Cosmological constant: relaxation vs multiverse

TL;DR

This paper explores a scalar field with a bottomless potential, analyzing how its dynamics can lead to a multiverse with a small cosmological constant, potentially resolving some anthropic selection issues.

Contribution

It introduces a model where rebounces in scalar field cosmology can select for small vacuum energy, offering a new perspective on the cosmological constant problem.

Findings

Rebounce dynamics can avoid singularities in scalar field cosmologies.

Probabilities in the multiverse can be better defined without usual ambiguities.

The model predicts a small cosmological constant more naturally than standard anthropic arguments.

Abstract

We consider a scalar field with a bottom-less potential, such as $g^3 \phi$, finding that cosmologies unavoidably end up with a crunch, late enough to be compatible with observations if $g \lesssim 1.2 H_0^{2/3} M_{\rm Pl}^{1/3}$. If rebounces avoid singularities, the multiverse acquires new features; in particular probabilities avoid some of the usual ambiguities. If rebounces change the vacuum energy by a small enough amount, this dynamics selects a small vacuum energy and becomes the most likely source of universes with anthropically small cosmological constant. Its probability distribution could avoid the gap by 2 orders of magnitude that seems left by standard anthropic selection.

Equations

$${\mathcal{L}}_{\varphi }=\frac{({\partial }_{\mu }\varphi {)}^2}{2}-V_{\varphi }(\varphi ),$$

$$\halign to=0.0pt{\@eqnsel\hskip\@centering$\displaystyle{#}$&\global\@eqcnt\@ne\hskip 2\arraycolsep\hfil${#}$\hfil& \global\@eqcnt\tw@\hskip 2\arraycolsep$\displaystyle{#}$\hfil&\llap{#}\cr 0.0pt plus 1000.0pt$\displaystyle{\frac{\ddot{a}}{a} &10.0pt\hfil${=&10.0pt$\displaystyle{-\frac{4\pi G}{3} (\rho+ 3p)\\ \ddot{\phi}&\hbox to0.0pt{\hss=&-3\frac{\dot{a}}{a}\dot{\phi}-V_{\phi}^{\prime}&(2)\cr}}}}}$$

$${\rho }_{\varphi }=\frac{{\dot{\varphi }}^2}{2}+V_{\varphi },p_{\varphi }=\frac{{\dot{\varphi }}^2}{2}-V_{\varphi }.$$

$$\displaystyle\frac{\partial P}{\partial t}=\frac{\partial}{\partial\phi}\bigg{(}\frac{M_{\rm Pl}^{2}}{4\pi}\frac{\partial H}{\partial\phi}P+\frac{H^{3/2}}{8\pi^{2}}\frac{\partial}{\partial\phi}(H^{3/2}P)\bigg{)}.$$

$$|V_{\varphi }^{\prime }|\ll H^2{M}_{\mathrm{Pl}}.$$

$$\frac{d{V}_{\varphi }}{dN}=V_{\varphi }^{\prime }\frac{d\varphi }{dN}=\frac{V_{\varphi }^{\prime 2}}{3H^2}\ll \rho \sim H^2{M}_{\mathrm{Pl}}^2.$$

$$t_{\mathrm{crunch}}=2{\int }_0^{a_{\max }}\frac{da}{aH}=\sqrt{\frac{\pi }{6}}\frac{M_{\mathrm{Pl}}}{\sqrt{-V_{\varphi }({\varphi }_{\mathrm{in}})}}\approx \sqrt{\frac{V_0}{-V_{\varphi }({\varphi }_{\mathrm{in}})}}3.6\times 10^{10}\mathrm{yr}.$$

$${\mathcal{L}}_{\mathrm{bath}}=\frac{\varphi }{f_{\varphi }}{F}_{\mu \nu }{\tilde{F}}_{\mu \nu }+y\varphi \bar{\Psi }{\gamma }_5\Psi .$$

$$\halign to=0.0pt{\@eqnsel\hskip\@centering$\displaystyle{#}$&\global\@eqcnt\@ne\hskip 2\arraycolsep\hfil${#}$\hfil& \global\@eqcnt\tw@\hskip 2\arraycolsep$\displaystyle{#}$\hfil&\llap{#}\cr 0.0pt plus 1000.0pt$\displaystyle{\frac{\ddot{a}}{a} &10.0pt\hfil${=&10.0pt$\displaystyle{-\frac{4\pi G}{3} (\rho+ 3p)\\ \ddot{\phi}&\hbox to0.0pt{\hss=&-(3\frac{\dot{a}}{a}+\Gamma)\dot{\phi}-V_{\phi}^{\prime}\\\dot{\rho}_{\rm bath}&=&-3\frac{\dot{a}}{a}(\rho_{\rm bath}+p_{\rm bath})+\Gamma\dot{\phi}^{2}&(9)\cr}}}}}$$

$$\rho ={\rho }_{\varphi }+{\rho }_m+{\rho }_{\mathrm{bath}},p=p_{\varphi }+p_{\mathrm{bath}}.$$

$$\dot{H}=-4\pi G\left(\rho +p\right)\le 0$$

$$\frac{d}{dt}\left(\begin{array}{c}{f}_S\\ f_M\\ f_A\end{array}\right)=\left[\left(\begin{array}{ccc}-{\kappa }_{SM}-{\kappa }_{SA}& 0& 0\\ {\kappa }_{SM}& -{\kappa }_{MA}& 0\\ {\kappa }_{SA}& {\kappa }_{MA}& 0\end{array}\right)+\left(\begin{array}{ccc}0& 0& {\kappa }_{AS}\\ 0& 0& {\kappa }_{AM}\\ 0& 0& -{\kappa }_{AS}-{\kappa }_{AM}\end{array}\right)\right]\cdot \left(\begin{array}{c}{f}_S\\ f_M\\ f_A\end{array}\right)$$

$$\left(\begin{array}{c}{f}_S\\ f_M\\ f_A\end{array}\right)\propto \left(\begin{array}{c}{\kappa }_{AS}{\kappa }_{MA}\\ {\kappa }_{AM}{\kappa }_S+{\kappa }_{AS}{\kappa }_{SM}\\ {\kappa }_{MA}{\kappa }_S\end{array}\right)$$

$$\frac{{\wp }_{\mathrm{rolling}}}{{\wp }_{\mathrm{random}}}\sim \frac{\min [1,(V_{\mathrm{class}}/V_{\mathrm{rebounce}}{)}^{V_{\mathrm{rebounce}}/V_0}]}{V_0/V_{\max }}.$$

$$V_{{\varphi }^{\prime }}=-g_{{\varphi }^{\prime }}^3{\varphi }^{\prime }-{\Lambda }^4\cos \frac{{\varphi }^{\prime }}{f_{{\varphi }^{\prime }}}$$

$${\mathcal{P}}_1(V)\sim \exp \left(-\frac{1}{2}\frac{V^2}{V_{\mathrm{rebounce}}^2}\right)with{V}_{\mathrm{rebounce}}=g_{{\varphi }^{\prime }}^3\frac{H_{\mathrm{infl}}}{2\pi }\sqrt{N_{\mathrm{infl}}}.$$

$$\mathcal{P}(V)\propto\left\{\begin{array}[]{ll}1&V<-V_{\rm rebounce}\\ e^{-(V+V_{\rm rebounce})^{2}/2\sigma^{2}}&V\geq-V_{\rm rebounce}\end{array}\right.$$

$${\mathcal{P}}_{\mathrm{obs}}(V)\propto \mathcal{P}(V){\mathcal{P}}_{\mathrm{ant}}(V),{\mathcal{P}}_{\mathrm{ant}}(V)\propto \int dt{\mathcal{V}}_{\mathrm{reg}}\frac{d^2{n}_{\mathrm{obs}}}{dtd\mathcal{V}}(V).$$

Full text

1. 1 Introduction

2. 2 Rolling: a bottom-less scalar in cosmology

3. 2.1 Cosmology with a falling scalar and thermal friction

4. 3 Rebouncing: a temporal multiverse

5. 4 Rolling and rebouncing: the hiccupping multiverse

6. 4.1 Possible hiccuping dynamics

7. 4.2 Probability distribution of the cosmological constant

8. 5 Conclusions

1 Introduction

The vacuum energy $V$ that controls the cosmological constant receives power-divergent quantum corrections as well as physical corrections of order $M_{\rm max}^{4}$, where $M_{\rm max}$ is the mass of the heaviest particle. In models with new physics at the Planck scale (e.g. string theory) one thereby expects Planckian vacuum energies, and the observed cosmological constant (corresponding to the vacuum energy $V_{0}\approx(2.3\,{\rm meV})^{4}$) can be obtained from a cancellation by one part in $M_{\rm Pl}^{4}/V_{0}\sim 10^{120}$. In tentative models of dimensionless gravity the heaviest particle might be the top quark ($M_{\rm max}\sim M_{t}$, see e.g. [1]), still needing a cancellation by one part in $M_{\rm max}^{4}/V_{0}\sim 10^{60}$.

A plausible interpretation of this huge cancellation is provided by theories with enough vacua such that at least one vacuum accidentally has the small observed cosmological constant. Then, assuming that the vacua get populated e.g. by eternal inflation, observers can only develop in those vacua with V\,\raise 1.29167pt\hbox{<\kern-7.5pt\lower 4.30554pt\hbox{\sim}}\,10^{3}V_{0} [2]. More quantitative attempts of understanding anthropic selection find that the most likely vacuum energy measured by a random observer is about 100 times larger that the vacuum energy $V_{0}$ we observe [2, 3, 4, 5] (unless some special measure is adopted, for instance as in [6, 7, 8, 9]). This mild remaining discrepancy might signal some missing piece of the puzzle.

Recently [10] (see also [11]) proposed a cosmological model that could make the cosmological constant partially smaller and negative. It needs two main ingredients:

1. a)

‘Rolling’: a scalar field $\phi$ with a quasi-flat potential and no bottom (at least in the field space probed cosmologically), such as $V_{\phi}=-g^{3}\phi$ with small g\,\raise 1.29167pt\hbox{<\kern-7.5pt\lower 4.30554pt\hbox{\sim}}\,H_{0}^{2/3}M_{\rm Pl}^{1/3} where $H_{0}$ is the present Hubble constant. \listpartThen, a cosmological phase during which the energy density is dominated by $V_{\phi}$ (with a value such that $\phi$ classically rolls down its potential) ends up when $V_{\phi}$ crosses zero and becomes slightly negative, starting contraction. During the contraction phase the kinetic energy of $\phi$ rapidly blue-shifts and, assuming some interaction with extra states, gets converted into a radiation bath, thus reheating the Universe and maybe triggering the following dynamics. 2. b)

‘Rebouncing’: a mechanism that turns a contracting universe into an expanding universe. Furthermore, to get a small positive (rather than negative) cosmological constant, the authors of [10] assume multiple minima and a ‘hiccupping’ mechanism that populates vacua up to some energy density $V_{\rm rebounce}$.

Hence, at this stage the Universe appears as hot, expanding and with a small positive cosmological constant, i.e. with standard hot Big-Bang cosmology. In this way, the cancellation needed to get the observed cosmological constant gets partially reduced by some tens of orders of magnitude, such that theories with $M_{\rm max}\sim\,{\rm MeV}$ no longer need accidental cancellations111Notice that the mechanism of [11] can relax vacuum energies up to $\sim\,{\rm TeV}^{4}$, while having a cutoff in the MeV range or lower. [11, 10]. However particles almost $10^{6}$ heavier than the electron exist in nature.

The authors of [10] restricted the parameter space of their model in order to avoid eternal inflation. However other features of the Standard Model, in particular light fermion masses, suggest that anthropic selection is playing a role [12]. The weak scale too might be anthropically constrained [13]. Taking the point of view that a multiverse remains needed, we explore the role that the above ingredients a) and b), assumed to be generic enough, might play in a multiverse context. Is an anthropically acceptable vacuum more easily found by random chance or through the mechanism of [10]?

In section 2 we consider in isolation the ingredient a), finding that all observers eventually end up in an anti-de-Sitter crunch, that can be late enough to be compatible with cosmological data. In section 3 we consider in isolation the ingredient b), finding that it modifies the multiverse structure, in particular leading to multiple cycles of a “temporal multiverse”.

Adding both ingredients a) and b), in section 4 we show that the mechanism of [10] can have a dominant multiverse probability of forming universes with an anthropically acceptable vacuum energy. In such a case, the small discrepancy left by usual anthropic selection (the measured vacuum energy $V_{0}$ is 100 times below its most likely value) can be alleviated or avoided. Conclusions are given in section 5.

2 Rolling: a bottom-less scalar in cosmology

A scalar potential with a small slope but no bottom is one of the ingredients of [10]. We here study its cosmology irrespectively of the other ingredients. We consider a scalar field $\phi$ with Lagrangian

[TABLE]

where the quasi-flat potential can be approximated as $V_{\phi}(\phi)\simeq-g^{3}\phi$ with small $g$. We consider a flat homogeneous universe with scale-factor $a(t)$ (with present value $a_{0}$) in the presence of $\phi$ and of non-relativistic matter with density $\rho_{m}(a)=\rho_{m}(a_{0})a_{0}^{3}/a^{3}$, as in our universe at late times. Its cosmological evolution is described by the following equations

[TABLE]

where $G=1/M_{\rm Pl}^{2}$ is the Newton constant; $\rho=\rho_{\phi}+\rho_{m}$ and $p=p_{\phi}$ are the total energy density and pressure with

[TABLE]

In an inflationary phase with negligible radiation and matter density $\rho_{m}$ the scale factor grows as $a\propto e^{N}$ and $\phi$ undergoes classical slow-roll $\dot{\phi}\simeq-V^{\prime}_{\phi}/3H$ i.e. $d\phi/dN\simeq-V^{\prime}_{\phi}/3H^{2}$ as well as quantum fluctuations $\delta\phi\sim H/2\pi$ per $e$-fold, where $H^{2}={8\pi}V/3M_{\rm Pl}^{2}$. We assume that all other scalars eventually settle to their minimum, such that we can assume $V=V_{\phi}$, up to a constant that can be reabsorbed in a shift in $\phi$.

Classical motion of $\phi$ dominates over its quantum fluctuations for field values such that $|V^{\prime}_{\phi}|\gg H^{3}$. The critical point is $\phi_{\rm class}\sim-M_{\rm Pl}^{2}/g$ which corresponds to vacuum energy $V_{\rm class}\sim g^{2}M_{\rm Pl}^{2}$. Classical slow-roll ends when $V_{\phi}\sim\dot{\phi}^{2}$: this happens at $\phi\sim\phi_{\rm end}\sim M_{\rm Pl}$ which corresponds to $V_{\phi}\sim V_{\rm end}\sim-g^{3}M_{\rm Pl}$. Such a small $V_{\phi}\approx 0$ is a special point of the cosmological evolution when $V_{\phi}$ dominates the energy density [11, 10]. The scale factor of an universe dominated by $V_{\phi}$ expands by $N\sim M_{\rm Pl}^{2}/g^{2}$ $e$-folds while transiting the classical slow-roll region.

Eternal inflation occurs for field values such that V_{\phi}\,\raise 1.29167pt\hbox{>\kern-7.5pt\lower 4.30554pt\hbox{\sim}}\,V_{\rm class}: starting from any given point $\phi<\phi_{\rm class}$ the field eventually fluctuates down to $\phi_{\rm class}$ after $N\sim|\phi|M_{\rm Pl}^{2}/g^{3}$ $e$-folds. The Fokker-Planck equation for the probability density $P(\phi,N)$ in comoving coordinates of finding the scalar field at the value $\phi$ has the form of a leaky box [14]

[TABLE]

This equation admits stationary solutions where $P$ decreases going deeper into the quantum region (while being non-normalizable), and leaks into the classical region.

A large density $\rho$ of radiation and/or matter is present during the early big-bang phase. The scalar $\phi$, similarly to a cosmological constant, is irrelevant during this phase. The variation in the scalar potential energy due to its slow-roll is negligible as long as

[TABLE]

Indeed

[TABLE]

Thereby the evolution of a scalar field with a very small slope $g^{3}$ becomes relevant only at late times when the energy density $\rho$ becomes small enough, \rho\,\raise 1.29167pt\hbox{<\kern-7.5pt\lower 4.30554pt\hbox{\sim}}\,V_{\phi}.

Fig. 1 shows the cosmological evolution of our universe, assuming different initial values of the vacuum energy density $V_{\phi}(\phi_{\rm in})$. If such vacuum energy is negative, a crunch happens roughly as in standard cosmology, after a time

[TABLE]

Unlike in standard cosmology the Universe finally undergoes a crunch even if $V_{\phi}(\phi_{\rm in})\geq 0$, because $\phi$ starts dominating the energy density (like a cosmological constant) and rolls down (unlike a cosmological constant). The crunch happens in the future for the observed value of the cosmological constant and for the value of $g^{3}=H_{0}^{2}M_{\rm Pl}$ assumed in fig. 1. Here $H_{0}$ is the present Hubble constant. For larger $g$ the crunch happens earlier. We do not need to show plots with different values of $g$ because, up to a rescaling of the time-scale, the cosmological evolution only depends on $g^{3}/H^{2}_{*}M_{\rm Pl}$ where $H_{*}$ is the Hubble constant when matter stops dominating (in our universe, $H_{*}$ is the present Hubble constant $H_{0}$ up to order one factors). This means that large enough values of the vacuum energy behave as a cosmological constant for a while.

Fig. 2 shows the time evolution of the dark-energy parameter $w=p_{\phi}/\rho_{\phi}$ for cosmologies that reproduce the present value of the matter and dark energy densities and for different values of the slope parameter $g$. The observed present value $w_{0}=-1.01\pm 0.04$ [15, 16] implies the experimental bound g\,\raise 1.29167pt\hbox{<\kern-7.5pt\lower 4.30554pt\hbox{\sim}}\,1.2H_{0}^{2/3}M_{\rm Pl}^{1/3}. This means that the anthropic restriction on the vacuum energy remains essentially the same as in standard cosmology (where vacuum energy is a cosmological constant), despite that all cosmologies (even for large positive cosmological constant) eventually end with a crunch.

2.1 Cosmology with a falling scalar and thermal friction

The mechanism of [10] employs some interaction that, during the crunch, converts the kinetic energy $\dot{\phi}^{2}/2$ into a thermal bath by particle production due to the evolution of $\phi$, thus reheating the Universe up to potentially high temperatures. The scalar $\phi$ can have interactions compatible with its lightness. Indeed, $\phi$ might be a Goldstone boson with derivative interactions e.g. to extra vectors $F_{\mu\nu}$ or fermions $\Psi$

[TABLE]

In the presence of a cosmological thermal bath of these particles $\phi$ acquires a thermal friction $\Gamma$ without acquiring a thermal mass [10]. For example $\Gamma\sim 4g_{\phi}^{6}T^{3}_{\rm bath}/\pi f_{\phi}^{2}$ in the bath of vectors with gauge coupling $g_{\phi}$. We study its effects during the expansion phase. The cosmological equations of eq. (8) generalise to

[TABLE]

where $\rho$ and $p$ are the total energy density and pressure

[TABLE]

Equation (2.1), dictated by energy conservation, tells the evolution of the energy density of the bath $\rho_{\rm bath}$ in view of the expansion of the universe and of the energy injection from $\phi$. The pressure $p_{\rm bath}$ equals $\rho_{\rm bath}/3$ (0) for a relativistic (non-relativistic) bath.

The presence of a bath can modify the expansion phase, even adding a qualitatively new intermediate period during which $\phi$ rolls down the potential acquiring an asymptotic velocity $\dot{\phi}\sim V^{\prime}_{\phi}/\Gamma(T_{\rm bath})$ while the bath, populated by the $\phi$ kinetic energy, acquires a corresponding quasi-stationary temperature $T_{\rm bath}\sim(g^{6}/H\Gamma)^{1/4}$. The final crunch gets delayed but it eventually happens as illustrated by the dashed curves in fig. 2, and as we now show analytically. Conservation of ‘energy’ gives a first integral of eq.s (8) well known as Friedmann’s equation, $H^{2}=8\pi G\rho/3$. By differentiating it and using (2.1) and (2.1) one obtains

[TABLE]

in which the contribution of $\Gamma$ cancels. In general, $\dot{H}$ is non-positive because the null-energy condition $\rho+p\geq 0$ is satisfied. At the turning point $H=0$ one has $\rho=0$ thanks to a cancellation between a positive $\rho_{\rm bath}$ and negative $\rho_{\phi}$: for our system this implies $p>0$ such that $\dot{H}$ is strictly negative and the Universe starts collapsing. This avoids the Boltzmann-brain paradox that affects cosmologies with positive cosmological constant [17, 18, 19]. The right panel of fig. 2 shows that interactions relax the observational bound on $g$ by an amount proportional to $f_{\phi}^{-1/3}$, for $g$ large enough.

3 Rebouncing: a temporal multiverse

It is usually assumed that anti-de Sitter regions with negative vacuum density collapse to a big-crunch singularity. The resolution of this singularity is not known (for example in perturbative string theory [20]), so it makes sense to consider the opposite possibility b): that collapsing anti-de-Sitter regions rebounce into an expanding space. The mechanism of [10] assumes that the vacuum energy density changes by a small amount $V_{\rm rebounce}$ in the process.

Following the usual assumptions that anti-de Sitter vacua are ‘terminal’, various authors tried to compute the statistical distribution of vacua in a multiverse populated by eternal inflation, in terms of vacuum decay rates $\kappa_{IJ}$ from vacuum $I$ to vacuum $J$ [21]. These rates are defined up to unknown multiverse factors, because eternal inflation gives an infinite multiverse, so probabilities are affected by divergences. Some measures lead to paradoxes (see for instance [17, 18, 19, 22]). Furthermore, even if the multiverse statistics were known, its use would be limited by our ability of observing only one event (our universe). Despite these drawbacks and difficulties many authors tried addressing the issue (see e.g. [21, 23, 24, 25, 26, 27]).

If vacua with negative cosmological constant are not terminal, multiverse dynamics would change as follows (see also [28, 29]). For simplicity we consider a toy multiverse with 3 vacua: $S$ (de Sitter), $M$ (Minkowski) and $A$ (anti de Sitter). The evolution of the fraction of ‘time’ spent by an ‘observer’ in the vacua is described by an equation of the form (see e.g. [21])

[TABLE]

If anti-de-Sitter vacua are terminal only the first term is present: then, in a generic context, the frequencies $f_{I}$ are dominated by decays from the most long-lived de Sitter vacuum [21]. If anti-de-Sitter vacua are not terminal, the second term containing the $\kappa_{AJ}$ recycling coefficients is present, allowing for a steady state solution

[TABLE]

where $\kappa_{S}=\kappa_{SA}+\kappa_{SM}$.

If anti-de-Sitter crunches rebounce due to some generic mechanism when they reach Planckian-like energies, the $\kappa_{AJ}$ coefficients might be universal and populate all lower-energy vacua. The mechanism of [10] needs a milder hiccupping mechanism, that only populates vacua with vacuum energy slightly higher than the specific AdS vacuum that crunched.

4 Rolling and rebouncing: the hiccupping multiverse

Finally we consider the combined action of the ingredients a) and b) of [10].

Due to a), observers in de Sitter regions unavoidably end up sliding down the $V_{\phi}$ potential until the vacuum energy becomes small and negative, of order $V_{\rm end}\sim-g^{3}M_{\rm Pl}$. This happens even if the vacuum energy is so big that quantum fluctuations of $\phi$ initially dominate over its classical slow-roll. This process can end de Sitter faster than quantum tunnelling to vacua with lower energy densities, as the vacuum decay rates are exponentially suppressed by possibly large factors.

Due to b), contracting regions with small negative vacuum energy density $\sim V_{\rm end}$ eventually rebounce, becoming expanding universes with vacuum energy varied by $\sim V_{\rm rebounce}$. As in the previous section, a temporal multiverse is created: different values of the vacuum energy are sampled in different cycles.

A new feature arises due to the presence of both ingredients: all cycles now last a finite time. A single patch samples different values of physical parameters (for example vacuum energies) with a given probability distribution. We refer to this as temporal multiverse.

A disordered temporal multiverse arises if the extra vacuum energy generated during the rebounce, $V_{\rm rebounce}$, is typically larger than $V_{\rm class}$, such that the small vacuum energy selected by the rolling mechanism at the previous cycle is lost. Small vacuum energy is not a special point of this dynamics, and the usual anthropic selection argument discussed by Weinberg [2] applies: the most likely value of the cosmological constant is 2 orders of magnitude above its observed value (for the observed value of the amount of primordial inhomogeneities, $\delta\rho/\rho\sim 10^{-5}$). This discrepancy by 2 orders of magnitude (or worse if $\delta\rho/\rho$ can vary) possibly signals that anthropic selection is not enough to fully explain the observed small value of the cosmological constant.

An ordered temporal multiverse arises if, instead, the contraction/bounce phase changes the vacuum energy density by an amount $V_{\rm rebounce}$ smaller enough than $V_{\rm class}$ such that, when one cycle starts, it proceeds forever giving rise at some point to an anthropically acceptable vacuum energy. We refer to this possibility as ‘hiccupping’. A small vacuum energy $\sim V_{\rm end}$ is a special point of this dynamics, such that, depending on details of hiccupping, the probability distribution of the vacuum energy can peak below the maximal value allowed by anthropic selection. In the absence of knowledge of a clear rebouncing mechanism (while some recent attempts have been done, see e.g. [30]), having one rather than the other possibility shoud be considered as an assumption.

Going beyond the two limiting cases discussed above, an intermediate situation can be broadly characterised by different scales: $V_{\rm rebounce}$ (the amount of randomness in vacuum energy at each rebounce); $V_{\rm class}\sim g^{2}M_{\rm Pl}^{2}$ (the critical value of $V_{\phi}$ above which $\phi$ starts fluctuating); $V_{\rm max}\sim M_{\rm max}^{4}$ (the maximal energy scale in the theory). The observed value $V_{0}$ of the vacuum energy can either be reached trough the rolling dynamics of [10] or by the usual random sampling the multiverse. The relative probability of these two histories is

[TABLE]

We ignored possible fine structures within each scale. If the mechanism of [10] provides the dominant source of anthropically-acceptable vacua (those with V\,\raise 1.29167pt\hbox{<\kern-7.5pt\lower 4.30554pt\hbox{\sim}}\,10^{3}V_{0}), the observed value $V_{0}$ of the vacuum energy density can have a probability larger than in the usual multiverse scenario.

4.1 Possible hiccuping dynamics

Let us now discuss the value of $V_{\rm rebounce}$ from a theoretical point of view. If the rebounce occurs when the contracting region heats up to temperatures $T_{\rm rebounce}$ (or, more in general, energy density $\sim T_{\rm rebounce}^{4}$), one can expect that scalars lighter than $T_{\rm rebounce}$ can jump to different minima (assuming that potential barriers are characterised by the mass). If the rebounce happens when contraction reaches Planckian densities, one expects $V_{\rm rebounce}\sim M_{\rm Pl}^{4}$. A very small value of $V_{\rm rebounce}\sim V_{0}$ could arise assuming that the contraction/rebounce/expansion phase triggers movement of some lighter fields $\phi^{\prime}$ with potentials such that vacua close by in field space have similar energies. ‘Ordered’ landscapes of this kind have been considered, for instance, in [31, 32, 33, 34, 35, 36, 37].222In our context, this property must only be obeyed by some lighter fields, not by the full landscape. An example of this hiccupping structure is provided by Abbott’s model [31], i.e. a light scalar $\phi^{\prime}$ with potential that can be (at least locally) approximated as

[TABLE]

with, again, a very small slope $g_{\phi^{\prime}}^{3}$, such that $g_{\phi^{\prime}}^{3}f_{\phi^{\prime}}\ll V_{0}$.333Such small slopes are typical, for instance, of relaxion models [32], where they can be generated dynamically by the clockwork mechanism, either in its multi-field [38, 39] or extra-dimensional [40, 41] versions. During a given cycle the field $\phi^{\prime}$ is trapped in a local minimum (that we may take at $\phi^{\prime}=0$) provided that $\Lambda$ is large enough to quench tunnelling. At the end of the cycle, during the contraction/rebounce/expansion phase, the barriers become irrelevant for some time and the field $\phi^{\prime}$ is free to diffuse from $\phi^{\prime}=0$ by thermal or de Sitter fluctuations. We focus on de Sitter fluctuations, given that a phase of the usual inflation with Hubble constant $H_{\rm infl}$ is probably needed to explain the observed primordial inhomogeneities. For $g_{\phi^{\prime}}\ll H_{\rm infl}$ the quantum evolution dominates (the classical rolling of $\phi^{\prime}$ gives a negligible variation in $V$, of order $\sim N_{\rm infl}g_{\phi^{\prime}}^{6}/H_{\rm infl}^{2}$) and the field $\phi^{\prime}$ acquires a probability density $\mathcal{P}_{1}\sim\exp(-2\pi^{2}\phi^{\prime 2}/H^{2}_{\rm infl}N_{\rm infl})$, where $N_{\rm infl}$ is the number of $e$-folds of inflation. Hence, when barriers become relevant again, the vacuum energy has probability density

[TABLE]

Quantum fluctuations happen differently in different Hubble patches: after inflation regions in different vacua progressively return in causal contact, and the region with lower vacuum energy density expands into the other regions. If $f_{\phi^{\prime}}\ll H_{\rm infl}$ there is a order unity probability that this is happening now on horizon scales, giving rise to gravitational waves [42] (and to other signals as in [43] if $\phi$ couples to photons). The field $\phi^{\prime}$ (for $\Lambda=0$) can be identified with $\phi$ provided that N_{\rm infl}\,\raise 1.29167pt\hbox{>\kern-7.5pt\lower 4.30554pt\hbox{\sim}}\,\left(2\pi M_{\rm Pl}/H_{\rm infl}\right)^{2} is large enough that $V_{\rm rebounce}\geq|V_{\rm end}|$. The above hiccup mechanism can be part of the scenario of [10], that tries avoiding the multiverse.

This hiccup mechanism preserves, on average, the value of $V$. Since the field $\phi$ classically rolls down a bit whenever a cycle starts with $V>V_{\rm end}$, $V$ gradually decreases and after a large number of cycles the probability distribution of $V$ becomes, for $V_{\rm rebounce}\gg|V_{\rm end}|$

[TABLE]

with $\sigma\simeq 1.3V_{\rm rebounce}$ according to numerical simulations. We refer to this as asymmetric hiccup.

An alternative speculative possibility is that the downward average drift of $V$ is avoided by some symmetric hiccup mechanism that gives a distribution of $V$ peaked around the special point of the dynamics $V=V_{\rm end}$ also for cycles with negative vacuum energy (e.g. thermalisation might cause loss of memory). In such a case $P(V)=P_{1}(V)$ may be peaked around some small scale.

4.2 Probability distribution of the cosmological constant

Finally, we discuss the probability distribution of the cosmological constant $\mathcal{P}_{\rm obs}(V)$ measured by random observers taking their anthropic selection into account. In our case $\mathcal{P}_{\rm obs}(V)$ is simply given by the product of $\mathcal{P}(V)$ times an astrophysical factor $\mathcal{P}_{\rm ant}(V)$ that estimates how many observers form as function of the vacuum energy $V$. As the volume of a flat universe is infinite, some regularising volume $\mathcal{V}_{\rm reg}$ is needed [44, 3, 45]:

[TABLE]

The temporal integral is over the finite lifetime of a single cycle. The quantity ${d^{2}n_{\rm obs}}/{dt\,d\mathcal{V}}$ is the observer production rate per unit time and comoving volume.444As the literature is not univocal, we adopt the following choice. For positive cosmological constants we take the observer production rate from the numerical simulations in [46], which qualitatively agree with the semi-analytical approach of [47]. For the observer model, we choose the “stellar-formation-rate plus fixed-delay” model [48], where the rate of formation of observers is taken as proportional to the formation rate of stars, with a 5 Gyr fixed time delay inspired by the formation of complex-enough life on Earth. For negative cosmological constants, we approximate the star formation rate as the zero cosmological constant rate supplemented by a hard cut-off at the crunch time of eq. (7). In doing so, we neglect a possible new phase of star formation during contraction since we assume a fixed time delay $\approx 5\,\mathrm{Gyr}$ for the formation of observers. The anthropic factor depends on the prescription adopted to regularise the number of observers. Following Weinberg [4] we consider the number of observers per unit of mass, which corresponds to ${\cal V}_{\rm reg}=1$ in eq. (18). This measure prefers vacuum-energy densities 2 orders of magnitude larger than the observed $V_{0}$. This unsatisfactory aspect of the standard spatial multiverse can be limited by choosing appropriate regularisation volumes, such as the causal-diamond measure (see, for instance, [6, 7, 8, 9]).

Without needing such choices, a temporal multiverse can give a probability distribution of $V$ peaked around its observed value. This needs an ordered landscape with small $V_{\rm rebounce}$. Figure 3 shows numerical result for $P_{\rm obs}(V)$ assuming $V_{\rm end}\ll V_{0}$:

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The red curve considers a disordered hiccup, or a ordered hiccup with $V_{\rm rebounce}\gg 10^{3}V_{0}$: they both give a flat $P(V)$ around $V_{0}$, reproducing the usual $\Lambda$CDM anthropic selection [2, 49]: vacuum energy densities 2 orders of magnitude larger than $V_{0}$ are preferred.

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The yellow and blue curves assume an ordered asymmetric hiccup, that cuts large positive values of $V$, but not negative large values.

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The green curve assumes an ordered symmetric hiccup, that cuts large (positive and negative) values of $V$. Assuming a small $V_{\rm rebounce}\sim V_{0}$ gives a $\mathcal{P}_{\rm obs}(V)$ peaked around the observed $V_{0}$, while the measure-dependent anthropic factor $\mathcal{P}_{\rm and}(V)$ becomes irrelevant, being approximatively constant in such a small $V$ interval.

5 Conclusions

The authors of [10] proposed a dynamical mechanism that makes the small vacuum energy density observed in cosmology less fine-tuned from the point of view of particle physics. This possibility was put forward as an alternative to anthropic selection in a multiverse. However, given that multiple vacua are anyhow needed by the mechanism of [10], and that a multiverse of many vacua is anyhow suggested by independent considerations, we explored how the ingredients proposed in [10] behave in a multiverse context.

A first ingredient of [10] is a scalar with a bottom-less potential and small slope that relaxes the cosmological constant down to small negative values. In section 2 we computed the resulting cosmology. In particular, we found that any universe eventually undergoes a phase of contraction, leading to a crunch, even starting from a positive cosmological constant. This avoids the possible Boltzmann-brain paradox generated e.g. by the observed positive cosmological constant. We calculated the parameter space compatible with present observations, with the novel behaviour starting in the future.

A second ingredient of [10] is a mechanism that rebounces a contracting universe into an expanding one and mildly changes its cosmological constant. In section 3 we explored how rebounces would affects attempts of computing probabilities in the multiverse. In particular a steady-state temporal multiverse becomes possible, as anti-de Sitter vacua are no longer terminal and rebounce into expanding regions.

In section 4 we combined both ingredients above. Any region now undergoes cycles of expansion, contraction and rebounce in a finite time555This turns out to be similar, in spirit, to the proposal of Ref. [50] based on the Ekpyrotic Universe, where periodic brane collisions give rise to cyclic big bangs that reheat the Universe during Abbott’s relaxation, thus solving its empty-Universe problem. However, in our case relaxation and reheating are intertwined: the latter is triggered by the contraction phase due to the former.. This temporal universe is not affected by issues that often plague the spatial multiverse. For instance, the Boltzmann-brain paradox and the youngness paradox [22] are avoided because there are no exponentially inflating regions nucleating habitable universes. In a part of its parameter space, the mechanism of [10] can provide the most likely source of universes with vacuum energy density below anthropic boundaries. One can devise specific models where the probability distribution of the vacuum energy improves on the situation present in the usual anthropic selection, where the most likely value of the cosmological constant seems 2 orders of magnitude above its observed value.

Acknowledgements

This work was supported by the ERC grant NEO-NAT. We thank Dario Buttazzo, Juan Garcia-Bellido, Luca Di Luzio, Antonio Riotto and Enrico Trincherini for discussions.