alpha$ in our universe, the quark mass difference can only vary downward by a factor of $\sim 2$. Greater changes lead to inverse $\beta$ decay. Larger values of the quark mass differences are unconstrained in this diagram, although other considerations come into play (see Figure 5). If the quark masses are held constant, then the fine structure constant can only become larger by a factor of $\sim 2$, but has no lower limit in this diagram.

2.2.8 Constraints on the Higgs Parameters

Instead of variations in the masses of the light quarks (and/or the electron), one can also consider possible changes in the parameters of the Higgs potential, e.g., the expectation value ${\cal V}$ = $\sqrt{-\mu^{2}/\lambda}$. As the value of ${\cal V}$ increases, the mass difference between neutrons and protons increases, so that neutrons are more likely to decay within nuclei. Larger values of ${\cal V}$ also increase the pion mass, which decreases the effective range of the strong force. Both of these effects lead to nuclei that are more unstable. The maximum allowed increase in the expectation value is estimated to be {\cal V}\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}5{\cal V}_{0} [21, 22], where the subscript corresponds to the value in our universe. For larger ${\cal V}$, the mass difference $\Delta{m}=m_{\rm n}-m_{\rm p}$ is larger than the typical binding energy of a nucleon within an atomic nucleus. As a result, neutrons can decay into protons within bound nuclei, thereby leaving hydrogen as the only truly stable nucleus. Somewhat tighter bounds are derived in Ref. [162] based on considerations of nuclear stability.

Although the range of the vacuum expectation values ${\cal V}$ is not overly restrictive, the observed value ${\cal V}_{0}\approx 246$ GeV and the maximum allowed value ${\cal V}_{\rm max}\sim 1200$ GeV remain small compared to the Planck scale (at $M_{\rm pl}\sim 10^{19}$ GeV) and/or the GUT scale (at $M_{GUT}\sim 10^{16}$ GeV). A problematic issue arises: In simple grand unified models, the Standard Model parameter $\mu^{2}$, which determines ${\cal V}$, has a naturalness problem. The quantum corrections are expected to be ${\cal O}(M_{GUT})$, so that the relevant terms must cancel to high accuracy in order to produce the observed value (see the discussion of [22] and references therein). Such models are fine-tuned in the sense that small changes in the other model parameters would presumably alter this precise cancellation and lead to typical values of $\mu^{2}$ and ${\cal V}$ that are much larger than those observed in our universe.

Additional constraints on the Higgs parameters arise from stability considerations. Sufficiently large changes to the Higgs potential could result in vacuum instability [140, 153, 476], which would have important consequences for the habitability of the universe. For example, the Higgs potential generally has more than one minimum. If the Higgs field resides in a higher energy minimum (a false vacuum state), then the field can tunnel into a lower (true) vacuum state sometime in the future. In order for the universe to remain viable, however, the vacuum must be either stable or sufficiently long-lived (if the false vacuum is metastable). The quantum tunneling rate depends on the shape of the Higgs Potential, which in turn depends on the input parameters. As one example, for the case where the Higgs potential has a quartic form, the highest order term $\lambda\phi^{4}$ must be positive to ensure vacuum stability. The coefficient $\lambda$ in the classical potential depends on the Higgs mass, but quantum corrections can modify its value and even render $\lambda<0$ [28, 95, 107]. These corrections depend on the Yukawa couplings, where that of the top quark makes the largest contribution. As a result, the shape of the Higgs potential and the fate of the cosmic vacuum state depend on the Higgs mass $m_{\scriptscriptstyle H}\sim 125$ GeV and the top quark mass $m_{\rm t}\sim 173$ GeV. The resulting constraints are determined by the form of the Higgs potential, which is not fully specified (and could have alternate forms in other universes). Recent work [28, 95, 107] indicates that vacuum stability requires the ratio $m_{\scriptscriptstyle H}/m_{\rm t}$ to be sufficiently large, where the measured values in our universe are close to the limit.

2.3 Constraints on the $\alpha$-$\beta$ Plane

The Standard Model of Particle Physics describes interactions at the fundamental level of quarks and leptons. At lower energies, however, the basic properties of atoms and molecules, and hence chemistry, are determined by the values of the fine structure constant $\alpha$ and the mass ratio $\beta=m_{\rm e}/m_{\rm p}$. Since the neutron and proton have similar masses, the neutron does not introduce a third parameter in this context. In this section, we review basic constraints on the constants $\alpha$ and $\beta$, and find the allowed region in the plane of parameter space.

Many authors (e.g., [61, 507]) have argued that both $\alpha\ll 1$ and $\beta\ll 1$ in order for chemistry to operate (in a manner roughly similar to chemistry in our universe). Several arguments imply that the $\alpha$ must be small. Since the kinetic energy of electrons in atoms scales as $K\propto\alpha^{2}m_{\rm e}$, the constant $\alpha$ must be smaller than unity in order for electrons to remain non-relativistic. In addition, as discussed in Section 7 (see also [8]), the fine structure constant must be smaller than unity in order for stars to function as nuclear burning objects. If the stars are required to have sufficiently high surface temperatures, the constraint on $\alpha$ is somewhat tighter [9]. Of course, if $\alpha$ becomes too large relative to the strong nuclear force, then large nuclei would cease to exist (F). Finally, for completeness, we note that the fine structure constant $\alpha$ must be less than unity in order for bulk matter to remain stable [352, 353]. All of these considerations restrict $\alpha\ll 1$. For purposes of this discussion, we thus adopt the particular bound \alpha\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}1/3.

Small values of the mass ratio $\beta$ are required for the existence of stable ordered structures, such as a solid or a living cell [507]. For the structure to be well ordered, the fluctuation amplitude of a nucleus must be much smaller than the distance between the atoms. This constraint requires that $\beta^{1/4}\ll 1$. Following [507], we enforce the constraint $\beta^{1/4}<1/3$ so that $\beta<1/81$. For completeness, note that the localization requires a large mass ratio, but that one could in principle have the electron heavier than the proton. As a result, a second window of allowed parameter space opens up for large mass ratios \beta\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}81.

The constants $(\alpha,\beta)$ also appear in the equations of stellar structure [136, 147, 265, 323, 427] and are thus constrained by stellar considerations. Although stellar masses in our universe can vary by a factor of $\sim 1000$, if $\alpha$ is too large, or $\beta$ is too small, then the minimum mass of a star would exceed the maximum stellar mass, thereby preventing the existence of working stellar entities. The minimum and maximum stellar masses are given in A. Combining equations (231) and (232), this constraint can be written in the form

[TABLE]

Stable nuclear burning stars can fail to exist for another reason: If the fine structure constant is too small, then the electrical barrier for quantum mechanical tunneling becomes too small and stars would burn all of their nuclear fuel at once [8]. The constraint required to avoid this circumstance can be written in the form

[TABLE]

where the numerical constant can be evaluated from the equations of stellar structure (see the Appendix of Ref. [9]).

The constants $(\alpha,\beta)$ determine, in part, how the gas in a forming galaxy can dissipate energy, cool, and collapse. This requirement places a limit/estimate for the mass scale of galaxies [449, 510], as outlined in A. Since the mass of the galaxy must be larger than the minimum mass of a star, we can combine equations (231) and (239) to derive a constraint of the form

[TABLE]

Note that this bound depends on the gravitational fine structure constant $\alpha_{G}$ in addition to $(\alpha,\beta)$. For the sake of definiteness in the following analysis, we fix $\alpha_{G}$ to be its value in our universe.

Figure 7 shows the allowed parameter space in the $\alpha$-$\beta$ plane subject to the constraints outlined above. The requirement that both $\alpha\ll 1$ and $\beta\ll 1$ limit the parameter space to the lower left quadrant of the figure, as delimited by the cyan and blue lines. The requirement that the minimum stellar mass is less than the maximum stellar mass requires $\beta$ to lie above the green curve with positive slope. In order for stars to exist as long-lived, stable, nuclear-burning entities, the mass ratio $\beta$ must lie above the green curve with negative slope. For completeness, note that the minimum point of the two green curves would be slightly rounded off if one uses results from a full stellar structure calculation. The requirement that galaxies are larger than the minimum stellar mass requires $\beta$ to lie above the red curve. This latter curve is so steep that it enforces an effective lower bound on $\alpha$, although the nuclear burning constraint is more restrictive for small values of $\beta$. For completeness, the figure also shows the limit where the galactic mass scale is equal to the typical stellar mass scale (marked by the purple curve).

In Figure 7, the location of our universe is marked by the star symbol. The allowed region of parameter space surrounding that point has a nearly triangular shape, where the base (range of $\alpha$) and altitude (range of $\beta$) span about 4 orders of magnitude. Notice that Figure 7 includes a second allowed region of parameter space in the upper central part of the diagram. This regime corresponds to the case where the electron is much heavier than the proton $m_{\rm e}\gg m_{\rm p}$. Universes with parameters in this region are likely to be quite different from our own, but the constraints enforced here do not rule them out as viable.

The constraints depicted in Figure 7 are based on the existence of known structures, including galaxies, stars, and atoms. However, another type of constraint can be placed on the fine structure constant based on purely theoretical considerations. The three gauge coupling constants of the Standard Model are energy dependent. If one enforces the requirement of Grand Unification — that the three constants have the same value at some large energy scale — then the value of $\alpha$ measured at low energy is highly constrained (see the recent review of [186]). These limits also assume that proton decay occurs at the GUT scale, with a new heavy $X$-boson, but that protons are stable on stellar timescales. The constraints on the fine structure constant obtained through this argument are more more restrictive than those presented in Figure 7 and are centered around the observed value. Previous estimates for the allowed range include 120\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}\alpha^{-1}\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}170 [199] and 85\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}\alpha^{-1}\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}180 [61]. At the present time, however, no experimental evidence exists for Grand Unified theories [186] and the Standard Model in its current form does not allow for unification (one needs to invoke new physics such as supersymmetry). As a result, the status of these tighter bounds on $\alpha$ remains undetermined.

2.4 Constraints on the Strong Coupling Constant

This section considers limits on the magnitude of the strong coupling constant $\alpha_{\rm s}$. One well-known constraint arises from the requirement that nuclei are stable against fission. This constraint is generally derived by using the Semi-Empirical Mass Function as a model for atomic nuclei [540] and then requiring that the binding energy of a nucleus is larger than the binding energy of two separated nuclei with half the particles [61, 507]. This consideration results in a limit on the strong force coupling constant as a function of the fine structure (electromagnetic) constant such that

[TABLE]

where the subscripts denote the values in our universe. The numerical coefficient depends on the largest nucleus that is required to have a bound state, where equation (26) uses the value corresponding to carbon-12.

Additional constraints on the strong coupling constant $\alpha_{\rm s}$ arise from the required ordering of atomic size and energy scales. In order for bulk matter to have its observed form in our universe, the size scale of atoms, given by the extent of electronic orbits, must be larger than atomic nuclei. Electron orbits have radii $r_{e}=1/(\alpha m_{\rm e})$, whereas nuclei have radii given approximately by $r_{N}\approx 1/(\alpha_{\rm s}m_{\rm p})$. The ordering of size scales thus implies the constraint

[TABLE]

where we have used $\beta=m_{\rm e}/m_{\rm p}$. Similarly, the energy scales for chemical reactions are much lower than those of nuclear reactions. If the opposite were true, then chemical reactions, which provide the basis for life, would instigate nuclear reactions and thereby change the elements that make up life forms during the course of biological processes. The required ordering of energy scales leads to the analogous constraint

[TABLE]

If equations (27) and (28) did not hold, it is possible that a universe could remain habitable, but it would be much different than our own. However, the previous section shows that $\beta\ll 1$ in viable universes, so that these constraints are less restrictive than that of equation (26).

Going in the other direction, the strong force cannot be too much greater than that realized in our universe without changing the manner in which nuclear processes occur. The most tightly bound nucleus is iron-56, which has a binding energy per nucleon of $E_{56}\approx 8.8$ MeV. If $\alpha_{\rm s}$ is too large, however, then the binding energy per nucleon would become larger than the nucleon mass, and energy levels of the nucleus would become relativistic. Although nuclear reactions can still take place under relativistic conditions, the way in which they occur in stars (and BBN) would be vastly different than in our universe. This consideration thus places an upper limit on the strength of the strong force. Here we invoke this constraint in the conservative form

[TABLE]

The constraints on the strong coupling constant outlined above are depicted in Figure 8, which shows the allowed region in the plane of parameters $(\alpha,\alpha_{\rm s})$. The range of the fine structure constant $\alpha$ is limited by the same constraints used in the previous section. First we require $\alpha\ll 1$ (see Section 2.3), so that the allowed region falls to the left of the blue line in the diagram. On the other hand, $\alpha$ must be large enough that stellar structure solutions exist [8, 9]. Working stars thus limit the allowed region to the right of the red line. Next we require that the binding energy per particle is small enough that the constituent particles in nuclei remain non-relativistic, so that the allowed region falls below the cyan line. Finally, the strong force must effectively compete with the electromagnetic force to prevent nuclear fission (see equation [26]), as marked by the green curve. The resulting region of parameter space spans a factor of $\sim 1000$ in both $\alpha$ and $\alpha_{\rm s}$.

For completeness, we note that many authors invoke tighter limits on the strong coupling constant through considerations of nuclei with mass number $A=2$ (see [189, 61, 446, 507] and many others; see also Section 2.2.4). If the strong force were somewhat stronger, diprotons would be bound, and the cross sections for nuclear reactions in stars would enhanced by an enormous factor. In spite of many claims of disaster, this enhancement would lead to only a modest decrease in the operating temperatures of stellar cores (from $\sim 15\times 10^{6}$ K down to $\sim 10^{6}$ K) and a modest decrease in stellar lifetimes (see Section 7 and [50]). If the strong force were somewhat weaker, then deuterium would no longer have a bound state, and the usual pathways for nucleosynthesis would be altered. Nonetheless, this scenario also allows stars to provide both nuclear processing and long-lived supplies of energy (see Section 7 and [15, 52]).

Estimates for the changes to the strong coupling constant required to make diprotons bound or deuterons unstable depends on the model of the nucleus. In the square well approximation for the nuclear potential, 6% increases in $\alpha_{\rm s}$ lead to stable diprotons whereas 4% decreases lead to unbound deuterium [163]. For nuclear potentials of Yukawa form [289, 430], the required increase (decrease) in $\alpha_{\rm s}$ becomes 17% (6%). Other authors find similar requirements [15, 446]. Bound states of the $A=2$ nuclei can also be altered with corresponding changes in quark masses, which result in different ranges for the strong force. If the sum of the light quark masses $(m_{\rm u}+m_{\rm d})$ is decreased by 25%, then diprotons become bound, whereas mass increases of 40% lead to unbound deuterium [54] (see also Section 2.2).

2.5 Additional Considerations

For completeness, this section considers additional constraints on the parameters of particle physics. Specifically, the issue of charge quantization is discussed in Section 2.5.1. We then present a constraint on the energy scale of Grand Unified Theories from the requirement that nucleons have sufficiently long lifetimes (Section 2.5.2).

2.5.1 Charge Quantization

Our understanding of the laws of physics remains incomplete. One important unresolved issue that could affect the habitability of the universe is the specification of electromagnetic charges on the fundamental particles. In our universe, charge is observed to be quantized. All free particles (notably protons, neutrons, and electrons) have charges that are integer multiples of the electron charge ($Q=ne$ for some $n\in\mathbb{Z}$). More generally, the charges for all Standard Model particles are integer multiples of the charge of the down quark $Q_{d}=e/3$.

Charge quantization is important for the operation of the universe, as it allows for the existence of atoms that are electrically neutral. In turn, neutral atoms allow for the construction of working stars and other bulk matter. However, in conventional quantum electrodynamics — including the Standard Model — electric charges are not specified by fundamental considerations, but rather are input parameters. On the other hand, as outlined below, both Grand Unified Theories and the removal of anomalies imply constraints on the charges of fundamental particles and can thus provide mechanisms for charge quantization.

Many types of Grand Unified Theories have been put forward [419]. One general feature of unification models is that the quark and leptons are incorporated into a larger symmetry group, so that their properties are related due to constraints on the theory. As one example, in the case of $\mathit{SU}(5)$ unification, the electric charge operator is the sum of diagonal $\mathit{SU}(2)$ and $U(1)$ generators (e.g., see [313] for a textbook treatment). Since the generators must be traceless, the sum of the eigenvalues of electric charge must vanish, which leads to a charge quantization condition of the form

[TABLE]

The neutrino has no charge in our universe, so that $Q({\bar{d}})$ = $-Q(e^{-})/3$ = $e/3$. This model not only implies charge quantization, but also provides the fractional charges measured for quarks. More generally, charge quantization must arise in any unified theory [231]. Although charge quantization is a highly desirable feature, Grand Unified Theories have not been experimentally verified, so it is not known if they provide the explanation for charge quantization in our universe. However, any universe described by a unified theory of this class will have its charges quantized.

Another way to achieve charge quantization is through the requirement that anomalies vanish in the theory. A full discussion of this topic requires a rather lengthy formalism and is beyond the scope of this review (for further detail, see Chapter 22 of [547]). Briefly, the conditions for anomaly cancellation lead to constraints on the sum of the particle charges, roughly analogous to that of equation (30), and such conditions imply charge quantization (see also [210, 405]).

Particle physics theories thus support two different classes of constraints — those arising from Grand Unification and those due to anomaly cancellation. Both considerations enforce charge quantization and thereby lead to viable universes. With the current state of the field, all anomalies must cancel to avoid the prediction of infinite quantities, whereas Grand Unified Theories are not yet experimentally necessary.

In addition to quantization of charge, our universe displays the related properties of charge conservation and charge neutrality. Conservation of charge follows from the symmetries of the Lagrangian of the underlying theory [399], so that the class of theories with this property is large and well-defined. The relative numbers of positive and negative charges in the universe are determined early in cosmic history through a number of processes, including baryogenesis and leptogenesis. At the present time, these mechanisms remain under study [182, 495], but observations indicate that the universe as a whole is close to neutral [409]. The excess charge density per baryon is bounded from above. One such estimate [118] takes the form Q_{0}/n_{\rm b}\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}10^{-26}e, where $n_{\rm b}$ is the number density of baryons, and where this limit holds for uniformly distributed excess charge. In principle, a universe could have a net electric charge and still obey charge conservation and charge quantization. Moreover, such a universe could remain viable provided that the excess charge is not too large (see also [369]). Although the range of $Q_{0}/n$ that allows for habitability requires further specification, it includes the value $Q_{0}/n=0$, which could be considered special — and perhaps even likely — in the space of all possible universes [49].

2.5.2 Constraint from Proton Decay

In any Grand Unified Theory, conservation of baryon number is necessarily violated [230], which allows for the possibility of nucleon decay (see also [335, 392, 419] and references therein). The requirement that nucleons are sufficiently long-lived thus places constraints on the theory. Since the number of possible theories — and operators that violate baryon number — is large, we consider only a representative example. For the simplest class of interactions that drive proton decay, the time scale can be written in the form

[TABLE]

where $C_{\rm p}$ is a dimensionless constant of order unity, $M_{\rm gut}$ is the GUT scale ($\sim 10^{16}$ GeV), and $\alpha_{\rm gut}$ is the coupling constant evaluated at that scale. Current measurements [4] indicate that the proton lifetime in our universe has a lower limit of \tau_{\rm p}\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}1.6\times 10^{33} yr for the decay channel $p\to e+\pi^{0}$ and \tau_{\rm p}\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}7.7\times 10^{33} yr for the channel $p\to\mu+\pi^{0}$.

If we require that protons (nucleons) live long enough for life to evolve, then we must enforce the limit $\tau_{\rm p}>N_{A}t_{A}$, where $t_{A}$ is the atomic time scale and $N_{A}$ is the number of atomic time scales required for successful biological evolution. Here we take $N_{A}\sim 10^{33}$, corresponding to an time scale of $\sim 1$ Gyr (see [368, 324, 465] and Section 7.3 for further discussion). This constraint can be written in the form

[TABLE]

where $\alpha$ is the fine structure constant in the low energy limit. In our universe, the quantity $(\alpha_{\rm gut}/\alpha)^{1/2}\beta^{-1/4}\approx 0.4$. This quantity is close to unity and the constraint of interest is not overly sensitive to its value. Moreover, the appropriate value of $N_{A}$ is not precisely known. As a result, we obtain the approximate bound $M_{\rm gut}>N_{A}^{1/4}m_{\rm p}\sim 10^{8}m_{\rm p}$. Any viable universe must have a significant hierarchy between the GUT scale and the mass of the proton in order to keep nucleons stable long enough for life to evolve. However, this minimum hierarchy (a factor of $\sim 10^{8}$) is much smaller than that realized in our universe (where $M_{\rm gut}\sim 10^{16}m_{\rm p}$). A similar situation arises for proton decay in supersymmetric theories: the anthropically preferred time scale is much shorter than the observed proton lifetime (see [45, 502] for further discussion). In addition, a number of scenarios have been put forth to allow for proton stability [392] (e.g., in theories with large extra dimensions), so that the bound implied by equation (32) is not ironclad.

3 Cosmological Parameters and the Cosmic Inventory

This section outlines the cosmological parameters that are required to describe a universe as a member of the multiverse. We start with a review of the cosmological parameters that are necessary to specify the current state of our own universe. However, some of these parameters have relatively little effect on structure formation and are not necessary for an arbitrary universe to be habitable. We thus define the subset of parameters that are relevant for considerations of fine-tuning across the multiverse. The section then outlines the flatness problem and related cosmological issues, and briefly describes how an early inflationary epoch can drive a universe to become spatially flat. We also discuss how inflationary models can provide cosmological perturbations and elucidate the relationship between the parameters of the inflaton potential and the amplitude of primordial fluctuations.

3.1 Review of Parameters

The current state of the universe can be characterized by a relatively small collection of parameters. The expansion of the universe is governed by the Friedmann equation

[TABLE]

where $a(t)$ is the scale factor, the curvature constant $k=0,\pm 1$, and the energy density $\rho$ includes contributions from all sources. If we are concerned only with the expansion and evolution of the universe as a whole — and not the formation of structure within it — then the current state of the universe can be specified by measuring all of the contributions to the energy density $\rho$ and the Hubble constant

[TABLE]

where all of these quantities are evaluated at the present epoch. Note that once $H_{0}$ and $\rho$ are specified, then the equation of motion (33) determines the curvature constant $k$. Following cosmological convention, we take $a=1$ at the present epoch. The total energy density $\rho$ contains a number of components, including matter $\rho_{\rm M}$, radiation $\rho_{\rm R}$, and vacuum energy $\rho_{\scriptscriptstyle\Lambda}$. The matter density $\rho_{\rm M}$ includes at least two contributions, from baryons ($\rho_{\rm b}$) and from dark matter ($\rho_{\rm dm}$). Notice also that the dark matter could have contributions from different types of particles, including neutrinos and some type of cold dark matter. Many candidates have been put forward, including the lightest super-symmetric partner and axions (e.g., see [40, 208, 307, 494] and references therein).

The various components of the energy density evolve differently in the presence of cosmic expansion. The matter components, both dark matter and baryons, vary with the scale factor according to $\rho_{\rm M}\propto a^{-3}$, whereas the radiation component varies according to $\rho_{\rm R}\propto a^{-4}$. Unfortunately, the behavior of the dark energy $\rho_{\scriptscriptstyle\Lambda}$ remains unknown [218]. Current observations indicate that the energy density of the vacuum energy evolves slowly over cosmic time, so that it acts like a cosmological constant. For simplicity, we assume here that $\rho_{\scriptscriptstyle\Lambda}\approx$ constant. One should keep in mind, however, that more complicated behavior is possible and could be realized in other universes even if $\rho_{\scriptscriptstyle\Lambda}$ is essentially constant in our own.

Instead of working in terms of energy densities themselves, one can also define a critical density,

[TABLE]

and write the energy densities as ratios

[TABLE]

where the subject identifies the component of the universe (dark matter, radiation, etc). The set of parameters ${\cal S}_{\rm exp}$ necessary to determine the expansion properties of the universe thus becomes

[TABLE]

With these parameters specified, note that the curvature constant $k$ is given by

[TABLE]

Notice also that the total mass density is determined

[TABLE]

The discussion thus far only accounts for the expansion of a universe, and implicitly assumes that space-time is homogeneous and isotropic. In order for structure to form, the universe in question must contain deviations from homogeneity. In our universe, the starting amplitude of these fluctuations is extremely small. Moreover, as discussed below, considerations of structure formation indicate that such fluctuations should be small in any successful universe. As a result, the expansion of the universe proceeds largely independently of the formation of structure on smaller scales.

The primordial fluctuations can be described in a number of ways. In our universe, these seeds of structure formation are found to be Gaussian-distributed adiabatic fluctuations to a high degree of approximation [17, 19, 489] (see also [155, 387]). Moreover, the fluctuations have a nearly scale-invariant spectrum. The theory of inflation (see below) tends to produce such a spectrum, but the scale-invariant hypothesis was proposed as a working model of the fluctuations much earlier [268, 561]. In any case, the spectrum of perturbations can be written in the form

[TABLE]

where $k$ is the wavenumber (equivalently, spatial scale) of the fluctuation. In our universe the spectrum is nearly independent of spatial scale so that $|n_{S}-1|\ll 1$.

In general, the universe must also contain a contribution to the fluctuations due to gravitational waves, often known as tensor modes. The dimensionless spectrum of tensor modes can be written in a form similar to that considered previously, i.e.,

[TABLE]

In our universe, tensor modes have not (yet) been observed, but we expect that $n_{T}\sim 0$ and the amplitude $A_{T}\ll A_{S}$.

The set of parameters ${\cal S}_{\rm per}$ required to specify the departures of the universe from homogeneity thus involves at least four parameters and can be written

[TABLE]

The number of parameters required for a full specification could be larger if the fluctuations are non-gaussian.

The experiments that determine the cosmological parameters in our universe rely heavily on observations of the cosmic microwave background [17, 487, 489]. These measurements depend on another cosmological parameter $\tau$, which is the scattering optical depth of the universe due to reionization. The optical depth $\tau$ = 0.09 in our universe and must be determined in order to make precise estimates for the other cosmological parameters of interest. In the present context, however, the scattering optical depth $\tau$ does not play an important role in structure formation. Reionization occurs only because structure — first galaxies and then massive stars — is able to form. In any case, we will not include the scattering optical depth as a relevant variable for purposes of studying fine-tuning.

The most important cosmological parameters are summarized in Table 1. This list contains quantities that define the cosmological inventory, the current expansion rate, and the characteristics of the primordial density fluctuations (see also [349, 332]. Note that the inventory is not complete, as one can consider the various types of stellar objects and gaseous phases that make up the baryonic component, as well as the radiation fields produced by a wide range of astrophysical processes [219]. On the other hand, not all of the parameters listed in Table 1 are important for discussions of fine-tuning. As discussed below, we can reduce the number of cosmological parameters to a minimal set.

We first note that the Hubble constant $H_{0}$, while vital to understanding the current state of our universe, essentially defines the current cosmological epoch. In considerations of other universes, however, we only need to consider whether or not structure formation occurs at any epoch. As a result, the Hubble constant, which varies with time, does not need to take on a specific value.

Next, our universe is observed to be nearly spatially flat. We can also argue that successful universes must be close to flat: Some solution to the flatness problem, either by inflation [251] or some other mechanism, is assumed to be operational in any viable universe (see Section 3.3 for further discussion). We can also assume that the horizon problem and monopole problem (unwanted relics) are not impediments to a successful universe. As a result, we can take $k=0$ and hence enforce the constraint

[TABLE]

In other words, only three of the density contributions are independent. The inventory of a universe is thus specified by three quantities. The values of $\Omega_{j}$, however, vary with time or equivalently scale factor. As outlined below, the early universe must emerge from the epoch of Big Bang Nucleosynthesis with an acceptable chemical composition, which in turn depends on the baryon to photon ratio $\eta$, which is (nearly) constant. We can thus use $\eta$ to specify the baryonic component. We can then use the ratio $\Omega_{\rm dm}/\Omega_{\rm b}$ to specify the dark matter content. Note that even though the $\Omega_{j}$ vary with time, the ratio of any two matter components does not. Alternately, we can define a dark matter parameter $\delta\equiv\eta\Omega_{\rm dm}/\Omega_{\rm b}$ so that we have symmetric definitions for the baryonic component $\eta$ and the dark matter component $\delta$. Finally, the dark energy density is assumed to be constant and can be specified through its value $\rho_{\scriptscriptstyle\Lambda}$. The inventory of the universe is thus specified by the reduced set of parameters

[TABLE]

The number of parameters necessary to specify the spectrum of density fluctuations can also be significantly reduced. Given that the tensor modes are subdominant in our universe, and that the spectrum of perturbations for both contributions is relatively flat, we can characterize the primordial fluctuations with a single parameter $Q\sim\sqrt{A_{S}}$, where $Q\approx 10^{-5}$ in our universe. The set of parameters for structure formation thus collapses to the form ${\cal S}_{\rm per}\to\left\{Q\right\}$. This simplification assumes that the index $n_{S}$ remains close to unity. For much larger (smaller) values of $n_{S}$, the spectrum of perturbations will have significantly more power on smaller (larger) spatial scales, and will lead to corresponding changes in structure formation. Unfortunately, a comprehensive assessment of the allowed range of the index $n_{S}$ has not been carried out. Nonetheless, before the value of $n_{S}$ was well-determined observationally, explorations of structure formation with a range of indices $n_{S}\neq 1$ (e.g., [46, 80]) did not find that the universe becomes uninhabitable.

With our present level of understanding of physics and cosmology, the parameters ${\cal S}_{\rm cos}=\left\{\eta,\delta,Q,\rho_{\scriptscriptstyle\Lambda}\right\}$ represent the most important dials that can be adjusted to specify the properties of a given universe. In a complete theory, the values of these parameters — or more likely the distributions of the parameter values — could be calculable from physics beyond the Standard Model. In the meantime, for this discussion, the values of $\left\{\eta,\delta,Q,\rho_{\scriptscriptstyle\Lambda}\right\}$ are left as free parameters.

The baryon to photon ratio $\eta$ is nonzero because the universe experienced an epoch of baryogenesis that broke the symmetry between matter and antimatter (unless $\eta\neq 0$ results from highly unusual initial conditions). Baryogenesis, in turn, requires three essential ingredients [457]: The first requirement is that baryon number is not conserved. The second is that both C and CP conservation must be violated, where ‘P’ is the discrete symmetry of parity and ‘C’ is that of charge conjugation. Finally, the universe must depart from thermal equilibrium during the epoch(s) when non-conservation of the aforementioned quantities takes place. Although these three features are known to be required for successful baryogenesis, an accepted theory of this process is not yet available (see [327] for additional detail and [180, 495] for more recent reviews). Grand Unified Theories [335], theories of quantum gravity [274], and other approaches allow for the violation of baryon number conservation, so that new physics should eventually predict the expected distribution of the baryon to photon ratio $\eta$.

The abundance of dark matter $\delta$ is also determined by processes taking place in the early universe. The simplest scenario occurs when the universe has a single dark matter species that is produced in thermal equilibrium. In that scenario, the abundance of dark matter is determined when the weak interactions become too slow to maintain statistical equilibrium, typically at cosmic times t\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}1 sec and temperature T\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}1 MeV [327]. The dark matter abundance depends on the particle properties (mass, interaction cross section, etc.), which are not known at the present time [307]. Again, a description of these properties requires extensions of the Standard Model, so that new physics may eventually predict the possible abundances of dark matter.

The value of the amplitude $Q$ of the primordial density fluctuations also cannot be predicted using known physics. In a large class of inflationary theories [355], quantum fluctuations in the inflaton field produce density perturbations, so that the amplitude $Q$ could be calculated in principle. In this case, a large number of inflationary scenarios are possible, so that the possible distribution of $Q$ is similarly enormous. In addition, even for a given set of inflaton fields, the spectrum of density fluctuations can depend on the initial conditions, i.e., the manner in which the universe enters into its inflationary epoch. Although the details are both complicated and unknown, the value of $Q$ is unlikely to have the same value in all universes, so that the density fluctuations must be described by a distribution of values across the multiverse. Moreover, this discussion assumes a scale-invariant spectrum of fluctuations within a given universe. In addition to the overall amplitude $Q$ of the spectrum having a distribution of values, the form of the fluctuation spectrum itself (see equations [40,41]) could also vary from universe to universe.

Finally, the value of the vacuum energy density $\rho_{\scriptscriptstyle\Lambda}$ is not understood in the context of known physics. This issue is essentially the cosmological constant problem [110, 546], which has a long history and no accepted resolution (see Section 4).

3.2 Constraints on the Cosmic Inventory

Figure 9 shows the evolution of the various density components of the universe. This figure is scaled to the observed values in our universe, where $a=1$ corresponds to the current cosmological epoch. The universe is radiation dominated at early times, transitions into a matter dominated era at intermediate times, and has just recently become dominated by its vacuum energy. When viewed across the relatively large span of cosmic time shown here, two things are evident: First, the fact that the universe is dominated by the vacuum energy at the present epoch is hard to discern – this near equality of matter and vacuum energy at the present epoch is a manifestation of the well-known cosmological constant problem [546, 413]. Second, the duration of the matter dominated era is relatively short. For some values of the cosmological parameters, the universe could move directly from its radiation era into a vacuum dominated era. Such a universe would have no period of matter domination and hence no structure formation. This scenario — without a matter dominated era — is an extreme version of cases where the vacuum energy density is too large relative to the primordial fluctuation amplitude to allow for structure formation (see Section 4 and Refs. [16, 206, 223, 224, 226, 238, 372, 383, 429, 545]).

We can derive a constraint that depends only on the cosmic inventory — independent of the fluctuation amplitude $Q$ — by requiring that the universe have a matter dominated era. Equivalently, the energy density of the vacuum must be smaller than the energy density of the universe at the epoch of equality. If we write the vacuum energy density in terms of an energy scale $\lambda$, i.e.,

[TABLE]

then the constraint takes the form

[TABLE]

where $\eta$ is the baryon to photon ratio and where $a_{R}=\pi^{2}/15$ is the radiation constant. In our universe, the energy scale of the vacuum $\lambda\sim 3\times 10^{-3}$ eV. For comparison, the right hand side of equation (46) is $\sim 4$ eV. The universe is thus safe by a factor of $\sim 1000$ for the energy scale $\lambda$ (and hence a factor of $\sim 10^{12}$ for energy density $\rho_{\scriptscriptstyle\Lambda}$).

The constraint of equation (46) is necessary but not sufficient. Even if the universe has a matter dominated era, structure formation can be suppressed if the vacuum energy density is too large relative to the amplitude of the primordial density fluctuations. This issue is taken up in Section 4 and provides stronger constraints on the energy density of the vacuum. On the other hand, the baryon to photon ratio $\eta$ can be larger in other universes (Section 5), which would allow for even larger values of the vacuum energy scale $\lambda$. If the baryon to photon ratio becomes too large, however, the epoch of matter domination will take place before Big Bang Nucleosynthesis. If we approximate the energy scale of BBN as $E_{\rm bbn}\sim\Delta{m}\sim m_{\rm e}$ [61, 122], this constraint can be written in the form

[TABLE]

where the numerical value corresponds to the parameters of our universe. Note that a universe in which matter domination occurs before the epoch of BBN would not necessarily be sterile, but it would represent a significant departure from the usual ordering of cosmological time scales.

3.3 The Flatness Problem

One of the classic fine-tuning problems in cosmology is sometimes known as the flatness problem. This issue can be illustrated by writing the equation of motion (33) for the scale factor in the form

[TABLE]

where the second equality defines the parameter $\chi(t)$. Note that, in general, the density parameter $\Omega$ is not constant in time. More specifically, for a radiation dominated universe, $\rho\propto a^{-4}$, so that $\chi\propto a^{2}$. Similarly, for a matter dominated universe $\rho\propto a^{-3}$ and $\chi\propto a$. The parameter $\chi$ thus increases as the universe expands for the case of both matter and radiation. On the other hand, if the universe is dominated by vacuum energy, then $\chi\propto a^{-2}$ and hence decreases with time.

In order for the universe to remain flat, equivalently for $\Omega$ to remain close to unity, the quantity $\chi$ must remain small. But if $\chi$ is small at a given epoch, then it must have been even smaller at an earlier epoch (for universes dominated by either radiation or matter). To leading order, for $\chi\ll 1$,

[TABLE]

In order for $\chi$ to be small at the present cosmological epoch, as observed, this quantity must have been extremely small at earlier times. When the cosmic age was $t_{\scriptscriptstyle{BBN}}\sim 1$ sec, near the beginning of Big Bang Nucleosynthesis, the parameter $\chi(t_{\scriptscriptstyle{BBN}})\sim 10^{-16}$. If we go all the way back to the Planck era, with age $t_{\rm pl}\sim 10^{-43}$ sec, the parameter $\chi(t_{\rm pl})\sim 10^{-60}$.

Figure 10 shows the evolution of the density parameter $\Omega$ as the universe expands. The scale factor is taken to be unity at the start of the evolution and five initial values of the parameter $\chi$ are used, specifically, $\log_{10}\chi(a=1)1$ = $-16,-15,\dots,-11$. The evolution in $\Omega$ is shown for universes with both positive (upper curves) and negative (lower curves) curvature. The density parameter $\Omega$ remains close to unity until the parameter $\chi$ becomes significant, and then $\Omega$ evolves rapidly. For positive curvature, the value of $\Omega$ becomes increasingly large and universe eventually recollapses. For negative curvature, $\Omega$ steadily decreases and the expansion becomes that corresponding to an empty universe.

The paradigm of inflation [27, 252, 354] was developed to alleviate this issue of the sensitive fine-tuning of the density parameter $\Omega$ (although other motivations, such as the monopole problem, were also important). Since this topic has been discussed extensively elsewhere [252, 253, 355], the present treatment will be brief. As outlined above, our universe requires the parameter \chi\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}10^{-60} at the Planck epoch in order to evolve into its present state. In contrast, the value of $\chi$ is expected to be of order unity at this time. If the universe experiences an epoch of rapid expansion due to vacuum domination, and $\chi\propto{a}^{-2}$ during that epoch, then successful inflation requires the scale factor $a$ to grow by a factor of $\sim 10^{30}$. This growth factor is generally expressed in terms of the number $N$ of e-foldings so that the requirement becomes

[TABLE]

The exact number of required e-foldings of the scale factor depends on the energy scale of the inflaton field, where lower energy scales lead to somewhat less stringent requirements. However, the number of e-foldings $N$ appears in the exponential, so that the required minimum number $N_{\rm min}$ generally falls in the range $N_{\rm min}$ $\approx 60-70$. Moreover, most inflation models tend to produce far more e-foldings so that $N\gg N_{\rm min}$.

Another problem facing our observed universe is the so-called horizon problem: Observations of the Cosmic Microwave Background show that the universe is isotropic to high precision. Without the aforementioned inflationary epoch, regions that are now observed in opposite sides of the microwave sky would have never been in causal contact. But they have the same temperature within about one part in $10^{5}$, where we can now measure the spectrum of these small deviations to high precision [17, 19, 487, 489]. Since one effect of the inflationary epoch is to accelerate the expansion and effectively move regions out of the horizon, such regions could have been in contact at earlier epochs provided that the duration of the inflationary epoch is long enough. The number of e-foldings of the scale factor required to alleviate the flatness problem (from equation [50]) is nearly the same as that required to alleviate the horizon problem [327, 66]. As a result, universes that emerge from an early inflationary epoch with sufficiently small $\chi$, which can survive to old ages, will also generally be close to isotropic. Our universe could in principle be habitable without the extreme level of isotropy that is observed, but smoothness and flatness tend to arise together in the simplest versions of inflation.

The paradigm of inflation alleviates the flatness problem — and hence the apparent fine-tuning of the density parameter $\Omega$ — as long as the universe can accelerate for the required number of e-foldings (equation [50]) and then subsequently evolve according to the standard, radiation dominated hot Big Bang model. In order for inflation to be successful [327, 497], a number of constraints must be met:

The universe must first be able to enter into an inflationary state, where the energy density is dominated by the vacuum and the scale factor accelerates (grows faster than linear with time). This superluminal phase must last long enough to solve the flatness, horizon, and monopole problems in our universe. Although other universes are not required to be be as isotropic as our own, so that the horizon problem is not as severe, all of these issues are addressed by the same large growth factors.

The likelihood that the universe can enter into an inflationary state remains under intense debate (for further discussion, see [97, 127, 130, 156, 232, 273, 282, 469, 496, 526]). The required fine-tuning for achieving successful inflation can be described in terms of the space of possible trajectories for the expansion history of the universe. Some authors argue that if the universe starts at the Planck epoch with reasonable assumptions, then successful inflation becomes “exponentially unlikely” [292]. In other words, given that the universe is required to have desirable late-term properties, only a small fraction of the possible starting conditions lead to acceptable cosmological histories. Other authors conclude that the paradigm of cosmic inflation remains on a strong footing [255]. The key question is whether or not the conditions required for successful inflation are more constraining than the cosmological problems that the paradigm seeks to alleviate. These conditions include both the necessary parameter values of the theory (e.g., the properties of the inflaton potential) and the requisite initial conditions. This issue remains unresolved. On a related note, the fraction of cosmological trajectories that lead to smooth universes at late times is dominated by those that are not smooth at early epochs [127], which changes the constraints on cosmological initial conditions.

During the inflationary expansion phase, quantum fluctuations in the inflaton field produce density perturbations in the background universe [47, 254, 388]. These perturbations must be sufficiently small in amplitude in order for the inflationary epoch to begin, and this condition is expected to hold in only a small fraction of realistic cosmologies [530]. Provided that inflation is successful, in the late universe these density fluctuations grow into galaxies, clusters, and other large scale structures. Moreover, these fluctuations must have an amplitude $Q$ that falls within the approximate range 10^{-6}\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}{Q}\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}10^{-2} in order for the universe to produce galaxies with acceptable densities and hence be habitable [13, 510, 511] (see Section 6). The relation between the amplitude $Q$ and the parameters that appear in the inflaton potential is described in Section 3.4.

After the universe has expanded by the required factor, it must leave its accelerating state and begin to expand in the usual subluminal manner. In order for the universe to become potentially habitable, first by producing heavy nuclei and later by forming galaxies and stars, it must become radiation dominated after inflation ends. The rapid, accelerated expansion of the inflationary phase leaves the universe with an extremely low temperature, of order $T\sim 10^{-60}T_{\rm pl}\sim 10^{-28}$ K, so that all of the energy is locked up in the vacuum. This vacuum energy must be converted into radiation and particles through a process known as reheating. The conversion must be efficient enough to reheat the universe to a sufficiently high temperature such that baryogenesis can take place. This minimum temperature is often taken to be the scale of the electroweak phase transition $\sim 100$ GeV. Successful reheating of the universe to this high temperature is by no means automatic, so this requirement represents another hurdle that a successful universe must negotiate.

3.4 Quantum Fluctuations and Inflationary Dynamics

As outlined in the previous subsection, an early epoch of inflation can potentially alleviate a number of cosmological problems, although achieving successful inflation is not without its own issues. Although alternate explanations exist, it is useful to illustrate inflationary dynamics in greater detail. Toward that end, this section describes the semi-classical dynamics for the evolution of the scalar field that drives inflation, generally called the inflaton field. Next we elucidate the relationship between the parameters that appear in the inflaton potential and the spectrum of fluctuations produced during the inflationary epoch.

The equation of motion for the evolution of the inflaton field $\phi$ form is generally written in the from

[TABLE]

where $H$ is the Hubble parameter and $V$ is the potential. During the inflationary epoch, the energy density of the universe is dominated by the potential of the inflaton, so that the Hubble parameter is given by

[TABLE]

Under a wide range of conditions, the first term in equation (51) and the last term in equation (52) can be neglected, and evolution takes place during what are called slow-roll conditions. The number of e-foldings of the inflationary epoch is then given by

[TABLE]

where the integrals are taken over the time interval (values of $\phi$) corresponding to the inflationary epoch.

To fix ideas, we can illustrate the scalar field dynamics of inflation with a simple power-law form for the potential [354],

[TABLE]

where $q_{4}$ is a dimensionless coefficient. With this choice, the number of e-foldings takes the form

[TABLE]

where $\phi_{0}$ ($\phi_{f}$) is the initial (final) value of the inflaton field. The requirement of sufficient inflation N\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}60 thus implies that the starting value of the inflaton field satisfies the constraint \phi_{0}\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}(60/\pi)^{1/2}\approx 4.4M_{\rm pl}. The requirement of sufficient inflation thus requires that the value of the scalar field $\phi$ is comparable to — and somewhat larger than — the Planck scale. As summarized below, however, constraints on the perturbation spectrum require the energy density to be well below the Planck scale.

In addition to allowing the universe to become connected and flat, the inflationary epoch also imprints density fluctuations on the otherwise smooth background of the universe [254]. To leading order, the spectra of cosmological perturbations produced by inflation can be written in the forms

[TABLE]

for scalar and tensor contributions, respectively [66]. These quantities are evaluated at the epoch when a perturbation with a particular length scale crosses outside the horizon [327]. Using the equation of motion (51) for the scalar field and the definition (52) of the Hubble parameter (in the slow-roll approximation), the quantities $H$ and ${\dot{\phi}}$ can be written in terms of the inflationary potential. For the particular potential given by equation (54), the expressions for the perturbation spectra take the forms

[TABLE]

The ratio of scalar to tensor perturbations is thus given by

[TABLE]

where $N$ is the number of e-foldings from equation (55). Since the perturbations that are constrained by measurements of the cosmic background radiation are those that left the horizon about $N\sim 60$ e-foldings before the end of inflation [327], the value of $\phi$ used to evaluate equation (57) is of order $\phi\sim\phi_{0}\sim 4M_{\rm pl}$. Using this result to evaluate the scalar perturbation ${\cal P}_{S}$ allows us to specify the amplitude $Q$ in terms of inflationary parameters,

[TABLE]

As outlined above (see Table 1), the primordial fluctuations in our universe have amplitude $Q\sim 10^{-5}$, so that the required value for the dimensionless constant $q_{4}$ must be extremely small, roughly $q_{4}\ll 10^{-14}$.

In the absence of special circumstances, however, the dimensionless parameter $q_{4}$ is expected to be of order unity. The requirement that its value must be incredibly small thus leads to a fine-tuning problem. More specifically, the problem is one of naturalness (see [515] and Section 1.1). The value of the parameter is expected to become of order unity due to quantum corrections unless the small required value is protected by a special symmetry. Moreover, one can show that the constraint $Q\sim\sqrt{q_{4}}$ and hence $q_{4}\ll 1$ holds more generally, and that any successful inflation model in this class must have a small parameter [11].

Given the tension between the requirement of a small value of the dimensionless constant ($q_{4}\sim 10^{-14}$) and its much larger expected value ($q_{4}\sim 1$), one would expect other universes to have larger values of $q_{4}$. As a result, it is natural (in both the technical and colloquial sense) for the amplitude $Q$ of cosmological fluctuations to be larger in other universes. Larger values of $Q$ lead to earlier structure formation and denser galaxies. Of course, the amplitude $Q$ could sometimes be smaller, so that the universe produces more rarefied galaxies. The consequences of these changes, and accompanying constraints, are discussed in Section 6.

Note that for viable universes the value of $Q$ is bounded from above: If the amplitude $Q$ becomes of order unity, fluctuations are close to non-linearity — and hence ready for collapse — as soon as they enter the horizon after inflation. In this case, a large fraction of the energy within the horizon could become locked up within black holes in the early universe. The analysis of [510] indicates that the resulting density of black holes could dominate the density of dark matter and baryons if Q\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}0.1. This bound assumes that the spectrum of density perturbations remains relatively flat, with index $n_{S}\sim 1$, down to the mass scales characteristic of the horizon just after inflation (see [121, 240] for further detail).

Finally, we note that although the tensor perturbations are often subdominant, they provide an important constraint on the energy scale of inflation (see also [348]). The requirement that {\cal P}_{T}\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}Q^{2} implies that the inflaton potential must obey the bound

[TABLE]

If we characterize the energy density of the potential by defining an energy scale $\mu\equiv V^{1/4}$, then the constraint becomes \mu\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}\sqrt{Q}M_{\rm pl}\sim 0.003M_{\rm pl} for our universe. In other words, for applications to our universe, the energy scale of inflation is bounded from above by the GUT scale and must be substantially below the Planck scale. In other universes with larger fluctuation amplitudes $Q$, the required hierarchy between the Planck scale and the inflation scale could be less pronounced.

3.5 Eternal Inflation

An important generalization of the inflationary universe paradigm is that (in many cases) most of the volume of the entire space-time is in a state of superluminal expansion, so that the inflationary epoch can be eternal [253, 356]. Sub-regions of space-time detach from the background and form separate ‘pocket universes’, which can evolve to become similar to our own. Moreover, this feature can be generic for some classes of inflationary theories [533]. The scenario of eternal inflation provides one specific mechanism for generating multiple universes and is thus of interest for the problem of fine-tuning and the multiverse.

To illustrate the manner in which inflation can be eternal, we consider the simple potential of equation (54). The scalar field obeys the equation of motion (51), so that its classical trajectory would be to slowly evolve to smaller $\phi$ and hence lower potential energy $V(\phi)$. The scalar field is said to ‘slowly roll downhill’. As the scalar field evolves down the potential, however, quantum fluctuations are superimposed on the classical motion. If these fluctuations are large enough, then parts of the space-time can remain in an inflationary state for an indefinitely long span of time. The conditions required for this scenario to operate are illustrated below.

Following the discussion of [252, 253], consider the time interval corresponding to one local Hubble time, i.e., $\Delta t$ = $H^{-1}$. At the start of that time interval, the scalar field will have a value $\phi_{0}$ corresponding to its average over a Hubble volume $V_{H}\sim H^{-3}$. During the Hubble time $\Delta t$, the scale factor grows by one factor of $e$ and the Hubble volume grows by a factor of $e^{3}$. At the same time, the scalar field will evolve. The change in the scalar field due to its classical trajectory can be denoted as $(\Delta\phi)_{\rm cl}$, whereas the change due to quantum fluctuations is $(\Delta\phi)_{\rm qm}$. The total change in the scalar field during one Hubble time is thus given by

[TABLE]

To leading order, the quantum fluctuations will have gaussian probability distribution, with width given approximately by $H/(2\pi)$ [493]. For small fluctuations, the scalar field will evolve to lower values. For sufficiently large fluctuations, however, the scalar field can move up the potential to higher values over the course of one Hubble time. In order for eternal inflation to operate, the probability of moving to larger values of $\phi$ must be high enough that at least one of $e^{3}\sim 20$ new Hubble volumes will have this property. This condition implies the constraint

[TABLE]

Using the equation of motion (51) and the definition (52) of the Hubble parameter, this constraint takes the form

[TABLE]

Although this required value of the field is larger than the Planck scale, the energy density is given by

[TABLE]

As a result, the energy density of the scalar field required for eternal inflation is much smaller than that given by the Planck scale. If the universe starts out at high temperatures close to the Planck scale, then it will be born with enough energy density to enter and maintain eternal inflation.

In this paradigm, at any given time, the energy density in most of the volume of the multiverse is dominated by the potential energy of the scalar field $\phi$ and is inflating. Some regions will evolve far enough down the potential so that quantum fluctuations do not push them back up to higher vacuum energy densities. These regions can experience inflation as described in the previous section, with decreasing $\phi$ following its classical trajectory to lower values of $V(\phi)$. Some subset of these regions will successfully convert vacuum energy into particles and radiation, and eventually evolve according to classical cosmological theory like our own. In this manner, the background space-time of the universe continually gives rise to new universes.

This evolutionary picture is illustrated in Figure 11, which shows a sequence of snapshots of the multiverse over time. The shaded regions depict the background energy density of the vacuum, which causes the universe to inflate. The volume of the multiverse in this rapidly expanding state grows exponentially with time, so that most of the space-time resides in this state dominated by vacuum energy. Some regions can evolve to lower-energy vacuum states and eventually become radiation dominated. These regions thus become universes that evolve according to some realization of cosmology, and are depicted as the square symbols in Figure 11. Although the background space expands exponentially, and the universes expand as well, this growth is suppressed in the diagram. The different universes can in principle have different realizations of the laws of physics and/or different values of the cosmological parameters. Only some fraction of these regions have suitable choices of the parameters for the universe to be habitable, as illustrated by the green regions in the figure. (As an aside: Many versions of eternal inflation result in the production of an infinity of universes, and could result in an infinite number of both red and green squares in Figure 11. Any assessment of the fraction of habitable universes depends on how the counting is carried out.)

4 The Cosmological Constant and/or Dark Energy

Although our universe can be specified by relatively few cosmological parameters (see Section 3), one of the necessary ingredients is a substantial energy density of the vacuum — often called dark energy. This quantity acts like a cosmological constant and is currently the dominant component of the cosmic inventory (see Table 1). The dark energy is driving the currently observed acceleration of the universe and will have enormous consequences in the future [104, 105, 390]. Moreover, the existence and nature of this counter-intuitive component poses an interesting and important problem for fundamental physics. On the other hand, astrophysical processes in our universe — for example, the formation of galaxies and other large scale structure — have been influenced more by dark matter than by dark energy thus far in cosmic history (see the discussion of [364]).

Even though the theory of general relativity allows empty space to have a nonzero energy density, its existence poses (at least) two coupled problems:

[A] If the vacuum energy density plays a significant role at the present cosmological epoch, its value must be exceedingly small relative to theoretically expected values. This extreme ordering of energy scales is one manifestation of the cosmological constant problem, and is an example of a Hierarchical Fine-Tuning problem (Section 4.1).

[B] If the vacuum energy density is large enough to affect the cosmological expansion, it acts to suppress the formation of dark matter halos and other cosmic structures. In a universe with too much energy density in the vacuum, cosmological structure is unable to separate itself from the expanding background and form gravitationally bound entities. This consideration constrains the allowed energy density of the vacuum, where the severity of the bound depends on the other cosmological parameters (Section 4.2).

In spite of being at the center of the cosmological constant problem, the vacuum energy density of the universe is not necessarily constant in time. In some models, this dark energy evolves smoothly, usually as a decreasing function of time, through the evolution of scalar fields [455, 563]. In other models, the vacuum energy density can evolve suddenly, e.g., through cosmological phase transitions [153, 538, 458, 562]. Although the dark energy could display a wide range of temporal evolution in other universes, for the sake of simplicity this treatment focuses on the case of constant dark energy.

4.1 The Cosmological Constant Problem

We start with a short review of the cosmological constant problem (for a more comprehensive discussion, see [546] and [87]). In General Relativity, the field equations take the form

[TABLE]

where the symbols have their usual meanings. In particular, the parameter $\Lambda$ is the cosmological constant, which is non-vanishing in general. Classically, the cosmological constant could have any value. The only other constant appearing in the field equation is the gravitational constant $G=M_{\rm pl}^{-2}$. On dimensional grounds, one expects the cosmological constant to be determined by $G$. In natural units, the quantity $\Lambda/G=\rho_{\scriptscriptstyle\Lambda}$ has units of energy density (the quantity measured in cosmology). We can thus write

[TABLE]

This expected value stands in sharp contrast to the value inferred from cosmological experiments, which indicate that $\rho_{\scriptscriptstyle\Lambda}\sim 10^{-10}$ eV4. The mismatch between these two values of the vacuum energy density is one manifestation of the cosmological constant problem.

Another way to present the problem is to start from the definitions of the basic units of time and length from the gravitational constant, i.e., the Planck time

[TABLE]

and the Planck length

[TABLE]

The universe is observed to be large compared to the Planck length, with size scale $R_{\rm univ}=c/H_{0}\approx 10^{28}$ cm $\sim 10^{61}\,\ell_{\rm pl}$. Similarly, the universe is inferred to be old compared to the Planck time, where the cosmic age $t_{\rm univ}\approx$ 13.7 Gyr $\sim 10^{61}\,t_{\rm pl}$.

Note that the magnitude of the cosmological constant problem depends on what quantity is under consideration. The observed vacuum energy density is smaller than the expected value by 122 orders of magnitude. But the age and size of the universe are larger than their expected values by ‘only’ 61 orders of magnitude. If we were to define an energy scale $\lambda$ for the vacuum, so that $\rho_{\scriptscriptstyle\Lambda}=\lambda^{4}$ (see equation [45]), then the observed value of the scale $\lambda\approx 3\times 10^{-3}$ eV is smaller than the expected value ($\lambda\sim M_{\rm pl}$) by 30 orders of magnitude. Of course, these discrepancies represent the same underlying hierarchy. On a related note, one could consider any function of the parameter (e.g., $\log\rho_{\scriptscriptstyle\Lambda}$ or even $\log[\log\rho_{\scriptscriptstyle\Lambda}]$), which would change the numerical value of the hierarchy, but not the underlying issue [403]. This choice of variable also affects the probability distributions of the parameters (see E).

Although classical general relativity is silent on the preferred value of the cosmological constant $\Lambda$, aside from the above dimensional considerations, quantum field theory indicates that the issue must be taken seriously. In general, the vacuum (although somewhat poorly named) has a non-zero energy density $\rho_{V}$, which can be written in the form

[TABLE]

The vacuum contribution to the energy-momentum tensor (left side of the equation) is proportional to the metric (indicated by the right side of equation) due to Lorentz invariance. The constant of proportionality is the energy density of the vacuum, and the sign convention is chosen so that if $\Lambda\equiv 8\pi\rho_{\scriptscriptstyle\Lambda}$, then we can identify $\rho_{V}\sim\rho_{\scriptscriptstyle\Lambda}$.

The key issue is that quantum field theory predicts that $\langle T_{\mu\nu}\rangle\neq 0$. In a manner roughly analogous to the zero-point energy of a quantum harmonic oscillator, all of the modes for all of the free fields produce a zero-point energy that contributes to the energy density of the vacuum. In the standard treatment of this effect (e.g., [129, 413]), the size of each component has the from

[TABLE]

where $M_{\rm max}$ is the maximum energy scale or cutoff scale. The mass $m$ of the field is assumed to be small such that $m\ll M_{\rm max}$. For completeness, we note that the arguments leading to equation (70) do not obey Lorentz invariance and do not produce the well-known equation of state $\langle{p}\rangle$ = $-\langle\rho\rangle$ for the vacuum. As a result, a more detailed treatment is necessary. For example, an argument using dimensional regularization [373] gives qualitatively similar results (specifically, a quartic dependence on a large mass scale, but with a logarithmic correction).

For quantum fluctuations in gravity, one expects $M_{\rm max}\sim M_{\rm pl}$ so that $|\langle T_{\mu\nu}\rangle|\sim M_{\rm pl}^{4}$. Even if one makes the argument that our current understanding of quantum gravity is too primitive to accept this result, the known particles of the Standard Model are expected to make vacuum contributions. In this case, the cutoff energies are comparable to scales appearing in the Higgs potential, $M_{\rm max}\sim 100$ GeV, so that the vacuum energy density is expected to be $\rho_{V}\sim 10^{8}$ GeV4, which is larger than the observed value $\rho_{\scriptscriptstyle\Lambda}$ by a factor of $\sim 10^{54}$. Many calculations of this type assume that supersymmetry holds so that the cutoff scale is the supersymmetric scale $M_{\rm ss}$ (which must be larger than $\sim 1$ TeV). For example, one class of potentials in M-theory implies that the vacuum energy density has the form $\rho_{\scriptscriptstyle\Lambda}\sim M_{\rm pl}^{2}M_{\rm ss}^{2}$ [6]. In any case, quantum field theory predicts many contributions to the vacuum energy density, where the known/expected contributions are larger than the observed vacuum energy density by at least 54 orders of magnitude, and perhaps much more.

For completeness, we note that vacuum energy density can arise on both sides of the field equation (65): If the original parameter from General Relativity $\Lambda\neq 0$, then the contribution originates on the left side of the equation and is usually called the cosmological constant. On the other hand, if the energy-momentum tensor has a contribution from quantum fluctuations (as in equation [69]), then the term arises on the right side of equation (65) and is usually called a vacuum energy density. Since the universe is (essentially) homogeneous and isotropic, the field equation (65) reduces to a single equation of motion for the scale factor (the Friedmann equation [33]), and both contributions provide a single term to the cosmic energy inventory. Current experiments indicate that $\rho_{\scriptscriptstyle\Lambda}\neq 0$, but make no distinction between contributions from ‘pure $\Lambda$’ and $\langle T_{\mu\nu}\rangle$.

Although many ideas have been put forth, the cosmological constant problem does not have a definitive solution at the present time [546]. Possible explanations include dynamical approaches using evolving scalar fields [36, 455], extra space-time dimensions [454], vacuum structure produced by parallel branes [527], and many others [488]. Before experiments discovered that our universe is accelerating [451], which indicates a nonzero vacuum energy density, one hope was that some physical mechanism or symmetry principle would show that the cosmological constant must be exactly zero. Given the currently measured $\rho_{\scriptscriptstyle\Lambda}\neq 0$, however, the cosmological constant problem now has two components [225]: We need an explanation for why its value so much smaller than the Planck scale and why it has the particular small value $\rho_{\scriptscriptstyle\Lambda}\sim(0.003{\rm eV})^{4}$.

Given the large number of possible mechanisms by which the universe could determine the magnitude of its vacuum energy density, it is likely that other regions of space-time (other universes in the multiverse) would have different values of the cosmological constant [545]. The value of $\rho_{\scriptscriptstyle\Lambda}$ (equivalently $\Lambda$ or $\lambda$) is thus expected to vary from universe to universe. In addition, unless the size of the cosmological constant is determined by as-yet-unknown physics, we expect many other universes to have much larger values of $\rho_{\scriptscriptstyle\Lambda}$ than that observed locally. As a result, the most common universes, with large $\rho_{\scriptscriptstyle\Lambda}$, will not be able to produce structure (see the following section) and will quickly evolve to become almost empty de Sitter space [126]. In this context, our universe is unusual in its profligate complexity.

4.2 Bounds on the Vacuum Energy Density from Structure Formation

Although the energy density $\rho_{\scriptscriptstyle\Lambda}$ of the vacuum is not predicted by any known theory, its value in our universe is nonetheless constrained by cosmological considerations. A relatively weak upper bound follows from the requirement that the universe must have a matter dominated era (see equation [46]). A stronger bound results from the requirement that galaxy formation occurs before the the expansion of the universe starts to accelerate due to the vacuum energy. This section reviews bounds of this type.

The original constraints of this type [43, 167, 457, 545] require that quasars form by redshift $z=4.5$ and can be written in the form

[TABLE]

where $\rho_{\rm eq}$ is the density at the epoch of matter/radiation equality and where the numerical coefficient is derived using a simple spherical infall model for galaxy formation [424]. For the the amplitude of density fluctuations observed in our universe, $Q=10^{-5}$, one obtains the upper limit \rho_{\scriptscriptstyle\Lambda}\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}200\rho_{\scriptscriptstyle\Lambda}(obs).

This bound on $\rho_{\scriptscriptstyle\Lambda}$ was subsequently generalized to estimate the likely values of the cosmological constant [372]. This updated treatment finds the probability of an observer measuring a given value of $\rho_{\scriptscriptstyle\Lambda}$, conditioned by the requirement that a given universe can produce observers. In this context, the formation of collapsed cosmological structures — essentially dark matter halos — serves as a proxy for the production of observers. Because values of $\rho_{\scriptscriptstyle\Lambda}$ much larger than the observed value (in our universe) tend to shut down structure formation, observers in any universe are likely to measure values of $\rho_{\scriptscriptstyle\Lambda}$ relatively close to the observed value. This conclusion continues to hold with alternate assumptions, for example using the amount of material in stars as a proxy for observers and using more detailed models of structure formation [423, 499].

The conclusion that typical observers will measure a value of $\rho_{\scriptscriptstyle\Lambda}$ roughly comparable to our own follows from assuming that the other cosmological parameters are held constant. Bounds of this type become substantially weaker if the amplitude $Q$ of primordial density fluctuations is allowed to vary [25, 363, 383]. Specifically, the bound of equation (71) scales as $Q^{3}$. For each factor of ten increase in $Q$, the limit becomes weaker by 3 orders of magnitude. Since $Q\sim 10^{-5}$ in our universe, larger values are not only possible, but likely [226] (see also Section 3.4). Constraints on the maximum allowed values of the amplitude $Q$ follow from requiring structure formation produce galaxies with favorable properties (see Section 6). Universes with larger $Q$ tend to form denser galaxies, which can disrupt habitability by scattering planets [510, 511] and by producing strong background radiation fields [13]. Nonetheless, the amplitude $Q$ can become as large as $Q\sim 10^{-2}$ without compromising habitability altogether. This increase in $Q$ allows the bound on $\rho_{\scriptscriptstyle\Lambda}$ to become a billion times less constraining. One thus obtains a limit of the form \rho_{\scriptscriptstyle\Lambda}\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}10^{11}\rho_{\scriptscriptstyle\Lambda}(obs).

The discussion thus far assumes that the transition between the radiation dominated and matter dominated eras occurs at the same epoch as in our universe. In general, however, the baryon to photon ratio $\eta$ can vary from universe to universe, although its value is constrained by the requirement that the universe emerge from the epoch of Big Bang Nucleosynthesis with an acceptable chemical composition (Section 5). Larger values of $\eta$ lead to earlier matter domination, and thereby give density fluctuations a longer time to grow. This generalization thus allows a universe to have even larger vacuum energy densities without compromising structure formation. Including this effect, the bound on $\rho_{\scriptscriptstyle\Lambda}$ takes the form

[TABLE]

where $A$ is a dimensionless constant of order unity and $a_{R}$ is the radiation constant. Since the baryon to photon ratio $\eta$ can increase by (at least) three orders of magnitude without rendering the universe inhospitable (see Section 5 and Figure 17), the bound on $\rho_{\scriptscriptstyle\Lambda}$ becomes weaker by an additional factor of one trillion [16].

The bounds discussed in this section are summarized in Figure 12, which plots the maximum allowed energy density $\rho_{\scriptscriptstyle\Lambda}$ of the vacuum as a function of the amplitude $Q$ of the density fluctuations. The allowed values of $Q$ (see Section 6) correspond to the area between the shaded regions on either side of the diagram and below the curves. Larger values of the fluctuation amplitude (right side) allow the universe to produce denser galaxies [424], which lead to solar system disruption [510, 511, 13] and even catastrophic black hole formation [444]. Smaller values of the fluctuation amplitude (left side) lead to rarefied galaxies, with gas that is too diffuse to cool, condense, and make structures [449, 510]. The red curve shows the bound corresponding to universes with the same matter content $(\eta,\Omega_{\rm dm}/\Omega_{\rm b})$ as our universe. The star symbol shows the location of our universe in the diagram. With the measured value of $Q$, $\eta$, and other properties of our universe, the vacuum energy density cannot be much larger than observed and still allow for structure formation to occur. Much larger values are possible for larger fluctuation amplitudes $Q$, where $\rho_{\scriptscriptstyle\Lambda}$ can be larger than the observed value by a factor of $\sim 10^{10}$ and still allow for structure formation.

If the baryon to photon ratio $\eta$ is increased, the bound on $\rho_{\scriptscriptstyle\Lambda}$ becomes substantially weaker, as shown by the blue curves in Figure 12. Considerations of BBN (Section 5) show that $\eta$ can become as large as $\eta=10^{-6}$ without rendering the universe uninhabitable [16]. In the extreme case with the largest allowed values of $(Q,\eta)$, the energy density of the vacuum would be larger than the value in our universe by a factor of $\sim 4\times 10^{22}$ (which lies off the top of the figure). Although such universes might avoid catastrophic black hole formation (with $Q<0.01$), the resulting galaxies would be much denser than those in our universe, and only a fraction of the galactic environment would allow for habitable planets (see Section 6). The upper bound on $\rho_{\scriptscriptstyle\Lambda}$ thus depends on the minimum number of habitable planets that are necessary for a habitable universe. As a result, a conservative (optimistic) approach allows $\rho_{\scriptscriptstyle\Lambda}$ to be larger than its observed value by a factor of $10^{10}$ ($10^{22}$). The corresponding allowed increase in the energy scale $\lambda$ (defined in equation [45]) is approximately a factor of 300 ($3\times 10^{5}$).

Notice also that as $\rho_{\scriptscriptstyle\Lambda}$ (equivalently $\lambda$) increases, the total mass contained within the late-time cosmological horizon decreases (see A). The number of stellar mass units inside this horizon is given by equation (241) and approaches unity when $\lambda\to m_{\rm p}$, i.e., when the energy scale of the vacuum is larger than the value in our universe by a factor of $\sim 4\times 10^{11}$. For the approximate limits on $\lambda$ given above, the number of stellar mass units within the horizon decreases from $N_{\star}\sim 10^{23}$ in our universe down to only about $N_{\star}\sim 10^{18}$ ($10^{12}$). As a result, for the largest possible contribution of dark energy consistent with structure formation, the horizon mass decreases to that of a large galaxy.

The hierarchical fine-tuning of the cosmological constant arises because the observed energy scale of the vacuum is small compared to the Planck scale. One way to make this problem less severe is for the universe to have a smaller Planck mass [16], corresponding to stronger gravity. Considerations of BBN (Section 5) and stellar structure (Section 7) allow for the gravitational constant to be larger by a factor of $\sim 10^{6}$, so that the Planck mass could be one thousand times smaller. Another resolution of the hierarchical fine-tuning issue would result if the possible energy scales (or energy densities) of the vacuum are distributed in a log-uniform manner (see E).

The discussion thus far considers only positive values $\rho_{\scriptscriptstyle\Lambda}>0$. As discussed above, however, contributions to this term can come from both geometric and matter sources. As a result, a negative contribution from the geometric side could be large enough to overcome the matter contribution, and thereby produce a cosmological constant term such that $\rho_{\scriptscriptstyle\Lambda}<0$. In this scenario, the vacuum energy acts to slow down the cosmic expansion and does not impede the formation of structure. On the other hand, the entire universe could collapse to a singularity in a finite time. For example, the total lifetime $t_{\rm s}$ for a flat ($k=0$) universe containing matter and a negative cosmological constant is given by

[TABLE]

In order for the universe to be habitable, it must live for a sufficiently long time [61, 545]. This lower limit on the cosmic lifetime $t_{\rm s}$ places an upper limit on the magnitude of the vacuum energy density. Although the time required habitability is not known, a minimum of $\sim 1$ Gyr is often invoked [141, 377, 408] (see also the discussion in Section 7). This choice leads to a bound on negative vacuum energy of the form

[TABLE]

where $\rho_{\rm crit}$ is the critical density in our universe today. Negative values of the cosmological constant are thus more constrained than positive values (compare with Figure 12).

5 Big Bang Nucleosynthesis

In order for a universe to become habitable, it must successfully produce complex nuclei. In our universe, nucleosynthesis occurs in multiple settings, including the early universe, stellar cores, supernova explosions, and spallation in the interstellar medium. In the first few minutes of its history, our universe processes about one fourth of its baryonic material into helium-4 during an epoch known as Big Bang Nucleosynthesis (BBN). Although stars also make helium, this early processing period is significant because BBN produces an order of magnitude more helium than the stars — and such a large helium-4 abundance is observed. In addition, BBN produces trace amounts of other light nuclei, especially deuterium, helium-3, and lithium-7. These light elements are generally not synthesized in stars, but rather are destroyed in stellar cores. By explaining the observed abundances of these isotopes, BBN thus provides an important confirmation of Big Bang theory [485, 542, 543] (see also [152] for a recent assessment).

On the other hand, helium-4 is readily produced in stars, and none of other nuclear species produced during BBN are known to be crucial for the eventual development of life. As a result, BBN does not represent a necessary constructive ingredient for a successful universe. Instead, BBN provides a constraint: It is important that the early universe does not emerge from its BBN epoch with an unacceptable chemical composition. The universe could end up sterile if it processes all of its protons and neutrons into heavier nuclei, thereby leaving no hydrogen behind to make water. Unfortunately, the minimum mass fraction of hydrogen required for habitability is not known. In this section, we review how different realizations of the basic parameters lead to different nuclear compositions following the BBN era.

The preceding discussion implicitly assumes that water is a necessary ingredient for the development of life. For completeness, one should keep in mind that that alternate solvents for biological operations are possible [368, 465], where ammonia (NH3) is a leading candidate [260]. Ammonia is a polar molecule, is made of common elements, supports numerous chemical reactions, and can dissolve most organic molecules. Methane and other hydrocarbons are also sometimes considered in this role [378]. However, ammonia, methane, and other hydrocarbons all require hydrogen (protons) as basic building blocks, so that the constraints on BBN remain, namely, to leave behind an ample supply of free protons.

5.1 BBN Parameters and Processes

We begin with a brief assessment of the parameters that play a role in BBN. The universe at this early epoch is radiation dominated, with the expansion governed by equation (33) with $\rho=\rho_{\rm R}$ and $k=0$. The gravitational constant $G$ (equivalently $\alpha_{G}$) determines the expansion rate, where $H={\dot{a}}/a$ $\propto G^{1/2}$. Baryons are a trace constituent of the universe at this time and are specified by the baryon to photon ratio $\eta\ll 1$ (where $\eta\approx 6\times 10^{-10}$ in our universe). The remaining parameters determine the nuclear reaction rates. The weak interactions play an important role by maintaining equilibrium early in the process (T\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}1 MeV, t\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}1 sec), and by allowing free neutrons to decay ($t\sim 1000$ sec). At the simplest level of description, the weak rates are set by the Fermi constant $G_{F}$. A full specification of BBN parameters requires the cross sections $\sigma_{jk}$ for all of the relevant nuclear reactions and the binding energies $B_{k}$ for all of the relevant nuclear species. The interaction rates ultimately depend on the strengths of the strong, weak, and electromagnetic forces. Unfortunately, however, no simple transformation exists between the fundamental parameters of the Standard Model (see Section 2) and the quantities that appear in numerical treatments of BBN [243, 485, 542, 543].

Although a detailed treatment of the BBN epoch requires the inclusion of dozens of nuclear reactions, and hence a full numerical treatment [485, 542], it is instructive to consider a simplified description [327]. At early times, the background temperature of the universe is above that of nuclear binding energies, and the weak interaction rate is fast enough to maintain equilibrium. As result, the nuclear species are in Nuclear Statistical Equilibrium (NSE), which determines their abundances as a function of temperature. The ratio of neutron density to proton density is of particular importance, and can be written in the form

[TABLE]

where the mass difference $\Delta{m}=m_{\rm n}-m_{\rm p}\approx 1.29$ MeV is a key parameter (see also Section 2.2.7). Note that the chemical potentials of the electron and neutrino are assumed to be small so that $(\mu_{e}-\mu_{\nu})/T\ll 1$.

Once the cosmic age reaches $t\sim 1$ sec and the temperature falls below $T\sim 1$ MeV (in our universe), the weak interaction rate becomes slower than the expansion rate, and the nuclear species fall out of equilibrium. In general this transition occurs when the weak interaction rate $\Gamma_{\rm w}$ becomes smaller than the expansion rate $H$. In the regime $T\gg\Delta{m}\sim m_{\rm e}$, the rates become equal [327] when $\Gamma_{\rm w}=H$, i.e.,

[TABLE]

In our universe, the temperature scale $T_{\rm wf}\sim 1$ MeV. After the temperature falls below this threshold, the neutron to proton ratio stops tracking its NSE value. At this juncture, the abundance ratio $n/p\sim 1/6$, but it continues to decrease slowly due to continued weak interactions and neutron decay.

With its large binding energy and low atomic number, helium-4 is the most abundant nuclear species produced during BBN. As a result, the most likely scenario where BBN leaves behind a sterile universe is for essentially all of the baryons to be processed into helium-4. Because the neutrons are outnumbered by the protons (in our universe) and interact readily due to their lack of charge, almost all of the free neutrons become locked up in helium-4 during the BBN epoch. A good estimate for the corresponding helium mass fraction $Y_{4}$ is thus given by

[TABLE]

where the subscripts refer to atomic mass numbers. In this expression, the neutron to proton ratio is evaluated at the time/temperature when most of the helium is produced. In our universe, for example, $T\sim 0.1$ MeV at this epoch, and the ratio $(n/p)_{\rm bbn}\sim 1/7$ is slightly smaller than its earlier value $(n/p\sim 1/6)$ when NSE was compromised. The corresponding helium mass fraction $Y_{4}\approx 0.25$. Note that the final version of this expression (77) generalizes the more familiar form [327] to include scenarios where the neutron abundance is greater than the proton abundance [245].

The simplest recipe for making a sterile universe thus requires $Y_{4}\to 1$, which in turn requires $(n/p)_{\rm bbn}\to 1$. Since the densities of neutrons and protons follow their NSE values until weak interactions become slower than the cosmic expansion rate, and since $n\approx p$ at high temperatures, a universe will be left with more neutrons — and greater helium abundance — as the weak interaction is made weaker [266]. Notice also that smaller values of the mass difference $\Delta{m}$ will also lead to more nearly equal populations of protons and neutrons (equation [75]). However, the condition $(n/p)_{\rm bbn}\to 1$ is necessary but not sufficient for a universe to process all of its baryons into helium and heavier elements. The interaction rates must be high enough for helium production to proceed to completion before the universe becomes too cool and diffuse. As a result, a viable regime of parameter space always exists for sufficiently small values of the baryon to photon ratio $\eta$ (see equation [79] below).

It is useful to identify the approximate conditions required for BBN to operate in a regime similar to that of our universe, where some light nuclei are produced, but not all of the protons are transformed into helium and heavier elements [122]. Given the equilibrium $n/p$ ratio from equation (75) and the approximate helium yield of equation (77), the interesting regime of BBN requires that the weak interaction rate competes with the cosmic expansion rate (see equation [76]) at a temperature roughly comparable to the mass differences between the light nuclei. The mass difference $\Delta{m}$ between the proton and neutron, as well as the binding energies of deuterium and helium-3, are of order 1 MeV, so that $T\sim 1$ MeV $\sim m_{\rm e}$. The requirement for interesting BBN thus becomes

[TABLE]

In our universe, the left hand side of the expression has a value $\sim 0.22$, relatively close to unity. For stronger versions of the weak interaction (larger $G_{F}$), decoupling of weak interactions occurs later (at lower temperature). The neutron to proton ratio follows NSE for a longer time and the fraction of neutrons becomes lower. With a diminished supply of neutrons, the universe supports fewer nuclear reactions, and BBN yields decrease in general [285]. In this limit, the universe could emerge from BBN with little nuclear processing, but this condition does not preclude stellar nucleosynthesis at later epochs, and the universe can remain viable.

In the opposite limit, where the weak force is less effective, the $n/p$ ratio is frozen out early and approaches unity. The neutron lifetime is long, so that neutron decay does not occur until well after the BBN epoch. In this regime, the helium fraction (from equation [77]) will approach unity if the nuclear reaction rate is fast enough. To prevent overproduction of helium, a universe needs sufficiently low density given by the constraint

[TABLE]

where all quantities are evaluated at the temperature (time) when the light elements are made. As a result, universes can avoid processing all of their neutrons into helium if the baryon to photon ratio is small enough. Even a universe with no weak interactions can remain viable [245, 266, 285], with 10% of the nucleons left over, provided that \eta\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}10^{-10} (see Section 5.3). In this weakless limit, one needs $\eta\sim 4\times 10^{-12}$ to maintain the same hydrogen content as our universe (where $\eta\approx 6\times 10^{-10}$).

For completeness, note that most treatments of BBN consider the universe to be completely homogeneous during the relevant span of time. This assumption is reasonable, given that our universe must evolve into its present state at a much later epoch. In other universes, however, inhomogeneities could be present during BBN [556] and would change the abundance patterns [33, 300]. Although this present discussion focuses on results for homogeneous BBN, the possibility of density variations should be kept in mind.

5.2 BBN Abundances with Parameter Variations

This section considers the abundances of the light elements produced during Big Bang Nucleosynthesis as the input parameters are varied. This ensemble of results illustrates the difficulty in rendering the universe sterile by using up all of the available protons (leaving none behind for water production). Here we present results calculated using the Kawano code [485] as well as the more recent BURST code [243, 244], both of which are descendants of the pioneering work of Wagoner [542].

The numerical codes are set up so that variations can be made for the baryon to photon ratio $\eta$, the gravitational constant $G$, and the neutron lifetime $\tau_{n}$. One of the key results from BBN studies is the determination of the baryon to photon ratio, which has a value of $\eta\approx 6\times 10^{-10}$ for our universe. In addition, the expansion rate can be varied and is equivalent to changes in the value of the gravitational constant $G$ and/or the number of light neutrino species. These codes also allow for variations in the neutron lifetime $\tau_{n}$ because its value (in our universe) was not well measured until recently. Finally, we also consider variations in the fine structure constant $\alpha$. As outlined below, this latter result is more approximate.

5.2.1 Variations in the Baryon to Photon Ratio

First we consider variations in the baryon-to-photon ratio $\eta$, where the results are shown in Figure 13. Historically, plots of this type have been made with the goal of determining the value of $\eta$ that is consistent with the primordial abundances of all of the light elements shown. The current estimated value is $\eta\approx 6\times 10^{-10}$, although some uncertainty arises due to the abundance of lithium. In this context, we are interested in what input parameters could potentially disrupt the future operations of the universe. As illustrated in Figure 13, however, the baryon to photon ratio can vary over many orders of magnitude and still allow the universe to have an acceptable chemical composition.

Figure 13 shows that for sufficiently small values of $\eta\sim 10^{-12}$, the mass fraction of helium-4 becomes smaller than that of deuterium. As $\eta$ increases, the nuclear reaction rates increase, and the mass fraction of helium-4 increases accordingly. The abundances of deuterium and helium-3 decrease with increasing $\eta$, as they are burned into helium-4. Note that in the limit of large $\eta$, the mass fraction of helium-4 reaches a limiting value of $Y_{4}\sim 0.3$. This limit corresponds to the regime where essentially all of the available neutrons are burned into helium-4. Notice also that the abundances of deuterium and helium-3 decrease with increasing $\eta$. With most of the neutrons incorporated into helium-4, few are left for the remaining light elements. Increasing values of $\eta$ also facilitate the burning of the lighter elements into heavier ones, which in turn leads to a slow increase in the abundance of lithium-7. Even with a thousand-fold increase in $\eta$, however, lithium remains a trace element after the BBN epoch.

5.2.2 Variations in the Gravitational Constant

Variations in the gravitational constant result in corresponding changes in the expansion rate of the universe, where $H\propto$ $G^{1/2}$. Figure 14 shows the results for varying the gravitational constant over a range of six orders of magnitude, from 100 times smaller than in our universe to $10^{4}$ times larger. For most of the range shown, the abundances of all of the light elements increase with $G$. For small values of $G$, with a slow expansion rate, the freezing out of weak interactions occurs later in cosmic history. As a result, protons and neutrons remain in NSE longer and the $n/p$ ratio is smaller. As $G$ increases, the expansion rate increases, freeze-out occurs earlier, and the $n/p$ ratio is larger. Since most of the neutrons are processed into helium-4, its abundance grows with increasing $G$. The abundances of deuterium and helium-3 also increase. For sufficiently large values of $G$, however, the expansion rate is so fast that not all of the neutrons can be made into helium-4. As a result, the mass fraction of helium-4 has a maximum value of $Y_{4}\sim 0.53$, which occurs at $G/G_{0}\sim 100$.

5.2.3 Variations in the Neutron Lifetime

Next we consider variations in the neutron lifetime $\tau_{n}$, which can be written in the form

[TABLE]

where $G_{F}$ is the Fermi constant, $m_{e}$ is the electron mass, $g_{A}\approx 1.26$ is the axial-vector coupling for nucleons, and the dimensionless parameter $\lambda_{0}\approx 1.636$ [327]. Variations in the neutron lifetime are thus equivalent to variations in the Fermi constant, which has a value of $G_{F}\approx$ (293 GeV)-2 in our universe.

Figure 15 shows the resulting abundances of the light elements as a function of $\tau_{n}$, which varies from 1 to $10^{6}$ sec (recall that $\tau_{n}\approx 885$ sec in our universe). As a rule, the abundances of all of the relevant nuclear species increase with increasing $\tau_{n}$, which corresponds to decreasing strength of the weak force. In the limit of small $\tau_{n}$, which corresponds to the weak force being stronger than in our universe, all of the abundances are low. In this limit, the weak force competes with the expansion of the universe down to low temperatures where the neutron to proton ratio (at freeze-out) becomes extremely small. The universe thus contains few neutrons due to both thermodynamic considerations and due to their rapid decay. With low neutron abundance, the universe can make few nuclei.

In the limit of long neutron lifetime $\tau_{n}$, the weak force is unable to keep up with the expansion of the universe even at early times and the freeze-out of weak interactions occurs at high temperature $T\gg\Delta m\sim 1$ MeV. In this limit, the freeze-out value of the neutron to proton ratio approaches unity. Since the neutron lifetime is long, nearly all of the neutrons can be incorporated into helium-4 before they decay. Moreover, because $p\approx{n}$ and $A=2Z$ for helium-4, nearly all of the protons are used up as well. Keep in mind that the abundances shown in Figure 15 are plotted on a logarithmic scale. Although $Y_{4}$ becomes large for the largest value $\tau=10^{6}$ sec considered here, the mass fraction of helium-4 is only $Y_{4}\approx 0.90$. Such a universe still retains 10% of its mass in protons, which are available for making water and for undergoing nuclear fusion in stars at later epochs (see also the discussion of Ref. [263]). This issue of overproduction helium-4 arises in the weakless universe [266], where $G_{F}\to 0$ and $\tau_{n}\to\infty$. As pointed out by the authors, universes with no weak interactions (or extremely long $\tau_{n}$) can avoid fusing all of their nucleons into helium-4 if they have a much smaller baryon to photon ratio $\eta$ (see Section 5.3).

The abundance trends shown in Figure 15 exhibit a relatively sharp transition between universes with little nuclear processing (left side of the figure) and those that emerge from the BBN epoch with a primarily helium composition (right side). This transition occurs near $\tau_{n}\sim 100$ sec, which is close to the observed value in our universe ($\tau_{n}=885$ sec). This proximity has been found previously [263], which also notes that the boundary can be expressed in terms of the value of $G_{F}$ or the mass of the $W$ boson. Notice also that the value of the Fermi constant $G_{F}=1/(\sqrt{2}{\cal V}^{2})$, where ${\cal V}$ is the vacuum expectation value of the Higgs. Changes in ${\cal V}$ can lead to changes in the masses of the fundamental particles, which in turn alter the structure of nuclei. The results shown in Figure 15 correspond to changes in the neutron lifetime only, but many other scenarios are possible (e.g., with different values of the Yukawa couplings that determine the masses of quarks and leptons, especially $m_{\rm u}$, $m_{\rm d}$, and $m_{\rm e}$).

5.2.4 Variations in the Fine Structure Constant

This section considers the effects of varying $\alpha$. Note that variations in the fine structure constant have two effects on nucleosynthesis. First, the value of $\alpha$ determines the size of the coulomb repulsion factors that appear in the nuclear reaction rates. The second effect is that, at the fundamental level, the value of $\alpha$ will determine, in part, the binding energy of the proton and neutron, and hence the mass difference $\Delta m$. This mass difference, in turn, affects the proton to neutron ratio during the BBN epoch. Unfortunately, it remains difficult to calculate this mass difference from first principles at the present time. The value of $\alpha$ also affects the binding energies of the composite nuclei being produced (such as helium-4). In our universe, the strong force has the dominant contribution to the binding energies. If we use the semi-empirical mass formula as a guide, then $\alpha$ must be increased (relative to the strong force) by an order of magnitude to compromise nuclear structure.

Unfortunately, no comprehensive studies of BBN currently include all of the effects outlined above. In the absence of such results, here we illustrate part of the issue by including variations in $\alpha$ into the coulomb repulsion factors in the nuclear reaction rates. In this treatment, the mass difference $\Delta m$, the nuclear binding energies, and the coefficients in the reaction cross sections all remain constant.

The effects of variations in the fine structure constant are shown in Figure 16, which plots the light element abundances over a range in $\alpha$. For large values of $\alpha$, the coulomb repulsion effects are strong, so that the abundances of all of the light elements except deuterium are suppressed. The deuterium abundance grows with increasing $\alpha$ because its production channel ($n+p\to d+\gamma$) is unchanged, whereas the reactions that lead to its destruction are suppressed. In fact, the deuterium abundance becomes much larger than that of helium-4 for sufficiently large values of the fine structure constant, namely $\alpha/\alpha_{0}>7$. As $\alpha$ is decreased, thereby removing the coulomb barrier for nuclear reactions, the helium-4 abundance increases. However, it reaches an asymptotic value $Y_{4}\sim 1/3$, which corresponds to essentially all of the available neutrons being incorporated into helium-4. Significantly, over the range of values shown in Figure 16, all of the universes remain viable.

5.2.5 Variations in both $G$ and $\eta$

The discussion thus far has considered the variation of only one parameter at a time. To illustrate the effects of multiple parameters, consider the case where both the baryon to photon ratio $\eta$ and the gravitational constant $G$ are allowed to vary. In order for the universe to remain habitable, the abundance of protons cannot become too small. We thus consider contours of constant hydrogen (proton) mass fractions in the $\eta$-$G$ plane. Figure 17 shows the result (from [16]) for an ensemble of BBN simulations performed with the BURST code [243, 244]. The plane of parameters in the figure spans a factor of a million in $\eta$ and a factor of 100 million in $G$. The proton mass fraction remains above 10% over almost the entire plane, with the exception of the extreme upper right corner (where G\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}3\times 10^{5}\,G_{0} and \eta\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}10^{-7}).

5.3 BBN without the Weak Interaction

As outlined above, universes can remain habitable in the face of extreme variations in the baryon to photon ratio $\eta$ (Figure 13), the gravitational constant $G$ (Figure 14), the neutron lifetime $\tau_{n}$ (Figure 15), and the fine structure constant $\alpha$ (Figure 16). The only parts of parameter space where the proton abundance becomes dangerously low is for extremely large values of $\eta$ and for long neutron lifetimes $\tau_{n}$. The latter regime arises when the weak force becomes substantially weaker than that of the Standard Model. Here we consider separately the case of the “weakless” universe [266], where the weak interaction is absent so that $\tau_{n}\to\infty$. By adjusting the parameters of particle physics and cosmology, Ref. [266] shows that one can obtain universes with properties that are roughly similar to our own (cf. [146]).

The BBN epoch plays out somewhat differently in a universe without weak interactions. In standard cosmology, weak interactions occur rapidly enough to maintain equilibrium at early times, and then later become slow compared to the expansion rate. In a weakless universe, by definition, weak interactions can never maintain equilibrium. As a result, the neutron to proton ratio is not determined by NSE (as in equation [75]) and is specified by other physics. If equal numbers of up and down quarks are produced during the earlier epoch of baryogenesis, then one expects $n/p\approx 1$ (see [266] for further discussion). In this case, all of the nucleons can be readily incorporated into helium-4, provided that the strong interactions are sufficiently rapid. In order to avoid overproduction of helium-4, and hence an accompanying hydrogen shortage, a smaller baryon abundance is necessary. For example, the value of $\eta\approx 4\times 10^{-12}$ produces a mass fraction of helium-4 that is comparable to that of our universe, $Y_{4}\approx 0.25$ [245, 266].

The possible ranges of the helium yields from Big Bang Nucleosynthesis in the weakless universe are shown in Figure 18 (from [245]) over a wide parameter space. The figure shows contours of constant helium mass fraction $Y_{P}$ in the $(\eta,n/p)$ plane, where $\eta$ is the baryon to photon ratio and $n/p$ is the neutron to proton ratio. Note that for the weakless universe, the ratio $n/p$ is determined by processes in the early universe, well before the start of the BBN epoch. One must thus consider a range of values as shown. The helium abundance arising from BBN has an acceptable level over most of the plane. For a universe that is both weakless and symmetric ($n/p\sim 1$), the preferred value of the baryon to photon ratio $\eta$ is somewhat smaller than the value in our universe. Although the baryon to photon ratio $\eta$ required to produce the helium mass fraction of our universe is only $\eta\approx 4\times 10^{-12}$, any value of \eta\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}10^{-10} will allow for Y_{P}\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}0.90 and hence a working universe.

In the weakless universe, the mass fraction of helium-4 is an increasing function of the baryon to photon ratio $\eta$. More specifically, the weakless universe emerges from its BBN epoch with mass fraction $X_{4}\approx$ 0.45, 0.90, and 0.99 for $\eta$ = $10^{-11}$, $10^{-10}$, and $10^{-9}$. The remaining baryonic mass is (relatively) evenly distributed among free protons, free neutrons, and deuterium [245], with trace quantities of tritium and helium-3. Weakless universes thus produce substantial amounts of deuterium, with mass fractions of order $10^{-1}$ (compared to $\sim 10^{-5}$ for standard BBN). In addition, the free neutrons that survive the BBN epoch can fuse with protons in dense regions of the interstellar medium, thereby producing more deuterium. At later times, the resulting inventory of deuterium provides an important fuel for stars: Without the weak force, free protons cannot fuse to make deuterium (through $pp\to{d}e^{+}\nu_{e}$), so that the standard $p$-$p$ chain for stellar nucleosynthesis is unavailable. Instead, stars must either burn free protons and any remaining free neutrons into deuterium or fuse the deuterium into helium-4 (see [245] and Section 7).

6 Galaxy Formation and Large Scale Structure

Galaxies are vital constituents of a habitable universe. They collect and organize the interstellar gas that is required to form stars, which provide the energy and heavy nuclei necessary for biological development. Of equal importance, galaxies must be massive enough to retain the heavy elements produced by stellar nucleosynthesis. In order to fulfill these roles, galaxies must first be able to form.

Galaxy formation involves a number of physical processes (see [71] for a recent review and [365] for a general overview). In our universe, the mass budget is dominated by weakly interacting dark matter. The first step is for the dark matter to decouple from the expanding background universe and begin to collapse into halos [168]. As outlined in Section 4, the collapse process must become sufficiently advanced before the epoch of vacuum domination, where this requirement leads to an upper limit on $\rho_{\scriptscriptstyle\Lambda}$ for a given matter inventory of the universe (see equation [72], Figure 12, and Refs. [16, 206, 372, 383, 545, 53]). The next requirement is that the baryonic component of the universe must be able to cool [248, 449, 553], so that the gas condenses into structures analogous to galactic disks. These cool and compact configurations are necessary for the subsequent formation of stars. Along with the initial collapse of dark matter into halos and condensation of cooling gas, dynamical processes sculpt the structure of the forming galaxies. Smaller halos merge together to form larger ones, while individual galactic structures undergo dynamical relaxation. The net result of these dynamical interactions is to produce dark matter halos with a nearly universal form [394]. The baryonic component, consisting of both stars and gas, forms spherical and disk-like structures [77, 78] that live at the halo centers.

In order for galaxy formation to be successful, the resulting structures must meet a number of constraints. The density fluctuations that produce the galaxies must become nonlinear, and detach from the expanding background, before the universe becomes vacuum dominated. The galaxies must be dense enough so that gas can cool, condense, and eventually form stars, yet remain rarefied enough that solar systems can survive. Finally, the galactic mass scales must be large enough to form many stars and have deep enough gravitational potential wells to retain the heavy elements produced by stellar nucleosynthesis. To assess the viability of galaxies in other universes, we thus need to determine their masses, densities, and other properties as a function of the cosmological parameters.

6.1 Mass and Density Scales of Galaxy Formation

Density fluctuations are produced in the ultra-early universe, but do not begin to grow into galaxies until the energy density becomes dominated by matter. At this epoch, when matter and radiation have equal energy densities, the cosmic background temperature is given by

[TABLE]

and the corresponding age of the universe is

[TABLE]

The total mass contained within the cosmological horizon at this epoch represents an important scale and can be written in the form

[TABLE]

where we have used $\eta=6\times 10^{-10}$ and $\Omega_{\rm M}/\Omega_{\rm b}=6$ to obtain the numerical values.

All of the mass scales smaller than the horizon mass $M_{\rm eq}$ begin to grow at the epoch of equality and become nonlinear at a later epoch denoted here as $t_{\rm col}$. The age of the universe at this time is given approximately by the expression

[TABLE]

where $Q$ is the amplitude of the primordial density fluctuations ($Q\approx 10^{-5}$ in our universe). The parameter $f_{\rm vir}<1$ is a dimensionless factor of order unity [510], and varies slowly with the halo mass (for $M\leq M_{\rm eq}$). If density fluctuations on all scales were to grow exactly linearly with the scale factor and start at the moment of matter domination, then we would have $f_{\rm vir}=1$. Notice also that larger mass scales collapse at later times given by $t\approx t_{\rm col}(M/M_{\rm eq})$ [510, 511].

To leading order, the density of the galactic halos is determined by the background density of the universe at the time $t_{\rm col}$ when the halo collapses. After the halo detaches itself from the cosmic expansion taking place in the background, its subsequent collapse proceeds in collisionless fashion, and results in a quasi-virialized structure with characteristic density that is larger than its pre-collapse value by a factor of $f_{\rm col}\sim 18\pi^{2}\sim 200$ (e.g., see [327, 422, 424, 436]). The characteristic density of the halo thus takes the form

[TABLE]

The characteristic density scales with the cube of the amplitude of primordial density fluctuations. However, as emphasized by many authors [372, 510, 511], the scaling depends on additional parameters $\rho_{\rm c}\propto T_{\rm eq}^{4}Q^{3}$, so that $\rho_{\rm c}=\rho_{\rm c}(Q,\eta,\Omega_{\rm M},\Omega_{\rm b})$. Inclusion of the factor $f_{\rm vir}$ leads to the characteristic density $\rho_{\rm c}$ increasing slowly with decreasing halo mass. On the other hand, larger halos with $M>M_{\rm eq}$ collapse later and will have lower characteristic densities, with the approximate scaling $\rho_{c}\propto(M_{\rm eq}/M)^{2}$.

6.2 Structure of Dark Matter Halos

Both numerical simulations and cosmological observations indicate that — to leading order — galactic halos have a nearly universal form. The pioneering theoretical study of this universality [394] found that the density profiles of these halo structures have the form $\rho\propto r^{-3}$ at large radii. However, subsequent work indicates a somewhat steeper slope [104, 105, 390], which arises from extending the analysis farther out in radius and evolving the halos into the future when they reach an asymptotic form (see also [81]). A good working model for galactic halos is provided by the Hernquist profile, originally put forth as a description for galactic bulges, which are analogous collisionless systems. The density distribution of this model has the simple form

[TABLE]

The profile is thus characterized by a density scale $\rho_{0}$ and a length scale $r_{0}$. The density profile of equation (86) can be integrated to obtain a finite mass $M_{0}=2\pi\rho_{0}r_{0}^{3}$. All dark matter halos with masses $M<M_{\rm eq}$ will virialize at approximately the same epoch and will have characteristic densities given by equation (85). As a result, we can make the identification

[TABLE]

For a given halo mass $M$, the corresponding length scale $r_{0}$ is then given by

[TABLE]

In addition to determining halo properties $(\rho_{0},r_{0})$ as functions of the cosmological parameters (equations [85], [87] and [88]), we also need to estimate the structure of the baryonic component of galaxies. The density profile of equation (86) provides a good working approximation to the final collapsed state of a wide variety of collisionless systems [116], including galactic bulges [277], dark matter halos [104, 390, 394], and even young embedded star clusters [437]. Moreover, elliptical galaxies have similar morphologies to bulges [77, 78]. As a first approximation, one can thus use equation (86) as a model for the baryonic component of the galaxy, including stars, albeit with different values of the density and radial scales [13]. Since baryons dissipate energy, the baryonic component collapses to higher densities, so one can write

[TABLE]

where one expects $F={\cal O}(10)$. The baryonic component of the galactic mass budget is a fraction of the total and is given by $M_{\rm b}=(\Omega_{\rm b}/\Omega_{\rm M})M$. With the density scale and mass specified, the radial scale for the baryonic component of the galaxy takes the form

[TABLE]

so that the density profile for the baryonic component of the galaxy can be written in the form

[TABLE]

As a reference point, for the value $\Omega_{\rm M}/\Omega_{\rm b}=6$ found in our universe, the baryonic density scale becomes

[TABLE]

where $\eta_{0}=6\times 10^{-10}$. For Q\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}10^{-2}, the density scale \rho_{\rm b}\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}10^{3}M_{\odot} pc*-3*, comparable to the mean density of a globular cluster [78]. The inner region $\xi<\xi_{\rm b}$, one fourth of the galactic mass, will have higher density and will thus be susceptible to black hole formation.

6.3 Bounds on the Amplitude of Primordial Fluctuations

from Planet Scattering

Galaxies that are too dense cannot support habitable solar systems because passing stars can strip planets out of orbit. As outlined above, large values of the amplitude $Q$ of the primordial density fluctuations lead to higher galactic densities, which in turn leads to more disruption. As a result, the requirement that planetary orbits must survive places a constraint on the allowed values of $Q$ for viable universes.

The first assessment of this effect [510] found that solar system disruption starts to occur when the amplitude $Q$ is only about an order of magnitude larger than that of our universe (see also [511, 363]). These assessments make two assumptions: First, the cross section for disruption of habitable orbits is taken to be the geometric cross section of the orbit, so that $\sigma_{\rm p*}\sim\pi$(1 AU)2. Second, the entire galaxy is assumed to have the (single) characteristic density given by equations (85) and (89). However, this calculation can be generalized [13] by allowing the galaxy to have a range of densities (e.g., with the profile of equation [86]) and by calculating the cross sections for disruption of planetary orbits as a function of encounter speed.

The cross section for disrupting Earth-like planets by stripping them from orbit about their host stars can be calculated from a large ensemble of numerical simulations [338, 347]. In this type of calculation, numerical integrations follow either single stars or binaries as they experience fly-by encounters with solar systems of a given architecture. The passing stars sample the stellar initial mass function as well as the observed distributions of binary orbital parameters. The encounters sample a range of pre-encounter velocities, impact parameters, and the angular variables necessary to specify the geometry of the interaction. All of the variables are then specified using a Monte-Carlo scheme, and numerical simulations are performed for each realization of the variables. The results can then be processed to find the the effective cross sections for any given post-encounter result of interest. Here we need the cross section for disrupting the orbit of a planet that is initially in a habitable orbit, where disruption includes both ejection from the system and increases in eccentricity beyond a specified threshold. The resulting cross section for disruption can be written in the form [13]

[TABLE]

Since dense galaxies have large velocity dispersion $\sim 10^{3}$ km/s, and the cross section decreases with encounter velocity, the value of $\sigma_{\rm p*}$ becomes significantly less than the geometric cross section. As a result, more solar systems survive. The Monte Carlo scheme used to calculate the cross sections samples over a distribution of encounter speeds, centered on a characteristic value (denoted as $v_{c}$). The reported cross section is thus related to the velocity-averaged value such that $\sigma_{\rm p*}=\langle\sigma v\rangle/v_{c}$.

With the interaction cross section specified, the requirement for solar system survival can be written in the form

[TABLE]

where $n_{*}$ is the number density of stars, $v$ is the encounter speed, and $\tau$ is the required lifetime of a habitable planet. Although this latter quantity remains unknown, $\tau=1$ Gyr is often taken as a benchmark value [324, 368, 465]. The number density of stars can be determined from the density profile of baryonic matter (e.g., equation [91]) by specifying the star formation efficiency (denoted here as $\epsilon_{\rm sf}$). Note that both the stellar density $n_{*}$ and the encounter velocity $v$ depend on the location within the galaxy.

Figure 19 shows the fraction of solar systems that can survive — not be disrupted by passing stars — as a function of the fluctuation amplitude $Q$. Survival curves are shown for galaxies with a range of masses, from $M=10^{10}M_{\odot}$ (left side of the diagram) to $M=10^{15}M_{\odot}$ (right side). Galaxies with masses comparable to the Milky Way correspond to the central black curve. The characteristic density of the galaxies increases with increasing $Q$, and the survival fraction decreases accordingly. In addition, the density decreases with increasing galactic mass. Although many solar systems can be disrupted, the survival faction remains above 0.10 (10%) over most of the range of parameter space. The most compromised galaxies are those with the smallest masses, whereas the most compromised universes are those with the largest amplitudes $Q$. The region on the right side of the diagram is shaded to indicate that universes with sufficiently large $Q$ tend to overproduce black holes, which leads to inhospitable galaxies. If the value of $Q$ approaches unity, black hole formation can take place immediately following the inflationary epoch [240]; for somewhat smaller values, $Q=0.01-0.1$, a large fraction of the total mass can be incorporated into black holes during the process of galaxy formation [444]. Note that the survival fraction never reaches zero. Even in the densest galaxies, the outer regions will be diffuse enough to allow for some planets to survive in habitable orbits.

6.4 Constraints from the Galactic Background Radiation

In addition to disruption by passing stars, habitable planets in dense galaxies face another hazard. As a galaxy becomes denser, the radiative flux provided by the background stars becomes more intense. If this galactic background radiation flux is larger than that received by a habitable planet from its host star, roughly comparable to the solar insolation received by Earth, then planets will cease to be habitable in any orbit [13]. This type of radiative disruption can occur in our universe in extreme environments, including the centers of galaxies [211] and sufficiently dense star clusters [519].

Consider, for example, an elliptical galaxy where the baryonic component can be modeled with a profile given by equation (91). At the dimensionless radial location $a=\xi_{\rm b}$ within the galaxy, the background radiation flux $F_{G}(a)$ due to starlight has the form

[TABLE]

where the angular brackets denote an average over the stellar population [13]. In order for habitable planets to survive with temperate climates, the background radiation flux from the galaxy $F_{G}$ must be smaller than the flux received by a habitable planet from its host star, i.e.,

[TABLE]

where $\varpi$ is the radius of the planetary orbit and $L$ is the luminosity of the host star. In the second approximate equality, $S_{\oplus}$ denotes the solar radiation flux received at Earth orbit. Since the value of the flux directly determines the surface temperature of the planet, the requirement of liquid water constrains the flux from the host star ($L/4\pi\varpi^{2}$) to be roughly equal to $S_{\oplus}$. The galactic flux $F_{G}$ is a strictly decreasing function of the radial variable $a$. The central regions of almost every galaxy will support radiation fields that violate the constraint of equation (96), leading to a situation analogous to that raised by Olber’s famous paradox [267, 552]. Conversely, the outer regions of every galaxy will be cool and dark enough to allow for habitable planets. This compromise is similar to that found earlier for disruption by scattering encounters. The fraction of the solar systems that survive in the face of strong radiation fields is comparable to, but somewhat larger than, the fraction that survive devastation by scattering events [13].

The presence of strong radiation fields in galaxies provides another possible channel for planets to be habitable. Although radiation in galactic centers will be too intense for planets to remain viable, and the outer regions will be cold like in our universe, there exists an intermediate zone where the background galactic flux is comparable to the solar flux, i.e., $F_{G}\sim S_{\oplus}$. Any planets residing in this intermediate regime can have appropriate surface temperatures for nearly any orbit (see Figure 20 for a schematic depiction of galactic structure in this scenario). The planets that reside too close to their host stars will be too hot, but all other planets will be warmed by the galactic background radiation and can have surface temperatures that support liquid water environments. The size and location of this Galactic Habitable Zone (GHZ) depend on the dimensionless ratio

[TABLE]

For values of the parameter in the range $X=1/2-100$, more than 10% of all solar systems reside in the GHZ. For the optimal case where $X\approx 3$, the GHZ contains approximately one fourth of the solar systems in the galaxy. If most stars have multiple planets, then galaxies in such universes could support more habitable planets than our own. As a result, our universe is not optimized for the support of habitable planets.

The value of the parameter $X$ is primarily determined by the column density of stars in the galaxy $N_{\ast}=n_{\ast}r_{\rm b}$, which must depend on the amplitude $Q$ of the primordial density fluctuations. The other quantities appearing in equation (97) are either constant or slowly varying, and depend on stellar (rather than galactic) properties. For galaxies like the Milky Way in our universe, the column density of stars $N_{\ast}\sim 10^{2}-10^{3}$. To allow the parameter $X\sim 3$, the column density must be much larger, $N_{\ast}\sim 10^{9}$. Since galactic density scales with the fluctuation amplitude $Q$ according to $n\sim\rho\sim Q^{3}$, and the galactic length parameter scales as $r_{\rm b}\sim Q^{-1}$, the column density scales as $Q^{2}$. Universes that optimize their galactic habitable zones thus need larger initial density fluctuations compared to our universe, where the preferred value is $Q\approx 10^{-2}$. As discussed above, the central regions of the resulting dense galaxies are susceptible to black hole formation.

The fluctuation amplitude $Q$ that optimizes the GHZ corresponds to an enhancement by a factor of $\sim 10^{3}$ if the other cosmological parameters are fixed. The characteristic density for galaxies scales as $\rho_{\rm c}\propto\eta^{4}Q^{3}$ (see equation [85]). As a result, universes can achieve the same high galactic density if the baryon to photon ratio $\eta$ is larger than that of our universe by a factor of $\sim 180$.

Under the dense conditions of the GHZ, planets and their orbits are subjected to a number of processes that are rare in our universe. First, note that when the composite parameter $X\sim 1$ (from equation [97]), the optical depth for planetary disruption is also of order unity (from equation [94]). As a result, a sizable fraction of the planets residing in the galactic habitable zone will be freely floating, rather than in orbit about a particular star. This condition does not necessarily preclude habitability, but planets would not have the same astronomical cycles — days and seasons — that characterize our calendar. Dynamical relaxation provides another potential issue. With a dense field of potential perturbers, planets (and stars) can random walk around the GHZ through a large number of distant scattering encounters [78]. However, the dynamical relaxation time for stars/planets in the GHZ is expected to be quite long, $t_{\rm relax}\sim 10^{5}$ Gyr. Although this time scale is much shorter than the relaxation time in our Milky Way galaxy, it much longer than the canonical time scale $\sim 1$ Gyr often invoked for habitability. Most planets will thus stay within the GHZ long enough for life to develop.

In addition to dynamical considerations, the dense environment of the GHZ poses other hazards for habitable planets. The supernova rate is expected to be comparable to that of galaxies in our universe. However, habitable planets will be closer to the explosions due to the enhanced stellar density. Since the galactic density scales with the cube of amplitude of the initial cosmological fluctuations, $\rho\propto Q^{3}$, the distance $d$ to the nearest destructive supernova will scale as $d\propto 1/Q$. Supernova can have a number of detrimental effects on potentially habitable planets, including stripping of the ozone layer from the atmosphere due to both cosmic rays and gamma rays. Although the lethal distance for this process is not well determined, estimates fall in the range $d$ = 8 – 50 pc for the atmosphere of Earth (see [229, 382, 483] and references therein). Thicker atmospheres can tolerate greater exposure (see also [144]). Gamma ray bursts present an analogous hazard [516]. Other risks include large fluctuations in the galactic background radiation due to close stellar passages and flaring activity on nearby stars [361]. These hazards arise due to the increased densities of galaxies. Note that galactic densities can also increase with the vacuum energy density $\rho_{\scriptscriptstyle\Lambda}$, as only the denser perturbations can collapse [521], which leads to analogous effects.

For completeness, note that our universe had a brief epoch of cosmic-wide habitability for redshifts 100\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}(1+z)\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}137 when the temperature of the cosmic background radiation was in the range $T$ = 273 – 373 K. During this early epoch, suitable rocky planets could have liquid water on their surfaces for virtually any orbit [366], much like the case of the GHZ. In addition, the concept of the Galactic Habitable Zone has been discussed previously in the context of our Galaxy [235, 236, 360]. This local version of the GHZ is based on the chemical makeup of the Galaxy as a function of radial position, where the mean metallicity for habitability is required to be within a factor of 2 of that of the Sun. In practice, the metallicity $\raisebox{0.86108pt}{--}\kern-5.50003ptZ$ decreases with galactocentric radius, so that the outer regions of the galaxy have low $\raisebox{0.86108pt}{--}\kern-5.50003ptZ$ and are less suited for life. However, the metallicity threshold for viable planets remains unknown. In addition, the degree of disruption provided by the environment of our Galaxy is generally much less severe than that considered above [308].

Supermassive black holes can provide another environment for a dense concentration of habitable planets [440]. In conventional solar systems, relatively few planets can reside within the habitable zone because orbits become dynamically unstable if they are too close together. The minimum orbital spacing for long-term stability is estimated to be $\sim 10R_{H}$, where $R_{H}$ is the mutual Hill radius given by

[TABLE]

Here, $a_{j}$ are the semimajor axes and $m_{j}$ are the masses of the orbiting bodies, and $M_{c}$ is the mass of the central object. For bodies orbiting a supermassive black hole, the central mass $M_{c}$ can be enormous (up to $\sim 10^{9}M_{\odot}$) so that $R_{H}$ is small and the orbits can be tightly spaced. Since black holes themselves have only feeble radiative emission, another power source is required. One can construct a hypothetical system in which many stars orbit the black hole on interior orbits, with an extensive collection of planets orbiting further out. The key feature of this scenario is that one can place a large number of bodies, stars on the inside and planets on the outside, in orbit about the same supermassive black hole.

6.5 Variations in the Abundances of Dark Matter and Baryons

The relative amounts of dark matter and baryons in the universe can affect the timing of galaxy formation and hence the resulting galactic properties. Here we consider the matter content of the universe to consist of two components, baryons and collisionless dark matter. The baryonic component is specified by the baryon to photon ratio $\eta$. For the dark matter, we consider an analogous parameter defined by $\delta=\eta\,\Omega_{\rm dm}/\Omega_{\rm b}$. If the dark matter particles had the same mass as baryons ($m_{\rm p}$), then $\delta$ would be the dark matter to photon ratio (but the mass of the dark matter particles is left arbitrary). In our universe, constraints from BBN indicate that $\eta\approx 6\times 10^{-10}$ and a host of cosmological measurements imply that $\delta\approx 6\eta$ [332]. In other universes, these quantities $(\eta,\delta)$ could have alternate values, which would lead to different properties for galaxies, as outlined below. (For completeness, however, we note that [88] provides an argument for comparable densities of baryons and dark matter; see also [555].)

This discussion roughly follows the arguments presented in [511] (see also references therein). The first consideration is that if the total mass content of the universe is too large, then the epoch of matter domination could occur before Big Bang Nucleosynthesis. As a rough estimate, the energy scale of BBN can be taken to be comparable to the electron mass $m_{\rm e}$, so that these two epochs coincide when

[TABLE]

For earlier matter domination, both BBN and galaxy formation would proceed in dramatically different fashion. Nonetheless, this regime of parameter space is not necessarily ruled out. The previous section shows that the abundances of light nuclei coming out of BBN are generally not problematic, even for values of $\eta$ much larger than that of our universe.

For somewhat smaller matter abundances, large scale structures (e.g., dark matter halos) can detach from the expanding background of the universe and become nonlinear before the epoch of decoupling. The first structures with roughly galactic mass scales become nonlinear for temperatures $T\sim 9Q(\eta+\delta)m_{\rm p}$ (see [511] for further detail). To leading order, the temperature of recombination is determined by solution to the Saha equation, which can be written in the form

[TABLE]

where $x$ is the ionization fraction and $B_{H}=m_{\rm e}\alpha^{2}/2$ is the binding energy of Hydrogen. This expression makes a number of simplifications [327, 422], including the assumption of ionization equilibrium and the neglect of Helium. Because the baryonic component is small $\eta\ll 1$, the relevant solutions correspond to low temperature $T\ll B_{H}$, so that equation (100) is dominated by its exponential term. As a result, the recombination temperature is slowly varying and is given approximately by the expression

[TABLE]

Using this result in conjunction with the requirement for structures to become nonlinear, the boundary in parameter space where structure grows before recombination takes the form

[TABLE]

In our universe, the epochs of recombination and photon decoupling occur after that of matter-radiation equality (the start of the matter dominated era). If the order is reversed, then the initial growth of structure will proceed differently. Instead of dark matter collapsing before the baryons (as in our universe), both matter components would collapse together starting from the epoch of equality. The boundary in parameter space where this crossover occurs has the approximate form

[TABLE]

Although the details of galactic collapse change with the ordering of the epochs of recombination and matter domination, collapse occurs either way.

The above considerations constrain the total matter content of the universe. The properties of galaxies can also be different if the relative contributions of baryons and dark matter are different. If the dark matter dominates the density of baryons by a large factor, then galactic disks will be stabilized. This trend acts to suppress the formation of molecular clouds and hence star formation. Unfortunately, there is not a clean estimate for the maximum value of the ratio $\delta/\eta$ that allows for star formation. However, the working estimate of [511] suggests that galactic disks will evolve differently when the dark matter parameter exceeds the limit

[TABLE]

Similarly, if the dark matter content is much lower than in our universe, then baryons dominate the process of galactic formation. Although galaxies are still able to form, they will be subject to enhanced levels of Silk damping [480], and will wind up with different characteristics. The threshold for this type of behavior is approximately given by

[TABLE]

The constraints of equations (99–105) divide up the possible parameter space for the matter components into distinct regions. The result is shown in Figure 21. In the regime of parameter space above the green curve, matter domination occurs before or during BBN. Although such universes may be able to support life, they would be quite different from our own, and this extreme region is shaded in the figure. The blue curves marks the boundary where structures can become nonlinear before the epoch of decoupling. Above this curve, forming galaxies remain coupled to the background radiation field, at least for the first part of their evolution. The magenta curve marks the boundary where decoupling occurs before the epoch of matter domination. The two diagonal lines in Figure 21 correspond to constant ratios of baryonic mass to dark matter $\eta/\delta$. Above the red curve, dark matter dominates (by a factor of 300). In this regime, dark matter halos provide stability to galactic disks (which are baryonic) and star formation is suppressed. Below the cyan curve (where $\eta=\delta$), baryonic matter dominates and Silk damping [480] will be enhanced. In this regime, the collapse of smaller scale structures will be suppressed. Although galaxies and their dark matter halos will have different properties over the parameter space delineated in Figure 21, the values of both $\eta$ and $\delta$ can vary by many orders of magnitude and still allow for habitable galaxies.

6.6 Gravitational Potential of Galaxies

One important role played by galaxies is that they retain and organize the heavy elements produced by their stellar components. If the heavy nuclei synthesized through stars are lost to the intergalactic medium, they cannot be incorporated into subsequent generations of stars, planets, and possible life forms. In our universe, the smallest dwarf galaxies lose interstellar gas through a combination of tidal and ram pressure stripping, as well as internal feedback from stars. This mass loss leads to inefficient star formation, and lower production rates of heavy elements. Moreover, the efficacy of galactic mass loss depends on the depth of the gravitational potential well.

For the halo density profiles of equation (86), the corresponding gravitational potential is given by

[TABLE]

where the scales $\rho_{0}$ and $r_{0}$ are defined in Section 6.2. Using equations (85), (87), and (88) to specify the parameters $(\rho_{0},r_{0})$, the depth of the potential well for a halo becomes

[TABLE]

The last approximate equality holds for the largest halos near the mass of the horizon at equality. Since the potential scale $\Psi_{0}$ defines the depth of the potential well and hence the escape speed from the galactic center, we have the result

[TABLE]

In our universe, the escape speed from galaxies is typically $v\approx 300$ km/s or $v/c\approx 10^{-3}$. The escape speeds from stellar surfaces are also of this order. This latter quantity sets the scale for stellar winds, supernovae, and other stellar processes involving the dispersal of heavy elements throughout the galaxy (and beyond). As a result, successful retention of heavy elements requires a minimum escape speed and hence a minimum value of $Q$. As a starting point, we take the minimum escape speed to be $\sim 100$ km/s, so that the minimum value of the fluctuation amplitude becomes

[TABLE]

This value is comparable to, but somewhat smaller than, the estimated minimum value of $Q$ based on cooling considerations [510], as addressed in the following subsection.

Notice that the bound of equation (109) is not sharp: Stellar processes will produce a distribution of ejection velocities, and gas stripping will operate with a range of efficiencies, so that decreasing values of $Q$ result in a larger fraction (but not all) of the heavy elements escaping. In addition, galaxies form with a distribution of masses and escape speeds, so that the largest structures could retain gases in universes with smaller amplitudes $Q$. Yet another complication is that ejected material is not necessarily gone forever: Exiled gas could cool and condense back into the galaxy at some later epoch, or even be accreted by neighboring galaxies. Nonetheless, for sufficiently small fluctuation amplitude $Q\ll Q_{\rm min}$, most heavy elements are expected to be lost, leading to a commensurate loss of habitability.

6.7 Cooling Considerations

Another constraint on the properties of galaxies arises from the requirement that galactic gas must cool promptly in order to make stars. In addition to providing an estimate for the masses of viable galaxies [449], this requirement of substantial cooling has been used to place a lower bound on the amplitude $Q$ of primordial fluctuations [510, 511] with the result that Q\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}10^{-6}. If one makes the further assumption that the gas not only cools on the free-fall collapse time scale, but also turns into stars on the same time scale, then an estimate for the star formation rate can be found [90, 91].

The cooling function for primordial (zero metallicity) gas involves a number of physical processes [5, 221]. The dominant mechanism varies with the temperature and density of the system. At high temperatures, bremsstrahlung is the most important process, whereas a full accounting for lower temperatures includes line cooling by neutral hydrogen, helium, and any heavier elements that are present. Both bremsstrahlung and line cooling become inefficient below a temperature $T\sim 10^{4}$ K. Molecular cooling lowers this temperature scale, but not substantially [5, 234, 259, 509]. As a result, gas generally cools down to this benchmark temperature relatively quickly, but further cooling is much slower.

In order to illustrate the effects of cooling, and obtain analytic expressions for the resulting constraint, we simplify the treatment to include only bremsstrahlung. For this extreme limiting case, the basic cooling rate has the form

[TABLE]

where each scattering interaction cools the gas by the energy increment

[TABLE]

The cooling time can then be written in the form

[TABLE]

where $v_{s}=(T/m_{\rm e})^{1/2}$ is the thermal speed of electrons, $\sigma_{T}$ is the Thomson cross section, and $A$ is a dimensionless constant of order unity (e.g., see [122]). The dynamics of the baryonic component of galaxies depends on the ratio of this cooling time scale to the gravitational collapse time

[TABLE]

The density in this expression is the characteristic density for collapsed structures found previously. Equating the cooling rate with the collapse rate implies

[TABLE]

The number density can be expressed in the form $n=\eta\rho/(\eta+\delta)m_{\rm p}$, where the density $\rho$ is specified through equation (85). After writing the cross section $\sigma_{T}$, the cooling energy increment $\epsilon$, and the sound speed $v_{s}$ in terms of fundamental constants, we find

[TABLE]

If we evaluate this expression by assuming that $f=0.01$ (corresponding to one of the denser galaxies) and that the temperature $T=10^{4}$ K (because cooling processes become much less effective for lower temperatures), the lower limit on the fluctuation amplitude becomes

[TABLE]

This limiting value of $Q$ is comparable to that determined using the full cooling curves [510, 511] and is slightly more stringent that the limit obtained by requiring that galaxies retain some fraction of their heavy elements (equation [109]). In approximate terms, we find that galaxies that can successfully cool, condense, and potentially form stars will also have deep enough gravitational potentials to keep the metals produced by stellar nucleosynthesis.

Although successful cooling is necessary for star formation to take place, we note that stars can be produced under a wide range of initial conditions. Stars form in galaxies with markedly different properties, with masses varying by factors of $\sim 10^{8}$ and mean densities varying over a corresponding range. In our universe, the star formation rate became substantial at redshift $z\sim 10$ and peaked at $z\sim 2$ when the universe was about 3.5 Gyr old [371]. Although its current rate is smaller by an order of magnitude, star formation takes place readily at the present cosmological epoch [379, 477], and is expected to continue (at a highly attenuated rate) for perhaps trillions of years [12]. As a result, star formation is not overly sensitive to the conditions provided by the background universe.

7 Stars and Stellar Evolution

Stars play two important roles regarding the habitability of our universe, and presumably others. First, they provide most of the energy [219] that is available to support biospheres on any conveniently situated planets. Second, they forge most of the heavy nuclei necessary for the development of complex structures [520, 522], ranging from planets themselves all the way down to biological entities.

7.1 Analytic Model for Stellar Structure

In order to estimate the range of parameter space that allows for the existence of stars, it is useful to have a working semi-analytic model for stellar structure. Toward this end, we utilize the model of [8, 9], which solves the equations of stellar structure [136, 427, 323, 265] subject to a number of approximations. As one simplification, the physical structure of the star is considered to be a polytrope with index $n$, which varies from $n=3/2$ for fully convective stars to $n=3$ for high-mass (radiative) stars. This approach uses a single reaction rate, which allows for only one nuclear reaction chain at a time. The resulting model [8, 9] reproduces the properties of stars in our universe to a reasonable degree of approximation (tens of percent), but allows for variations in the fundamental constants over a wide range of values (ten orders of magnitude).

Since a full description is given elsewhere [8, 9], this section presents only an outline of the model. The pressure is a function of the density so that

[TABLE]

The density can be expressed in terms of a dimensionless function $f(\xi)$ such that

[TABLE]

The function $f(\xi)$ is a solution to the Lane-Emden equation [136, 427]. The mass of the star can then be written in terms of the dimensionless integral parameter $\mu_{\star}$,

[TABLE]

where the dimensionless radius of the star $\xi_{\star}$ is of order unity. The temperature $T(r)$ within the star specified using the change of variable

[TABLE]

where $m_{j}$ and $Z_{j}$ are the masses and charges of the reactants. Recall that the Gamow energy $E_{G}\approx 493$ keV for hydrogen fusion in our universe.

For a single nuclear reaction chain, the cross section can be separated into factors through the ansatz

[TABLE]

where $S(E)$ is a slowly varying function of energy. The model then defines a nuclear reaction parameter that encapsulates all of the nuclear physics in the star,

[TABLE]

where $\langle\Delta E\rangle$ is the mean energy generated per nuclear reaction and $E_{0}$ is the energy (temperature) where the reaction rate is maximized for a given star [323, 427]. In our universe, the parameter ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}\approx 2\times 10^{4}$ cm5 s*-3* g*-1* for fusion via the proton-proton chain under typical conditions in stellar interiors. With the nuclear parameters specified, the stellar luminosity is given by

[TABLE]

where $\Theta_{\rm c}=\Theta(\xi=0)$. In the integral, the function $\Theta=\Theta_{\rm c}f^{-1/3}$, so that the expression is specified by the solution to the Lane-Emden equation up to the value of $\Theta_{\rm c}$.

Note that the definition (122) of the nuclear parameter ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$ includes a factor of the fine structure constant. This approach thus implicitly assumes that nuclear yield and reduced cross section scale with $\alpha$. This choice of scaling is conservative, in that it results in a smaller region of allowed parameter space than the alternate scaling (without the factor of $\alpha$): For values of $\alpha$ larger than in our universe, the total nuclear reaction rate is exponentially suppressed by the coulomb factor, so this scaling choice has little effect on the allowed parameter space. For smaller values of $\alpha$, the value of ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$ is smaller than it would be otherwise, so that the allowed parameter is smaller.

Using the energy generation equations, in conjunction with the equations of energy transport, one can solve for the central temperature of the star, or equivalently the central value $\Theta_{\rm c}$ of the temperature parameter. This quantity is determined by the solution to the integral equation

[TABLE]

In this expression, $\langle m\rangle$ is the mean mass of the particles that make up the star and $\kappa_{0}$ is the benchmark value of the stellar opacity. In addition, $\ell_{\star}$ is a dimensionless parameter of order unity and is determined by the luminosity integral over the structure of the star, as characterized by the polytropic index $n$. Note that both sides of the equation are dimensionless. For the typical parameter values in our universe, the right hand side of this equation has a value of approximately $10^{-9}$.

Equation (124) illustrates the parameters that determine stellar structure solutions. In the crudest approximation, the opacity scale $\kappa_{0}\sim\sigma_{T}/m_{\rm p}$, where $\sigma_{T}$ is the Thomson cross section, so that $\kappa_{0}\sim\alpha^{2}$. The polytropic index $n$ varies over a small range $(3/2-3)$ as the stellar mass varies by a factor of 1000 (in our universe). Moreover, the dimensionless integrals $\mu_{\star}$ and $\ell_{\star}$ are always of order unity. If we fix the particle masses, then the remaining parameters are the fine structure constant $\alpha$, the gravitational constant $G$ (equivalently $\alpha_{G}$), and the composite nuclear parameter ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$. For a given stellar mass $M_{\ast}$, we thus have a three dimensional parameter space $(\alpha,G,{\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}})$ that determines stellar properties across the multiverse.

With the central temperature $\Theta_{\rm c}$ determined through equation (124), the equations of stellar structure specify the remaining properties of the star. The stellar radius $R_{\ast}$ is given by

[TABLE]

The stellar luminosity $L_{\ast}$ takes the form

[TABLE]

The photospheric temperature $T_{\ast}$ of the star is then determined from the outer boundary condition so that

[TABLE]

where $\sigma_{\rm sb}$ is the Stefan-Boltzmann constant.

The above solutions specify stellar properties $(T_{c},R_{\ast},L_{\ast},T_{\ast})$ for a given stellar mass $M_{\ast}$. For a given set of fundamental parameters $(\alpha,G,{\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}})$, stars can exist over a finite range of masses (see A). If a star has too little mass, then its central temperature cannot become hot enough to sustain nuclear fusion. If a star has too much mass, then its equation of state is dominated by radiation pressure and the structure becomes unstable. As a result, only part of the parameter space allows for stable, long-lived nuclear burning stars. This parameter space is determined by the values of the parameters $(\alpha,G,{\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}})$ for which equations (124 – 127) have solutions.

The allowed parameter space for working stars is shown in Figure 22 (adapted from [8]; see also [9, 50]). For a given value of the composite nuclear parameter ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$, the gravitational structure constant $\alpha_{G}$ must lie below the curve in order for stable stars to exist. Curves are shown in the figure for a wide range of values for ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$, from that in our universe (lower black curve) to a value $10^{21}$ times larger (upper black curve). Note that the vertical axis spans 40 orders of magnitude. Moreover, the largest value ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}/{\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}_{0}=10^{21}$ is larger than that expected for a universe in which diprotons are stable and nuclear burning can take place through the strong interaction (see Section 7.5). The intermediate curves correspond to values of ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$ that increase by factors of 1000, from bottom to top. The value of the nuclear burning parameter ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$ thus acts to change the scale in the plot, but a large region of parameter space allows for working stars.

7.2 Minimum Stellar Temperatures

In addition to having sustained nuclear reactions, stars must also be hot enough to drive chemical reactions and thereby host potentially habitable planets [319, 320]. This requirement places constraints on the allowed range of parameters for viable universes.

The surface temperature $T_{\rm P}$ of a planet is determined by balancing the incoming radiation from the star and the exhaust heat from the planet,

[TABLE]

where $d$ is the radius of the planetary orbit (taken to be circular). The efficiency factor $f_{T}$ takes into account both the radiation reflected away from the planet and the heat retained by the atmosphere.

The surface temperature $T_{\rm P}$ of the planet must be larger than the temperature required to drive chemical reactions and hence support biological operations. This required temperature (energy scale) is a small fraction of the atomic energy scale and can be written [61] in the form

[TABLE]

where $E_{\rm atom}=\alpha^{2}m_{\rm e}c^{2}$ is the characteristic energy scale for atoms. In our universe, $E_{\rm atom}\approx 27$ eV, whereas chemical reactions take place at room temperature where $E_{\rm chem}$ = $kT\approx 0.026$ eV, so that the chemical conversion factor $\epsilon_{\rm c}\sim 10^{-3}\ll 1$. Using the requirement that the planet is warm enough, $kT_{\rm P}\geq E_{\rm chem}$, in conjunction with the requirement that the orbit must lie outside the star, $d\geq R_{\ast}$, one can derive the constraint

[TABLE]

Using the solutions for the stellar radius and stellar luminosity from equations (125) and (126), the ratio $L_{\ast}/R_{\ast}^{2}$ can be evaluated for a given stellar mass,

[TABLE]

Because the right hand side of this expression increases with mass $M_{\ast}$, we can derive a constraint by evaluating it using the maximum stellar mass for a given universe.

We thus need to determine the maximum stellar mass for a given set of fundamental constants. As the mass of a star increases, the fraction of its internal pressure that is provided by radiation pressure (instead of gas pressure) increases. Let $f_{\rm g}$ denote the fraction of the pressure provided by the ideal gas law, so that $(1-f_{\rm g})$ is the fraction provided by radiation. The star becomes unstable when the radiation pressure dominates [427]. If we use the critical value $f_{\rm g}\approx 1/2$ to specify the pressure fraction for which the stellar mass is maximum, we find

[TABLE]

The minimum temperature constraint from equation (130) can now be evaluated using equation (131) to determine the ratio $L_{\ast}/R_{\ast}^{2}$ and using equation (132) to specify the maximum stellar mass. The resulting bound becomes

[TABLE]

This expression includes the temperature parameter $\Theta_{\rm c}$ evaluated at the stellar center, where this quantity is determined by the stellar structure solution (124) evaluated using the maximum stellar mass (132),

[TABLE]

Now we can simplify the expressions further. Let $\langle m\rangle=m_{\rm p}$, $n=3/2$, and use the definition of $E_{G}$, so that the central temperature is given by

[TABLE]

and the constraint takes the form

[TABLE]

This constraint on the fundamental constants is required for stars to have surface temperatures hot enough to support viable biospheres.

7.3 Stellar Lifetime Constraints

For a universe to be habitable, at least some of its stars must live long enough for biological evolution to take place. Stellar lifetime increases as stellar mass decreases, so we can constrain the fundamental parameters by considering stars with the lowest mass. The minimum mass necessary for sustained nuclear fusion to take place has been derived previously [8, 265, 323, 427] and takes the form

[TABLE]

If we invert equation (137), it determines the maximum temperature $T_{\rm nuc}$ that can be obtained with a star of a given mass, where this temperature is an increasing function of stellar mass. By using the minimum stellar mass from equation (137) to specify the mass in equation (124), we obtain the minimum value of the stellar ignition temperature. This central temperature, or equivalently the value of $\Theta_{\rm c}$, is determined by solving the following equation

[TABLE]

The quantities on the right hand side of the equation have been grouped to include pure numbers, constants that set units, dimensionless quantities from the polytropic solution, particle masses, and finally the stellar parameters that depend on the fundamental constants. Note that this expression has been simplified by setting $\langle m\rangle=m_{\rm ion}=m_{\rm p}$ and by using the polytropic index $n$ = 3/2.

The stellar lifetime $t_{\ast}$ is determined by the available supply of nuclear fuel and can be written in the form

[TABLE]

where $f_{\rm c}$ is the fraction of the stellar material that is available for fusion and where ${\cal E}$ is the nuclear conversion efficiency (recall that ${\cal E}\approx 0.007$ in our universe). Stars with masses comparable to the Sun have access to only a fraction $f_{\rm c}\approx 0.1$ of their nuclear fuel during the main sequence phase, whereas smaller stars have larger $f_{\rm c}$ [339, 12].

If the fine structure constant changes, then atomic structure and atomic time scales are different. We thus want to measure stellar lifetimes in units of the the time scale for atomic reactions, where this latter quantity is given by

[TABLE]

This atomic time scale has the value $t_{\rm A}\sim 2\times 10^{-17}$ sec in our universe. On the other hand, the time required for biological evolution to develop complex life forms (observers) on Earth was $\sim 1$ Gyr, which corresponds to $\sim 10^{33}$ atomic time units. Given our current sample size of one, the expected characteristic time scale for biological evolution has enormous uncertainty [408, 490], and many interpretations are possible [132, 133]. For the sake of definiteness, this treatment considers the terrestrial value of 1 Gyr as a fiducial time scale, although it does not represent a definitive limit (see also [141, 377]). In other words, we use the reference value of $10^{33}$ atomic time units as a starting point.

The ratio of the stellar lifetime to the atomic time scale can be written in the form

[TABLE]

In other universes, the largest possible value of this ratio, corresponding to the smallest, long-lived stars, is thus given by

[TABLE]

where we have grouped the various factors as before. Note that equation (142), as written, depends on the temperature parameter $\Theta_{\rm c}$, which is specified via equation (138). We can thus combine equations (138) and (142) to solve for the ratio of time scales, and set it equal to the minimum required for life to develop (here we use $t_{\ast}/t_{\rm A}>10^{33}$ as described above).

The requirements of a minimum stellar lifetime (this section, 7.3) and a minimum photospheric temperature (Section 7.2) place further constraints on the allowed parameter space for stars. This trend is illustrated in Figure 23, which shows the viable regime of parameter space for stars with the value of the nuclear burning parameter ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$ for hydrogen burning stars in our universe. The black curves delineates the portion of the $\alpha$-$\alpha_{G}$ plane that allows for working stars. In order for the stars to live long enough (using the benchmark value of $10^{33}$ atomic time scales), the parameter space is limited to lie below the red curve. In order for the stars to have a sufficiently hot surface temperature, the parameter space must fall to the left of the blue curve. Finally, the figure enforces an additional requirement, that the stellar mass scale must be less than the characteristic mass scale for a galaxy (see A). This constraint limits the parameter space to the right of the green curve. These additional constraints thus remove about half of the original parameter space shown in the diagram.

Figure 23 also provides constraints on the ratio of the strengths of the gravitational and electromagnetic forces. In our universe, this ratio is notoriously small, with $\alpha_{G}/\alpha\sim 10^{-36}$. In order for working stars to exist (black curve), this ratio can be larger by a factor of $\sim 10^{4}$. If we also require that the stars live for $10^{33}$ atomic time scales (red curve), then the ratio can only be larger by a factor of $\sim 100$. Both of these factors are small compared to the observed ratio. As a result, viable universes are constrained to have an extreme hierarchy of force strengths, provided that the nuclear burning parameter ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$ is comparable to that in our universe. This degree of hierarchy between gravity and electromagnetism can be made much less extreme for much larger values of ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$, as shown in Figure 22. Specifically, if the parameter ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$ is increased by a factor of $10^{21}$, which corresponds to nuclear reactions occurring via the strong force instead of the weak force, then the ratio $\alpha_{G}/\alpha$ can be larger by a factor of $\sim 10^{24}$ (see also Section 7.5).

7.4 The Triple-Alpha Reaction for Carbon Production

In our universe, the production of carbon takes place through a somewhat convoluted process known as the triple alpha reaction [147, 323]. Because of the importance of carbon to life forms and the delicate nature of this requisite reaction, carbon production is often used as an example of fine-tuning [23, 49, 61, 122, 280]. Even for our own universe, the exact determination of the reaction rate and its dependence on fundamental parameters have been elusive (see [169] for a recent review).

The basic complication for carbon production, and hence the necessity of the triple alpha reaction, arises because our universe has no stable nucleus with atomic mass number $A=8$. After a star has processed the hydrogen in its central core into helium, it adjusts its internal structure by condensing so that the central temperature and density increase. The seemingly obvious next step would be to fuse the helium nuclei (alpha particles) into 8Be nuclei. Unfortunately, this channel is unavailable because 8Be is unstable. In the absence of this natural stepping stone, helium fusion must take place through the triple alpha process [147, 323],

[TABLE]

where three helium nuclei combine to forge carbon. This reaction relies on the formation of a transient population of 8Be [459]. More specifically, helium nuclei continually combine to make 8Be, which decays back into its constituent particles with a half-life of about $\tau\sim 10^{-16}$ sec. Because they are mediated by the strong force, these reactions occur rapidly enough to maintain nuclear statistical equilibrium (NSE). This small and transient population of 8Be is large enough to support the additional reaction

[TABLE]

In order for this latter reaction to operate fast enough, however, it must take place in a resonant manner, which requires the 12C nucleus to have a resonance at a particular energy. The existence and energy level of this resonance was predicted by Hoyle [286]. Subsequent experiments measured the resonance in the laboratory [188] and provided a remarkable confirmation of this paradigm for carbon synthesis (see the review of [213]).

The resonance in question is the $0^{+}$ excited state of the 12C nucleus and has an energy of 7.6444 MeV, where this energy lies just above that of a 8Be nucleus and an alpha particle considered separately (given by the left side of equation [144]). The key question is thus how precisely specified this energy level must be in order for stars to produce a significant amount of carbon. To address this issue, we start with some definitions:

Although the reaction 4He + 4He $\to$ 8Be is not energetically favored, unstable 8Be can be formed, with the reaction rate (e.g., see [204]) controlled by the energy difference

[TABLE]

where $E_{8}$ and $E_{4}$ are the ground state energies of 8Be and 4He. The ground state of the carbon nucleus is denoted here as $E_{12}$ and the excited state is $E_{12}^{\star}$. The energy difference between the excited carbon nucleus and the reactants in equation (144) is then given by

[TABLE]

Finally, one can define the energy scale $E_{R}$ of the resonant reaction according to

[TABLE]

The currently measured value of this energy level is $(E_{R})_{0}\approx 379.5$ keV. With the above definitions, the resonant reaction rate $R_{3\alpha}$ is given by

[TABLE]

where $\Gamma_{\gamma}$ is the radiative width of the Hoyle state. Given that the resonance energy $E_{R}$ appears in the exponential term, its value determines the net reaction rate for the entire process. Here we let the resonance level vary up or down by an increment $\Delta E_{R}$ given by

[TABLE]

where the subscript denotes the value in our universe.

If the energy level of the 12C resonance is higher, $\Delta E_{R}>0$, then carbon production is suppressed. At a given temperature, a higher energy level for the resonance leads to a lower reaction rate for helium burning (carbon production), so the stellar core adjusts to a higher temperature in order for nuclear reactions to support the star. At the same time, however, this higher temperature allows any extant carbon nuclei to fuse into oxygen through the reaction

[TABLE]

The energy level of 16O lies at 7.1187 MeV, which is below the combined energy of the reactants 12C and 4He (with energy 7.1616 MeV). The basic problem is that if the rate of burning carbon into oxygen becomes large compared to the rate of making carbon, then any carbon produced by stellar nucleosynthesis will immediately be transformed into oxygen. No carbon would be left for making life forms and other structures.

Although the general trend of nuclear evolution in stars is to forge ever larger nuclei (like burning carbon into oxygen), the case of the triple alpha reaction is on a different footing. In general, the production of larger nuclei involves larger coulomb barriers, which in turn requires higher temperatures in the stellar core. Because carbon production is suppressed by the requirement of passing through unstable 8Be, the temperature required to make carbon is higher than it would be otherwise and is comparable to the temperature required to destroy it (e.g., through equation [150]). This coincidence renders carbon production sensitive to the exact energy levels of the 12C nucleus.

The sensitivity of the energy level of the carbon resonance has been explored in a number of papers. One straightforward approach is to vary the energy level of the 12C nucleus, but keep all other parameters the same. At the fundamental level, changes in the excited state of nuclei are determined by changes in the strengths of the fundamental forces, especially the strong and electromagnetic interactions [202, 203, 204, 205, 333, 380, 381]. Changes in these interaction strengths would affect all nuclear structures, not just the energy level of the 12C resonance of interest here. Nonetheless, variations in the resonance energy provide a good starting point for understanding the sensitive dependence of carbon production on its value.

It is important to note that carbon production also occurs through a non-resonant reaction. The non-resonant contributions are larger than the resonant terms [401] for temperatures lower than $T\sim 3\times 10^{7}$ K (for $\alpha+\alpha\to$ 8Be) and $T\sim 7\times 10^{7}$ K (for $\alpha$ + 8Be $\to$ 12C). As a result, if the resonance level is changed so that carbon is not efficiently produced, the non-resonant reactions become important.

The first such treatment [362] considered the evolution of stars with $M_{\ast}=20M_{\odot}$ as well as shell helium burning in Asymptotic Giant Branch (AGB) stars. They found that increasing the energy of the $0^{+}$ energy level of 12C by $\Delta{E}_{R}=+60$ keV does not affect the amount of carbon produced in stellar interiors, but a larger increase of $\Delta{E}_{R}=+277$ keV leads to significant changes in nuclear burning patterns and relatively little carbon is produced. In AGB stars, an increase in the resonance level also leads to reduced carbon production, but the strength of the thermal pulse is increased. Stronger pulsations can increase the amount of (nuclear processed) material that is transferred to the outer layers of the star and ultimately distributed into the interstellar medium. Finally, if the resonance level of the carbon nucleus is decreased, then the amount of carbon produced can be increased significantly.

Subsequent studies [159, 406, 470] find similar results to those outlined above, although stars with different masses show somewhat different dependence on the value of $\Delta{E}_{R}$. For example, stars with masses of $M_{\ast}$ = 15 and 25 $M_{\odot}$ were considered in [470]. For an increase in the resonance level of $\Delta{E}_{R}=+100$ keV, the carbon yield from 15 (25) $M_{\odot}$ stars decreased by a factor of 7 (17). On the other hand, for a decrease in the energy of the resonance, $\Delta{E}_{R}=-100$ keV, carbon production increased by a factor of $\sim 3$ (7.5).

Figure 24 provides an overview of results from more recent stellar evolution calculations [288] that determine the abundances of intermediate alpha elements, including carbon and oxygen, as a function of the change $\Delta{E}_{R}$ in the resonance energy. The figure shows results calculated using the MESA stellar evolution code [420, 421] for stellar masses in the range $M_{\ast}=15-40M_{\odot}$. The curves show the expectation values for the mass in a given element produced per star during the course of stellar evolution. These values are obtained by integrating the yields over the stellar mass range, weighted by a stellar initial mass function of the form $dN/dm\sim m^{-2.3}$ [460]. Results are shown for yields of carbon (blue), oxygen (orange), neon (green), magnesium (red), and silicon (magenta). The sum of the mass in these elements is given by the black dashed curve. The stellar models depicted in the figure began with metallicity $\raisebox{0.86108pt}{--}\kern-5.50003ptZ=10^{-4}$ for computational convenience. The black symbols on the curves for carbon and oxygen show that expectation values of the mass in those elements at the start of the simulations.

Several trends are evident from Figure 24: The carbon yields are a steeply decreasing function of the resonance energy for values near that of our universe. Nonetheless, the carbon yields are larger than the starting values (for $\raisebox{0.86108pt}{--}\kern-5.50003ptZ=10^{-4}$) for energy increments as large as $\Delta{E}_{R}\approx+480$ keV. As a result, somewhat larger changes in the energy level ($\Delta{E}_{R}$ \lower 2.15277pt\hbox{;\buildrel>\over{\sim};}+500 keV) can seriously compromise carbon production. In contrast, decreasing the energy level results in an increase in mean carbon production, with $\Delta{E}_{R}=-200$ keV providing a factor of $\sim 10$ enhancement. Carbon production continues down to energy increments $\Delta{E}_{R}\approx-300$ keV, where carbon production ceases to be energetically favorable (in addition, the temperature dependence of the triple alpha reaction rate ceases to allow for stellar stability at similar values of $\Delta{E}_{R}$ [288]). The oxygen yields are roughly anti-correlated with those of carbon. Decreasing the resonance energy level $\Delta{E}_{R}$ results in less oxygen production. Increasing the resonance level up to about $\Delta{E}_{R}\sim+100$ keV leads to enhanced oxygen yields, but for even larger increases in the resonance level the oxygen production plateaus and then declines. The oxygen abundance falls below its starting value in these simulations for \Delta{E}_{R}\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}+280 keV. Oxygen, which is necessary to make water, becomes scarce for larger values.

For the five alpha elements shown in Figure 24, each species has the maximum abundance for a range of $\Delta{E}_{R}$. For increasing values of the resonance increment, the peak values occur for carbon, neon, oxygen, magenta, and then silicon. With the exception of neon, these peaks are found in order of increasing atomic number. As the resonance energy (given by $\Delta{E}_{R}$) increases, the operating temperature of the stellar core increases, and larger elements can be synthesized. Notice also that the sum of the mass (black dashed curve) contained in these five alpha elements is nearly constant: Most of the mass in metals outside the iron core (which becomes either the neutron star or black hole produced by the ensuing supernova explosion) is contained in these five species, and their mass contributions show nearly zero-sum behavior.

In approximate terms, these results can be summarized as follows: Changing the carbon resonance energy by increments of $\sim 300$ keV changes the carbon and oxygen yields (in opposite directions) by roughly an order of magnitude. The total range in the triple alpha resonance energy for viable carbon production is given by $-300$ keV \lower 2.15277pt\hbox{;\buildrel<\over{\sim};}\Delta{E}_{R}\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}+500 keV, with the range for oxygen somewhat smaller. Notice that the nuclear yields depend on stellar mass (not shown – see [288]). Although the stellar mass function is remarkably robust within our universe, it could vary in other universes, and such variations would affect the masses in the chemical species averaged over the stellar population. Finally, note that the mean yields of carbon and oxygen are equal at a resonance level just below that of our universe (for $\Delta{E}_{R}\approx-35$ keV).

As illustrated in Figure 24, stellar evolution calculations indicate that the ratio of carbon yield to oxygen yield is of order unity for massive stars in our universe. This finding is consistent with the carbon-to-oxygen (C/O) ratio of [C/O]$\approx 0.44$ found in the Galaxy, the ratio [C/O]$\approx 0.67$ observed for the Solar Wind [539], and the value [C/O]$\approx 0.51$ measured for the Solar System as a whole [114]. Similarly, the C/O ratio has been measured for nearby stellar populations in our universe, including all stars in the Solar neighborhood [100] and those that are known to host extrasolar planets [398]. The observed distribution is relatively wide, with C/O ratios varying by at least a factor of $\sim 3$, and falling in the range [C/O]$\approx 0.25-0.75$. Although these values produce a consistent picture, one should keep in mind that intermediate mass stars also provide a substantial contribution to the carbon inventory of the universe.

In contrast to observed stellar abundances, the C/O ratio for Earth is anomalously low, with estimates ranging from [C/O]$\approx 0.01$ [32] down to [C/O]$\approx 0.002$ [374]. Earth is thus severely depleted in carbon relative to the Sun. Although increasing the carbon-12 resonance energy lowers both the carbon abundance and the C/O ratio, the change must be larger than $\Delta{E}_{R}\approx+300$ keV in order to make these values smaller than the observed carbon abundance and C/O ratio of Earth. As a result, the range of resonance energy increment $-300$ keV \lower 2.15277pt\hbox{;\buildrel<\over{\sim};}\Delta{E}_{R}\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}+300 keV produces cosmic carbon abundances that are larger than those of Earth — which is the only place in the universe where life is known to exist. Unfortunately, the minimum carbon abundance required for habitability remains unknown, but these considerations suggest that the corresponding maximum value of the resonance energy must be larger than $\Delta{E}_{R}\approx+300$ keV. A related unresolved issue is the relationship between cosmic abundances, stellar abundances, and the chemical compositions of planets [74, 384], which form within circumstellar disks associated with forming stars (see also Section 8). For example, the chemical compositions of planetessimals (building blocks of planets) are affected by both the C/O ratio of the system and by their radial location within the disk [304]. In our Solar System, the planet Venus has a relatively low C/O ratio comparable to that of Earth [386], whereas the majority of asteroids are carbonaceous (C-type), with a carbon-rich composition close to solar [404] (after accounting for the depletion of hydrogen, helium, and other volatiles).

The above considerations indicate that the change $\Delta{E}_{R}$ in the energy level of the carbon-12 resonance can vary over a total range of $\sim 800$ keV and still allow the universe to be viable. The next question is whether this range is large or small. The criterion for an enhanced nuclear cross section is that the particle energies are near a resonance, where the resonances are given by the energy levels of the nucleus. In general, excited nuclear states are spaced at intervals of order $\sim 1$MeV [135]. More specifically, the carbon nucleus has excited states at $E$ = 4.44, 7.65, 9.64, 12.7, and 15.1 MeV, so that the energy intervals are about 3 MeV. Given the allowed range of $\Delta E_{R}$, the chance of being sufficiently near resonance is about 1 part in 4.

In addition to carbon and oxygen, the abundances of other elements will be affected by changes to the carbon-12 resonance level. As one example, the radionuclides 26Al and 60Fe, with half-lives of order 1 Myr, provide an important energy source during the epoch of planet formation. The abundances of these radioactive species are particularly sensitive to variations in $\Delta{E}_{R}$ [523].

Another class of research considers the time variation of the constants of nature in our universe [336, 528, 529], including constraints from carbon production of Population III stars [194]. Possible variations in the constants of physics lead to corresponding variations in stellar properties [532]. In this setting, however, the constants are constrained to have only small variations. Over the age of the universe, the fine structure constant has variations smaller than $(\Delta\alpha)/\alpha\sim 10^{-5}$. Such variations are thus too small to affect carbon production at earlier epochs in our universe.

The above discussion indicates that changes in the carbon resonance over a range of $\sim 800$ keV allow for stars to produce carbon and oxygen at acceptable levels, within an order of magnitude of values in our universe, but larger changes are potentially problematic. As a reference point: In order to move the resonance energy by an increment of 100 keV, the corresponding change in the nuclear force strength (e.g., [470]) is estimated to be $\sim 0.5\%$ and/or the change in the electromagnetic interaction is $\sim 2-4\%$ (see also [150, 151, 194, 204, 205]). On the other hand, the 8Be nucleus only fails to be bound by 92 keV. As a result, the degree of change to nuclear physics required to compromise carbon production is significantly larger than that needed to allow for stable 8Be. Detailed calculations using Lattice Chiral Effective Field Theory confirm that bound states of 8Be require the strengths of the nuclear and/or electromagnetic forces to vary by only about $\sim 1-2\%$ [204]. Moreover, a stable isotope with mass number $A=8$ removes the need for the triple alpha process altogether [14, 279]. In universes with stable 8Be, stars can burn helium into beryllium, and later burn the beryllium into carbon. With stable 8Be, the necessary carbon producing reactions can take place at later evolutionary stages within the same star or in different stars in later generations. Although stellar evolution is modified in universes with stable 8Be, the epoch of Big Bang Nucleosynthesis will continue to produce light nuclei with the usual abundances [149, 150, 468]. The conditions of low density, high entropy, and short evolutionary time scale prevent BBN from producing elements of larger atomic numbers.

If stars fail to make carbon through the triple alpha reaction, they tend to produce other alpha elements with larger atomic numbers. In this case, some carbon can still be synthesized through spallation reactions with high energy cosmic rays [478]. Typical reactions include

[TABLE]

and many others. The cross sections for such reactions are of order $\sigma\sim 100$ mb [479, 544]. For the cosmic ray flux of our Galaxy, the rate at which oxygen is converted into carbon in the interstellar medium is only of order $3\times 10^{-8}$ per Hubble time per target nucleus. These rates can be enhanced in supernova remnants [417], in planetary atmospheres [407], and by the inclusion of high energy photons [305]. Since stars in other universes readily synthesize alpha elements, and supernovae are expected to accelerate cosmic rays, spallation will always produce nonzero abundances of carbon. Nonetheless, the expected mass fractions are much lower than those observed in our universe (unless the cosmic ray flux is significantly enhanced).

For completeness, we note that if the universe fails to make enough carbon, another — highly speculative — channel for life could exist. The reason for low carbon abundance is not that it cannot be synthesized, but rather that it is immediately processed into heavier elements (first oxygen, then larger nuclei). The stars of sufficiently high mass are thus likely to make appreciable amounts of silicon (Figure 24). The idea of silicon based life has long been the subject of science fiction, but scientists have not ruled out the possibility. The molecular diversity of life is becoming understood [411] and life is known to thrive in extreme environments [247, 453]. Recently, some steps toward realizing silicon-based life have been carried out [312]. Both carbon and silicon can produce large molecules that can (in principle) carry large amounts of biological information [412]. On the other hand, carbon readily produces chemical bonds with many other atoms, whereas silicon tends to interact with many fewer species. As a result, carbon allows for much greater chemical versatility, which in turn increases its efficacy as a basis for biology. In addition, chemical reactions with silicon are generally slower than those with carbon and the chemical bonds (e.g., Si-Si) are weaker. Nonetheless, if life could exist with a silicon architecture, then observers could still arise in carbon-poor universes, and life would not depend on the triple alpha process.

7.5 Effects of Unstable Deuterium and Bound Diprotons on Stars

Another standard example of possible fine-tuning arises for nuclei with atomic mass number $A=2$. In our universe, deuterium is stable, whereas diprotons and dineutrons do not have stable bound states. A common assertion in the existing literature of fine-tuning constraints is that small changes in the strength of the strong nuclear force, in either direction, would render the universe lifeless (for example, see [61, 163, 171, 186, 189, 280, 430, 446, 467, 507, 511] and references therein). The required change in the strong force is estimated to be $\sim 15\%$.

If the strong force were stronger, then diprotons and dineutrons could be bound, and nuclear reactions in stars could take place through the strong interaction, in contrast to the case of our universe where weak interactions play an important role. Given the much larger reaction rates, the concern is that stellar lifetimes could become too short. On the other hand, if the strong force were somewhat weaker, then deuterium would not be bound, and no stable $A=2$ nuclei would exist. In our universe, both BBN and stellar nucleosynthesis rely on stable deuterium as a stepping stone on the path to larger nuclei. Here the concern is that no heavy nuclei can be made in the absence of stable deuterium. In spite of the ubiquity of these arguments, recent calculations of stellar structure and evolution show that stars can serve as power sources for habitability in universes with stable diprotons [50, 93] (see also [8, 9]) and in universes with unstable deuterium [15, 52]. These scenarios are discussed in the following subsections.

7.5.1 Universes with Stable Diprotons

In universes with stable diprotons or dineutrons, nuclear reactions can be mediated by the strong interaction (without requiring the weak interaction). This behavior stands in contrast to our universe, where nuclear reactions in stars must convert four protons into a helium nucleus. The net reaction, proceeding through either the p-p chain or the CNO cycle [147, 323, 427], necessarily involves the transformation of two protons into two neutrons and hence requires the weak force. Nuclear reactions involving only the strong force generally have much larger reaction rates, and can potentially alter the course of stellar evolution.

The magnitude of the change, from weak to strong nuclear reactions, can be illustrated by the case of deuterium burning in stars in our own universe. The nuclear reaction parameter ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$ (see equation [122]) for deuterium burning can be written in the form

[TABLE]

where the value is scaled to the deuterium abundance in our universe [50]. This value should be compared to that of ordinary hydrogen burning (${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}\approx 2\times 10^{4}$ in the same units), which involves the weak interaction. Although nuclear physics in alternate universes allows for a range of possible binding energies and reaction cross sections for $A=2$ nuclei, a good starting assumption is that nuclear burning rates in universes with stable diprotons are roughly comparable to the case of deuterium. If the mass fraction of nuclear fuel is of order unity, comparable to hydrogen in our universe, then the nuclear reaction parameter for diprotons becomes ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$ $\sim 10^{22}$ cm5 s*-3* g*-1*, which is larger than the value for the p-p reaction (in our universe) by a factor of $\sim 10^{18}$.

Before analyzing the effect of changing the nuclear reaction rate by such a large factor, we note that an analogous process takes place in our universe. Star formation does not involve nuclear reactions, so that stars are not born with the proper configurations required for fusion to take place [477]. Instead, most stars (those with masses M_{\ast}\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}7M_{\odot}) are born with large stellar radii ($\sim 3-4$ times larger than their main sequence values) and central temperatures that are too low to sustain hydrogen burning reactions [491, 492]. Young stars derive their energy from gravitational contraction and evolve over millions of years without hydrogen fusion. As the stars condense, their central regions eventually become hot enough ($T_{\rm c}\sim 15\times 10^{6}$ K) for hydrogen burning to occur. However, well before the central temperature reaches this benchmark value, deuterium fusion takes place when the central temperature becomes $T_{\rm c}\sim 10^{6}$ K (with ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$ given by equation [152]). Nothing disastrous occurs when stars reach the configuration where deuterium burning takes place. They briefly derive additional energy from this nuclear process, which briefly delays their evolution. Because of the low deuterium abundance, this phase is short compared to other stellar timescales (of order 0.1 – 1 Myr), and the stars subsequently continue their contraction toward the main sequence.

In universes with stable diprotons, a number of different nuclear reaction chains could be realized. The reaction networks will depend on the values of the cross sections, the binding energies, and the abundances of protons, diprotons, deuterium, and the helium isotopes (in the stellar interior). As one example, consider the care where the universe emerges from its BBN epoch with protons as the dominant nuclear species. The first stage of nuclear burning in stars involves the production of diprotons ($p+p\to$ 2He + $\gamma$), which takes place through the strong interaction. The diprotons can then capture free electrons to become deuterium, and subsequent reactions are the same as those in the standard p-p chain [147, 265]. Note that electron capture involves the weak force, but proceeds more rapidly than the usual reaction (equation [154]) due to the absence of a coulomb barrier. Moreover, the starting reaction takes place at a lower temperature than for stars in our universe (as outlined above). As a result, the final step of the p-p chain (3He

• 3He $\to$ 4He $+2p$) does not occur promptly: The star first builds up a abundance of 3He while the core operates at $T_{\rm c}\sim 10^{6}$ K, and then produces 4He in a later stage of nuclear burning. Other scenarios are possible, but in general the nuclear burning parameter ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$ is expected to be greatly enhanced.

Stellar structure is relatively insensitive to the value of ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$. Figure 22 shows the allowed regions of the $(\alpha,\alpha_{G})$ parameter space for stars with a wide range of nuclear burning parameter ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$, extending up to values $10^{21}$ times larger than that of p-p burning in our universe. Stable, long-lived stellar configurations are thus possible over an enormous range of the parameter ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$. In fact, as the value of ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$ increases, the allowed region of the $(\alpha,\alpha_{G})$ parameter space becomes larger. Stellar structure solutions thus exist for stars operating through the strong interaction.

The insensitivity of stars to the nuclear parameter ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$ arises for a number of (coupled) physical reasons: Stars are supported by ordinary gas pressure, which balances gravity for any given stellar mass, largely independent of the energy generation mechanism. The internal structure of the star adjusts itself so that the nuclear reaction rate is whatever it needs to be to provide the required pressure. If the nuclear parameter becomes larger (smaller), then the star compensates by making its central temperature and density lower (higher) to maintain the same pressure support. In addition, the nuclear reactions take place via quantum mechanical tunneling. The normal operating state of the star is such that the central temperature is far lower than that required for protons to directly overcome their mutual coulomb barrier (and thereby drive nuclear reactions directly). If such conditions did not prevail, the star would resemble a nuclear bomb rather than have a stable long-lived structure. In any case, the nuclear reaction rate is exponentially sensitive to the central temperature, so that small changes in $T_{\rm c}$ can compensate for large changes in ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$.

This robust nature of the stellar structure solutions is exemplified by equation (124), which specifies the central temperature required for sustained nuclear reactions. The dimensionless integral $I(\Theta_{\rm c})$ that determines the nuclear burning temperature is a decaying exponential function of the temperature parameter $\Theta_{\rm c}\propto{T_{\rm c}}^{-1/3}$, and can be approximated with a fitting function of the form [9]

[TABLE]

where the numerical values correspond to stellar models with polytropic index $n=3/2$ and radiative energy transport. The combination of equations (124) and (153) shows that the central temperature parameter $\Theta_{\rm c}$ depends only logarithmically on the value of the nuclear burning parameter ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$. Increasing ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$ by a factor of $10^{17}$ thus only increases $\Theta_{\rm c}$ by a modest factor (an increment of $\sim 17\log(10)/3\approx 13$), enough to lower the central temperature of the star down to $T_{\rm c}\approx 10^{6}$ K.

In addition to sustaining nuclear fusion reactions, stars in universes with stable diprotons (large ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$) must have sufficiently long lifetimes in order to support habitable planets. The relative insensitivity of stellar structure to the value of ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$ indicates that stars in such universes can indeed be long-lived. As shown in equation (139), the stellar lifetime $t_{\ast}$ depends only linearly on the value of the central temperature parameter $\Theta_{\rm c}$ (which depends logarithmically on ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$), but depends inversely on the square of the stellar mass ($t_{\ast}\propto M_{\ast}^{-2}$). Moreover, the range of stellar masses extends down to lower values as the parameter ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$ increases. As a result, the longest-lived stars in a universe with bound diprotons have main sequence lifetimes measured in trillions of years, comparable to the longest-lived stars in our universe (this result follows from equations [138], [139], and [153]; see also Ref. [50]). For completeness, note that the smallest stars in our universe will outlast the Sun by factors of $\sim 10^{3}$ [339]. Stars in universes with stable diprotons thus live long enough to support habitability.

To illustrate the effects of stable diprotons on stellar structure and evolution, we have used the MESA stellar evolution code [420] to explore the case where the nuclear reaction rates are enhanced by a factor of $10^{15}$. In this scenario, the stars start with the same chemical composition as the Sun. The first stage of nuclear processing converts free protons into diprotons, which capture electrons to become deuterium, and then add protons to make 3He. Because of the lower central temperatures ($\sim 10^{6}$ K), the resulting 3He does not burn promptly, but rather accumulates in the stellar core. After the protons are exhausted in the central regions, the stars adjust their structure and process the 3He into 4He. This second stage of evolution completes the p-p chain of nuclear reactions. For larger stars (M_{\ast}\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}1M_{\odot}), the CNO cycle occurs alongside the aforementioned nuclear processes. For sufficiently small stars (M_{\ast}\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}0.03M_{\odot}), the central cores cannot become hot enough to support the second stage of nuclear burning, and the stellar cores end with a 3He composition. The main sequence for these stars is shown as the red curve in Figure 25. For comparison, the blue curve shows the zero-age main sequence using the standard nuclear reaction rates. Compared to hydrogen burning stars in our universe, these stars have a larger range of masses, redder photospheres, and a comparable range of luminosities. The longest-lived stars have lifetimes of trillions of years, comparable to those in our universe, and much longer than the current cosmic age.

7.5.2 Universes with Unstable Deuterium

In universes without stable deuterium, the standard nuclear reaction chains in both BBN and stellar interiors must be altered. For example, in the p-p chain that powers most stars, the production of helium takes place by first combining two protons into a deuterium nucleus,

[TABLE]

which then reacts further to produce helium through the reactions

[TABLE]

[TABLE]

along with other branches [147, 265, 323, 427]. Without the first step of deuterium production in equation (154), the subsequent reactions of the p-p chain cannot take place. This difficulty has been noted by many authors [61, 186, 280, 430, 446, 467, 507, 511]. Although the absence of stable deuterium compromises the p-p chain, stars have access to other sources of energy generation and other channels of nucleosynthesis.

Even in the absence of any nuclear processing, stars can still generate energy through gravitational contraction. As outlined above, most stars are formed with somewhat extended configurations and with central temperatures that are too cool to sustain nuclear reactions. Young stars thus convert gravitational potential energy into luminosity, grow smaller in radius, and their central cores become hotter and denser. In our universe, this pre-main-sequence contraction phase continues until the temperature at the center reaches the proper nuclear ignition temperature for the given stellar mass (typically about $T_{\rm c}\approx 1-2\times 10^{7}$ K). The onset of nuclear reactions halts further contraction, and the star subsequently derives its energy from nuclear power rather than from gravity. In the absence of nuclear reactions, this contraction phase continues far longer. The longest-lived stars have a luminosity comparable to that of the Sun and can live for up to a billion years [15], which could be long enough for biological evolution to take place on favorably situated planets.

The final fate of gravitationally contracting stars depends on the stellar mass. Stars with sufficiently small masses can be supported by the degeneracy pressure of non-relativistic electrons and end their lives as white dwarfs. The maximum stellar mass that can become a white dwarf given by the Chandrasekhar mass [136], where $M_{\rm ch}$ $\approx 5.6M_{\odot}$ for stars with a pure hydrogen composition (as expected in the absence of nuclear reactions). Stars with progenitor masses above the Chandrasekhar mass cannot be supported by electron degeneracy pressure and continue to their contraction until their cores reach enormously high temperatures and densities, as shown in Figure 26 (from [15]). The stellar models shown in the figure (calculated using a modified version of the MESA stellar evolution package [420, 421]) are evolved until the central temperature reaches $T_{\rm c}\sim 10^{10}$ K and the central density reaches $\rho_{\rm c}\sim 10^{11}$ g cm*-3*. In actuality, the stars condense further to even higher temperatures and densities. Under such extreme conditions, nuclear reactions are no longer suppressed: recall that $T\sim 10^{10}$ K corresponds to $\sim 1$ MeV, the typical energy scale for nuclear reactions. At such temperatures, protons have enough energy to overcome their coulomb barrier and can undergo nuclear reactions without tunneling. As a result, many types of nuclear reactions take place readily in the final death throes of these stars. This process of explosive nucleosynthesis can supply the universe with heavy nuclei, even in the absence of stable deuterium.

Universes without stable deuterium have additional channels for stellar nucleosynthesis. If explosive nucleosynthesis — or any other process — can produce a small amount of carbon, then stars can operate through the CNO cycle. In this process, carbon acts as a catalyst to synthesize helium through a chain of reactions:

[TABLE]

[TABLE]

[TABLE]

Although the Sun generates only a few percent of its power through the the CNO cycle, this nuclear process becomes dominant for somewhat larger stars (with M_{\ast}\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}1.5M_{\odot}). Other branches of the CNO cycle exist, but equation (156) accounts for most of the helium production. Notably, none of the branches of the CNO cycle involve deuterium, so that its instability is not an impediment to the production of heavier elements.

Given that stars can generate power through the CNO cycle, the key question is how much carbon is required to start the process. Figure 27 shows the main sequences for stars burning hydrogen via the CNO cycle for low metallicities in the range $\raisebox{0.86108pt}{--}\kern-5.50003ptZ$ = $10^{-10}-10^{-6}$ (shown as the solid curves in the diagram). These main sequences were calculated [15] using the MESA stellar evolution package [420, 421] with the p-p chain removed from the nuclear reaction network. For the sake of definiteness, the nuclear composition is taken to be the same as in our universe, except for the overall normalization set by the metallicity $\raisebox{0.86108pt}{--}\kern-5.50003ptZ$. Compared to the case of our universe, these stars are somewhat hotter and brighter, and the allowed range of stellar masses is truncated at the low-mass end. These trends become more pronounced as the metallicity becomes lower. Nonetheless, even for $\raisebox{0.86108pt}{--}\kern-5.50003ptZ=10^{-10}$, the main sequence appears relatively normal. In addition, the stellar lifetimes for these stars (not shown; see [15]) are as long as $t_{\ast}\sim 10^{11}$ yr or about ten times the current age of our universe. As a result, stars with such low metallicity can live long enough to support habitable planets. For even lower values of metallicity, $\raisebox{0.86108pt}{--}\kern-5.50003ptZ=10^{-12}-10^{-14}$, only high mass stars can sustain nuclear burning and the main sequence becomes shorter. However, subsequent stellar generations will attain higher metallicities, so that the stellar population will behave similarly to that of our universe.

Stars without stable deuterium can generate energy and heavier nuclei through yet another process involving three nucleons. This triple nucleon process is roughly analogous to the triple alpha process that drives carbon production in our universe, with some differences. Even though the deuterium nucleus is unstable, the reaction $p+p\to d$ that produces deuterium can still take place, even though the product will fall apart a short time later. If the deuterium producing reactions were fast enough, they could maintain nuclear statistical equilibrium, and the star would have a standing population of transient deuterium (analogous to the standing population of unstable 8Be that leads to the triple alpha process). The reaction that produces deuterium involves the weak force and does not operate quickly enough to maintain equilibrium. When the deuterium nuclei are unstable, however, they generally first decay into protons and neutrons, instead of the original two protons. The neutron is also unstable, but its half-life is of order ten minutes, which is enormous compared to the short time scales of nuclear reactions in the stellar core. The star thus maintains a standing population of neutrons, which can then interact quickly enough to make heavier elements through the chain of reactions

[TABLE]

[TABLE]

Note that the final reaction takes place entirely through the strong force. The triple nucleon reaction thus leads to the net process

[TABLE]

The products interact further to make helium,

[TABLE]

where this final reaction is the same as that utilized in the p-p chain.

Figure 27 shows the main sequences for stars burning hydrogen into helium through the triple nucleon process (shown as the dashed curves). These calculations were carried out [15] using the MESA code [420, 421], where the CNO cycle reactions have been removed and the p-p reactions are replaced by the triple nucleon reactions described above. All of the reaction rates and yields are the same as in our universe, except that deuterium is unstable and decays on a short time scale given by the decay width $\Gamma(d)$. Results are shown in Figure 27 for decay widths in the range $\Gamma(d)=1-1000$ eV. The lower end of this range corresponds to a half-life of about $10^{-16}$ sec, which is comparable to that of unstable 8Be in our universe (in analogy to the triple alpha process). A number of trends are evident from this H-R diagram. As the decay width increases (so that the half-life of deuterium decreases), the stars become increasingly hotter, and the main sequence become shorter as the minimum stellar mass required for nuclear ignitions increases. We note that working stars exist for even larger decay widths $\Gamma(d)=10^{4}-10^{5}$ eV (not shown), although only the most massive stars are operational.

7.6 Stellar Constraints on Nuclear Forces

In our universe, the weak interaction plays an important role in stars: Most of the stellar energy is generated by fusing together four protons into a helium nucleus, which requires the conversion of two protons into neutrons via the weak interaction. Nonetheless, stars can still function in universes where the weak interaction is either weaker or stronger than in the Standard Model [285], including scenarios where the weak interaction is absent altogether [266, 245] (see Section 7.6.1). On the other hand, the weak interaction cannot become too strong without compromising stellar evolution; an upper bound is derived in Section 7.6.2. The weak interaction is also constrained [122] by the requirement that neutrinos must be optically thick in supernova explosions (Section 7.6.3).

7.6.1 Stellar Evolution without the Weak Interaction

Another scenario for stellar evolution arises in universes where the weak interaction is absent [245, 266], or much weaker than that of our universe [285]. As outlined in Section 5, the weakless universe emerges from its BBN epoch with a substantial mass fraction in helium-4, and with the remainder of its baryons distributed among free protons, free neutrons, and deuterium. Given this chemical composition and the absence of weak interactions, stellar evolution must rely on unconventional nuclear reaction chains. Nonetheless, stars can still function in such universes, and their properties are roughly similar to those of ordinary stars.

In our universe, with its weak interaction, free neutrons decay to protons, so that stars begin their evolution with a store of hydrogen fuel (helium does not play a role in early evolution). The first step of the reaction chain is to synthesize deuterium through the reaction $p+p\to{d}+e^{+}+\nu_{e}$, which involves the weak force. This reaction cannot take place in a weakless universe. However, if free neutrons remain, stars can make deuterium through the reaction $p+n\to{d}$, which involves only the strong force. Some fraction of the primordial free neutrons can be used up, through this same reaction, before the neutrons are incorporated into stars. Whether the neutrons are fused before star formation, or in stellar cores, the net result is a substantial supply of deuterium, in addition to that produced during BBN. This deuterium is available for the next steps of the reaction chain, including $p+d\to$ 3He $+\gamma$, $d+d\to$ 3He $+n$, and $d+d\to$ 4He $+\gamma$, and others. The first of these reactions is the standard next step in the $p$-$p$ chain in ordinary stars [147, 265, 323]. The final two reactions are highly suppressed in our universe due to the small primordial abundance of deuterium and its rapid reaction rate in stars. Given the enhanced abundances of deuterium in the weakless universe, stars can generate energy primarily through deuterium burning.

Numerical simulations have recently been carried out to illustrate how stellar evolution takes place in a weakless universe [245]. These calculations use the MESA stellar evolution package [420, 421], which has been modified to incorporate the required nuclear reaction chains. For example, interactions involving the weak force have been removed, whereas a standing population of free neutrons and their corresponding reaction pathways have been added. Additional nuclear reactions that are rare in our universe (given its chemical composition) are also included, primarily those involving deuterium (which is more plentiful in a weakless universe).

The results of stellar evolution calculations for the weakless universe are shown in Figure 28. Stellar properties are plotted for three different compositions emerging from BBN (see Section 5.3), corresponding to baryon to photon ratios $\eta$ = $10^{-11}$ (red curves), $10^{-10}$ (green), and $10^{-9}$ (blue). The initial neutron to proton ratio is taken to be unity, $n/p=1$, for all cases considered here. As $\eta$ increases, the helium abundance increases, whereas the mass fractions of deuterium and other nuclear species decrease.

The upper left panel shows the effective surface temperature of the stars as a function of stellar mass. Over the range of compositions shown, and the entire range of stellar masses, these temperatures fall in the interval 3000 K \lower 2.15277pt\hbox{;\buildrel<\over{\sim};}T_{\rm eff}\lower 2.15277pt\hbox{;\buildrel<\over{\sim};} 10,000 K (recall that the Sun has $T_{\rm eff}\approx 5800$ K). Compared to stars in our universe, these objects have comparable surface temperatures, but span a somewhat narrower range. The lower left panel in the Figure shows the central temperature of the stars versus mass. As expected, the central temperature is a slowly increasing function of stellar mass, and also increases with decreasing deuterium abundance. The most significant feature is that the central temperatures are of order $10^{6}$ K, characteristic of deuterium burning. For comparison, the central temperature of the Sun is $\sim 1.5\times 10^{7}$ K, the value characteristic of hydrogen burning.

The upper right panel of Figure 28 shows the stellar luminosity as a function of mass. The luminosity displays the approximate power-law form $L_{\ast}\propto M_{\ast}^{2}$, which is less steep than the dependence in our universe for hydrogen burning stars. The range of luminosities is somewhat smaller than in our universe: The luminosity for massive stars is comparable, but that for low-mass stars is larger by factors of 10 – 100. The lower right panel shows the stellar radius as a function of mass, which displays a nearly linear dependence $R_{\ast}\propto M_{\ast}$. This finding is consistent with the outer boundary condition for stars, $L_{\ast}$ = $4\pi R_{\ast}^{2}$ $\sigma_{\rm sb}T_{\rm eff}^{4}$, given the relatively constant surface temperatures and the mass-luminosity relation. For massive stars, the stellar radii are somewhat larger than those of our universe, consistent with their cooler photospheric temperatures. As a result, massive stars in the weakless universe look much like the red giants of our universe.

In spite of their non-standard nuclear reactions, stars in a weakless universe have roughly standard properties, including luminosities and lifetimes. One can understand this similarity as follows: The net result of nuclear burning — both with and without the weak interaction — is the production of helium-4, which has a binding energy of about 28 MeV. Instead of starting the process with protons, stars in the weakless universe begin with deuterium, which has a binding energy of 2.2 MeV. The energy supply in weakless stars is smaller than that of ordinary stars, but only by $\sim 10\%$.

7.6.2 Stellar Constraint on the Weak Interaction

The previous section showed that stars can function and universes can remain viable in the absence of the weak interaction. The requirement of working stars can also be used to place an upper bound on the strength of the weak interaction. Nuclear burning in stellar interiors produces a large flux of neutrinos, which freely stream out of the stars. If the weak interaction cross section were much larger, then the neutrinos could be optically thick, and stellar structure could be altered accordingly [285]. It remains possible for stars to function with optically thick neutrinos, but their evolution would change significantly.

In this setting, the optical depth for neutrinos can be written in the form

[TABLE]

where the integral is taken over the extent of the star and $\langle n\rangle$ is defined via the mean value theorem. The cross section for neutrino interactions is of order $\sigma_{\nu}\sim G_{F}^{2}E_{\nu}^{2}$, where $E_{\nu}$ is the energy of the stellar neutrinos (typically $E_{\nu}\sim 1$ MeV for stars in our universe [41]). The mean number density of particles in the star can be written $\langle n\rangle\approx$ $3M_{\ast}/(4\pi m_{\rm p}R_{\ast}^{3})$. The stellar radius has the form given by equation (125), so that $R_{\ast}\approx$ $GM_{\ast}m_{\rm p}/T_{\rm c}$, where $T_{\rm c}$ is the central nuclear burning temperature (in natural units). Finally, the stellar mass can be expressed in terms of the fundamental mass scale for stars (see A) so that $M_{\ast}=X\alpha_{G}^{-3/2}m_{\rm p}$, where $X$ is a dimensionless parameter of order unity. The combination of these factors allows the optical depth to be written in the form

[TABLE]

The requirement that neutrinos can freely stream out of stellar interiors thus implies a constraint on the weak interaction of the form

[TABLE]

where we have used $E_{\nu}$ = 1 MeV and $T_{\rm c}$ = 1 keV to evaluate the bound and where $(G_{F})_{0}$ = (292 GeV)-2 is the value in our universe.

7.6.3 Supernova Constraint on the Weak Interaction

In order for neutrinos to play a role in supernova explosions, they must have a sufficiently high interaction rate. In order of magnitude, the neutrino interaction rate must be comparable to the free-fall collapse time for the inner region of the star [122]. This condition can be written in the form

[TABLE]

where $n$ is the number density of particles that the neutrinos interact with, $\rho$ is the total mass density of collapsing stellar material, and $c_{1}$ is a dimensionless constant of order unity. The parameter $c_{1}$ determines how closely the time scales for neutrino interactions and stellar collapse must match (and $c_{1}$ could have a range of values). The speed $v=c=1$ and the cross section can be written in the form

[TABLE]

where $E_{\nu}$ is the neutrino energy and the dimensionless constant $c_{2}$ depends on the types of particles in the stellar material. The density is close to nuclear densities, so we can express it in the form $\rho=c_{3}m_{\rm p}m_{\pi}^{3}$, where $c_{3}$ is another dimensionless constant, and $m_{\pi}$ is the pion mass. The neutrino energies are of order 1 MeV, so we can write $E_{\nu}=c_{4}m_{\rm e}$. Combining these expressions thus yields the requirement that must be met in order for neutrinos to affect supernovae,

[TABLE]

If we define $\alpha_{\rm w}=G_{F}m_{\rm p}^{2}$ and use the values of the constants $c_{j}$ corresponding to equality in our universe, then we find the approximate constraint

[TABLE]

In order for neutrinos to help enforce supernova explosions, the approximate equality of equation (166) must be met. If the weak interactions are ineffective, then all of the neutrinos can leave the system without interacting and the explosion stalls. On the other hand, successful detonation seems to require that the neutrinos escape the stellar core and reach the outer layers that are driven off by the explosion. For neutrino interactions that are too strong, the neutrinos scatter many times before leaving the core and distribute their momenta among many nuclei. If this momentum deposition occurs deep in the gravitational potential, then Type II supernovae could still fail. Moreover, two-dimensional and three-dimensional effects, along with stellar mass and composition, also influence the explosion. Unfortunately, our current understanding core-collapse supernovae remains incomplete (see [297, 298] for recent reviews), so that we do not know how closely the two sides of equation (166) must approach equality.

For completeness, we note that supernovae from massive stars are not the only sources of elements heavier than iron. In our universe, neutron star mergers provide the primary environments for the rapid neutron-capture process required to synthesize such large nuclei [214, 215], and the gravitational radiation from such events can now be observed [2]. Explosive nucleosynthesis can also take place in white dwarfs, which result from the deaths of lower mass stars. Explosion of these objects takes when they accrete enough mass to exceed the Chandraskehar limit [295, 402], and more rarely via collisions. In any case, core-collapse supernova explosions are not strictly necessary for heavy element production, although they do provide the requisite neutron stars for mergers in our universe. On the other hand, neutron stars can also be produced by mergers of white dwarfs [456], so that many evolutionary channels are possible in principle.

7.6.4 Supernova Constraints on the Nucleon Potential

The successful launch of supernova explosions depends on a number of factors. As discussed above, one requirement is that neutrinos must be sufficiently optically thick to help drive out the explosion. Another possible constraint arises from the form of the nuclear potential. In addition to the general Yukawa form described by equation (6), the nucleon-nucleon potential has a repulsive core that acts on short distance scales because the wavefunctions of the nucleons do not readily overlap [201]. If the nuclear potential had a somewhat different form, then supernovae could be compromised.

After a massive star produces a degenerate iron core, its central regions rapidly collapse and reach densities comparable to that of nuclear material. In successful explosions, the repulsive core of the nucleon-nucleon potential prevents the central region of the star from becoming so dense that the material falls within its event horizon and becomes a black hole. Although some supernovae produce black holes in our universe, the majority result in explosions that distribute heavy elements into the galaxy and leave behind neutron stars. In other universes, if the nuclear potential has different properties, more (or perhaps all) core-collapse supernovae could result in failed explosions and black hole formation. Such universes would still produce heavy elements via AGB stars, Type-Ia supernovae, and collisions within the diminished population of neutron stars, but they could have much lower metallicity than ours.

The short-range repulsive force can be modeled by the exchange of vector mesons. The relevant particles are the $\omega$ and $\rho$ mesons, both with masses $m\sim 770-780$ MeV, corresponding to a Compton wavelength of $\sim 0.25$ fm. This scale is comparable to the repulsive core of the nucleon potential, which has an estimated size $\sim 0.4-0.5$ fm [201]. The vector mesons can be described in terms of their constituent quarks, e.g., the $\omega$ particle can be written in the form

[TABLE]

Larger quark masses would thus imply heavier vector mesons and a smaller spatial extent of the repulsive core of the nuclear potential. In a universe with such properties, when massive stars collapse at the end of their lives, they would reach higher central densities and would be susceptible to black hole formation.

A detailed assessment of the properties of the nuclear potential necessary for supernova explosions has not yet been carried out. In approximate terms, we know that supernovae generally result in nuclear densities and the production of neutron stars. Since neutron stars have masses $M\sim 1M_{\odot}$ and radii $R\sim 10$ km, whereas black holes of the same mass have radii $R_{\rm bh}=2GM/c^{2}\sim 3$ km, a density enhancement of $\sim 20-30$ during stellar collapse should be sufficient to favor black hole production. This enhancement would result from a factor $\sim 3$ decrease in the spatial extent of the repulsive core in the nuclear potential, and hence a factor of $\sim 3$ increase in the masses of both the vector mesons and their constituent quarks. This constraint on the light quark masses is roughly comparable to those found in Section 2.2.

The description given here in terms of meson exchange is highly approximate, and a full treatment using effective field theory [201, 381] should be carried out. The change in the length scale of the repulsive core can also be thought as a change in the equation of state of dense nuclear matter. If the equation of state becomes sufficiently soft [475], then black hole formation is enabled. Moreover, near the densities where the repulsive core becomes important, nucleons begin to break up into their constituent quarks. The quarks are not truly free until they reach much larger densities, so a proper treatment must include the physics of the transition and its complications. For reference, note that recent lattice QCD calculations find that the quark/hadron phase transition occurs at energies $T_{c}\approx 173$ MeV [504].

8 Planets

Planets represent the smallest astrophysical objects that are necessary for the development of life (as we know it). In order for a given universe to become habitable, the laws of physics must allow for the production of planets with a number of basic properties, as outlined in this section. These bodies must be small enough in mass so that they are not degenerate. This requirement is also necessary (but not sufficient) for the planet to have a solid surface [293]. On the other hand, the planets must have enough mass to retain a gaseous atmosphere and must have enough particles to support a biosphere with sufficient complexity. One also expects planets to be smaller in mass than their host stars and galaxies. These considerations place constraints on the allowed range of parameter space $(\alpha,\alpha_{G})$.

In addition to constraints on the properties of planets themselves, the existence of habitable planets also involves a number of environmental constraints. Potentially habitable planets in our universe are subject to well-known requirements: Planets must be the right distance from their host stars to allow for liquid water, the planet mass must be roughly comparable to Earth, and the stars must have sufficiently long lifetimes [320, 334, 367, 368, 465]. Additional requirements must be enforced in other universes: As discussed above, planets must reside in galaxies that are dense enough to cool and make stars and planets, yet remain diffuse enough to allow for the survival of habitable orbits (Section 6). Planets must be made of heavy elements, which require successful stellar nucleosynthesis (Section 7), along with mild constraints on Big Bang Nucleosynthesis (Section 5). At the more fundamental level, nuclei themselves must be stable (Section 2). In addition, if habitable planets require particular nuclear structures, such as iron being the most stable nucleus and/or particular types of radioactivity, then even tighter constraints on the fundamental parameters (e.g, $\alpha$) can be derived [462, 463].

8.1 Mass Scale for Non-Degenerate Planets

In order for an astronomical body to function as a planet, it must be small enough in mass to remain non-degenerate. This requirement allows the planet to have a solid surface — or liquid in the case of so-called water worlds.

The standard way to formulate this constraint on planetary mass is to require electromagnetic forces to dominate gravitational forces on the scale of the planet [61, 435, 549]. Equivalently, the electromagnetic energy $E_{\rm em}$ must be larger than that of self-gravity $E_{\rm g}$. Consider a planet with mass $M_{P}$ and radius $R_{P}$. This constraint can be written in the form

[TABLE]

where $N$ is the number of atoms in the planet, $\ell$ is the effective distance between charges, and $f_{p}$ is a dimensionless constant of order unity that depends on the internal density distribution of the planet [136], where typical values fall in the range $f_{p}=3/5-3/4$. The mass of the planet and the number of atoms are related so that

[TABLE]

where $A$ is the mean atomic weight for the constituent atoms. With these specifications, the mass of the planet must obey the upper limit

[TABLE]

This scale is comparable to the mass of Jupiter, where $M_{J}\sim 300M_{\oplus}$. In our universe, planets with such large masses are observed to have low density and are inferred to be primarily made of gaseous material. In order for a planet to be rocky, and have a solid surface, the mass must be significantly smaller, with current observations indicating $M\sim 10M_{\oplus}$ [557]. As a result, the limit of equation (170) is necessary, but not sufficient. In addition to hard surfaces, habitable planets must also have surface gravities weak enough to allow for the survival of flora and fauna [290]. Both of these considerations suggest that the expected masses for habitable planets are closer to the lower bound considered in the following section.

8.2 Mass Scale for Atmospheric Retention

The surface gravity of a planet must be sufficiently strong in order for it to retain its atmosphere [61, 434, 435]. At the same time, the surface of the planet, and hence the atmosphere, must be hot enough to support the chemical reactions necessary for life. This latter requirement can be written in the form

[TABLE]

where $T$ is the surface temperature and $E_{\rm chem}$ is the energy required for chemical reactions to occur (from equation [129]). Recall that the chemical conversion factor $\epsilon_{\rm c}\sim 10^{-3}$.

In order for a planetary atmosphere to remain intact, its constituent molecules cannot evaporate on short timescales. As a result, the surface temperature must correspond to an energy scale less than the gravitational binding energy of an atmospheric molecule, i.e.,

[TABLE]

where $A_{\rm atm}$ is the mean atomic weight of the molecules (recall that $A_{\rm atm}\approx 29.6$ for terrestrial air). The constraint of equation (172) is approximate: The evaporation of planetary atmospheres depends on a number of additional factors, including the planetary structure (which determines the surface gravity), the planetary magnetic field, the location within its solar system, and spectral properties of the host star [410].

The surface temperature of the planet must be hot enough to support chemistry (equations [171, 129]) and cold enough to suppress evaporation (equation [172]). In order to evaluate the planetary radius, one can assume that the atoms in the (solid) planet subtend a volume $\sim a_{0}^{3}$, where $a_{0}=(\alpha m_{\rm e})^{-1}$ is the Bohr radius. These combined constraints then result in a lower bound for the planetary mass

[TABLE]

Keep in mind that the atomic weight of the atmosphere $A_{\rm atm}$ can be different from that of the bulk of the planet $A$.

Note that the maximum mass for non-degenerate planets from equation (170) is significantly larger than the minimum mass from equation (173). If we let $A_{\rm atm}$ = $A$, then the two mass scales are related such that $M_{P{\rm min}}$ $\approx$ $\sqrt{3}\epsilon_{\rm c}^{3/2}M_{P{\rm max}}$. Since the chemical conversion factor $\epsilon_{\rm c}\sim 10^{-3}\ll 1$, one finds that $M_{P{\rm min}}\ll M_{P{\rm max}}$, as required.

For completeness, notice also that planetary bodies with too little mass will not have enough self-gravity to become quasi-spherical. In sufficiently small bodies, like most of the asteroids and Kuiper Belt objects in our Solar System, imperfections in the planetary surface (effectively mountains) can be as large as the planet itself. In general, the requirement of retaining an atmosphere is stronger than that of maintaining a spherical shape [61, 549].

8.3 Allowed Range of Parameter Space for Planets

With the above constraints specified, this section estimates the allowed range of parameter space that supports potentially habitable planets. We first note that planets must be larger than the mass scale of equation (173) to retain their atmospheres and smaller than the mass scale of equation (170) to remain non-degenerate. Both of these mass limits can be written as dimensionless parameters times the mass scale

[TABLE]

where $A$ is the atomic weight of the planetary material. (Note that we can define a single characteristic planetary mass scale by setting $A=1$; see equation [237]). The lower limit (173) is smaller than the upper limit (170) by a factor of $\epsilon_{\rm c}^{3/2}\sim 10^{-9/2}\ll 1$, so that both limits can be simultaneously satisfied.

Notice also that the scale of equation (174) that characterizes planetary masses differs from the corresponding mass scale for stars (see A and equation [228]) by a factor of $\alpha^{3/2}$. If one requires that planets are less massive than their host stars, then the fine structure constant obeys the constraint \alpha\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}1.

The planetary mass scale from equation (174) must be larger than the minimum mass required for a planet to become habitable. One specific constraint is that planets must be large enough to support a working biosphere [13]. Since we do not know the minimum information content of a biosphere, this limit is necessarily speculative. As a starting point, we can use the values in our universe to obtain a working estimate: The biosphere of Earth has 500 to 800 billion tons of carbon, which implies a possible information content of $Q_{B}\approx 4\times 10^{40}$ bits. The corresponding mass of our biosphere is $M_{B}=12Q_{B}m_{\rm p}\approx 10^{-10}M_{\oplus}$. In general, the host planet must be larger in mass than its biosphere by a large but unknown factor $f_{\rm bio}$. If we require the planetary mass scale to be larger than that necessary to support a biosphere, the following constraint must be met

[TABLE]

Although the limiting value of $f_{\rm bio}$ is not known, only the outer layers of a planet are hospitable for biology. On our planet, the biosphere has a thickness of order 10 km, whereas the planet has radius $R_{\oplus}\approx 6400$ km. As a result, we expect $f_{\rm bio}\gg 1000$.

One can derive another (relatively weak) constraint on the structure constants by requiring that planetary masses are smaller than the masses of their host galaxies. This constraint is necessary, but not sufficient, for planet formation to take place. Using the planetary mass scale (237) and galactic mass scale (239) from A, this constraint takes the form

[TABLE]

where $\beta$ is the electron-to-proton mass ratio. Since $\alpha_{G}\ll\alpha$, this constraint only becomes significant for extremely small values of the fine structure constant $\alpha$.

The constraints described above limit the allowed parameter space for universes with differing values of the structure constants $\alpha$ and $\alpha_{G}$, as shown in Figure 29. In order for planets to be smaller in mass than their host stars, the fine structure constant $\alpha$ must be small, as delimited by the vertical green line. In order for planets to be smaller in mass than their host galaxies, $\alpha$ must be larger than the limit given by the cyan line (from equation [176]). Finally, the magenta line shows the limit required for planets to simultaneously be small enough to remain non-degenerate and contain enough mass to support a biosphere of sufficient complexity (from equation [175], where the planet is assumed to be $f_{\rm bio}=10^{4}$ times more massive than its biosphere). The star symbol shows the location of our universe in the diagram. The allowed range in $\alpha$ spans six orders of magnitude, whereas the range in $\alpha_{G}$ spans more than eleven orders of magnitude (although one should keep in mind that the upper limit on $\alpha_{G}$ depends on biosphere requirements and is thus uncertain). In any case, the existence of planets does not require highly specialized values of $(\alpha,\alpha_{G})$. Notice also that the allowed range of parameter space for planets (from Figure 29) is larger than the corresponding range for the existence of working stars (from Figure 23).

8.4 Planet Formation

The discussion thus far has considered the structure of planets – and the their prospects for survival – but not the processes that form these bodies. We have thus implicitly assumed that if planets can exist, they can somehow be made. Although a comprehensive theory of planet formation remains under construction, current observations of extra-solar planets suggest that planet formation takes place readily. Here we briefly review the current state of the field [37] and identify the basic ingredients necessary for planet formation to occur.

The field of exoplanets has developed rapidly over the past two decades. The first confirmed planet in orbit about another star was discovered in 1992 associated with a pulsar [558]. The detection of the first planet orbiting a main sequence star, 51 Pegasi, was reported in 1995 [375], and planetary discoveries have steadily piled up. Exoplanets have now been discovered by radial velocity methods, transits of their host stars, microlensing, and by direct imaging. The first planet with a mass comparable to Earth and an orbit consistent with liquid water was reported in 2014 [438], and about twenty so-called habitable planets are now known [316]. The inventory of exoplanets is thus enormous, with thousands of detections already reported [284]. Moreover, current projections indicate that the Galaxy contains more planets than stars and that the fraction of Sun-like stars with Earth-like planets is sizable, namely $\eta_{\oplus}\approx 0.1$ [426].

Planets form in the nebular disks that are found ubiquitously around young stars. This nebular hypothesis for the origin of planets [317, 337, 560] predates by centuries the actual observations of circumstellar disks. These structures were unambiguously discovered in the 1980s through their infrared radiation signatures and other indirect evidence. They can now be imaged directly using submillimeter interferometry and their properties can be measured with exquisite precision (see [271] for a recent review). These disks have roughly solar system size scales, with typical outer radii in the range 10 – 100 AU. The total mass in the disk represents the material available for planet formation and varies with time. Current observations indicate that disk mass can be as large as 10 percent of the stellar mass at the end of the star formation process and drops rapidly over the next 3 – 10 Myr [276].

Within these disk environments, planets can form through two conceptually different channels. First, planet formation can take place from the top down, through gravitational instabilities in massive circumstellar disks [85, 115, 439]. Second, planet formation can take place from the bottom up, through the accumulation and agglomeration of rocky building blocks into ever larger bodies [432]. If this process occurs fast enough, these growing rocky cores can accrete large amounts of gas from the nebula and become giant planets (like Jupiter). In other cases, the gas in the disk can be removed before runaway accretion takes place, and the planet is left with a rocky core, along with ices and other volatiles. This latter scenario produces planets like Uranus and Neptune, and apparently occurs $\sim$ten times more often than the successful production of Jovian planets.

Although both of the aforementioned mechanisms are likely to take place, the core accretion (bottom up) scenario is thought to dominate. This paradigm accounts for the observed mass distribution of planets, which is weighted toward bodies of lower mass. Moreover, many of the observed Jovian planets are close to their stars, where they can be observed using both radial velocity and transit techniques. As a result, their mass and radius can be measured, and hence their densities, and their internal structure can be inferred. This program indicates that many Jovian planets have high metallicity, consistent with the presence of large rocky cores. These considerations suggest that sufficiently high metal content is a necessary ingredient for planet formation. In fact, in the current theoretical calculations of giant planet formation, the surface density of solid material is the most important variable in the problem [294].

In considering the potential viability of a universe, it is important to keep in mind that the chemical composition of planets is not necessarily the same as that of the host star — or the universe as a whole. Moreover, all of the planets in a given system will generally not have the same composition [74, 304, 384]. In our Solar System, for example, the giant planets are enriched in carbon relative to the Sun, whereas Earth is significantly depleted [32, 374]. A related issue is that the composition of the planetary surface layers, which are most relevant for biospheres, is generally not the same as the bulk composition. Planetary surfaces are often sculpted during their late formation stages, leading to both chemical enrichment and depletion. Relevant processes include continuing impacts of minor bodies, which provide one channel for water delivery to otherwise dry planets, and cataclysmic collisions, thought to be the mechanism that formed our moon [20, 72].

Our universe produces planetary bodies with a wide range of sizes by utilizing a variety of processes (as outlined above). Even our own Solar System contains many objects that are too large to have rocky surfaces (the four giant planets) and a multitude of objects that are too small to retain atmospheres (including $\sim 10$ dwarf planets and thousands of smaller bodies). Since planets of all sizes are readily produced, it is likely for some to fall within the mass range required for habitability. In other words, universes that support planetary structures (see Figure 29) are likely to produce some bodies with favorable properties.

8.5 Planets and Stellar Convection

Another potential fine-tuning constraint arises from the requirement that both convective and radiative stars are necessary for a successful universe. If all stars were convective, then massive stars would mix all of their nuclear burning layers as they evolve, and thereby erase the classic “onion skin” structure found during the advanced stages of nuclear burning. Such mixing would cycle all of the elements with intermediate mass numbers (such as carbon and oxygen) into the stellar core, where they would be processed into larger nuclei. This scenario could potentially leave the universe with little carbon and oxygen. On the other hand, if all stars were radiative, then some authors have claimed (starting with [131]) that planet formation would be compromised. This section re-examines the argument that both convective and radiative stars are necessary for a universe to be viable.

As a general rule, low mass stars remain convective over much of their lifetimes, whereas larger stars develop a radiative structure [147, 265, 323, 427]. The boundary in stellar mass between these two regimes can be expressed in terms of fundamental constants [61, 122] and has the form

[TABLE]

where $M_{\star}$ is the mass scale of a typical star. If one requires that the mass threshold $M_{\rm r/c}$ is comparable to the typical stellar mass $M_{\star}$, then the following approximate equality must hold:

[TABLE]

Because of the large exponents on $\alpha$ in equations (177) and (178), modest changes in the fine structure constant could raise or lower the mass threshold and lead to a universe where all stars are either convective or radiative.

Although suggestive, the mass threshold in equation (177) is not sharp. Most stars — including the Sun — are convective in their early phases of evolution and develop radiative cores later on. As a result, the question of whether stars have convective versus radiative structure depends on time. In addition, when massive stars enter into their advanced stages of nuclear burning, various layers of the star become convective and radiative as the stars evolve (this complicated behavior is readily seen in stellar evolution calculations of intermediate and high mass stars using modern computational methods [420, 421]). In any case, it is overly simplistic to use a back of the envelope estimate such as equation (178) to delineate the boundary between convective and radiative energy transport in stellar interiors.

Putting aside the above complications, suppose that the mass threshold $M_{\rm r/c}$ is raised, so that all stars become convective. In the worst case scenario, all of the massive stars would remain completely convective over their entire lifetimes. After being produced during intermediate evolutionary stages, carbon and oxygen would be mixed deep into the stellar interior and processed into even larger nuclei. By the time the iron core is fully developed and the star explodes as a supernova, most if not all of the carbon and oxygen could be depleted.

Although this scenario would lead to a universe different from our own, with different cosmic abundances of the elements, habitability is not necessarily compromised. Stars with intermediate masses, roughly comparable to the Sun and somewhat larger, burn their helium into carbon and oxygen, but do not produce heavier elements. In universes with convective massive stars, these intermediate mass stars could provide enough carbon and oxygen to support habitability. In fact, in our universe, a substantial fraction of the carbon inventory is produced during the Asymptotic Giant Branch (AGB) phases of intermediate mass stars [106, 250, 278]. AGB stars in other universes could thus provide the necessary carbon, even if more massive stars maintain convective cells and burn through their supply.

If the mass threshold of equation (177) is lower, then all stars would be radiative. In spite of the original claim [131] (see also [61, 122, 507, 511] and others), planet formation does not rely on stars having a convective internal structure. As outlined above, planets form in circumstellar disks, which are found nearly ubiquitously around young stars, and which are decoupled from the internal structure of the star. Both mechanisms of planet formation, core accretion and gravitational instability, rely on physical processes that take place within the circumstellar disk, and do not depend on the stellar interior. Taking an observational perspective, we have now detected planets around stars without convective zones, including higher mass stars (see [303] and many others), degenerate white dwarfs [531, 564], and even pulsars [558]. As a result, both observations and theoretical considerations definitively show that convective stars are not necessary for planet production.

9 Exotic Astrophysical Scenarios

Our local bubble of parameter space is suitable for habitability, and the discussion thus far has focused on delineating the boundaries of this region. Given the wide range of particle physics and cosmological parameters that could be realized across the multiverse, it becomes possible for other universes to utilize unconventional pathways and power sources. As a result, additional bubbles of habitability could exist with cosmic properties markedly different from our own. This section explores some of these possibilities:

Dark matter annihilation provides a promising channel of detecting this elusive material [145], but the expected contribution to the galactic energy budget is minimal. In other universes, however, with denser galaxies and/or different dark matter properties, this source of energy can be substantial (Section 9.1). Similarly, dark matter is expected to collect and annihilate within stellar bodies in our universe, especially degenerate stellar remnants such as white dwarfs and neutron stars. This channel of energy production could also be enhanced in universes with denser galaxies and more interactive dark matter (Section 9.2). Note that the dark matter abundance is determined in the early universe at age $t\sim 1$ sec and depends on the cross section for self-interactions [327]. In order for other universes to have comparable dark matter abundances but different cross sections, other cosmological parameters must vary, or the dark matter abundance must be determined by non-thermal (out of equilibrium) processes [40, 307, 314]. Next we note that black holes can provide a source of power through Hawking evaporation. Although this process is completely negligible in our universe, it could become important in the regime where gravity is much stronger relative to the electromagnetic force (Section 9.3). If the weak force is less effective, then compact objects composed of degenerate dark matter can play the role of stars (Section 9.4). Finally, even in the absence of nuclear reactions, universes can still produce dark matter halos, galaxies, and even stars that shine via gravitational contraction (Section 9.5). These latter universes would not be habitable, but would nonetheless bear an eerie resemblance to our own.

9.1 Dark Matter Halos as Astrophysical Objects

Dark matter halos are essentially inert in our present-day universe, as their evolutionary time scales are much longer than the current cosmic age [12]. Here we consider the case where dark matter can be more interactive than in our universe, either through enhanced densities or larger cross sections. For the sake of definiteness, we consider the dark matter halos to have the form of a Hernquist profile (see equation [86], Section 6.2, and Refs. [277, 394]).

In our universe, primordial density fluctuations, inferred from observed inhomogeneities in the Cosmic Background Radiation, have amplitude $Q\approx 10^{-5}$. These fluctuations could be larger in other universes, with the consequence that galaxies can form earlier and become denser (see Section 6 and references therein). Here we define the relative amplitude

[TABLE]

where $Q_{0}\approx 10^{-5}$ is the value in our universe. The parameters $\rho_{0}$ and $r_{0}$ that specify the properties of dark matter halos vary with the fluctuation amplitude that specifies the initial conditions. For dark matter halos with density profiles given by equation (86), the dependence of $\rho_{0}$ and $r_{0}$ on the amplitude $Q$ has been derived previously (see equations [85] and [88]), where these results are based on the standard paradigm for galaxy formation [553]. The resulting scaling laws take the form

[TABLE]

The fluctuation amplitude can be larger by more than a factor of $\sim 1000$ (Section 6.3), so that the halo densities can be enhanced by many orders of magnitude.

9.1.1 Power from Dark Matter Annihilation

The annihilation rate per particle $\Gamma_{1}$ at a given radial location within the halo has the form

[TABLE]

where $\langle\sigma{v}\rangle_{\rm dm}$ is the cross section for dark matter annihilation and $m_{\rm dm}$ is the mass of the particle. The total annihilation rate and hence the luminosity are determined by integrating over the entire halo. The luminosity due to dark matter annihilation is given by

[TABLE]

Typical values in our universe for Milky-Way-like galaxies are $\rho_{0}\sim 10^{-25}$ g cm*-3* and $r_{0}\sim 65$ kpc [77, 78], whereas typical dark matter properties are $\langle\sigma{v}\rangle_{\rm dm}\sim 10^{-27}$ cm3 s*-1* and $m_{\rm dm}\sim 100m_{\rm p}$ [208, 307]. For these parameters, the luminosity (182) evaluates to $L_{\rm dm}\approx 500L_{\odot}$. Since the luminosity of the halo scales as $q^{3}$, other universes can have an enhancement of order $\sim 10^{9}$ (using $q$ = 1000). The dark matter luminosity thus becomes $L_{\rm dm}\sim 5\times 10^{11}$ $L_{\odot}$, comparable to the stellar luminosity of a moderate-sized galaxy in our universe. The luminosity $L_{\rm dm}$ results from annihilation of dark matter and corresponds to a mass loss rate given by

[TABLE]

The radiation flux at a location $\xi=a$ within the galaxy can be written in the form

[TABLE]

The angular integral can be evaluated to obtain

[TABLE]

where we have defined a dimensionless integral function of the position $a$, i.e.,

[TABLE]

Near the galactic center, where $a\ll 1$, the integral approaches the form $I\to-\log(a)$. In the outer parts of the halo where $a\gg 1$, the integral $I\sim a^{-1}$, so that the flux $F_{G}\sim a^{-2}$.

The typical value of $F_{G}\sim L_{\rm dm}/(4\pi r_{0}^{2})\sim 4\times 10^{-12}$ erg s*-1* cm*-2* $\sim 3\times 10^{-18}S_{\oplus}$ (where $S_{\oplus}$ is the radiation flux received by Earth from the Sun). The radiation flux from dark matter annihilation scales as $F_{G}\sim q^{5}$, so that dense galaxies in other universes can be enhanced by a factor of $10^{15}$ or more. Inner parts of such galaxies, or somewhat denser galaxies resulting from $q\approx 3200$, can thus have background radiation fields equal to the value $S_{\oplus}$ due to Solar irradiance on Earth.

9.1.2 Time Evolution of Dark Matter Halos

For a given mass, the halo properties are determined by specification of the parameters $\rho_{0}$ and $r_{0}$. In order to provide a quantitative description of the time evolution of the dark matter halo, we need another constraint on the halo structure in addition to the equation (183) that determines the mass loss rate. Here we make the approximation of adiabatic compression [473], which assumes that the specific angular momentum of orbits (here, for dark matter particles) remains constant as the mass changes. In this context, the approximation is equivalent to assuming that that composite parameter

[TABLE]

This constraint, in conjunction with the specification of mass loss through equation (183), determines the time evolution of the halo properties. We first define a dimensionless time variable

[TABLE]

where $M_{0}$ and $L_{0}$ are the initial mass and luminosity of the halo respectively. The second equality defines the characteristic time scale for time evolution of dark matter halos. For typical parameters $\langle\sigma{v}\rangle_{\rm dm}\approx 10^{-27}$ cm3 s*-1* and $m_{\rm dm}\approx 100m_{P}$, this time scale become $t_{\rm dm}\approx 3\times 10^{21}$ yr for a dark matter halo with initial properties similar to those of the Milky Way.

However, the characteristic timescale for the evolution of dark matter halos scales as $t_{\rm dm}\sim q^{-3}$. Using the larger value $q=1000$, the time scale can be shorter by a factor of $\sim 10^{9}$. Given the fiducial time scale for halo evolution of $\sim 10^{22}$ yr, denser halos in other universes could evolve on time scales of $\sim 10^{13}$ yr without changing the properties of the dark matter.

The time-dependent density, potential, and radial scale are given by

[TABLE]

and the corresponding solutions for the time evolution of the halo mass and luminosity have the forms

[TABLE]

The column density of the dark matter halo, integrated from spatial infinity to a radial location $\xi$, is given by the expression

[TABLE]

The optical depth of the halo to its radiation field (that generated by the dark matter annihilation) is thus of order

[TABLE]

where $\sigma_{\rm rad}$ is the cross section for interactions between the annihilation photons and the remaining dark matter particles. For values in our universe, the optical depth $\tau\sim 10^{-16}$. As a result, the halo does not have a photosphere – it is optically thin to the radiation it generates, so that photons freely stream outwards.

On the other hand, the halo could contain a gaseous component. Here we assume that the mass in baryons is a fraction $f$ of the dark matter density and that the gas has the same form for its density distribution (see equation [86]). The optical depth of the baryons to the radiation produced via dark matter annihilation is given approximately by

[TABLE]

where the numerical estimate assumes properties comparable to those of our Galaxy.

The optical depth scales as $q^{2}$, and hence can be larger by a factor of $\sim 10^{6}$. Even with this level of enhancement, the annihilation products (gamma rays) would have an optical depth that is much less than unity for interactions with dark matter. However, the halo would become optically thick due to its baryon content. In this case, the annihilation products interact with hydrogen gas and the radiation field would be processed to longer wavelength (lower energy), which is more compatible for supporting habitable planets.

Figure 30 shows the allowed parameter space where dark matter halos produce enough radiation to support habitable planets, where this energy is produced via dark matter annihilation. The plane of parameter space includes the amplitude $Q$ of the primordial density fluctuations and the annihilation cross section for dark matter particles. For larger $Q$, galaxies form earlier and become more compact, thereby leading to enhanced radiation flux due to annihilation (see equation [185]). Similarly, larger cross sections lead to enhanced radiation fields. The flux due to dark matter annihilation is larger than the radiation received by Earth from the Sun for the parameter space above the blue curve. If dark matter annihilation is too efficient, however, dark matter halos will evolve too quickly. The green curve shows the boundary where the potential well of the halo becomes too shallow ($v<100$ km/s) over a time scale of 10 Gyr, which is comparable to the current age of our universe. If the amplitude $Q$ is too large, above the red curve in the figure, then black hole formation can proceed catastrophically. The figure shows that a wide range of parameters allows for dark matter to provide the energy necessary for habitability. On the other hand, this regime is far removed from the parameters of our universe, which is marked by the orange arrow at the origin.

Notice that Figure 30 does not include an upper limit on the annihilation cross section for dark matter, although $\langle\sigma{v}\rangle_{\rm dm}$ cannot be made arbitrarily large. This cross section depends on the strength of the weak force. In our universe the strong force dominates on the scale of atomic nuclei, so that the weak force provides only a perturbative effect on nuclear structure. Increases in the cross section $\langle\sigma{v}\rangle_{\rm dm}$ correspond to increases in the weak coupling constant, which cannot become too large without compromising nuclear reactions in stars and even the existence of bound states. In the low energy limit, the weak coupling constant $\alpha_{\rm w}$ = $G_{F}m_{\rm p}^{2}\sim 10^{-5}$, which is much smaller than the strong coupling constant $\alpha_{\rm s}\sim 10$ (see Section 2.1). As a result, the Fermi constant $G_{F}$ cannot increase by more than a factor of $\sim 10^{6}$, and the interaction cross section $\langle\sigma{v}\rangle_{\rm dm}\propto G_{F}^{2}$ cannot increase by more than a factor of $\sim 10^{12}$. More stringent bounds are likely, but require an in-depth analysis of nuclear structure. These considerations thus limit the parameter space shown in Figure 30. This effect is roughly comparable to the requirement that the halos live long enough, as depicted by the green curve.

9.2 Dark Matter Capture and Annihilation in White Dwarfs

White dwarfs provide another channel through which dark matter can be processed in a galactic halo [12]. These dense stellar remnants can accrete dark matter particles, which accumulate in the stellar core. Once the population is sufficiently large, the system reaches a steady state where the rate of annihilation in the stellar core is balanced by the rate of particle accretion from the halo. Although this process is expected to produce (at most) modest luminosities in our universe, this channel of energy generation can be significant in alternate universes with larger $Q$ and denser galaxies. Notice also that current experiments for the direct detection of dark matter [26, 34] put tight constraints on interactions between dark matter and baryons in our universe, but the relevant cross sections could be different in other universes.

White dwarfs are expected to be optically thick to dark matter particles. Here we assume that cross section of interaction between dark matter particles and the baryonic stellar material is $\sigma_{\rm dm}\sim 10^{-38}$ cm2. The density of a white dwarf is of order $\rho_{\rm wd}\sim 10^{6}$ g cm*-3*, which corresponds to a number density $n_{\rm wd}\sim 10^{30}$ cm*-3*. With the radius of a white dwarf, $R_{\rm wd}\sim 10^{8}$ cm, we find the optical depth to be

[TABLE]

The rate of capture of dark matter particles by the star is thus given by

[TABLE]

where $\sigma_{\rm wd}$ is the cross section of the star for the capture of dark matter. The cross section is enhanced over the geometric cross section of the star through gravitational focusing so that

[TABLE]

The dark matter particles will collect inside the star until the annihilation rate and the capture rate become equal, so that a steady state is reached. The resulting luminosity of the star, produced by the capture and subsequent annihilation of dark matter, is thus given by

[TABLE]

This luminosity is about $10^{17}$ Watt for white dwarfs in our galaxy. Including the billions of white dwarfs in the galaxy, the total luminosity produced through this channel is of order 1 $L_{\odot}$.

The power generated by dark matter capture and annihilation in white dwarfs is comparable to that generated via direct (particle on particle) annihilation. This approximate equality arises due to the similarity in opacities (cross section per unit mass) of the two processes, i.e.,

[TABLE]

With typical values for the parameters, both opacities are of order $10^{-14}-10^{-13}$ cm2 g*-1*.

In other universes, with larger values of $Q$, the galactic halos will be denser and the luminosity of individual white dwarfs generating energy via this mechanism will be larger by a factor of $q^{3}$. As discussed above, the initial fluctuations could be enhanced by a factor as large as $q=1000$, thereby increasing the luminosity to $L_{\rm\ast dm}\sim 10^{-2}L_{\odot}$. The total radiation flux from the galactic background will thus be given by equation (95), where the luminosity per star is taken to be $L_{\rm\ast dm}$. Since this luminosity scale is smaller (on average) than that of main sequence stars, the radiation fields from ordinary stars would dominate as long as they are actively burning nuclear fuel. In the absence of nuclear power, however, the luminosity of the galaxy is only smaller by a factor of $\sim 10^{2}-10^{3}$, so that the prospects for habitability are not completely diminished.

For completeness, we note that the dense conditions found in white dwarf interiors are conducive to pycnonuclear reactions, where nuclear fusion takes place at low temperature via quantum tunneling [113, 472, 461]. In this process, which is too slow to be important in our universe [475], the quantum mechanical zero point energies of the nuclei allow them to overcome their mutual Coulomb repulsion and interact. Although hydrogen can fuse in white dwarfs and thereby produce some additional helium, larger nuclei are not synthesized, even over time scales that vastly exceed the current age of the universe [12]. In other universes with different fundamental parameters, however, such pycnonuclear reactions could compete with dark matter annihilation as an additional energy source for white dwarfs. This issue should be considered further.

9.3 Black Holes as Stellar Power Sources

Black holes are expected to exist in any universe. Given that our universe forms black holes on both stellar and galactic scales, one also expects that the formation of black holes will not be completely suppressed. As a result, black holes will provide a power source through Hawking evaporation [272]. Such radiation is completely negligible in our universe at the present epoch, but could play a role in universes with stronger gravity and other favorable parameters. Black hole properties depend primarily on the gravitational constant $G$ (equivalently $\alpha_{G}$), whereas habitability involves atomic energies and time scales, and thus depends on the fine structure constant $\alpha$.

Black holes can exist over a wide range of masses, but they require a production mechanism. For this illustrative treatment, we focus on the case of stellar black holes, where the mass scale is approximately given by $M_{\rm bh}\approx M_{\star}=\alpha_{G}^{-3/2}m_{\rm p}$, comparable to the Chandrasekhar mass (see equation [228]). Black holes have effective temperatures given by

[TABLE]

where $M_{\rm bh}$ is the mass of black hole, which is taken to be $M_{\star}$ in the second equality. This temperature must be larger than that required to drive chemical reactions (from equation [129]). Black holes must satisfy the constraint

[TABLE]

The lifetime of a black hole with initial mass $M_{\rm bh}$ takes the form

[TABLE]

where $g_{\ast}$ is the number of effective degrees of freedom in the radiation field produced by Hawking evaporation and where the second equality assumes $M_{\rm bh}=M_{\star}$. In order for black holes to serve as engines of habitability, their lifetime must be sufficiently long. If we measure time in terms of atomic time scales, and require the benchmark number of such time units $N_{\rm bio}=10^{33}$, we find the following constraint

[TABLE]

In general, these stellar black holes have a relatively small luminosity, given by

[TABLE]

If we require that the total power output is larger then a fiducial minimum value $L_{\rm min}$, then we obtain an additional constraint. Unfortunately, the value of $L_{\rm min}$ remains unknown. For the sake of definiteness, we take the (arbitrary) value $L_{\rm min}$ = $10^{17}$ erg/sec and scale the result for other universes. This benchmark value corresponds to 10 gigawatts, enough power to run a relatively large city. For universes with varying $\alpha$, we scale the minimum power $L_{\rm min}$ as follows: The energy levels of atoms vary $\sim\alpha^{2}$ and the atomic time scales vary according to $t_{A}\sim\alpha^{-2}$. In order for the black hole luminosity to provide the same number of atomic reactions over the lifetime of the system, the scaling law becomes $L_{\rm min}$ = $L_{{\rm min}0}(\alpha/\alpha_{0})^{4}$, where the subscript denotes values in our universe.

The allowed region of parameter space that satisfies both the temperature constraint (199) and the lifetime constraint (201) is shown in Figure 31. The red line provides the upper limit on $\alpha_{G}$ by requiring that the black holes live for $N_{\rm bio}=10^{33}$ atomic time scales, taken here to be the time required for biological evolution. The blue curve provides the lower limit on $\alpha_{G}$ by requiring that the black hole temperature is high enough to support chemical reactions and hence life. The shaded portion of the diagram depicts the parameter space for which black holes are both hot enough and sufficiently long-lived to play the role of stars. The dashed line represents the additional requirement that the power output of the black holes must be larger than the fiducial minimum value $L_{\rm min}$. This constraint requires $\alpha_{G}$ to lie above the dashed curve. Our universe, marked by the star symbol, falls well outside the region where black holes are an important power source.

As illustrated in Figure 31, the allowed range of parameters for which black holes can play the role of stars spans many orders of magnitude and thus appears reasonably large. However, this diagram assumes particular values for the time scales necessary for biological evolution (red curve) and the minimum luminosity for a biosphere (dashed curve). One should keep in mind that these choices are uncertain, and could be either much larger or smaller than the fiducial values used here. Moreover, the viable region for black holes shown in Figure 31 does not overlap with the region of the plane corresponding to working stars from Figure 23. This mismatch indicates that universes are unlikely to have both nuclear burning stars and stellar black holes providing power sources for biology.

The discussion thus far has focused on black holes with masses comparable to stars $M_{\rm bh}\sim M_{\star}$. Another possibility is for universes to produce primordial black holes with masses much smaller than stars [120]. To illustrate how such objects could play the role of stars, we define a fiducial mass scale

[TABLE]

which represents the minimum mass for a black hole to live for the current age of the universe (comparable to the time required to support habitability). The black hole temperature (199), lifetime (201), and luminosity (203) can be written in the forms

[TABLE]

where $g=G/G_{0}$ is the gravitational constant scaled to the value in our universe. Next we impose the constraints that the surface temperature must be sufficiently hot, $T_{\rm bh}>300$ K, and that the objects must live longer than $10^{10}$ yr. The mass and scaled gravitational constant then obey the constraints

[TABLE]

The smallest masses leads to the largest luminosities. For the gravitational constant in our universe ($g=1$), the maximum black hole luminosity compatible with these constraints is $L_{\rm bh}\approx 360$ MW, enough power to run a small city. Even larger luminosities are possible for weaker gravity (smaller $g$). Primordial black holes thus provide another channel for alternate universes to generate power.

9.4 Degenerate Dark Matter Stars

This section considers the possibility that an alternate universe can produce stellar objects composed entirely of dark matter and supported by the degeneracy pressure of the constituent particles. Such dark matter stars can generate energy through the process of dark matter self-annihilation. Unlike the case of dark matter halos (Section 9.1) where the interaction rate is too slow in our universe to play a significant role, degenerate dark matter stars are so dense that their annihilation rates are too rapid. Although the formation of these types of stars is problematic, this section outlines the parameter space necessary for degenerate dark matter stars to play the role of hydrogen burning stars in our universe (see also [8]).

The properties of these stellar bodies can be determined using arguments analogous to those used for white dwarfs. The equation of state for a degenerate star is that of an $n=3/2$ polytrope, where the leading coefficient $K$ in this setting is given by

[TABLE]

where $m_{\rm dm}$ is the mass of the dark matter particle. The mass-radius relation has the form

[TABLE]

and the central density is given by

[TABLE]

These stars will have a maximum mass that can be supported by the degeneracy pressure of the dark matter particles. This mass scale is the analog of the Chandrasekhar mass and can be written in the form

[TABLE]

where the parameter $\mu_{\star}\approx 2.714$ for an $n=3/2$ polytrope (Section 7). Note that this expression does not include general relativistic corrections [475]. For dark matter particles with mass $m_{\rm dm}=100m_{\rm p}$, the Chandrasekhar mass scale $M_{\rm ch}\approx 0.0007M_{\odot}$. Notice also that if we consider stars that are a fraction of the Chandrasekhar mass, so that $M_{\ast}=XM_{\rm ch}$, then the central density is given by $\rho_{\rm c}\sim X^{2}m_{\rm dm}^{4}$ (in natural units).

The luminosity of the stars is determined by the annihilation rate of the constituent dark matter particles. The annihilation rate per particle $\Gamma_{1}$ is given by

[TABLE]

where $n$ is the number density of particles in the star. The corresponding total annihilation rate $\Gamma_{T}$ integrated over the volume of the star is given by

[TABLE]

where $\gamma_{\star}\approx 1.128$. As a result, the total annihilation rate is given approximately by $\Gamma_{T}\sim 0.4$ $N_{T}\Gamma_{1}$, where $N_{T}$ is the number of particles in the star and $\Gamma_{1}$ is evaluated at the stellar center. The corresponding stellar luminosity is then given by

[TABLE]

For the parameters of our universe, this luminosity is enormous, and these degenerate stars will be short-lived. In order for this scenario to produce stellar objects that are useful for habitability, the annihilation cross section must be smaller, as discussed below.

We want to consider stars, with masses below the Chandrasekhar mass $M_{\rm ch}$, that satisfy two constraints. The first requirement is that the star is has a high enough surface temperature to drive chemical reactions on suitably situated planets, which implies that

[TABLE]

where the efficiency $\epsilon_{\rm c}\sim 10^{-3}$ (see Section 7). After some simplification, this constraint can be written in the form

[TABLE]

where the numerical constants have been combined into a single constant $B_{1}\approx 52,000$. We can write the mass of the star in terms of the fundamental mass scale for degenerate stars with particle mass $m_{\rm dm}$, i.e.,

[TABLE]

With this ansatz, the constraint for surface temperature becomes

[TABLE]

Next we require that the stellar lifetime is sufficiently long. If the degenerate star starts its evolution with initial mass $M_{0}$ and later has a mass $M_{\ast}(t)\ll M_{0}$, then its age $t_{\ast}(M_{\ast})$ is related to its current mass $M_{\ast}$ through the expression

[TABLE]

where $L_{\ast}$ is the luminosity of the star when it has mass $M_{\ast}$. We want the stellar age $t_{\ast}$ to exceed the time required for biological evolution. This constraint can be written in the form $t_{\ast}>N_{\rm bio}t_{A}$, where $t_{A}$ is the time scale for atomic processes and where we expect $N_{\rm bio}\sim 10^{33}$. We thus obtain the requirement

[TABLE]

As before, we write the stellar mass in terms of the fundamental mass scale and we combine the dimensionless constants into a single composite $B_{2}$, so that we obtain

[TABLE]

Both constraints of minimum temperature (equation [217]) and long stellar lifetime (equation [220]) must be met in a viable universe. More specifically, these expressions define the parameters necessary for dark matter stars to play the role of ordinary stars. To explore this parameter space, we set $X=0.1$, corresponding to stars that are comparable to, but smaller than, the maximum mass limit. The cross section for dark matter annihilation can be written in the approximate form

[TABLE]

where the Fermi constant $G_{F}=1/(\sqrt{2}{\cal V}^{2})$ (and where $G_{F}\approx$ (293 GeV)-2 in our universe). For a given stellar mass, the parameter space is specified by the fine structure constant $\alpha$, the strength of the weak force specified by ${\cal V}$, and the mass $m_{\rm dm}$ of the dark matter particles.

Figure 32 shows the constraints derived above in the plane of parameter space $({\cal V},\alpha)$ for different masses of the dark matter: $m_{\rm dm}$ = 0.01 $m_{\rm p}$ (red curves), 1 $m_{\rm p}$ (blue curves), and 100 $m_{\rm p}$ (green curves). The curves with shallow slope depict the constraint that the stellar surface temperature is high enough to allow suitably situated planets to have temperatures that support biological (chemical) processes. Viable universe must fall below these curves (for a given mass $m_{\rm dm}$). The curves with steeper slope depict the constraint that the stellar lifetime is long enough to allow for biological evolution, where we use the benchmark value $N_{\rm bio}=10^{33}$. All of the curves shown use stellar masses with $X=0.1$, roughly comparable to, but smaller than the Chandrasekhar mass. The shaded region in the diagram is not allowed for viable universes. It is bounded on the right by the constraint that the weak scale parameter must be smaller than the Planck scale ${\cal V}<M_{\rm pl}$. The region is bounded from above by requiring that the fine structure constant $\alpha\ll 1$.

This analysis shows that a sizable region of parameter space allows for degenerate dark matter stars to live long enough and have sufficiently high luminosities to serve as engines of biological evolution. However, the formation mechanism(s) for these bodies remains unknown. In principle, dark matter can have self-interactions, which allow for energy dissipation and the formation of bound structures [175, 391]. On the other hand, the annihilation cross sections can be large enough to prevent the stars from achieving hydrostatic equilibrium [160]. Another possibility is for bound structures to be produced through phase transitions in the early universe [556]. Nonetheless, the formation of dark matter stars represents a formidable obstacle.

9.5 Nuclear-Free Universe

As another non-conventional scenario, it is interesting to consider universes in which no nuclear reactions can take place [364, 448]. In this case, the universe is assumed to include the strong force so that it supports protons as bound states of quarks. Unlike our universe, however, no further nuclear processing can take place, so the universe retains a pure hydrogen composition. Such a universe would be far too simple to support life, or anything resembling biochemistry. On the other hand, when viewed from an astronomical perspective, this specter universe would look superficially much like our own:

On the largest scales, galaxy formation takes place through the collapse of dark matter to form halos [449, 553]. The interplay between cosmic expansion and gravitational collapse is largely independent of nuclear considerations, so that large scale structure would be essentially unchanged. After the dark matter halos form, baryonic gas must cool and condense, and cooling processes depend on chemical composition. In our universe, however, most of galaxy formation takes places with primordial abundances of the elements. As a result, the main difference between the nuclear-free universe and our own is the presence of primordial helium. However, the cooling from higher temperatures is dominated by bremsstrahlung processes [5, 221], so that heavy nuclei (and molecules) provide only higher order effects.

After galaxies are in place, stars are produced on smaller scales. The star formation process itself does not depend on nuclear considerations [379, 477]. On the other hand, the initial conditions for star formation depend on cooling processes in the dense interstellar medium, where molecules and dust play an important role, and these starting conditions affect the distribution of stellar masses [64]. The cooling processes of the nuclear-free universe will be much like those of our universe during the epoch when the first generation of stars was forming [5, 509]. We thus expect the distribution of stellar masses to be similar to those of Population III stars. Although the initial mass function for these early stars is not known, it is thought to skew towards stars of higher mass compared to the stellar mass distribution of the present day [101, 471, 500].

The vast majority of stars are born with large radii and relatively cool central temperatures. Nuclear burning does not take place until after an extended phase of pre-main-sequence contraction, which depends on stellar mass and typically takes millions of years [147, 265, 427]. In the absence of nuclear reactions, the gravitational contraction phase continues much longer — until the stars reach an end state analogous to either white dwarfs or black holes [15], depending on the initial mass of the object. Note that the cooling time for white dwarfs is longer than the current age of our universe [559], so that these stellar remnants can continue to provide power over times scales t\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}10 Gyr after the end of the contraction phase.

Just as in our universe, the starting state of the star formation process is expected to have substantial amounts of angular momentum. As collapse proceeds, most of the infalling material gets channeled onto a circumstellar disk, which subsequently transfers material onto the star. The formation of this disk is significant, as it allows for planets to form. As outlined in Section 8.4, planet formation can take place via two channels. The core accretion paradigm requires heavy elements to make large rocky cores [432] and would not operate in a nuclear-free universe. However, planets could still form in principle through gravitational instabilities in the circumstellar disks [85, 115, 439]. The natural mass scale of secondary objects formed in this manner is roughly $M_{S}\sim 10M_{\rm Jup}$, significantly larger than most planets in our universe. By default, these planets would be composed of hydrogen, which would cool and take a solid form.

For stars with masses less than the Chandrasekhar limit, stellar evolution ends with all of the energy leaking out of the star, which eventually reaches the radius of a zero-temperature white dwarf. The total energy radiated over the course of the stellar lifetime is thus given by

[TABLE]

where we expect the dimensionless constant $f_{\rm wd}\approx 3/7$ in the long-time limit where the star becomes an $n=3/2$ polytrope. The final equality holds because the final radius $R_{\rm wd}$ (comparable to the radius of Earth) is much smaller than the starting radius $R_{1}$ (several times the radius of the Sun). The total energy $E_{T}$ produced is thus the binding energy of a white dwarf, which can be written in the form

[TABLE]

If we write the stellar mass in the form $M_{\ast}=XM_{\star}$ (see equations [228] and [267]), the above expression can be written

[TABLE]

where $\beta=m_{\rm e}/m_{\rm p}$. In our universe, the energy generated by a star via hydrogen burning on the main sequence is $E_{H}\sim f_{core}M_{\ast}{\cal E}$, where $f_{core}\approx 0.1$ is the fraction of the mass in the stellar core, and where ${\cal E}=0.007$ is the nuclear efficiency for hydrogen fusion. Since $\beta=1/1836$, low mass stars in the nuclear-free universe will produce about 10 times less energy than those in our universe.

For non-rotating stars with masses above the Chandrasekhar limit, gravitational contraction continues until the star is smaller than its event horizon, and the object becomes a black hole. Over the course of its evolution, a high mass star radiates an energy comparable to its total mass energy so that

[TABLE]

If the star has appreciable angular momentum, then not all of the mass would immediately fall within the event horizon. Instead, the material with the highest specific angular momentum would form an accretion disk surrounding the collapsed central object. Note that the size of this disk structure (many Schwarzschild radii, perhaps 10 – 100 km) is much smaller than the sizes of circumstellar disks that form planets (10 – 100 AU). This disk could then dissipate energy, transfer angular momentum, and eventually channel more of the mass into the black hole, with the total radiated energy approaching that of equation (225).

Compared to our universe, the nuclear-free universe thus has galaxies and other large scale structures that are essentially the same, stars that generate less total energy by factors of $\sim 10$, and many fewer planets that have larger masses. On the other hand, such a universe would have no rocky planets, no nuclei heavier than protons, and nothing approaching biological complexity. This specter universe would thus be quite similar to ours on cosmological scales and completely different on terrestrial scales.

10 Conclusion

An intricate network of constraints must be satisfied in order for a given universe to be viable. This section provides a guide through the labyrinth by organizing the results of this review in several different ways: Section 10.1 presents a straightforward summary of the most important results. Section 10.2 identifies general trends emerging from this collection of constraints. The relation to anthropic arguments is addressed in Section 10.3, and parameter variations that potentially allow universes to be more habitable are outlined in Section 10.4. This overview concludes with a brief discussion of open issues (Section 10.5) and the insights gained from this endeavor (Section 10.6).

10.1 Summary of Fine-Tuning Constraints

The first step in determining the degree of fine-tuning of our universe — and others — is to delineate the range of parameter space for which observers can arise. This issue is made difficult because we have no definitive determination of what parameters are allowed to vary and what requirements must be enforced to ensure habitability. For this latter issue, this treatment considers a universe to be viable if it can successfully produce complex structures, including composite nuclei, planets, stars, and galaxies. The genesis of these entities puts additional constraints on the universe itself. Although a definitive assessment remains elusive, this review suggests the following constraints:

The masses of light quarks are among the most constrained parameters of the Standard Model of Particle Physics. The mass of the down quark can only vary by a factor of $\sim 7$, while the mass of the up quark can vary by several orders of magnitude (see Figure 5). The ranges are asymmetric, however, favoring lighter quarks, rather than quarks with larger masses. Note that previous treatments [21, 22, 25, 162, 185, 186, 280] generally obtain even tighter constraints on the quark masses by invoking the (unnecessary) constraints that deuterium must be bound and diprotons cannot be bound.

The mass difference between the proton and neutron $\Delta{m}$ = $m_{\rm n}-m_{\rm p}$ depends on both the mass difference between the light quarks $\delta{m}=m_{\rm d}-m_{\rm u}$ and the fine structure constant $\alpha$. Although $\delta{m}$ cannot be made too small, the parameter space allows $\delta{m}$ to vary by a factor of $\sim 10$ for the fine structure constant in the range $0\leq\alpha/\alpha_{0}\leq 2$ and $\Delta{m}=1-4$ MeV (see Figure 6).

The fine structure constant $\alpha$ and the ratio $\beta$ of the electron mass to the proton mass can vary by several orders of magnitude while allowing for stable atoms and working stars (Figure 7). The corresponding allowed region for $\alpha$ and its strong force counterpart $\alpha_{\rm s}$ is somewhat smaller (Figure 8), but still spans many orders of magnitude.

In order for the universe to evolve to its present state, the initial value of the total energy density $\Omega$ must be extremely close to unity, so that the space-time of the early universe must be spatially flat to one part in $\sim 10^{60}$ (e.g., Figure 10). This type of tuning can be explained if the early universe experiences an inflationary epoch [252] or its equivalent. Successful inflation is far from guaranteed, however, and the initial conditions required to achieve such a solution to the flatness problem can introduce additional fine-tuning issues [47, 311, 508, 535, 536].

The energy density $\rho_{\scriptscriptstyle\Lambda}$ of the vacuum (equivalently, the cosmological constant) can vary by many orders of magnitude (Figure 12) and still allow galaxies and clusters to form, contrary to many previous claims. If the amplitude $Q$ of the primordial density fluctuations varies over its allowed range, $\rho_{\scriptscriptstyle\Lambda}$ can be larger than its observed value by a factor of $\sim 10^{10}$. Universes with even larger values of $\rho_{\scriptscriptstyle\Lambda}$ can produce structure if the baryon to photon ratio $\eta$ increases. The bound is proportional to $\eta^{4}$ (see equation [72]) so that the upper limit increases by an additional factor of $\sim 10^{12}$. For universes with large $(\eta,Q)$, the resulting galaxies would be much denser than those in our universe, so only a fraction of the solar systems (residing in the outer galaxy) would remain viable.

Big Bang Nucleosynthesis generally does not cause a universe to lose its potential habitability. In order for BBN to render a universe lifeless, the early universe must process almost all of its nucleons into helium and heavier elements — leaving no hydrogen behind to make water. However, the relevant cosmological parameters can vary over many orders of magnitude without over-producing helium, including the baryon to photon ratio (Figure 13), the gravitational constant (Figure 14), the neutron lifetime (Figure 15), and the fine structure constant (Figure 16). Even without the weak force, universes can emerge from BBN with viable compositions if the baryon to photon ratio is smaller than that of our universe [245, 266].

The range of allowed amplitudes $Q$ for the primordial density fluctuations is approximately given by 10^{-6}\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}Q\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}10^{-2} (see Figure 19). The density of forming galaxies scale with the value of $Q$ such that $\rho_{\rm c}\sim Q^{3}$. For smaller values of $Q$, galactic gas has difficulty cooling, so that star formation is suppressed (or at least delayed). For larger values of $Q$, galaxies become so dense that planets can be stripped out of their orbits by passing stars, background radiation fields become more intense than the solar flux received by Earth, and galactic black hole formation becomes problematic. Nonetheless, the allowed range of $Q$ spans about four orders of magnitude.

The baryonic and dark matter inventories of the universe can also vary by many orders of magnitude without disrupting habitability (Figure 21). For large values of $\delta/\eta$, which sets the mass ratio of dark matter to baryons, galactic disks become stabilized, which slows down star formation; for smaller values of the ratio $\delta/\eta$, Silk damping becomes important and suppresses structure formation on small scales. Although galaxies and their universes can have different properties in the separate regimes delineated within Figure 21, most of the parameter space remains habitable.

Stable hydrogen burning stars can exist over a wide range of parameter space, specified here by the fine structure constant $\alpha$, the gravitational structure constant $\alpha_{G}$, and a composite parameter ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$ that specifies the nuclear reaction rate (Figure 22). The allowed parameter space is reduced by requiring stars to have sufficiently high surface temperatures and long lifetimes, but still spans many orders of magnitude (Figure 23). The ratio $\alpha_{G}/\alpha$ is notoriously small ($\sim 10^{-36}$) in our universe. Within the range of the $(\alpha,\alpha_{G})$ plane allowed by working stars, this ratio can be larger by a factor of $\sim 10^{4}$, but still remains small compared to unity ($\sim 10^{-32}$). Planet properties do not depend on the nuclear reaction parameter ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$, and the allowed range of the $(\alpha,\alpha_{G})$ parameter space for viable planets is even larger than that for working stars (Figure 29).

In addition to operating over a wide range of parameters, stars and stellar structure are less sensitive to nuclear considerations than suggested by previous claims. Stars can continue to make substantial amounts of carbon as long as the energy of the triple-alpha resonance is not raised by more than $\sim 500$ keV. If the energy of the resonance is lowered, then stars actually make more carbon (see Figure 24). Moreover, the 8Be nucleus fails to be bound by only 92 keV, so that changes to its binding energy of this magnitude can lead to stable beryllium and remove the need for the triple-alpha reaction altogether. Finally, the carbon found in the Earth is depleted relative to Solar and cosmic abundances by a factor of $\sim 100$ [32, 374], so that habitability does not necessarily require a full carbon inventory.

Stars can also operate with both stable diprotons and unstable deuterium, again contrary to many previous claims. If diprotons are stable and nuclear reaction rates are enormously larger (by factors of $\sim 10^{16}$), then the central temperatures in stars decrease from $T_{\rm c}\approx 15\times 10^{6}$ K down to about $T_{\rm c}\approx 10^{6}$ K, but stars otherwise function normally. As the nuclear reaction rates increase, the allowed parameter space for the structure constants $(\alpha,\alpha_{G})$ increases substantially (Figure 22). In the opposite case, where deuterium is unbound and cannot provide the usual stepping stone toward larger nuclei, stars continue to operate through a variety of processes. Gravitational contraction is sufficient as an energy source, and explosive nucleosynthesis can produce heavy nuclei at the end of stellar lifetimes. A triple nucleon process allows stars to burn hydrogen into helium in the absence of stable deuterium. Finally, the CNO cycle continues to operate, and allows for working stars, even with metallicity as small as $\raisebox{0.86108pt}{--}\kern-5.50003ptZ\sim 10^{-14}$. Through these four stellar processes, stars can provide both energy and nucleosynthesis in universes without stable deuterium.

In universes with parameters that are significantly different from those of our universe, new types of astrophysical processes can contribute to the generation of energy (Section 9). For the case of dense galactic halos and larger cross sections for weak interactions, the energy generated by dark matter annihilation can compete with stellar radiation as a power source for habitable planets (Figure 30). Dark matter can also collect inside stellar remnants (such as white dwarfs) and subsequently annihilate. This channel of power generation is also enhanced with denser halos and larger weak interaction cross sections. In the regime of stronger gravity and weaker electromagnetism (smaller $\alpha$), black holes can be bright enough and sufficient long-lived to serve as hosts for habitable planets (Figure 31).

In order to delineate the allowed ranges for the fundamental and cosmological parameters, we needed to express the mass scales of astrophysical objects in terms of the fundamental constants. These mass scales are collected in A. The number of dimensions of space represents another physical property that could in principle vary from universe to universe [56, 192, 193, 446, 506, 554]. Although this quantity rests on a somewhat different footing from the other parameters reviewed in this paper, a number of arguments suggest that three spatial dimensions ($\mathfrak{D}=3$) are strongly preferred (B). Constraints on the fundamental parameters due to requirements imposed by chemistry and working biomolecules are not as developed as those arising from physics. Work to date suggests that chemical constraints on the fine structure constant $\alpha$ and the mass ratio $\beta$ are not as stringent as those arising from physics (C). Considerations of stellar structure can be used to place global upper bounds on the gravitational constant and on the ratio $\alpha_{G}/\alpha$. Although the hierarchy between gravity and electromagnetism can be made smaller by several orders of magnitude, it remains large (D). A complete description of fine-tuning requires knowledge of the probability distributions from which the parameters are sampled; this part of the problem is much less developed than the constraints outlined above (E). Finally, variations in the fundamental constants can lead to unstable nuclei and a smaller periodic table. These constraints are outlined in F using the Semi-Empirical Mass Formula for nuclear structure.

The constraints reviewed in this paper are summarized in Table 2. For each quantity of interest, the table lists the range of that quantity that allows for a viable universe, where the result is expressed in decades. If $x$ is a parameter, with minimum and maximum allowed values $x_{\rm min}$ and $x_{\rm max}$, then the range (in decades) is defined by

[TABLE]

Note that, in general, the ranges of allowed parameters are asymmetric with respect to the values in our universe.

The table is organized into three sections. The top portion includes fundamental parameters associated with particle masses, as specified by the Standard Model of Particle Physics. The central portion considers the dimensionless strengths of the four fundamental forces, where the values in our universe are evaluated in the low energy limit. The bottom portion includes the cosmological parameters. Note that the ranges of the up quark mass, the gravitational constant, the strength of the weak force, and the energy scale of the vacuum energy do not have well-defined lower limits. The up quark mass and the strength of gravity cannot vanish, however, so the results given in Table 2 correspond to the parameter space presented in the Figures, but the ranges could be even larger. On the other hand, universes could be viable without the weak force and without any vacuum energy; for these parameters, Table 2 lists the number of decades for which the parameters could be larger.

Notice also that the allowed range for the strong coupling constant $\alpha_{\rm s}$ would be much smaller if one imposes the constraints that diprotons remain unbound and/or deuterium remains bound. Although these requirements are not necessary for functioning stars (see Section 7.5), they would limit variations in $\alpha_{\rm s}$ to $\pm 15\%$, corresponding to a range of only $D(\alpha_{\rm s})\sim 0.06$. The allowed range of the strong coupling constant (and the fine structure constant) would also be much smaller if one requires the triple alpha reaction (Section 7.4). The triple alpha constraint limits variations in $\alpha_{\rm s}$ to $\sim 2\%$, corresponding to a range of only $D(\alpha_{\rm s})\sim 0.01$. However, smaller variations in the fundamental constants allow for stable 8Be nuclei, which provide an alternate channel for carbon production and obviate the triple alpha constraint.

With the allowed ranges of the fundamental and cosmological parameters specified in Table 2, we are left with the question of whether these ranges of possible variations are large or small. Given that the concept of fine-tuning evokes the image of tuning the dials of a radio, as one point of comparison we consider the tuning required to capture a radio station. The AM carrier frequencies lie in the range $f_{\scriptscriptstyle AM}$ = 535 to 1605 kHz, where the spacing between stations is 10 kHz. To find a particular radio station, the electronics must be accurate to about 1 part in 100. Similarly, the FM radio band extends from $f_{\scriptscriptstyle FM}$ = 88 to 108 MHz, where the central carrier frequencies for individual stations are assigned at 200 kHz intervals. Finding an FM station thus requires tuning the frequency to 1 part in 500. The final two lines of Table 2 give the ranges (in decades) required for successful AM and FM radio reception, $D(f)$ = 0.0043 and 0.00087, respectively. These ranges are much smaller than those required of the fundamental parameters. In fact, the allowed ranges for all of the parameters listed in the table are enormously larger than the entire range of radio frequencies. Tuning a radio thus requires far more precision than tuning a universe.

Limits on the cosmological density parameter $\Omega$ are notably absent from Table 2. As discussed in Section 3, the value of $\Omega$ must be either (a) tuned exquisitely to one part in $10^{60}$ at the Planck epoch, or (b) driven to its observed value $\Omega\rightsquigarrow 1$ by some mechanism (e.g., an inflationary epoch in the early universe or its analog). Although the universe must indeed be spatially flat to high precision, almost all universes could have this property according some measures on the space of initial conditions [126, 130, 127]. The allowed range of $\Omega$ in the early universe is thus extremely small, perhaps $D(\Omega)\approx-60$, but its implications for fine-tuning depend on the (as yet unknown) likelihood of the universe achieving successful inflation or the equivalent, and/or on the resolution of the cosmological measure problem [156, 232, 469].

Finally, note that solutions to specific fine tuning problems in stellar astrophysics require additional physical processes and are summarized in Table 3. The top part of the table addresses the triple alpha problem for carbon production, which occurs through a resonant reaction in our universe. Possible solutions include varying the resonance level $\Delta{E}_{R}$ and/or increasing the binding energy of beryllium-8. Spallation reactions can produce some carbon by breaking down larger alpha elements, although the abundances are small. The table then lists processes relevant for universes with unstable deuterium. Energy can be generated by gravitational contraction via small stars, and heavy elements can be produced through explosive nucleosynthesis in larger stars. In addition, stars can burn hydrogen through triple nucleon reactions (if the deuterium half-life $\tau_{1/2}(d)$ is long enough) and through the CNO cycle (if the metallicity $\raisebox{0.86108pt}{--}\kern-5.50003ptZ$ is high enough). Stars can function with the nuclear burning parameter ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$ (see equation [122]) varying over more than 21 orders of magnitude, including values appropriate for universes with stable diprotons. Finally, in universes without the weak interaction, stars can burn deuterium over $\sim$Gyr lifetimes (if the baryon to photon ratio $\eta$ is small enough).

10.2 General Trends

Another way to summarize the degree of fine-tuning of the universe is to organize the results described above into the following general trends:

$\bullet$ The allowed parameter space is large. Most of the relevant parameters can vary by several orders of magnitude and still allow for the development of complex structures, from atoms to stars to galaxies: The parameters that are allowed to vary include the masses of the light quarks and leptons $(m_{\rm u},m_{\rm d},m_{\rm e})$, the structure constants $(\alpha,\alpha_{\rm s},\alpha_{\rm w},\alpha_{G})$, and cosmological parameters $(\eta,\delta,\rho_{\scriptscriptstyle\Lambda},Q)$.

$\bullet$ Particle physics is more sensitive than astrophysics. Particle physics considerations are more constraining than astrophysical considerations: The allowed mass range for the light quarks (shown in Figure 5) is smaller than the corresponding ranges for the astrophysical parameters, including the fluctuation amplitude $Q$ (Figure 19), the baryon to photon ratio $\eta$ and its dark matter counterpart $\delta$ (Figures 13 and 21), and the energy density of the vacuum $\rho_{\scriptscriptstyle\Lambda}$ (Figure 12). Stars can operate over a wide range of parameter space and can produce heavy nuclei through many channels (Section 7). The key issue is that stable nuclei (especially carbon) must exist (see F). Finally, although only preliminary work has been carried out, chemistry seems to be less confining than physics (see [322] and C).

$\bullet$ Large hierarchies remain $\rightsquigarrow$ gravity must be weak. Even if the parameters of physics and cosmology can deviate from their values in our universe by orders of magnitude, ‘unnaturally small’ ratios are still required: For example, the cosmological constant can vary over a wide range, but must be small compared to the Planck scale (Section 4.2). Similarly, the ratio $\alpha_{G}/\alpha$ of the gravitational structure constant to the fine structure constant can vary by several orders of magnitude, but must remain small compared to unity (D). Both of these ratios are extremely small due to the required weakness of gravity, equivalently, the large value of the Planck mass. These hierarchies for the physical parameters ultimately lead to the enormous ranges of mass and size scales observed in the universe (Figure 3). In general, the universe exhibits more Hierarchical Fine-Tuning than it does Sensitive Fine-Tuning. A summary of remaining hierarchies in the universe is provided by Table 4.

$\bullet$ Multiple variations are important. More possibilities for working universes arise if more than one parameter is allowed to vary: For example, if all other parameters are fixed, then the vacuum energy density $\rho_{\scriptscriptstyle\Lambda}$ can only be larger than its observed value by a modest factor. If the amplitude $Q$ of the density fluctuations is larger, the allowed range in $\rho_{\scriptscriptstyle\Lambda}$ increases by a a factor of $\sim 10^{9}$. If the baryon to photon ratio is larger, the allowed range increases by a factor of $\sim 10^{12}$.

$\bullet$ Our universe does not lie at the center of parameter space. The ranges for viable parameters are often asymmetric and are sometimes constrained in only one direction. If the triple alpha resonance level is raised, then carbon production decreases, but stars produce more carbon if the resonance level is lower. The fine structure constant $\alpha$ cannot be too large without compromising nuclear structure, but larger ratios of the strong-to-electromagnetic force lead to a wider variety of stable nuclei. Stars no longer function if the gravitational constant is increased by more than a factor of $\sim 10^{6}$, but $G$ can become arbitrarily small and working stellar solutions still exist. Similarly, large values of the cosmological constant $\Lambda$ compromise structure formation, but essentially all smaller values are allowed. Finally, limits on the light quark masses $(m_{\rm u},m_{\rm d})$ are asymmetric: Larger masses lead to a shorter range for the strong force, and result in tight constraints from nuclear structure. Smaller quark masses lead to a longer range for the strong force, and the limits are not nearly as stringent.

$\bullet$ Universes have multiple pathways. Viable universes are not required to be exactly like our own — they can in principle achieve habitability through alternate routes: Although our universe has $\beta\ll 1$, so that the electron mass is much smaller than the proton mass, the opposite ratio $\beta\gg 1$ could also allow for working atoms. If the nuclear parameters are different so that carbon production via the triple alpha reaction becomes compromised, then some universes can support stable beryllium-8 nuclei and would not need the triple alpha process [14]. Another possible scenario is that of a Cold Big Bang [23], where the cosmological parameters $(\eta,\delta,\Lambda,Q)$ differ from those in our universe by orders of magnitude. Yet another alternate universe could have no weak interactions [266, 228]; the weakless universe requires smaller values of $\eta$ to avoid overproducing helium during BBN, but can remain habitable. With sufficiently large departures of dark matter properties and gravity, other universes can generate enough energy to sustain habitable planets through dark matter annihilation or even black hole radiation (Section 9). These alternate pathways expand the range of parameter space for viable universes.

$\bullet$ The fine structure constant must be small. Although the value of $\alpha$ can vary over a wide range, a number of independent lines of argument indicate that the fine structure constant must remain much less than unity: Stable long-lived stars burn their nuclear fuel through a quantum mechanical tunneling process, which would shut down if $\alpha$ is too large (Figure 22). An even stronger upper limit on $\alpha$ arises from the requirement that the photospheric temperatures of stars are hot enough to support chemical reactions (Figure 23). If the value of $\alpha$ becomes of order unity, then large atomic nuclei would not exist [62, 163], and the periodic table would be much smaller (see also F). Even if the nuclei remain bound, the electrons in atoms would become relativistic for larger values of $\alpha$. Although such atoms can remain in existence, contrary to many previous claims, the atomic energy levels and hence the chemical properties would be markedly different (see [241, 242, 353, 450] and references therein). The value of $\alpha$ must remain less than unity in order for planets to have smaller masses than their host stars (equation [174]). Finally, smaller values of $\alpha$ result in relatively small changes to biomolecules, but larger values of $\alpha$ are disallowed (C and [322]). All of these considerations constrain the fine structure constant to be small, such that $\alpha\ll 1$. As a result, physicists in any viable universe should be able to understand quantum electrodynamics, as QED will always lie in the perturbative regime.

$\bullet$ Myths and unnecessary constraints: A number of constraints that are often enforced on viable universes are not as severe as some previous work suggests: In universes with stable diprotons, where nuclear reactions proceed through the strong force only, stars operate with somewhat lower central temperatures, but do not burn through their nuclear fuel in a catastrophic manner (Figure 22). Similarly, in universes with unstable deuterium, stars can generate energy and synthesize heavy elements through alternate channels, including gravitational contraction, explosive nucleosynthesis, the CNO cycle, and the triple-nucleon process (Figure 27 and [15]). A possible constraint on the weak coupling constant arises from the requirement that core collapse supernovae must have optically thick neutrinos (so that $\alpha_{\rm w}^{4}$ $\sim\alpha_{G}$; see Section 7.6.3), but Type Ia supernovae can provide heavy elements [266] even if all massive stars collapse to form black holes. Another commonly invoked constraint is the requirement that some stars must be convective in order to produce planets [131]; we now know that planet formation is independent of stellar convection (Section 8.5). The relaxation of these unnecessary constraints results in a significantly larger parameter space for working universes.

10.3 Anthropic Arguments

Anthropic arguments are related to — but not equivalent to — the fine-tuning discussion considered in this paper. Although anthropic arguments have been put forth with a range of definitions (classic references include [61, 122, 131, 132]), the basic consideration is that the universe must have the proper version of physical law in order to develop observers. As discussed throughout this review, the fundamental constants and cosmological parameters must lie within specified ranges in order for the universe to develop interesting structures such as galaxies, stars, and planets. Since the universe must produce these astronomical entities in order to support observers, at least those of familiar form, the finding that the fundamental constants lie within the aforementioned specified ranges is not by itself surprising. If the ranges of allowed parameters were sufficiently small, then the finding that our universe has the proper parameters becomes interesting. In the opposite limit, if the universe could develop structure while its parameters vary over enormous ranges, then the observed properties of the universe do not provide a strong constraint. As a result, the range of parameters for viable universes must be small in order for anthropic arguments to carry weight. In other words, some fine-tuning of the universe is necessary — but not sufficient — for the efficacy of anthropic arguments.

One point of contention is the degree to which anthropic arguments can predict the values of the fundamental parameters, while another difficulty is the extent to which such arguments can be falsified [198]. Discussion of these issues is complicated by the vast literature on the subject, where some reviews describe at least 30 different definitions of the anthropic cosmological principle [86, 498]. A full review of this subject is beyond the scope of this contribution (for example, see [42, 61, 86, 122, 164, 363, 380, 548] and references therein). Here we briefly outline the nature of anthropic arguments and elucidate their relation to the degree of fine-tuning (see also [55] for a history of the subject).

To illustrate the nature of anthropic arguments, let’s consider a generic example. Suppose for, example, that $X$ is a fundamental constant, and the existence of stars requires that the value of $X$ lies in the range $X=X_{0}\pm\delta{X}$, where $X_{0}$ is the observed value of the parameter in our universe. If the value of $\delta{X}$ is small, then we can say that the existence of stars provides an anthropic argument for the value of $X$. If the value of $X_{0}$ is not yet measured, then this argument provides an anthropic prediction for the value of $X$. Note that in order for such an argument to carry much weight, the allowed range of the parameter (and hence $\delta{X}$) must be small in some sense. In other words, the parameter $X$ must be fine-tuned at some level. As a result, the fine-tuning of fundamental parameters is a prerequisite for anthropic arguments.

In spite of its seeming simplicity, the above class of arguments has a number of complications: First, we have no consensus on how small $\delta{X}$ must be in order for an anthropic argument to be meaningful (or for the parameter $X$ to be fine-tuned). The range can be measured in absolute terms, as a relative change $\delta{X}/X_{0}$, as a factor $f=(X_{0}+\delta{X})/X_{0}$ by which the variable can change, and so on. In addition, more than one parameter could vary from universe to universe. In some cases, for example, the range $\delta{X}$ must be small if all of the other fundamental parameters are held constant, but can be much larger for other (allowed) values of those parameters.

Even if the anthropic argument is successful, so that the argument confines the parameter $X$ to a small range, we are left with the question of how much the argument actually explains: If, for example, $X$ must be very close to its observed value in order for stars to work, do we now know why $X$ has such a value? A more fundamental argument that specifies the value of the parameter $X$ would clearly be preferable. The anthropic argument thus provides only a partial explanation for the value of $X$.

In spite of the complications outlined above, and many others, anthropic arguments have been used to explain the observed values of both fundamental constants and cosmological parameters in our universe. These studies include constraints on the cosmological constant, starting with the original work of Weinberg [545], which has subsequently been generalized to include variations of additional parameters [16, 226, 310, 363, 383, 270, 423, 537]. Additional studies have presented anthropic arguments for the energy scale of electroweak symmetry breaking [21, 54, 302], big bang nucleosynthesis [370], stellar nucleosynthesis including the triple-alpha resonance [302, 362, 380], the parameters of aluminum-26 decay [463], the mass of the proton [415], the masses of neutrinos [512], the finite age of the universe [142], and for the existence of three generations in the Standard Model [237, 291, 467].

To illustrate the difficulties faced by anthropic arguments, it is useful to consider a concrete example. Perhaps the most widely discussed anthropic argument is that invoked for the cosmological constant. As outlined in Section 4, in order for structure formation to successfully take place, the energy density of vacuum must obey the bound of equation (72). For fixed values of the parameters $(Q,\eta,\Omega_{\rm M},\Omega_{\rm b})$, the cosmological constant cannot be much larger than its observed value. This argument was put forth [545] a decade before astronomical observations provided unambiguous evidence for a nonzero value of $\rho_{\scriptscriptstyle\Lambda}$ [451, 452]. This example is often considered not only as an explanation for the observed value of $\rho_{\scriptscriptstyle\Lambda}$, but also as a successful anthropic prediction. On the other hand, the upper bound on $\rho_{\scriptscriptstyle\Lambda}$ is proportional to the product $\eta^{4}Q^{3}$, and both of these parameters can be larger than the values realized in our universe by several orders of magnitude (Figure 12). To summarize the situation: An anthropic bound on $\rho_{\scriptscriptstyle\Lambda}$, derived before the observations, provides an interesting upper limit if all of the other parameters are held fixed. For alternate values of $(\eta,Q)$ that allow for viable universes, the bound is weaker by a factor of $\sim 10^{21}$. For values of $\rho_{\scriptscriptstyle\Lambda}$ smaller than that observed, universes are not only viable, but perhaps even “better” than our own (see the following section). Finally, constraints on the allowed range of $\rho_{\scriptscriptstyle\Lambda}$ are not the same as a fundamental explanation for its value.

The Anthropic Cosmological Principle, which requires our universe to contain observers, could be considered at odds with the Copernican Principle, which holds that no physical locations are privileged. The Copernican Revolution is generally considered an important milestone in the history of science [329, 330]. Considerations of the multiverse suggest that our universe could be one out of many, so that our place in the cosmic order must be reassessed. Nearly five centuries ago, Copernicus argued that Earth does not occupy a privileged location within the Solar System [154]. Subsequent astronomical discoveries have continually degraded our status. The Solar System does not lie at the center of the Galaxy. Given that the observable universe is homogeneous and isotropic, the Milky Way does not occupy a special location. Continuing this trend, the idea of a Copernican Time Principle [12] suggests that the current cosmological epoch does not have special significance. With the emergence of the multiverse paradigm, with the possibility that far-away disconnected regions of space-time exist, our universe no longer occupies a privileged location within the cosmic archipelago. The multiverse thus extends the Copernican Principle. At the same time, the existence of multiple universes allows for Anthropic Selection, which provides a mechanism for our universe to develop observers and hence be privileged. However, whether or not our universe is special depends on how the assessment is made: While Earth does not lie at the center of the Solar System, it does reside the proper distance from the Sun to allow for liquid oceans on its surface. In a similar vein, our universe is not special in an a priori sense, but it does exhibit the proper version of laws of physics to support the existence of observers. It remains to be seen how this change of cosmic status, that our universe could be but one of many, will affect the status of the Copernican Principle.

10.4 Is our Universe Maximally Habitable?

Discussions of fine-tuning often implicitly assume that our universe is optimized for the development of observers [341]. However, it is interesting to revisit this panglossian assumption [38] and ask if different choices for the fundamental constants or the cosmological parameters could lead to universes that are even more favorable to the development of life, or at least the production of complex cosmic structure (see also [393]). In the realm of extra-solar planetary systems, researchers are now considering whether or not Earth is the best prototype for a habitable planet [275, 428]. Asking the same question on a cosmic scale, we find that several parameter choices could lead to possible improvements of the universe:

$\bullet$ Smaller vacuum energy: The cosmological constant could vanish (equivalently, $\rho_{\scriptscriptstyle\Lambda}=0$). In this case, the universe would never enter into a late-time phase of accelerated expansion. This scenario would favor structure formation, as the matter dominated era would never end. In addition, ever larger cosmological structures would continue to enter the horizon, so that large scale structures of ever larger masses could be constructed [12, 143, 190, 328, 390]. Even if the energy density of the vacuum is nonzero, it could be much smaller than the value realized in our universe, so that structure formation could continue over a longer span of time.

$\bullet$ Larger primordial fluctuations: The amplitude $Q$ of the primordial density fluctuations could be larger. If some type of inflation occurs in the ultra-early universe, then the probability of producing density fluctuations with a given amplitude could be an increasing function of $Q$, so that larger values of $Q$ are more natural [226, 503]. In addition, universes with larger values of $Q$ produce denser galaxies [510, 511], and allow structure formation to occur with larger values of the cosmological constant [363, 383, 16]. With the right choice of the amplitude, roughly $Q\sim 10^{-2}$, galaxies can be dense enough that starlight from the background galaxies allows for planets to have habitable temperatures from almost any orbit [13], thereby producing a Galactic Habitable Zone (see Figure 20). This scenario allows the universes in question to support even more potentially habitable planets than our own. For fluctuation amplitudes as large as $Q=10^{-2}$, galaxies are susceptible to the overproduction of supermassive black holes [444], so that the optimum value of $Q$ is somewhat lower.

$\bullet$ More baryons: The ratio $\eta$ of baryons to photons could be larger. Such enhancements lead to earlier matter domination, which allows for large scale structures to grow more easily. Such universes could tolerate a greater range of fluctuation amplitude $Q$ or vacuum energy density $\rho_{\scriptscriptstyle\Lambda}$; for some parameter choices, they could produce denser galaxies and support more habitable planets (analogous to increases in $Q$).

$\bullet$ More stars: The range of stellar masses in our universe spans a factor of $\sim 1000$, but could be much larger in other universes with other versions of the fundamental constants. For example, if the fine structure constant is smaller, the range of stellar masses is larger, with a maximum stellar mass range of $\sim 18,000$ for $\alpha\sim 1/6300$ (see A). Stars can also have longer nuclear burning lifetimes with other choices of parameters (Section 7.5.1).

$\bullet$ Stable beryllium-8: If the strong force were slightly stronger than in our universe, then 8Be could be a stable isotope [14, 204, 258]. In this type of universe, carbon can be produced through the reaction of equation (144) without the need for the triple alpha reaction (which is more sensitive to stellar conditions [147, 323] and the underlying parameters [194, 202, 204, 406, 470]). Carbon production could thus take place within the same star that produced the 8Be, or much later in a different star [14]. In addition to allowing for more branches of the nuclear reaction network to produce carbon, this type of universe would have a more orderly nuclear inventory. In our universe, the most abundant isotopes beyond hydrogen are 4He, 16O, 12C, and 20Ne (in that order). All of these nuclei are made up of alpha particles, with numbers $N_{\alpha}$ = 1, 4, 3, and 5, respectively. The $N_{\alpha}=2$ nucleus is thus conspicuously absent, but a ‘more logical’ universe could make all of its most common isotopes with integer numbers of alpha particles.

$\bullet$ Weaker gravity: If the strength of gravity is weaker than in our universe, then the cosmos would expand more slowly, so that life would have more time to emerge and evolve [364]. In addition, stars and planets would be larger in size, allowing for larger animals ([434], equation [244]) and more interesting topography [61]. Perhaps more importantly, for weaker gravity, the range of values for the fine-structure constant that allows for working stars would be wider (see Figures 22 and 23).

10.5 Open Issues

This review has discussed the possible parameters from particle physics and cosmology that can vary across the multiverse and can potentially affect the habitable properties of the constituent universes. The main focus has been to delineate the allowed ranges for these parameters that allow a universe to develop complex structures, including nuclei, planets, stars, and galaxies. Although this enterprise has made steady progress, and now has a vast literature, many open issues remain:

Although this review has discussed the parameters that are allowed to vary, this determination is not definitive. At the present time, all of the individual parameters that appear in the Standard Model of Particle Physics are considered as independent. Many workers hope that a more fundamental theory, such as string theory or its descendants, would contain fewer fundamental parameters, so that the currently considered quantities (masses, mixing angles, and coupling constants) would be derived from a smaller set [186, 280, 315, 466, 467, 501]. Until a more fundamental theory is in place, however, we are left with the rather large number of parameters outlined in Section 2.

Similarly, the cosmological parameters have been allowed to vary separately, as outlined in Section 3, but these values could in principle be determined by physics beyond the Standard Model. For example, the amplitude $Q$ of the primordial density fluctuations could be determined by the properties on an inflaton field, a high energy (perhaps near the GUT scale) scalar field that is not yet discovered. The baryon to photon ratio $\eta$ is considered here as a freely varying parameter, but its value must ultimately be set by the physics of baryogenesis (which is driven by out of equilibrium, CP and baryon number violating processes at high energy). Similarly, the value of the vacuum energy density $\rho_{\scriptscriptstyle\Lambda}$ could be specified by additional physics in the gravitational sector, but remains undetermined in the current state of physics.

On a related note, the parameters considered here are allowed to vary independently of each other. It remains possible — perhaps even likely — that a more fundamental understanding of physics would require parameter values to vary together [315, 467]. As one example, the strengths of the coupling constants might have fixed ratios in a fundamental theory, but the overall strength could scale up and down (since the coupling constants are also energy dependent, this scaling would have to apply at a specific energy, or in the limit zero temperature). Grand Unified Theories generally have this property (starting with [231]). As another example [79], the parameters $X_{sm}$ of the Standard Model could vary with the scale $\lambda$ of the vacuum energy according to a renormalization group equation,

[TABLE]

where the exponents $p_{\scriptstyle X}$ are chosen so that the flow reaches a fixed point at the defining energy scale (e.g., the Planck scale).

Many of the results presented herein are represented as allowed ranges of parameters in a plane of two variables (see Figures 5 – 31). If the possible variations are coupled, as outlined above, universes would sample only a subset of the plane. The viable range of parameter space would then correspond to a curve passing through the planes presented here. On the other hand, most of the current exploration of parameter space starts with the values realized in our universe and changes the parameter values until some failure point is reached. In principle, there could exist distant islands of parameter space, far removed from values in our universe, that allow for viable universes (e.g., Figure 7).

As discussed in Section 1, a full assessment of fine-tuning requires not only a specification of the allowed parameter space, but also the underlying probability distribution for universes to realize a given set of parameters (E). As outlined in this review, the field has made substantial progress in delineating the allowed ranges of parameters, in the context of particle physics, cosmology, and stellar astrophysics. In contrast, our understanding of the underlying probability distributions remains in its infancy. An important challenge for the future is to develop a priori determinations of the underlying distributions from which the fundamental parameters are sampled.

Another remaining challenge is to develop a rigorous theory for the creation of individual universes and the mechanism through which they select their vacuum states, which then determine the laws of physics for that region. The launch of a universe onto an expanding trajectory — one that separates itself from the background space-time of the rest of the multiverse — involves a full theory of quantum gravity, which remains elusive. In addition, we need a rigorous assessment of the possible vacuum states of the universe (compare [44, 45, 70] with [92, 281, 309, 467, 501]), and the manner in which an individual universe settles into such a state. Eternal inflation provides one specific mechanism that allows for the fundamental constants and cosmological parameters to vary from universe to universe (Section 3.5). Bouncing cosmologies provide another example. In this latter scenario, the universe expands and re-collapses, and then begins a new phase of expansion [65, 340]. This process continues in cyclic fashion, where each reincarnation of the universe can result in a new realization of the fundamental/cosmological parameters [29, 59]. The idea of cosmological natural selection [486] provides yet another mechanism for producing multiple universes. In this hypothetical scenario, singularities at black hole centers act as sources of new universes, which are hidden from the parental universe by the event horizon and can have different realizations of the fundamental parameters.

Another complication that arises in assessments of fine-tuning is the sheer number of possibilities for different types of universes with different types of physics. In addition to the variations discussed above, many generalized versions of the laws of physics (and alternate cosmological models) have been put forth for applications in our universe. Although most of these scenarios are highly constrained by experimental data, these generalizations could be realized in other regions of space-time. For example, the constants of nature could vary with time [58, 75, 139, 513, 528], including variations in the effective value of the gravitational constant [99]. More extreme departures from the standard theory of general relativity are also possible, as described by a wide range of modified gravity theories [24, 117, 148, 306, 400], including the particular paradigm of Modified Newtonian Dynamics [68, 376, 385]. The equivalence principle, which is well established experimentally, could be violated in other universes [161]. In addition to taking on different values, the cosmological ‘constant’ could evolve over cosmological history [563], including being an increasing function of time [111]. The vacuum energy contribution could also conspire to produce a steady-state universe [82]. Although the flux of magnetic monopoles in our universe is highly constrained [10, 418, 524], they arise naturally in unified gauge theories [176, 514], are readily produced cosmologically [433], and could be abundant elsewhere. Other cosmological defects are highly constrained in our universe [18], but domain walls, cosmic strings, and global texture could instigate structure formation in other universes [321, 525, 534]. The global geometry of space-time could also vary, including having a compact topology [60, 157, 331, 344]. Neutrinos in our universe have relatively small masses and do not contribute significantly to the current cosmic density, but larger neutrino masses and other variations in the dark matter inventory are possible [431, 512]. One can also envision universes with different numbers of light quarks [296], photons with nonzero mass [246], and/or different numbers of particle generations [237, 291, 467]. Other generalizations could lead to additional (fifth) forces [209, 396], which could have astrophysical consequences [346]. All of these possible variations in the laws of physics — and many others — are likely to affect the potential habitability of other universes and should be addressed in the future.

10.6 Insights and Perspective

The consideration of counterfactual universes necessarily lies near the boundary of science. Some authors have considered this enterprise to lie safely within the scientific realm, while others have adamantly taken the opposite point of view (e.g., see [86, 119, 128, 165, 198, 216, 217] for further discussion). Instead of continuing the debate as to whether or not fine-tuning arguments and alternate universes should be officially considered as part of science, perhaps a better question is whether or not the results are useful. Taking the affirmative position, we wrap up this review with a brief summary of what assessments of fine-tuning reveal about physics and astrophysics in our universe. By determining the parameter variations necessary to render the cosmos devoid of life, we gain a greater understanding of how the universe operates.

Stars are more robust than most people realize: The range of fundamental constants that allow for working stars is enormous (Section 7). Stars exist while the fine structure and gravitational constants vary by many orders of magnitude. The nuclear reaction rate can change by factors of $10^{21}$ and stars will still function. One can eliminate the weak interaction altogether (Figure 28), make deuterium unstable (Figure 27), or make diprotons stable (Figure 22), and the stars will still shine. And not only will stars continue to operate as stable nuclear-burning entities, their power output, surface temperatures, and lifetimes are commensurate with values considered appropriate for habitability (Figure 23). The limiting feature for a viable universe — including ours — is not the astrophysical engines that synthesize heavy elements and generate energy, but rather the parameters of particle physics that allow for the existence of stable nuclei (see Figures 5 and 36).

Although they represent a significant problem for particle physics theories, the observed hierarchies of the fundamental parameters are a distinguishing feature of the cosmos [122], and extreme hierarchies are required for any universe to be viable: The strength of gravity can vary over several orders of magnitude, but it must remain weak compared to other forces so that the universe can evolve and produce structure, and stars can function. This required weakness of gravity leads to the hierarchies of scale that we observe in the universe (Figure 3). In addition, any working universe requires a clean separation of the energies corresponding to the vacuum, atoms, nuclei, electroweak symmetry breaking, and the Planck scale (Figure 35). In other words, the hierarchies of physics (see also Figure 4) lead to the observed hierarchies of astrophysics, and this ordering is necessary for a habitable universe. Moreover, these required hierarchies can occur with reasonably high probability if the underlying distribution of energy scales has a log-random (scale-free) form (E).

Finally, in spite of its biophilic properties, our universe is not fully optimized for the emergence of life. One can readily envision more favorable universes (Section 10.4). Possible improved cosmic properties include a larger primordial fluctuation amplitude $Q$, larger baryon to photon ratio $\eta$, smaller fine structure constant $\alpha$, stable beryllium-8, and less dark energy (smaller $\rho_{\scriptscriptstyle\Lambda}$). Such variations could lead to more galaxies, stars, and potentially habitable planets, which provide more opportunities for biological development.

The universe is surprisingly resilient to changes in its fundamental and cosmological parameters, whether such variations are realized in other regions of space-time or are merely gedanken in nature. Considerations of these possible variations thus improve our understanding and alter our interpretation of observed aspects of physics and astrophysics – in our universe and others.

Acknowledgments: This review benefited from discussions and input from many colleagues. I would especially like to thank Konstantin Batygin, Juliette Becker, Tony Bloch, Sean Carroll, Gus Evrard, George Fuller, David Garfinkle, Evan Grohs, Alex Howe, Lillian Huang, Dragan Huterer, Gordy Kane, Jake Ketchum, Martin Rees, Frank Timmes, and James Wells. I also thank the anonymous referee for many useful suggestions. This work was supported by the University of Michigan and in part by the John Templeton Foundation through grant ID55112 Astrophysical Structures in Other Universes.

Appendix A Mass Scales in terms of Fundamental Constants

This Appendix provides a summary of the astrophysical mass scales discussed in the main text and introduces some additional masses (see also the previous treatments from [61, 103, 131, 415, 416, 435, 441, 549]). Each quantity is expressed in terms of fundamental constants.

We start the discussion with stars. A characteristic stellar mass scale $M_{\star}$ [427] can be written in the form

[TABLE]

where the value of this quantity in our universe is $M_{\star}\approx$ $3.7\times 10^{33}$ g $\approx 1.85M_{\odot}$. Although this scale mass is relatively close to the mass of the Sun, it is important to keep in mind that $M_{\star}$ is an order of magnitude larger than the typical mass of stars in our universe. The stellar initial mass function has a nearly log-normal form with a characteristic mass of about $M_{C}\approx 0.2M_{\odot}\sim M_{\star}/10$ [137]. It is also important to keep in mind that the range of stellar masses spans a factor of $\sim 1000$ in our universe [147, 323, 427], and such a wide distribution of masses cannot be fully characterized by a single value.

The minimum mass necessary to sustain nuclear fusion is determined by the requirement that stars can achieve a sufficiently high central temperature in the face of degeneracy pressure [147, 265, 427]. This minimum mass can be written in the form

[TABLE]

where $T_{\rm nuc}$ is the temperature required for sustained nuclear fusion. For hydrogen burning reactions in our universe, the required temperature $T_{\rm nuc}\approx 10^{7}$ K, although a more general estimate can be derived. Nuclear reaction rates in stellar cores depend on quantum mechanical tunneling of the particles through the Coulomb barrier. With this requirement, in conjunction with the Boltzmann distribution of thermal speeds, the reaction rate is proportional to an exponential factor of the form $\exp[-3\Theta]$, where $\Theta=(E_{G}/4kT)^{1/3}$ and $E_{G}$ is the Gamow energy ($E_{G}\approx\pi^{2}\alpha^{2}m_{\rm p}{c^{2}}$ for proton reactions). Semi-analytic stellar models [8, 9] show that the parameter $\Theta$ must be of order unity, with a typical value in the range $4-7$. Putting together these considerations, and setting $\Theta=\Theta_{X}$, the nuclear burning temperature can be written in the form

[TABLE]

so that the minimum stellar mass scale becomes

[TABLE]

The final approximate equality assumes $\Theta_{X}\approx 6$.

Stars become unstable if the pressure contribution from radiation exceeds the gas pressure by a sufficiently large margin [147, 427]. The maximum stellar mass is given by the expression

[TABLE]

where $f_{\rm g}\approx 1/2$ denotes the fraction of the pressure provided by the ideal gas law.

Next we consider the Chandrasekhar mass, which represents the maximum stellar mass that can be supported by the degeneracy pressure of non-relativistic electrons [136]. This mass scale is the upper limit for white dwarfs and can be written in the from

[TABLE]

where $Z$ and $A$ are the mean atomic number and atomic weights of the stellar material. The mass scale thus depends on chemical composition, where $Z\approx{A}/2$ for the carbon and oxygen composition expected for white dwarfs in our present-day universe.

High mass stars (M_{\ast}\lower 2.15277pt\hbox{;\buildrel>\over{\sim};}8M_{\odot} in our universe) leave behind neutron stars as remnants. To leading order, the expected mass scale for neutron stars is comparable to the Chandrasekhar mass from equation (233). The full story is more complicated, however, and involves the subtleties of nuclear physics. In approximate terms, the maximum mass for a neutron star is given by the requirement that the object cannot exceed nuclear densities, where the inter-particle spacing is determined by the Compton wavelength of the pion, which sets the range of the strong force. These considerations [103] result in the mass scale

[TABLE]

where $m_{\pi}$ is the pion mass (about 140 MeV) and the dimensionless parameter $f$ is the factor by which the star is larger than the Schwarzschild radius for the same mass.

The minimum mass accessible to a neutron star is given by the requirement that the object has a lower energy state than the white dwarf configuration of the same mass [475]. The neutron star is smaller in radius than the equivalent white dwarf, so that its gravitational binding energy is greater; in contrast, the neutrons in the neutron star are free, rather than bound into nuclei, and this freedom costs energy. Given that degenerate objects have larger radii for lower masses, a crossover point occurs when the gravitational binding energy per nucleon is equal to the nuclear binding energy per nucleon. The resulting mass scale is given by

[TABLE]

where $\alpha_{\rm s}\approx 0.2$ is the analog of the fine-structure constant for the strong force (so that the binding energy for a nucleon is given $E_{B}\approx\alpha_{\rm s}^{2}m_{\rm p}$).

In order to make stars, interstellar gas must condense and fragment. A number of studies have considered the minimum mass subject to opacity limited fragmentation [442, 481]. In this scenario, parcels of gas continue to condense until they become sufficiently optically thick that they can no longer radiate away the energy released during contraction. This type of analysis results in a variety of different expressions for the minimum fragmentation mass, depending on the assumptions. If we find the point where a gas parcel contains one Jeans mass, has optical depth unity, and can radiate away is energy of self-gravity on a free-fall time scale, the resulting fragmentation mass takes the form

[TABLE]

The mass scale for rocky planets is determined, in part, by the requirement that the body is supported by electromagnetic forces rather than degeneracy pressure (see Section 8). This constraint leads to the characteristic mass scale for planets [9, 61, 103], which takes the form

[TABLE]

In our universe, this mass scale is roughly comparable to that of Jupiter. Significantly, since $\alpha\ll 1$ in our universe, and we expect \alpha\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}1 more generally, planets supported by electromagnetic forces have smaller masses than their host stars.

The mass scale of the cosmological horizon at the epoch of equality between the matter and radiation components [13, 510] is given approximately by the expression

[TABLE]

In this expression, $\eta$ is the baryon to photon ratio, whereas $\Omega_{\rm b}$ and $\Omega_{\rm M}$ are the energy densities of baryons and matter, respectively.

One characteristic mass scale for galaxies is given by the requirement that the gas cooling time is comparable to the free-fall collapse time for cosmological structures. In terms of the structure parameters, the resulting mass scale [449, 510] can be written in the form

[TABLE]

For completeness, we note that mass scales for sub-galactic systems can also be defined. Here the result depends on the nature (and abundance) of dark matter, and whether the primordial fluctuations are adiabatic or isothermal [123]. In spite of these complications, for our universe one generally finds $M_{\rm sub}\sim 10^{5}-10^{6}M_{\odot}$, close to the Jeans mass of the baryons at the epoch of recombination. Similarly, at the earlier epoch of mass-radiation equality, we can write the Jeans mass in the form

[TABLE]

where the final expression ignores all of the dimensionless constants or order unity.

One can also define a mass scale for entire universe, provided that it contains a non-zero contribution of vacuum energy $\rho_{\scriptscriptstyle\Lambda}=\lambda^{4}\neq 0$. At late times, the universe develops a horizon with size given by the inverse Hubble parameter, which approaches a constant (e.g., [79]). The mass scale of the universe thus becomes

[TABLE]

Ignoring the dimensionless coefficient of order unity, the number $N_{\star}$ of ‘stellar mass units’ in the universe is given by $(m_{\rm p}/\lambda)^{2}$. In our universe, $\lambda\approx 0.0023$ eV, so that $N_{\star}\approx 10^{23}$.

We can also derive a mass scale for a matter dominated universe, with no dark energy, although the result is a function of time. From the Friedmann equation (33), the density of the universe is given by $\rho=3H^{2}/(8\pi G)$, where $H$ is the Hubble parameter. The inverse $H^{-1}$ is essentially the horizon distance, so that the corresponding mass scale at time $t$ becomes

[TABLE]

A viable universe must live for a sufficiently long time, which places a lower limit of the value of $t$. Here we require the universe to live for as long as the minimum stellar lifetime imposed in Section 7, so that $t>10^{33}t_{\rm A}$, where $t_{\rm A}$ = $\hbar/(\alpha^{2}m_{\rm e}{c^{2}})$ is the atomic time scale and where the numerical factor corresponds to the equivalent of 1 Gyr. The mass scale of a viable matter dominated universe is thus bounded from below such that

[TABLE]

A crude estimate for the mass scale of life forms can be found by invoking the following requirements [434, 435]: The life forms are assumed to (a) be composed of molecules, (b) reside on planets with the masses and radii discussed in Section 8, and (c) become as large as possible without breaking apart if they fall down on the planetary surface. This set of constraints leads to a mass estimate of the form

[TABLE]

where $\epsilon_{\rm c}$ is the chemical conversion factor for life forms introduced in Section 7. In order to describe observed energy scales of chemical reactions, the conversion factor $\epsilon_{\rm c}\sim 10^{-3}$. In this context, however, equation (244) provides a better description for animal life on our planet if the efficiency $\epsilon_{\rm c}$ is of order unity (see also [414]). Analogous considerations for plant life limit the height of trees to $\sim 100$ meters on Earth [326], about 4 times larger than a blue whale.

The mass scales presented in this Appendix are plotted in Figure 33 as a function of the fine structure constant $\alpha$. The maximum stellar mass (blue curve) and the Chandrasekhar mass (yellow curve) are independent of $\alpha$ and follow horizontal lines in the figure. The minimum stellar mass (red curve) increases with $\alpha$ and eventually becomes larger than the maximum stellar mass for $\alpha/\alpha_{0}\approx 68$ or equivalently $\alpha\approx 1/2$. The maximum planetary mass scale (green curve) increases with $\alpha$ in parallel to the minimum stellar mass, but the planetary mass is always much smaller. The galaxy mass scale is a steeply increasing function of $\alpha$ (cyan curve). For sufficiently small values of the fine structure constant, $\alpha/\alpha_{0}<6\times 10^{-4}$ or $\alpha<4.6\times 10^{-6}$, the galactic mass scale is smaller than the smallest star, so that such universes are not viable. For completeness, the minimum fragment mass is shown as the magenta curve, which decreases slowly with $\alpha$. For $\alpha<1.6\times 10^{-4}$, the smallest fragment mass is larger than the smallest star, but is unlikely to inhibit star formation (given that stars form in cloud cores that are much larger in mass [477]). Note the the mass scale (241) for the universe is much larger than the mass range depicted in Figure 33, whereas the mass scale for life forms (244) is much smaller.

The shaded region in Figure 33 depicts the allowed range of stellar masses as a function of the fine structure constant (see also Section 7). For large values of $\alpha$, the range of stellar masses decreases and then vanishes altogether (due to the lack of viable stellar structure solutions). The range of stellar masses also narrows for sufficiently small values of the fine structure constant. As $\alpha$ decreases, the mass scale for galaxies decreases rapidly, and galaxies become smaller than stars, first only the high mass stars, but eventually stars of all possible masses. These trends underscore two important issues: First, the range of $\alpha$ that allows for working stars is relatively wide, spanning about five orders of magnitude. Second, the value of $\alpha$ that allows for the largest range of stellar masses is smaller than the value in our universe by a factor of $\sim 46$, i.e., for $\alpha\sim 1/6300$. For this value, stars span a range of masses corresponding to a factor of $\sim 18,000$ (compared to only $\sim 1000$ in our universe). As a result, our universe is not optimized to support the widest possible range of stellar masses. On the other hand, for universes with a wider range of stellar masses, the mass hierarchy between galaxies and stars is smaller.

Another way to summarize the mass scales of the universe is to write all of the quantities in terms of the fundamental constants and in units of the stellar mass scale $M_{\star}$. The resulting set of mass scales is presented in Table 5. For each quantity of interest, the table lists the expression for the mass scale in terms of fundamental and/or cosmological parameters, where the result is scaled by $M_{\star}=\alpha_{G}^{-3/2}m_{\rm p}$. The top portion of the table includes the masses of the universe and galaxies, which have masses much larger than the stellar scale $M_{\star}$ for our universe. Note that the weakness of gravity (small $\alpha_{G}$), and to a lesser extent the small value of $\beta$, allows galaxies to contain billions of stellar mass scales. Similarly, the small energy scale of the vacuum $\lambda$ (compared to the proton mass $m_{\rm p}$) allows the universe to contain a large number of stellar masses (trillions of billions). The central portion of the table includes mass scales resulting from stellar evolution, which are comparable to $M_{\star}$. Finally, the bottom portion of the table includes planets and life forms, which have masses much less than $M_{\star}$. On these scales, small values of the structure constants $(\alpha,\alpha_{G})$ allow stars to host many planetary masses and for planets to host an enormously large number of lifeforms.

Appendix B Number of Space-Time Dimensions

The number of space-time dimensions represents another property of the universe that could vary from region to region within the multiverse. Developments in string theory and M-theory indicate that the fundamental manifold of space-time must have at least 10 dimensions. In most versions of the theory, these extra dimensions only manifest themselves at sufficiently high energies and small size scales that are currently outside the reach of experimental probes. In our universe, only three of the spatial dimensions have become large, whereas the remaining dimensions are compactified. Given the possible existence of alternate universes with higher numbers of dimensions, we are left with the question of why our universe has only three large spatial dimensions (and one temporal dimension). This Appendix briefly reviews some of the arguments for the standard $3+1$ structure of space-time.

We first note that most authors agree that habitable universes should have only one time dimension [56, 84, 506, 507]. If space-time had more than one temporal dimension, then closed time-like loops could be constructed. Such loops, in turn, allow for observers to revisit the “past” and thereby affect causality. In addition to violations of causality, multiple time dimensions can lead to violations of unitarity, tachyons, and ghosts [212]. Although space-times with multiple time variables can be constructed, the corresponding universes would have properties vastly different from the types of potentially habitable universes discussed herein. As a result, they will not be considered further.

Notice also that the question of the number of space-time dimensions rests on a different footing than the other issues considered in this review. If one considers variations in the strength of the gravitational constant or the mass of the electron, for example, the basic form of the laws of physics would remain the same. Moreover, many of the physical structures relevant for habitability — such as atoms and stars — could retain their general form, albeit with different specific properties. If the number of dimensions varies, however, more drastic changes to physics arise. The first obvious change is to the number of phase space dimensions, which could be much larger or much smaller than in our universe. As a result, the structure of orbits depends sensitively on the number of dimensions — compare orbits in 1, 2, and 3 spatial dimensions in our universe. These differences arise not only from the additional degrees of freedom available in spaces of higher dimension, but also due to changes in the functional form of force laws. As long as the forces, such as gravity and electromagnetism, are described by a version of the Poisson equation, or its generalizations, the force laws will change with the number of dimensions.

One basic requirement for a viable universe is that the force laws must allow for stable orbits. The orbits in question include both the motion of planets around their host stars and the motion of electrons around atomic nuclei. Although the latter orbits are more complicated due to quantum mechanics, their stability leads to the same constraint on the number of spatial dimensions. Ehrenfest is often credited with starting this line of inquiry [192, 193] by presenting constraints from the stability of classical orbits and wave propagation. A host of additional considerations have since been invoked, including the necessity of even numbers of dimensions based on topological considerations [554], the existence of quantum mechanical solutions for atomic structure [249], orbits in Schwarzschild space [505], and the structure of white dwarf stars [138]. Here we outline the basic arguments from the stability of orbits (B.1) and the stability of atoms (B.2). A number of previous reviews provide additional detail (e.g., see [56, 61, 506, 507]).

As outlined below, the stability of orbits and atoms places an upper bound on the number of spatial dimensions, namely $\mathfrak{D}\leq 3$. Nonetheless, these stability constraints would allow universes to have lower numbers of dimensions $\mathfrak{D}$ = 1 or 2. The standard argument against universes of lower dimension is that they are too simple to allow for the complexity necessary for a universe to become habitable. This argument has been given many times [61, 446, 554], but is briefly summarized here: A universe with $\mathfrak{D}=2$ would be confined to a plane, much like the classic satirical novella Flatland [1]. Orbits of both planets and electrons would also be confined to planes. The resulting atomic structures (electron orbitals) would necessarily be much simpler than those in our universe, so that chemistry would be compromised. Another issue arises with the construction of networks, such as neural pathways or electrical circuits: A sufficiently complicated network, confined to a two-dimensional surface, would result in wires crossing, whereas higher dimensional spaces allow more possibilities. Universes with only a single spatial dimension $\mathfrak{D}=1$ would be even simpler and hence not viable.

Universes with $\mathfrak{D}=1,2$ are also disfavored by considerations of signal propagation. The properties of wave equations depend sensitively on the number $\mathfrak{D}$ of spatial dimensions [61, 192, 193]. For example, consider a simple wave equation of the form

[TABLE]

and where $\nabla_{\mathfrak{D}}$ is the Laplacian operator in $\mathfrak{D}$ dimensions. For the case of one- and two-dimensional wave equations, the properties of the wave solutions depend on the entire domain over which the wave propagates. This property implies that disturbances can propagate at any speed $v\leq{c}$. Slower waves, emitted earlier, can thus be overtaken by faster waves produced later, which acts to impede the transmission of sharply defined wave signals [257]. On the other hand, for $\mathfrak{D}=3$, only the boundary of the domain (here the spherical surface at $r=ct$) determines the wave solution. As a result, waves in three spatial dimensions must propagate at the speed $c$ appearing in the wave equation (245). Universes with $\mathfrak{D}=3$ are thus distinguished from those of lower dimension in terms of their capability for transmission of wave signals and information processing.

B.1 Stability of Classical Orbits

The stability of classical orbits has been studied in many contexts, and places a constraint on the number of spatial dimensions $\mathfrak{D}$ such that

[TABLE]

One can illustrate this constraint as follows. Suppose that one considers the force due to a point mass (or a point charge for classical orbits mediated by the electric force). For the case where the force carrier is massless, the force law is given by a version of the Poisson equation and takes the familiar power-law form

[TABLE]

In equation (247), $F$ is the magnitude of the force (assumed to be attractive), $B$ is a constant determined by the strength of the coupling constant, and the power-law index $q$ is related to the number of spatial dimensions $\mathfrak{D}$ according to $\mathfrak{D}=q+1$. For a classical orbit with a force law of the form (247), the orbital frequency is given by

[TABLE]

where $m$ is the mass of the test particle in question. One way to determine the stability of the orbit is to consider the epicyclic frequency $\kappa$, i.e., the frequency of oscillation of the particle’s motion about the guiding center of the orbit [78, 389]. This quantity is given by

[TABLE]

This result shows that for $\mathfrak{D}>4$, the epicyclic frequency is imaginary and orbits are unstable. For $\mathfrak{D}=4$, the epicyclic frequency vanishes, but further analysis shows that the orbit is also unstable. Stable orbits thus require $\mathfrak{D}<4$, consistent with equation (246), which implies $\mathfrak{D}=$ 1, 2, or 3 (assuming the number of dimensions is discrete).

The above result can be generalized to include massive force carriers and other departures from a power-law force law. We can write the force law in the generalized form

[TABLE]

where $g(r)$ is a dimensionless function of radius. For example, the function $g(r)\sim\exp[-\mu r]$ for the Yukawa potential. With this generalized form (250), the epicyclic frequency becomes

[TABLE]

If the function $g(r)$ is a decreasing function of radius, then the constraint on the number of dimensions becomes more restrictive.

B.2 Stability of Atoms: Bound Quantum States

Next we consider the stability of atoms. In the semi-classical limit, the stability problem for electronic orbits reduces to an analog of that considered in the previous section for classical orbits [192, 193], and one obtains the same constraint $\mathfrak{D}<3$ (from equation [246]). Since atoms are quantum mechanical, semi-classical arguments do not provide an adequate proof of stability or instability (see the discussion of [56]), and one must start from basic quantum mechanical principles.

The stability of a hydrogenic atom represents the simplest case, which we outline here, both for three dimensional space and higher values of $\mathfrak{D}$. The problem can be formally stated as follows: The ground state energy of a quantum mechanical system is given by the expectation value

[TABLE]

where $H$ is the Hamiltonian operator and the minimum is taken over the space of all properly normalized trial wave functions $\psi$ (so that $\langle|\psi|^{2}\rangle=1$). The system is stable if the ground state energy, as defined here, is finite, i.e., is bounded away from negative infinity [351].

Historically, two different (but related) approaches to this problem have been taken. The direct way to proceed is to start from the Schrödinger wave equation (SWE) for the problem and directly construct the ground state wave function. Of course, the SWE results from applying the calculus of variations to the problem posed in equation (252) subject to the constraint $\langle|\psi|^{2}\rangle=1$. Another option is to proceed indirectly by using a series of inequalities to show that the ground state energy $E_{0}$ is bounded. For the hydrogen atom in three dimensional space, the SWE can be readily solved, and the direct method provides a simple way to prove stability. For more complicated systems, including the classic problem of the stability of bulk matter, solutions to the SWE are generally not available, but the indirect method provides a way forward [191, 350].

To illustrate the problem, consider a hydrogenic atom in $\mathfrak{D}$-dimensional space. After suitable definitions, the SWE can be written in dimensionless form, and the angular part of the wave function can be separated. The remaining radial part of the wave function $R(\xi)$ is then governed by the equation

[TABLE]

If we take the dimensionless radius to be $\xi=r/r_{0}$, where $r_{0}$ is a constant scale, the remaining dimensionless parameters include the energy,

[TABLE]

the eigenvalue of the angular part of the Laplacian operator in $\mathfrak{D}$ spatial dimensions,

[TABLE]

and the depth of potential

[TABLE]

For $\mathfrak{D}\neq 4$, one can define the radial scale $r_{0}$ so that $Z=1$. For our universe with $\mathfrak{D}=3$ spatial dimensions, $r_{0}=a_{0}/2$, where $a_{0}=\hbar^{2}/(me^{2})$ is the Bohr radius. For the ground state with $\ell=\lambda=0$, equation (253) admits the well-known solution $\psi=\exp[-Z\xi/2]$; the corresponding ground state energy eigenvalue is finite and has the value $\epsilon=-Z^{2}/4$. For higher dimensions with $\mathfrak{D}>3$, however, the energy spectrum extends to $\epsilon\to-\infty$, so that the energy has no lower bound, and the hydrogen atom is unstable [249]. This result indicates that universes with more than three (large) dimensions are not habitable.

The result described above rests on the assumption that the electric force in higher dimensions obeys Gauss’s law, so that the potential $V(\xi)\propto\xi^{2-\mathfrak{D}}$. For completeness, note that one could assume that Maxwell’s equations are no longer valid in higher dimensions, and that the electric potential retains its form $V\propto 1/\xi$ in spaces with arbitrary dimension (here $\xi$ is the radial coordinate in $\mathfrak{D}$ dimensional space). In that case, the hydrogen atom could be stable in higher dimensions (see [134] and references therein for further discussion). Another option is to consider non-Euclidean spaces. For example, the higher dimensional space $\mathbb{R}^{3}\times{S}^{1}$ allows for stable hydrogenic atoms [102].

Another way to consider stability is to write the energy eigenvalue of the ground state in the form

[TABLE]

where $\mathfrak{D}>2$. To show that atoms are stable, one can show that the energy $\epsilon$ is bounded from below. If the above expression is finite for all trial wavefunctions $\psi$, then the ground state energy must be finite (as codified in equation [252]).

For three dimensional space, one can use the Sobolev inequality [351] to write the constraint in the form

[TABLE]

where $S$ is a dimensionless constant. By defining the probability density ${\cal P}\equiv|\psi|^{2}$, and using Hölder’s inequality to rewrite the first term [351], one obtains the bound

[TABLE]

where $K$ is another dimensionless constant. Finally, applying the calculus of variations to the integral, subject to the constraint $\int{\cal P}\,dV=1$, one finds a lower bound on the energy of the form

[TABLE]

where $C>0$ is a dimensionless constant. This argument shows that the ground state energy is larger than some value (namely that given by equation [260]) by showing that the true expression for the energy is larger than a series of energy functionals evaluated for any trial wavefunction (so the energy must be larger than the energy functional evaluated with the “correct” wavefunction).

We can now show how this argument fails for higher dimensions: The energy eigenvalue in $\mathfrak{D}$-dimensional space can be written in the form

[TABLE]

Assume here that the ground state has zero angular momentum, so that the wavefunction has no angular dependence. The volume element can be written

[TABLE]

Now we consider a particular trial wavefunction of the form

[TABLE]

where $A$ is the normalization constant. This form is motivated by the ground state of the hydrogen atom in three dimensional space. The parameter $\mu$ defines the effective width of the wavefunction and is left arbitrary for now. After finding the normalization constant and evaluating the integrals from equation (261), we find the following expression for the energy

[TABLE]

where we have defined $L\equiv 1/\mu$. Equation (264) illustrates the basic issue of stability in higher dimensions. If $\mathfrak{D}>4$, then the negative second term has a higher power of the length scale $L$ in the denominator. As a result, the energy of the trial wavefunction can become arbitrarily large and negative as $L$ is made increasingly smaller. In contrast, for $\mathfrak{D}=3$, we have $\epsilon\sim 1/L^{2}-Z/6L$, so that the first positive term dominates for small $L$. We can summarize this result with the following heuristic argument [351]: The first term represents the squeezing of the electron into a smaller volume, but the wavefunction develops an effective “pressure” contribution ($\propto 1/L^{2}$) that is strong enough to overcome the tendency for energy gain due to the electron going deeper into the potential. For higher dimensions, however, the second term dominates, indicating that this “pressure” is not sufficient to overcome the change in energy. As a result, it becomes energetically favorable for electrons to fall ever deeper toward/into the nucleus, the energy has no lower bound, and atoms are unstable.

Note that the mathematical arguments presented in this Appendix are abridged. One should refer to the primary references for a more rigorous treatment. Finally, given the preference for three spatial dimensions, the next step is to determine the manner in which an initially higher dimensional universe relaxes into a configuration with (only) three large dimensions and to assess its probability (e.g., [98, 318]).

Appendix C Chemistry and Biological Molecules

The discussion thus far has considered fine-tuning issues in physics by requiring that the fundamental constants and cosmological parameters support the existence of galaxies, working stars, stellar nucleosynthesis, habitable planets, complex nuclei, and stable atoms. These constraints are necessary but not sufficient [124]. Additional requirements are needed in order for a universe to actually develop observers. Specifically, a viable universe must be able to construct the essential molecules required for life [197]. Although the exact requirements are not known [76], these biomolecules are likely to include water with suitable properties, proteins, DNA, and RNA (or equivalent types of complex molecules). We are thus excluding discussion of more exotic life forms such as Hoyle’s Black Cloud [287]. Life in its familiar form also requires fatty acids, sugar, starch, and cellulose. Unlike physical considerations reviewed in the main text, the possible fine-tuning in biologically relevant molecules is relatively unexplored. This Appendix provides a brief overview of current work.

In order for life to exist, biological molecules must function properly in the context of their host organisms. One recent summary of the requirements suggests that life requires proteins, metabolic networks, gene regulatory networks, and signal transmission networks [541]. Biology thus requires that particular molecules (e.g., proteins) function properly, which requires the right types of chemistry. These chemical requirements, in turn, place constraints on the laws of physics. At the fundamental level, the link between chemistry and physics is provided by the Schrödinger equation, which can be written in the dimensionless form

[TABLE]

where $(r,t)$ are dimensionless variables defined by $r\to r/\ell$ and $t\to t/\tau$, where the reference values are defined via $\ell=\hbar/m_{\rm e}c$ and $\tau=\hbar/m_{\rm e}c^{2}$. In equation (265), the mass $m$ is that of the particle described by the wave equation, whereas the electron mass $m_{\rm e}$ appears because it is used to define the dimensionless units.

One important issue is the manner in which the sizes and other properties of atoms change with the values of the fundamental constants. Suppose we let $\alpha\to S\alpha$, where $S$ is a dimensionless scaling factor. If the reference length is rescaled according to $\ell\to\ell/S$, then both the potential and the kinetic term in equation (265) are scaled by a factor of $S^{2}$, so that the energy changes by this same factor. Different values of $\alpha$ thus result in changes to the atomic energy levels. If one includes the hyperfine structure (which depends on the proton mass and hence $\beta$), then the atomic structure changes as well. As a result, one does not expect chemistry to operate in the same manner with different values of $\alpha$ and $\beta$ (see [39] for further discussion). Moreover, although it is called the “fine structure constant”, $\alpha$ is one of the most important parameters for the specification of atomic structure: For quantum systems where $m=m_{\rm e}$ and hyperfine corrections are ignored, the constant $\alpha$ is the only parameter appearing in the dimensionless Schrödinger wave equation (265).

To illustrate the difficulty of assessing fine-tuning in molecules, one can consider the ionized Hydrogen molecule H${}_{2}^{+}$, one of the simplest cases with a single electron in orbit about two protons. Even for this molecule, an analytic solution for the energy levels is not available. One can proceed using the method of Linear Combination of Atomic Orbitals, where the electron wave functions for the two protons are found separately, and then combined, taking into account the overlap integrals and the proper normalization (details are given in most advanced physical chemistry texts – see [39]). Even in this highly simplified case, the solutions for the energy levels and the internuclear potential are rather cumbersome to write down. The resulting energy is shown as a function of internuclear separation in Figure 34 for three different choices of the fine structure constant $\alpha$. The minimum energy state corresponds to the equilibrium separation of the two protons and thus defines the size of the molecule. The blue curve shows the result for the value of $\alpha$ in our universe. For comparison, the figure shows the corresponding energies for $\alpha$ values that are 10% larger (red curve) and 10% smaller (green curve). Figure 34 shows that as the fine structure constant decreases, the molecular size increases. More specifically, 10% changes in the fine structure constant $\alpha$ lead to corresponding $\sim 10$% changes in the nuclear separation, as determined by the minimum of the energy curve.

Relatively little work has been carried out concerning fine-tuning of molecular structures larger than H${}_{2}^{+}$. One pioneering study [322] considers how variations in the fine structure constant $\alpha$ and the electron to proton mass ratio $\beta$ can affect the structure of biologically important molecules (see also [517, 518]). Both of these dimensionless constants are small compared to unity. This study shows that the quantum chemistry [39] of hydrogen, carbon, nitrogen, and oxygen correspond to the asymptotic limit where $\alpha,\beta\to 0$. As a result, even smaller values of these two parameters leave biologically important molecules, such as water, with properties similar to those in our universe. For sufficiently larger values of these parameters, however, the necessary biomolecules could cease to function properly.

The water molecule, one of the essential ingredients for life as we know it, provides a working example for how molecular structure changes with the values of $\alpha$ and $\beta$. The equilibrium bond angle in the water molecule decreases with increasing values of the fine structure constant $\alpha$. If this angle becomes too small, then water will no longer have its tetrahedral bonding capabilities between hydrogen atoms. Since this property of water is essential for water to expand upon freezing, the loss of this property alters the characteristics of ‘liquid water environments’ and hence changes the prospects for habitability. Increases in $\alpha$ also reduce the polar nature of the water molecule and reduce the strengths of its bonds. If the mass ratio $\beta$ is increased, the energy for the water producing reaction

[TABLE]

increases accordingly. This increase in energy, in turn, increases the thermodynamic stability of water, which partially offsets the negative effects of increasing $\alpha$.

Current work does not provide definitive bounds on $(\alpha,\beta)$ due to the constraint of working water molecules (and other molecules such as H2, O2, and CO2). However, as a benchmark [322], an increase in $\alpha$ by a factor of 7 decreases the strength of O-H bond in water by an increment of 7 kcal mol*-1*. An increase in $\beta$ by a factor of 100 increases the strength of that same bond by 11 kcal mol*-1*. For comparison, burning one mole of hydrogen into water releases about 60 kcal. As a rough summary of these findings: An increase in $\alpha$ by an order of magnitude and/or an increase in $\beta$ by two orders of magnitude change the energetics of the reaction (266) by $\sim 10$ percent. Keep in mind that decreases in the parameters ($\alpha,\beta$) have much smaller effects.

Relatively little work has been carried out to study the effects of changes in the fundamental constants on larger molecules. Ultimately, we would like to know how variations in the fundamental parameters (e.g., $\alpha$ and $\beta$) alter the nature of biologically important structures such as DNA [197]. These biomolecules are complex and involve extremely long chains of base pairs. Their successful operation requires precisely controlled interactions that involve molecular recognition at the relevant binding sites. The characteristic distance between adjacent sugars and/or phosphates in DNA molecules is about $\ell\approx 6$ Å, whereas working DNA chains require the distance $\ell$ to fall between about 5.5 and 6.5 Å  [112]. This variation represents a tolerance of about $\sim 15\%$. However, if biological evolution were to take place under different conditions where $\ell$ is significantly larger or smaller, it is not known if DNA molecules could operate with new spacings, or how far the distance $\ell$ could vary. Moreover, the variations in the fundamental constants required to produce such changes has not been calculated (although the required variations are of order 10% for the Hydrogen molecule, as shown in Figure 34).

In any case, sufficiently severe changes to the fundamental parameters $(\alpha,\beta)$ would lead to significant changes to chemical properties and reactivity, and hence to biology. However, as the authors [322] state “$\dots$ the broad $(\alpha,\beta)$ sensitivity determined here for the chemistry of life-supporting molecules is not as spectacular as the narrow constraints on these fundamental constants established previously in physics and cosmology”.

Appendix D Global Bounds on the Structure Constants

This Appendix uses the equations of stellar structure to derive global bounds on the fundamental parameters that determine the properties of stars (see [9, 50, 30]). The relevant variables for this problem include the fine structure constant $\alpha$, the electron to proton mass ratio $\beta$, the nuclear burning parameter ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$, and the gravitational constant $G$ (equivalently $\alpha_{G}$). We can also find global bounds on the ratio $\alpha_{G}/\alpha$.

These results follow from the analytic model of stellar structure developed previously (see Section 7 and [8, 9]). We start by writing all of the constraints in dimensionless form. The stellar mass can be written in terms of the fundamental stellar mass scale $M_{\star}$ (from equation [228]) so that

[TABLE]

The fine structure constant, electron to proton mass ratio, nuclear parameter, and gravitational constant can be written in terms of their values in our universe,

[TABLE]

where the subscripted quantities are the values in our universe.

The equation that sets the central temperature of the star necessary for a long-lived stable configuration can then be written in the form

[TABLE]

where the dimensionless constant $A$ is given by

[TABLE]

All of the quantities in the above expression correspond to the values in our universe.

The condition that stars have a minimum temperature can be written

[TABLE]

where the dimensionless constant $B$ is given by

[TABLE]

The condition that stars have a sufficiently long lifetime takes the form

[TABLE]

where the constant $C$ is given by

[TABLE]

where $N_{\rm life}$ is the number of atomic time scales required for a functioning biosphere (and where we use $N_{\rm life}$ = $10^{33}$ to obtain the numerical value).

The maximum allowed value of the stellar mass defines a maximum value of the parameter $X$, i.e.,

[TABLE]

Finally, the minimum stellar mass can be written in terms of the minimum value of $X$ such that

[TABLE]

where the constant $D$ is defined by

[TABLE]

The bounds for a minimum stellar temperature (equation [271]) and a minimum stellar lifetime (equation [273]) can be combined and written in the form

[TABLE]

The outer parts of the composite inequality (278) lead to the constraint

[TABLE]

In order for the stellar structure equation (269) for the central temperature to have a valid solution, the parameter $\Theta_{\rm c}$ must be bounded from below so that

[TABLE]

The previous two equations thus imply the bound

[TABLE]

Next we derive an upper limit on the gravitational structure constant $\alpha_{G}$. Using the same procedure, we also find a corresponding upper limit on the ratio $\alpha_{G}/\alpha$ of the structure constants. If we combine the stellar temperature equation (269) with the minimum value of the stellar mass parameter $X$ from equation (276), we obtain the inequality

[TABLE]

This result can be simplified to obtain the form

[TABLE]

We also require that the minimum stellar mass is less than the maximum stellar mass, $X_{\rm min}\leq X_{\rm max}$. This condition can be used to obtain a bound on the scaled fine structure constant $a$,

[TABLE]

Combining the previous two equations then yields the inequality

[TABLE]

which can be rewritten in the form

[TABLE]

Note that we have replaced the value of the function $\Theta_{\rm c}^{4}I(\Theta_{\rm c})$ with is maximum value.

Similarly, we can make an analogous argument to find a limit on the ratio $g/a$, which results in the upper bound

[TABLE]

Using the definition of the integral function $I(\Theta_{\rm c})$ [8], we can find a bound on the function of the form

[TABLE]

where $J_{0}$ is given by the integral

[TABLE]

where $f(\xi)$ is the solution to the Lane-Emden equation for polytropic index $n$. Note that we can also write the expression for $J_{0}$ in the form

[TABLE]

As long as the polytropic index $n>2/3$, the factor in square brackets is less than unity, whereas the remaining part of the expression is just $\mu_{\star}$, so that we obtain the bound

[TABLE]

Given the upper limit on $I(\Theta_{\rm c})$, we can find an upper limit on functions of the form

[TABLE]

which is bounded by

[TABLE]

Using this result to evaluate the bounds of equations (286) and (287), we find the limits

[TABLE]

The bounds of $\alpha$ and $\alpha_{G}$ thus scale linearly with the value of the nuclear burning parameter ${\mathcal{C}_{\scriptscriptstyle{\kern-0.50003pt\star}}}$, and also depend on the mass ratio $\beta$. As discussed in Section 2.3, the electron to proton mass ratio is constrained to be much less than unity, with an approximate bound b\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}23 or equivalently \beta\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}1/3 (see also Section 10.2). With this additional information, the global bounds can be written in the form

[TABLE]

One should keep in mind that these are upper bounds on the structure constants. Tighter limits can in principle be found. Nonetheless, although the ratio $g/a$ can be made larger by many orders of magnitude, it always falls well short of the hierarchy of 36 orders of magnitude found in our universe.

Appendix E Probability Considerations

The main focus of this review has been to delineate the ranges over which parameters can vary and still allow a universe to be potentially habitable. In order to make a complete assessment of the degree of fine-tuning, however, the probability for universes to obtain particular parameters must be considered. Unfortunately, the relevant probability distributions are neither specified by theory nor measured by experiments at the present time. As a result, discussion of this topic, which is subject to much uncertainty, is relegated to this Appendix.

Note that (at least) two additional probabilistic complications must be taken into account [51, 269, 511]: First, the allowed ranges for working parameters generally do not have sharp boundaries. For example, one constraint on the fine structure constant is that stars must live ‘long enough’. Although this constraint is undoubtedly necessary, it is not known how long it takes for life to evolve. For the sake of definiteness, we have taken the required time to be $10^{33}$ atomic oscillations, equivalent to 1 Gyr in our universe, but other reasonable choices could be made. Moreover, the time required for biological evolution probably does not have a single minimum value. Instead, life is unlikely to evolve on short time scales, and more likely to evolve if given more time. In this context, we would like to know the probability $P(t)$ that life can arise on a suitable planet within a given time $t$. Unfortunately, again, we do not know the probability distribution for this eventuality. Of course, this probability distribution is likely to depend on the properties of both the planet and its environment. Because of these complications, the allowed regions of the parameter spaces (shown in the figures in this review) must be subject to some (as yet unknown) quality weighting. One should also keep in mind that the most favorable part of the parameter space (that with the highest weighting) does not necessarily correspond to the parameter values in our universe.

Second, in addition to the range of allowed values, which must be weighted as outlined above, we need to know the a priori probabilities for universes to obtain given values of the parameters. At the present time, we do not even know (with certainty) what parameters are allowed to vary in a fundamental theory, much less the probability for a universe to realize them. As a result, the probability distributions for a universe to attain given values of the parameters considered herein currently remain highly uncertain. As one example [511], the a priori probability distribution for the vacuum energy density $\rho_{\scriptscriptstyle\Lambda}$ was assumed to be flat (independent of $\rho_{\scriptscriptstyle\Lambda}$) because only a small portion of the possible range can produce structure. Other choices are possible (e.g., see [372, 464] and many others).

Even though we do not know the probability distributions sampled by the relevant parameters, we can consider some general trends. As noted in Section 10.1, for most of the parameters traditionally considered in fine-tuning discussions, the allowed ranges span a couple to several orders of magnitude (Figures 5, 7, 8, 12, 17, 19, 21, 22, 23, and 29). These ranges are summarized in Table 2. For our universe at the present epoch, the full range of allowed energy/time/mass scales spans a factor of $\sim 10^{61}$, or 61 decades (see Section 4.1 and equations [67] and [68]). Similarly, the range of mass and size scales allowed by physical considerations spans about $80\times 15$ decades (Figure 3). If the parameter probabilities were distributed evenly in logarithmic space across a comparable range (of order 100 decades — see also the discussion below), then the allowed ranges of parameter space correspond to a few percent (maybe $\sim 1-10\%$) of the total. In contrast, if the parameters were distributed uniformly over the full range — and the viable parameters do not lie at the upper end of that range — then the allowed range would represent an incredibly small fraction of the total.

The cosmological constant provides the canonical example of this probability issue. The observed value of energy density of the vacuum $\rho_{\scriptscriptstyle\Lambda}=\lambda^{4}\sim 10^{-10}$ eV4 is smaller than the maximum (and perhaps expected) scale $M_{\rm pl}^{4}$ by $\sim 120$ orders of magnitude. The energy density $\rho_{\scriptscriptstyle\Lambda}$ can be much larger than the observed value (by a factor of $\sim 10^{10}$ or more; see Section 4) and still allow for a working universe, but even the maximum allowed energy density is $\sim 100$ orders of magnitude smaller than the benchmark. If the possible values of $\rho_{\scriptscriptstyle\Lambda}$ are distributed uniformly, then the chances of realizing such a small value are only $\sim 10^{-100}$ and hence highly improbable. On the other hand, if the possible values of $\rho_{\scriptscriptstyle\Lambda}$ are distributed in a log-uniform manner, the probability of obtaining a workable value of $\rho_{\scriptscriptstyle\Lambda}$ is given by

[TABLE]

where $\rho_{min}$ is the minimum allowed value of the vacuum energy density. The numerical result on the right assumes that the minimum value is equal to the observed value $\rho_{obs}$. Any smaller choice leads to a larger probability. For example, if the energy density could be 120 orders of magnitude smaller than that observed (that energy scale corresponds to a wavelength comparable to the cosmological horizon size), then $P_{\Lambda}\sim 13/24\sim 1/2$. Even in the worst case scenario, approximately 1 out of 12 universes would inherit a value of the cosmological constant that is compatible with habitability.

This same state of affairs applies to other fundamental constants and cosmological parameters. The strengths of the four forces span a range of about 40 orders of magnitude (note that because the coupling constants run with energy, this value depends on the energy scale of interest, and this estimate applies in the low energy limit). The range of masses measured for the quarks spans only about 5 or 6 orders of magnitude. However, if one considers masses for the full collection of existing particles, from the Higgs mass at $125$ GeV down to estimated neutrino masses of order $\sim 10^{-3}$ eV, the range becomes about 14 orders of magnitude. Cosmological parameters, such as the baryon to photon ratio $\eta\sim 10^{-10}$ and the fluctuation amplitude $Q\sim 10^{-5}$, could also vary over $\sim 10$ orders of magnitude. Given these ranges for the tunable parameters (tens of orders of magnitude), in conjunction with the ranges of values that allow for habitable universes (typically an order of magnitude or a few – see Table 2), log-uniform distributions imply that successful realizations would be chosen with probabilities of order $1-10\%$. Taken together, these results suggest that: If the fundamental parameters are sampled from log-uniform distributions, then hierarchical fine-tuning issues are largely alleviated. On the other hand, if the parameters are sampled from a random distribution with a large high-end cutoff, then hierarchical fine-tuning problems remain troublesome, even pernicious.

As is well known in probability theory [482], in the absence of any additional defining information, the principle of maximum entropy [299, 474] implies that the underlying probability distribution should be flat or uniform: $P(x)\approx$ constant. On the other hand, we get a different result for problems where the relative change in parameters is important, rather than their absolute values. In this case, the preferred probability distribution has the form

[TABLE]

which corresponds to the probability being distributed in a log-uniform manner. This probability density function, sometimes called the Jeffreys prior [301], arises in problems that do not have a single well-defined fundamental scale (except for possible cutoffs). More generally, power-law distributions arise in problems that are scale-free or self-similar [48], and are found in a wide variety of observed phenomena, including earthquake magnitudes, crater sizes, intensity of solar flares, and populations of cities [395].

The key issue is whether the tunable parameters of physics and cosmology are distributed in a uniform or a log-uniform manner. The parameters could sample a more complicated distribution, of course, but it is useful to consider this simple dichotomy as a starting point. This choice is equivalent to asking whether the parameters are governed by a single well-defined scale, or if they are essentially scale-free. Significantly, this question has no definitive answer at the present time.

Nonetheless, one piece of evidence is provided by Figure 4, which shows that the quark masses are in fact distributed in a (nearly) log-uniform manner (for further discussion, see [183, 296]). These data are consistent with a log-uniform sampling of the particle masses. This finding, in turn, argues for the lack of a well-defined mass scale and hence some alleviation of hierarchical fine-tuning issues.

Another example of a possible log-uniform distribution is provided by the energy scales of the universe, as shown in Figure 35. This plot shows the spacing of six fundamental energy scales (values are taken from Table 1 of [125] and from [419]; five of the scales are also discussed in [79]): The largest value is the Planck scale, where gravity exhibits quantum behavior ($E_{\rm pl}=M_{\rm pl}=10^{19}$ GeV). The next step on the hierarchy is the Grand Unified Scale (taken here to be $E_{\rm gut}=10^{16}$ GeV) where the other three forces become unified. The Fermi scale for electroweak interactions is next ($E_{\rm w}=G_{F}^{-1/2}\sim 250$ GeV), followed by the QCD scale ($E_{\rm qcd}=\Lambda_{\rm qcd}\sim 100$ MeV). Atomic energy scales are set by the fine structure constant and are given approximately by $E_{\rm atom}\sim\alpha^{2}m_{\rm e}\sim 10$ eV. Finally, the energy scale of empty space, given by the dark energy or cosmological constant, is the lowest scale on the diagram ($\rho_{\scriptscriptstyle\Lambda}^{1/4}=\lambda\sim 10^{-3}$ eV). As shown in Figure 35, these scales span a wide range of energies extending over more than 30 decades. Although the spacing is not perfectly even, this collection of energy scales is roughly consistent with a log-uniform distribution.

The biggest apparent difference between a log-uniform distribution and that of the energy scales shown in Figure 35 is the large gap between the electroweak scale and the GUT scale. This well-known feature is sometimes denoted as The Particle Desert. But such a gap is not unlikely: If the six energy scales are sampled from a log-uniform distribution over the range shown, an interval as large as the observed desert will arise about 54 percent of the time (note that this percentage will vary with the chosen range of possible energy scales). Nonetheless, the presence of an additional energy scale within the gap would go a long way toward making the distribution even closer to log-random. One candidate for such an additional scale arises from the paradigm of supersymmetry, which posits a basic relationship between fermions and bosons, where each known particle has a superpartner of the opposite type [256, 397]. With the current deficit of experimental data, the scale of supersymmetry could lie anywhere within the energy interval corresponding to the particle desert. One collection of arguments suggests an energy scale of $\sim 1$ PeV [550], while other models put the energy scale closer to that of unification [35].

Finally, note that one can turn the argument around: If the tunable parameters of physics are sampled from a uniform distribution, then the probability of attaining certain small values required for a successful universe (like the observed energy density $\rho_{\scriptscriptstyle\Lambda}$ of the vacuum) would become uncomfortably small. On the other hand, if the underlying distributions are log-uniform, the probabilities for realizing parameters consistent with a habitable universe are no longer problematic. This result could thus be considered as evidence in favor of scale-free and hence log-uniform distributions. Such distributions are also suggested by renormalization group treatments (see equation [227]). Nonetheless, the construction of credible probability distributions for particles masses, energy scales, and other fundamental parameters represents a formidable challenge for the future.

For completeness, note that the dark energy density $\rho_{\scriptscriptstyle\Lambda}$ could be negative, whereas this discussion and Figure 35 consider only the scale $\lambda=|\,\rho_{\scriptscriptstyle\Lambda}|^{1/4}$, which takes on only positive values. One must thus consider a more complicated treatment (instead of using only a log-random distribution) to take into account negative $\rho_{\scriptscriptstyle\Lambda}$. In addition, the parameter value $\lambda=0$ is allowed, but $\lambda=0$ lies an ‘infinite distance’ away from the observed $\lambda\neq 0$ value according to the log-random distribution.

Appendix F Nuclei and the Semi-Empirical Mass Formula

As outlined in the main text, the stability of nuclei is an important consideration for the potential habitability of other universes. One general trend emerging from the entire collection of constraints is that the requirement that bound nuclear states exist can be more restrictive than the requirement that those nuclei can be synthesized in stars or during Big Bang Nucleosynthesis. This Appendix explores the possible ranges of allowed nuclear structures in of other universes through the use of a standard version of the Semi-Empirical Mass Formula or SEMF [540]. This formalism has already been invoked in Section 2.4 to place limits on the strong coupling constant (see equation [26] and Figure 8). Here we consider the range of allowed nuclei.

Within the context of this model, the nuclear binding energy $E_{\rm b}$ can be written in the form

[TABLE]

The corresponding mass $M(A,Z)$ of the nucleus is then given by

[TABLE]

The coefficients $(V,A,C,B)$ and the function $\delta_{\rm qp}(A)$ are chosen to provide a good fit to the observed nuclear binding energies in our universe, but are based on physical considerations as outlined below. The contributions in equation (298) include the volume term due to nearest neighbor interactions, the surface term which corrects for the decrease in such interactions for nucleons near the surface, the Coulomb term due to electromagnetic repulsion, the asymmetry term which favors equal numbers of protons and neutrons, and finally the quantum pairing term $\delta_{\rm qp}$ which favors even numbers of nucleons. Different authors provide different fits for the binding energy, and hence the coefficients, but the values do not vary by wide margins. For the sake of definiteness, we use the following choices [63]:

[TABLE]

In the present context, we apply the SEMF across a range of universes with different values of the fundamental parameters. To provide a simplified treatment, we ignore the pairing term, which favors nuclei with even numbers of neutrons and protons. This term models variations in the binding energy curve as the values of $(Z,A)$ vary from odd to even integers. Here we are interested in the gross properties of the nuclear binding energy curves and thus ignore these “smaller scale” variations.

Although the constants from equation (300) are fit to experimental data, they have a well motivated theoretical basis. In particular, the Fermi gas model allows for a semi-quantitative assessment of the terms appearing in equation (298). The coefficients of the volume term $V$, the surface term $S$, and the asymmetry term $B$ are all proportional to the Fermi Energy $\varepsilon_{F}$, with coefficients of order unity [63]. The Fermi energy is given by

[TABLE]

where $n$ is the number density of nucleons in the nucleus and $m$ is the nucleon mass. In our universe, the benchmark nuclear density $n_{0}\sim 0.15$ fm*-3* is set by the strength and range of the strong force. Here we can parameterize $n_{0}$ through the ansatz

[TABLE]

where the mass scale $\mu\approx 106$ MeV will reproduce the observed nuclear density $n_{0}$ in our universe. This scale is close to the pion mass (as expected). The number density $n$ appearing in the definition of the Fermi energy (301) is the density of the protons, or neutrons, considered as separate components of the Fermi gas. Under the approximation that $Z\approx N\approx A/2$, the density $n\approx n_{0}/2$ and the Fermi energy can be written in the form

[TABLE]

We can thus define scaled versions of the coefficients such that

[TABLE]

The Coulomb term arises from the electromagnetic repulsion of the constituent protons. This term is determined by the potential energy of the charge distribution. For a uniform charge density, this potential energy takes the form

[TABLE]

where $R$ is the size of the nucleus, which is generally written as $R=A^{1/3}r_{0}$, where $r_{0}\sim\mu^{-1}$. The Coulomb term thus takes the form

[TABLE]

As a result, we can define a scaled version of the coefficient

[TABLE]

The scaled version of the SEMF becomes

[TABLE]

When written in this form, the Fermi energy $\varepsilon_{F}$ defines the overall energy scale for nuclear binding energies. The scaled coefficients $(a_{V},a_{S},a_{C},a_{B})$ are dimensionless quantities of order unity. The only remaining parameter is the ratio ${\cal R}$ of the Coulomb coefficient to the Fermi energy,

[TABLE]

which determines the shape of the nuclear binding energy curve.

One can find the atomic number $Z_{max}$ of the most bound nucleus for a given value of the atomic weight. The function $Z_{max}(A)$ is determined by the condition

[TABLE]

Using this value for $Z_{max}(A)$, we can find the maximum nuclear binding energy as a function of atomic number $Z$. The result, expressed in terms of the binding energy per nucleon $[E_{\rm b}(A)/A]_{Z_{max}}$ is shown in Figure 36 for varying values of the strength of the Coulomb term. The curves show how the shape of the binding energy curve changes with the value of the ratio ${\cal R}$ of the Coulomb term to the Fermi energy. As ${\cal R}$, equivalently $\alpha$, increases, the binding energy curve becomes more peaked and nuclei with large atomic numbers become increasingly unbound. However, the range of bound nuclei remains relatively large, well beyond the atomic number of iron, as long as the Coulomb term does not increase by more than about a factor of $\sim 4$ (corresponding to the cyan curve in Figure 36). The peak of the curve (where the energy per particle is maximum) specifies the atomic number of the most bound nucleus and varies by a factor of $\sim 2$ as the ratio ${\cal R}$ varies by a factor of 16.

We can determine another benchmark value of the ratio ${\cal R}$ by requiring the existence of nuclei for the most abundant elements found in terrestrial life forms. These elements include carbon, hydrogen, nitrogen, oxygen, phosphorus, and sulfur (CHNOPS) and make up 98% of known biomolecules. The largest of these nuclei, sulfur, has atomic number $Z=16$ and mass number $A=32$. By requiring the stability of nuclei up to this size, the ratio ${\cal R}$ cannot be more than $\sim 6$ times larger than the value in our universe. As a result, the fine structure constant is bounded from above such that \alpha\lower 2.15277pt\hbox{;\buildrel<\over{\sim};}6\alpha_{0}.

Note that the constraint that a universe must support large stable nuclei is necessary but not sufficient. Even if bound nuclear states exist, they can still be unstable to radioactive decay. If the half-lives for either $\alpha$-decay or $\beta$-decay are too short, then the nuclei would no longer be useful for habitability. A full assessment of the possible radioactivity for nuclei in other universes has not been carried out, but will place further constraints on this scenario.

In this formulation of the problem, the Fermi energy $\varepsilon_{F}$ sets the overall scale of the binding energy curve. The entire binding energy thus scales up and down with changes in $\varepsilon_{F}$, which in turn depends on both the nucleon mass $m$ and the scale $\mu$ that determines nuclear densities. The typical nuclear binding energies are 5 – 10 MeV per particle for the value of $\varepsilon_{F}$ (about 36 MeV) in our universe. This value is bounded from above by the requirement that the nucleons remain non-relativistic. The value is bounded from below by the requirement that nuclear energy levels are much larger than atomic energy levels, which drive chemical reactions and hence biological function. The parameters of the problem must thus obey the ordering

[TABLE]

Our universe displays a well defined ordering of these scales, with atomic energy levels $\sim$eV, nuclear binding energies $\sim$MeV, and nucleon masses $\sim$GeV. As outlined above, the ratio ${\cal R}$ cannot vary by more than a factor of $\sim 4$ without the universe losing too much of its periodic table. This constraint, in turn, implies that the ratio of the first two terms in equation (311) cannot vary by more than a factor of $\sim 16$, so that the hierarchy between atomic and nuclear energy levels will be maintained within any viable universe.

Although the Semi-Empirical Mass Formula provides a useful framework to consider possible variations in nuclear structure across the multiverse, it is certainly too simple to capture the entire nuclear landscape. In this model, the nucleons are considered to be self-contained entities so that the interactions can be attributed to the exchange of a single pion. A more complete treatment, including the complexities of QCD, are now being carried out for specific nuclei [194, 204, 333, 380]. Although a full exploration of the nuclear parameter space remains to be carried out, such calculations will undoubtedly add structure and nuance to the simplistic picture presented here (see also [296, 466]).