The Coincidence Problem and the Swampland Conjectures in the Ijjas-Steinhardt Cyclic Model of the Universe

TL;DR

This paper examines how the cyclic universe model addresses the coincidence problem, showing that scalar fields consistent with Swampland conjectures can naturally produce short cycles that explain the observed matter-dark energy density coincidence.

Contribution

It demonstrates that scalar field models aligned with Swampland conjectures can resolve the coincidence problem within cyclic universe scenarios.

Findings

Shorter cycle lifetimes naturally solve the coincidence problem.

Swampland-conforming scalar fields enable finite cycle durations.

The model aligns cyclic cosmology with string theory constraints.

Abstract

In the Ijjas-Steinhardt cyclic model, the universe passes through phases dominated by radiation, matter, and a dark energy scalar field, with the value of the scale factor increasing with each cycle. Since each cycle terminates in a finite time, it is straightforward to calculate the fraction of time that the universe spends in a state for which the matter and dark energy densities have comparable magnitudes; when this fraction is large, it can be taken as a solution of the coincidence problem. This solution of the coincidence problem requires a relatively short lifetime for each cycle, but unlike in the case of phantom models, there is no fixed upper bound on this lifetime. However, scalar field models satisfying the Swampland conjectures yield sufficiently short lifetimes to provide a satisfactory resolution of the coincidence problem.

Equations

$$w=p_{\mathrm{DE}}/{\rho }_{\mathrm{DE}},$$

$${\rho }_{DE}={\rho }_{DE0}(a/a_0{)}^{-3(1+w)},$$

$${\rho }_M={\rho }_{M0}(a/a_0{)}^{-3}.$$

$$r\equiv \frac{{\rho }_{DE}}{{\rho }_M}=\frac{{\rho }_{DE0}}{{\rho }_{M0}}{\left(\frac{a}{a_0}\right)}^3.$$

$${\left(\frac{\dot{a}}{a}\right)}^2=\frac{\rho }{3}.$$

$${\left(\frac{\dot{a}}{a}\right)}^2=\frac{1}{3}\left[{\rho }_{M0}{\left(\frac{a}{a_0}\right)}^{-3}+{\rho }_{DE0}\right].$$

$${\left(\frac{1}{3}\frac{\dot{r}}{r}\right)}^2=\frac{1}{3}{\rho }_{DE0}\left(\frac{1}{r}+1\right)$$

$${\sinh }^{-1}\sqrt{r}=\sqrt{(3/4){\rho }_{DE0}}t.$$

$$r_1<r<r_2.$$

$$f=\frac{{\sinh }^{-1}\sqrt{r_2}-{\sinh }^{-1}\sqrt{r_1}}{\sqrt{(3/4){\rho }_{DE0}}{t}_{cyc}},$$

$$f=\frac{{\sinh }^{-1}\sqrt{r_2}-{\sinh }^{-1}\sqrt{r_1}}{{\sinh }^{-1}\sqrt{r_0}}\left(\frac{t_0}{t_{cyc}}\right).$$

$$f=1.3\left(\frac{t_0}{t_{cyc}}\right).$$

$$\Delta \varphi <d\sim \mathcal{O}(1),$$

$$\lambda \equiv |V^{\prime }|/V>c\sim \mathcal{O}(1),$$

$$\ddot{\varphi }+3H\dot{\varphi }+V^{\prime }=0,$$

$$\Delta \varphi \sim \ddot{\varphi }{t}^2\sim V^{\prime }{t}^2.$$

$$\Delta \varphi \sim \lambda {\Omega }_{\varphi },$$

$$V(\varphi )=V_0\exp (-\lambda \varphi ),$$

Full text

Cosmological data union08 ; hicken ; Amanullah ; Union2 ; Hinshaw ; Ade ; Betoule indicate that roughly 70% of the energy density in the universe is in the form of a negative-pressure component, called dark energy, with approximately 30% in the form of nonrelativistic matter (including both baryons and dark matter). The dark energy component is often parametrized by its equation of state parameter, $w$, taken to be the ratio of the dark energy pressure to its density:

[TABLE]

where $w=-1$ corresponds to a cosmological constant. If $w$ is roughly constant, the density of the dark energy, $\rho_{DE}$, scales as

[TABLE]

where $a$ is the scale factor, and $\rho_{DE0}$ and $a_{0}$ are the dark energy density and scale factor, respectively, at the present. (We will use zero subscripts throughout to refer to present-day values). Current observations constrain $w$ to be relatively close to $-1$.

The matter density, in contrast, scales as

[TABLE]

This leads to a problem: while the matter and dark energy densities today are within nearly a factor of two of each other, at early times, $\rho_{M}\gg\rho_{DE}$, and in the far future we expect $\rho_{DE}\gg\rho_{M}$. It would appear, then, that we live in a very special time: this is the well-known coincidence problem.

Many solutions have been proposed for the coincidence problem, but we will concentrate here on one particular approach: models in which the universe can be shown to spend a significant fraction of its lifetime in a “coincidental” state. In Ref. Scherrer it was suggested that the coincidence problem could be resolved in the context of phantom dark energy models. In such models, $w<-1$, and the universe terminates in a singularity at a finite time Caldwell2 ; Caldwell3 , so that the fraction of time for which the dark energy and matter densites are relatively close can be a significant fraction of the universe’s (finite) lifetime.

This result was extended to phantom models with a time-varying equation of state in Ref. Kujat and to scalar field models with a linear potential in Ref. Avelino . In the latter models, the dark energy density can evolve toward negative values, causing the universe to cease expanding and recollapse into a final singularity Felder ; Kallosh , so the universe has a finite lifetime and can spend a significant fraction of that lifetime in a state with roughly equal abundances of matter and dark energy. Both the linear models and the phantom dark energy model were further examined as solutions to the coincidence problem in Ref. BA .

Applications of this approach to cyclic phantom models were explored by Chang and Scherrer CS , who examined the particular model of Ilie et al. Ilie . (See also similar models in Refs. Crem ; Xiong and the much earlier model of Ref. BD , a more conventional oscillating model with positive curvature and a cosmological constant. While Ref. BD does not attempt to solve the coincidence problem, it does make an argument for the “largeness” of the cosmological constant relative to the radiation/matter components). In cyclic phantom models, the universe goes through repeated cycles of matter/radiation domination followed by a phantom dark energy phase. Within each cycle, there is a significant period in which the dark energy and matter densities are comparable. Since these cycles repeat endlessly, it is not surprising that we find ourselves in an epoch in which the dark energy and matter densities are of the same order of magnitude.

Here we propose a similar argument in the context of the cylic model recently put forward by Ijjas and Steinhardt IS . In the Ijjas-Steinhardt (IS) model, the universe undergoes alternating periods of expansion and much shorter periods of contraction. While the overall scale factor increases with each cycle, the relative values of $a$ within a single cycle remain unchanged from one cycle to the next, so the universe appears to repeat the same expansion behavior within each cycle. This model has several interesting features. It resolves many of the same problems as inflation, such as the monopole, flatness, and horizon problems, and the evolution of the universe remains purely classical at all times, without an initial or final singularity.

Conceptually, the discussion of the coincidence problem in the IS model most closely resembles that of Ref. CS for cyclic phantom models. However, the IS model differs significantly from these models in lacking a final singular state. The dark energy dominated state in the IS model can be arbitrarily long, making the fraction of time spent in a state with roughly equal matter and dark energy densities very small. We will show that this problem can be remedied in the context of the Swampland conjectures, which force a relatively early termination of the expanding phase of the universe relative to the present day.

In the IS model, the universe contains the standard components of radiation, nonrelativistic matter (baryons and dark matter) and dark energy, which is assumed to be in the form of a scalar field (quintessence). The dark energy drives the accelerated expansion of the universe at late times, but the universe eventually undergoes a transition to an ekpyrotic contracting phase, driven by the same scalar field. This contracting phase lasts a very brief time compared to the expanding phase. At the conclusion of the contracting phase, the epkyrotic field is converted into matter and radiation, and another cycle begins.

Because the evolution of the universe is cyclic, with a finite lifetime for each cycle, it is possible to calculate the fraction of time that the universe spends in a coincidental state, defined to be a state for which the ratio of the density of dark energy to the density of matter lies within some fixed range close to 1. Note that the periodicity of the universe is crucial in this argument; a universe undergoing a single expansion to a final de Sitter phase spends an infinite time with $\rho_{DE}\gg\rho_{M}$, and the fraction of its lifetime spent in a coincidental state is effectively zero (to the extent that this fraction is defined at all). While not strictly required by the IS model, we will take the final expanding state to be close to de Sitter, with $w_{DE}\approx-1$, in agreement with current observational data.

Let $\rho_{DE}$ be the dark energy density, and $\rho_{M}$ be the nonrelativistic matter density, and define the coincidence ratio $r$ as in Ref. Scherrer :

[TABLE]

We will then define a coincidental state to be one for which $r$ lies sufficiently close to one, where the definition of “sufficiently close” is, of course, somewhat arbitrary.

We assume a flat Friedman-Robertson-Walker model and work in reduced Planck units ($\hbar=c=8\pi G=1$), so that the evolution of the scale factor is given by

[TABLE]

The era of radiation domination constitutes a minuscule fraction of the lifetime of each cycle, so we can neglect it and consider only the time during which matter and dark energy are dominant, with the density of the latter assumed to be nearly constant. Then Eq. (5) becomes

[TABLE]

In terms of the coincidence ratio $r$ defined in Eq. (4), we have

[TABLE]

This equation can be integrated exactly to give

[TABLE]

Now we can calculate the fraction of each cycle that the universe spends in a coincidental state, with $r$ near 1. As already noted, we neglect the radiation-dominated portion of each cycle. We will also neglect the time spent in the contracting phase, as this is assumed to be much smaller than the time over which the universe is expanding IS . Since each cycle in the IS model is identical to all of the others, modulo an overall expansion factor, we can simply calculate the coincidence fraction for the current cycle. Following Ref. Scherrer , we will define a coincidental state to correspond to

[TABLE]

Then the fraction of time $f$ that the universe spends in this coincidental state is

[TABLE]

where $t_{cyc}$ is the lifetime of the current cycle, i.e., the time at which expansion ceases and contraction commences. Since the cycles repeat indefinitely, $f$ is also the fraction of the entire lifetime of the universe that is spent in a coincidental state. We can rewrite Eq. (10) in terms of the present day ratio of dark energy density to dark matter density, $r_{0}$, and the present age of the universe, $t_{0}$:

[TABLE]

The choices for $r_{1}$ and $r_{2}$ are somewhat arbitrary. Here we will define the state of the universe to be “coincidental” if the matter and dark energy densities are within an order of magnitude of each other, i.e., $r_{1}=1/10$ and $r_{2}=10$. Then taking $r_{0}\approx 7/3$, we obtain

[TABLE]

Note that the derivation of Eqs. (10)-(12) assumes that $t_{cyc}>t_{2}$, where $t_{2}$ is the value of $t$ when $r=r_{2}$. For our choice of $r_{2}$, this corresponds to the limit $t_{cyc}/t_{0}>1.9$, so $f<0.7$. In what follows, this limit will always be satisfied.

As we would expect, $f$ varies inversely with $t_{cyc}$. As an illustrative example, Ijjas and Steinhardt discuss the case where $t_{cyc}=10t_{0}$, which gives $f=0.13$, so the universe spends 13% of its time in a state for which the matter and dark energy densities are within an order of magnitude of each other. For this particular case, $f$ is not so large that the argument for this solution of the coincidence problem is compelling, but neither is $f$ so absurdly small that the argument fails completely.

One might hope to invert this argument to make a Bayesian estimate of the likelihood of a particular value of $t_{cyc}$ based on the fact that we do happen to observe $\rho_{M}\sim\rho_{DE}$; such an argument would tend to favor relatively smaller values of $t_{cyc}/t_{0}$. (For an example of such arguments, see Ref. BA ). The problem arises because the IS model does not include a prescription for $t_{cyc}$, so $t_{cyc}$ could, in principle, be arbitrarily large. It then becomes impossible to define a reasonable prior on the distribution of $t_{cyc}$.

As an alternative, we can place an upper bound on $t_{cyc}$ by requiring the IS model to satisfy the Swampland conjectures. These conjectures arise in the context of attempts to derive a quantum theory of gravity within string theory. A variety of general constraints on such models have been proposed (see, e.g., Refs. Arkani ; OV for some of the earliest work in this area). Here we consider specifically the constraints examined in Refs. Brandenberger ; limit1 ; Akrami ; limit2 ; Garg . Because these references discuss the motivations for these conjectures in detail, here we will simply summarize them.

Consider a scalar field $\phi$, with potential $V(\phi)$, that serves as the dark energy. The Swampland conjectures we consider here provide an upper bound on the total distance $\Delta\phi$ over which $\phi$ can evolve, and a lower bound on the logarithmic derivative of $V$ with respect to $\phi$. Specifically, they argue for the existence of constants $d$ and $c$, both of order unity, such that, in reduced Planck units,

Conjecture 1:

[TABLE]

Conjecture 2:

[TABLE]

where $V^{\prime}$ is the derivative of $V$ with respect to $\phi$.

Conjecture 1 is longstanding OV and has a great deal of theoretical support (although see Ref. Scalisi for mechanisms allowing $d$ to be somewhat larger than $\sim\mathcal{O}(1)$). Conjecture 2 is more recent Obied . Conjecture 2 is inconsistent with standard $\Lambda$CDM and is in tension even with quintessence models. The problem arises because observations favor $w$ near $-1$ at moderate redshifts, but $w$ near $-1$ generally translates into values of $\lambda$ less than 1. Agrawal et al. limit1 show that for any quintessence model consistent with observations, $\lambda$ can be at most $0.6$, implying that c\ {\raise-2.15277pt\hbox{\buildrel<\over{\sim}}}\ 0.6. Akrami et al. Akrami and Raveri, Hu, and Sethi limit2 find similar limits on this parameter: $c<0.54$ and $c<0.51$, respectively, at the 95% confidence level. As the value of $c$ in the second Swampland conjecture is not specified exactly, it is fair to say that quintessence models are in moderate tension with this conjecture but not absolutely inconsistent at this point.

Now consider the implications of these Swampland conjectures for the IS model. If we express the IS model in terms of “thawing” quintessence models CL , in which the scalar field is initially at rest in a potential $V(\phi)$ with $w$ near $-1$ and slowly rolls down the potential, then the onset of rolling is determined by the value of $\lambda$: larger $\lambda$ corresponds to an earlier transition to rolling behavior. This is easy to see if we examine the quintessence equation of motion:

[TABLE]

where the dot denotes time derivative, and $H\equiv\dot{a}/a$. For quintessence evolution (in which both the scalar field and matter densities enter into the value of $H$), all three terms in Eq. (15) are of the same order of magnitude (unlike the case of slow-roll evolution in inflation). Then at early times, the distance travelled by the field $\phi$ is

[TABLE]

Then using the fact that $t^{2}\sim 1/(\rho_{M}+\rho_{\phi})$, and the quintessence energy density is initially $\rho_{\phi}=V$, we have

[TABLE]

where $\Omega_{\phi}$ is the fraction of the total density in the form of quintessence. (Ref. limit1 provides a more exact derivation of $\Delta\phi$ for the case of an exponential potential.) Eq. (17) illustrates the constraints that the Swampland conjectures place on the IS model. Conjecture 2 places a lower bound on $\lambda$ which, from Eq. (17), gives a lower bound on $\Delta\phi$. Then the de Sitter expansion phase cannot continue indefinitely because the shape of the potential must change before $\Delta\phi$ violates Conjecture 1.

To say anything more quantitative requires a particular model for the quintessence field driving the expansion phase of the IS model. As a specific example, we will examine the quintessence model with an exponential potential:

[TABLE]

with $\phi=\dot{\phi}=0$ at $t=0$. (Note that the value of $V_{0}$ is arbitrary, as it can be absorbed into a translation of $\phi$). In this model, $\phi$ begins frozen at $\phi=0$, but eventually rolls down the potential, so that $w$ increases from $-1$ as the field thaws. Quintessence with an exponential potential has been widely studied Wetterich ; RatraPeebles ; FerreiraJoyce1998 ; CLW ; Liddle , but it is particularly important with regard to the Swampland conjectures, because, as noted in Ref. limit1 , it allows for the largest value of $V^{\prime}/V$ for thawing quintessence consistent with current observations.

In order to serve as a quintessence field in the IS model, $V(\phi)$ must change sign at some value of $\phi$. Hence, we will assume that $V(\phi)$ evolves sharply away from an exponential potential and takes on a negative value after $\phi$ has evolved over a distance $\Delta\phi$. At this point, the universe begins its contracting phase.

We have numerically integrated Eq. (15) for a range of values of $\lambda$, terminating the evolution at a given value of $\Delta\phi$, which we take to correspond to the end of the expansion phase. We take the time at which the expansion ceases to be $t_{cyc}$ in Eq. (12). This allows us to plot $f$ as a function of $\Delta\phi$ and $\lambda$, as shown in Fig. 1.

As expected, smaller values of $\Delta\phi$ and larger values of $\lambda$, both favored by the Swampland conjectures, result in a shorter cycle time, increasing the likelihood of finding ourselves in a “coincidental” state today. The precise value of $f$ depends on the (unknown) Swampland limits (Eqs. 13 and 14). If, for example, we take $d\approx 1$ and $c\approx 0.6$, we find that $f>0.42$, a reasonable solution to the coincidence problem.

Of course, this is only one of many possible forms for $V(\phi)$ in the IS model, but it gives a good qualitative estimate of the way that the coincidence fraction is likely to depend on the Swampland parameters. It might seem intellectually perverse to invoke the Swampland conjectures in the context of the IS model, as one of the most interesting aspects of the IS model is that it corresponds to a universe in which one never needs to invoke quantum effects to describe the evolution, while the Swampland conjectures arise from attempts to model quantum gravity. Nonetheless, there is no obvious reason to believe that the two ideas are incompatible.

Finally, note that this paper makes two separate claims, with two very different degrees of disputability. The argument that the IS model can help to resolve the coincidence problem is quite strong. Given a value for $t_{cyc}$, one can calculate the exact fraction of time that we would expect to find ourselves in a coincidental state. If this is a substantial fraction of unity, then the IS model can be taken as a plausible solution to the coincidence problem. Our second argument, that the Swampland conjectures provide an upper bound on $t_{cyc}$, is on shakier ground, if only because the Swampland conjectures are, in fact, conjectures, and at this point lack specific values for the bounds specified in Eqs. (13) and (14). However, the first argument does not rely on the second. If one could find some other way to place an upper bound on $t_{cyc}$ in the IS model, then the coincidence problem could be ameloriated without recourse to the Swampland conjectures.

Acknowledgments

R.J.S. was supported in part by the Department of Energy (DE-SC0019207).