The Coincidence Problem and the Swampland Conjectures in the Ijjas-Steinhardt Cyclic Model of the Universe
Robert J. Scherrer
gr-qc astro-ph.CO hep-th


















    (
            [w_DEDepartment of Physics and Astronomy, Vanderbilt University,
            Nashville, TN    37235
            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.
            The Coincidence Problem and the Swampland Conjectures in the
            Ijjas-Steinhardt Cyclic Model of the Universe
                Robert J. Scherrer
                July 24, 2019
            ==========================================================================================================
            
            
            
            Cosmological data <cit.>
            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:
            
            w=p_ DE/ρ_ DE,
            
            where w=-1 corresponds to a cosmological constant.
            If w is roughly constant, the density of the dark energy, ρ_DE, scales as
            ρ_DE=ρ_DE0 (a/a_0)^-3(1+w),
            
            where a is the scale factor, and ρ_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
            ρ_M = ρ_M0 (a/a_0)^-3.
            
            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, ρ_M ≫ρ_DE,
            and in the far future we expect ρ_DE≫ρ_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. <cit.>
            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 <cit.>,
            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. <cit.> and
            to scalar field models with a linear potential in Ref. <cit.>.
            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 <cit.>, 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. <cit.>.
            
            Applications of this approach to cyclic phantom models were explored by Chang and Scherrer
            <cit.>, who examined the particular model of
            Ilie et al. <cit.>. (See also similar models in Refs. <cit.>
            and the much earlier model of Ref. <cit.>, a more
            conventional oscillating model with positive curvature and a cosmological constant.  While Ref. <cit.>
            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 <cit.>.  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. <cit.> 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 ρ_DE≫ρ_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≈ -1, in agreement
            with current observational data.
            
            Let ρ_DE be the dark energy density, and ρ_M be the nonrelativistic
            matter density, and define the coincidence ratio r as in Ref. <cit.>:
            
                r ≡ρ_DE/ρ_M = ρ_DE0/ρ_M0a/a_0^3.
            
            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 (ħ = c = 8π G = 1),
            so that the
            evolution of the scale factor is given by
            ȧ/a^2=ρ/3.
            
            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. (<ref>) becomes
            ȧ/a^2=1/3
                ρ_M0a/a_0^-3+ρ_DE0
            .
            
            In terms of the coincidence ratio r defined in Eq. (<ref>), we have
            1/3ṙ/r^2 = 1/3ρ_DE01/r + 1
            This equation can be integrated exactly to give
            
                sinh^-1√(r) = √((3/4)ρ_DE0) t.
            
            
            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 <cit.>.  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. <cit.>, we will
            define a coincidental state to correspond to
            
            r_1 < r < r_2.
            
            Then the fraction of time f that the universe spends
            in this coincidental state is
            
            f = sinh^-1√(r_2) - sinh^-1√(r_1)/√((3/4)
            ρ_DE0) t_cyc,
            
            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. (<ref>) 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:
            
            f = sinh^-1√(r_2) - sinh^-1√(r_1)/sinh^-1√(r_0)t_0/t_cyc.
            
            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 ≈ 7/3, we obtain
            
            f = 1.3t_0/t_cyc.
            
            Note that the derivation of Eqs. (<ref>)-(<ref>) 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
            ρ_M∼ρ_DE; such an argument would tend to favor relatively smaller
            values of t_cyc/t_0.  (For an example of such arguments, see Ref. <cit.>).
            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. <cit.> for some of the earliest work in this area).
            Here we consider specifically the constraints examined in
            Refs. <cit.>.  Because these references
            discuss the motivations for these conjectures in detail, here
            we will simply summarize them.
            
            Consider a scalar field ϕ, with potential V(ϕ), that serves as the dark energy.
            The Swampland conjectures we consider here provide an upper bound on the total distance Δϕ over which ϕ
            can evolve, and a lower bound on the logarithmic derivative of V with respect to ϕ.
            Specifically, they argue for the existence of constants d and c, both of order unity, such that,
            in reduced Planck units,
            
            Conjecture 1:Δϕ < d ∼𝒪(1),
            Conjecture 2:λ≡ |V^'|/V > c ∼𝒪(1),
             where V^' is the derivative of V with respect to ϕ.
            
            Conjecture 1 is longstanding <cit.> and has a great deal
            of theoretical support (although see Ref. <cit.> for mechanisms allowing
            d to be somewhat larger than ∼𝒪(1)).
            Conjecture 2 is
            more recent <cit.>.  Conjecture 2 is inconsistent with standard ΛCDM 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 λ less than 1.  Agrawal et al. <cit.> show that for any quintessence model consistent
            with observations, λ can be at most 0.6, implying that c  0.6.  Akrami et al. <cit.>
            and Raveri, Hu, and Sethi <cit.> 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 <cit.>, in which the scalar field
            is initially at rest in a potential V(ϕ) with w near -1 and slowly rolls down the potential, then the onset
            of rolling is determined by the value of λ:  larger λ corresponds to an earlier transition to rolling
            behavior.  This is easy to see if we examine the quintessence equation of motion:
            ϕ̈+ 3Hϕ̇ + V^' =0,
            
            where the dot denotes time derivative, and H ≡ȧ/a.  For quintessence evolution (in which both the scalar field and matter densities
            enter into the value of H), all three terms in Eq. (<ref>) 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 ϕ is
            
                Δϕ∼ϕ̈ t^2 ∼ V^' t^2.
            
            Then using the fact that t^2 ∼ 1/(ρ_M + ρ_ϕ), and the quintessence energy density is initially ρ_ϕ = V,
            we have
            Δϕ∼λΩ_ϕ,
            
            where Ω_ϕ is the fraction of the total density in the form of quintessence.  (Ref. <cit.>
            provides a more exact derivation of Δϕ for the case of an exponential potential.)  Eq.
            (<ref>)
            illustrates the constraints that the Swampland conjectures place on the IS model.
            Conjecture 2 places a lower bound on λ which, from Eq. (<ref>), gives a lower bound
            on Δϕ.  Then
            the de Sitter expansion phase
            cannot continue indefinitely because the shape of the potential must change before Δϕ 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:
            
            V(ϕ) = V_0 exp(-λϕ),
            
            with ϕ = ϕ̇=0 at t=0.  (Note that the value of V_0 is arbitrary,
            as it can be absorbed into a translation of ϕ).  In this model, ϕ begins frozen at ϕ = 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 <cit.>, but it is particularly important
            with regard to the Swampland conjectures, because, as noted in Ref. <cit.>, it allows for the largest
            value of V^'/V for thawing quintessence consistent with current observations.
            
            In order to serve
            as a quintessence field in the IS model, V(ϕ) must change sign at some value of ϕ.
            Hence, we will assume that V(ϕ) evolves sharply away from an exponential potential and
            takes on a negative value
            after ϕ has evolved over a distance Δϕ.  At this point, the universe begins
            its contracting phase.
            
            We have numerically integrated Eq. (<ref>) for a range of values of λ,
            terminating the evolution at a given value of Δϕ, 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. (<ref>).  This allows us to plot f as a function of
            Δϕ and λ, as shown in Fig. 1.
            
            As expected, smaller values
            of Δϕ and larger values of λ, 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. <ref> and <ref>).  If, for example, we take d ≈
            1 and c ≈ 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(ϕ) 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. (<ref>) and
            (<ref>).  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).
            
            
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