It is suggested that dark energy in a brane world can help reconcile an
infinitely cyclic cosmology with the second law of thermodynamics. A cyclic
cosmology is described, in which dark energy with constant equation of state
leads to a turnaround at finite future time, when entropy is decreased by a
huge factor equal to the inverse of its enhancement during the initial
inflation. Thermodynamic consistency of cyclicity requires the arrow of time to
reverse during contraction. Entropy reduction in the contracting phase is
infinitesimally smaller than entropy increase during expansion.Comment: 11 pages late

Baum, Lauris

Frampton, Paul H.

English

arXiv

It is speculated how dark energy in a brane world can help reconcile an infinitely cyclic cosmology with the second law of thermodynamics. A cyclic model is described, in which dark energy with w&lt;-1 equation of state leads to a turnaround at a time, extremely shortly before the would-be big rip, at which both volume and entropy of our Universe decrease by a gigantic factor, while very many independent similarly small contracting universes are spawned. The entropy of our model decreases almost to zero at turnaround but increases for the remainder of the cycle by a vanishingly small amount during contraction, empty of matter, then by a large factor during inflationary expansion

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arXiv:hep-th/0610213v2  5 Nov 2006hep-th/0610213November 2006Turnaround in Cyclic CosmologyLauris Baum and Paul H. FramptonDepartment of Physics and Astronomy,University of North Carolina at Chapel Hill, NC 27599-3255, USAAbstractIt is speculated how dark energy in a brane world can help reconcile an infinitelycyclic cosmology with the second law of thermodynamics. A cyclic model is de-scribed, in which dark energy with w < −1 equation of state leads to a turnaroundat a time, extremely shortly before the would-be Big Rip, at which both volumeand entropy of our universe decrease by a gigantic factor, while very many indepen-dent similarly small contracting universes are spawned. The entropy of our modeldecreases almost to zero at turnaround but increases for the remainder of the cycleby a vanishingly small amount during contraction, empty of matter, then by a largefactor during inflationary expansion.One of the oldest questions in theoretical cosmology is whether an infinitely oscillatoryuniverse which avoids an initial singularity can be consistently constructed. As realized byFriedmann [1] and especially by Tolman [2, 3] one principal obstacle is the second law ofthermodynamics which dictates that the entropy increases from cycle to cycle. If the cyclesthereby become longer, extrapolation into the past will lead back to an initial singularityagain, thus removing the motivation to consider an oscillatory universe in the first place.This led to the abandonment of the oscillatory universe by the majority of workers.Nevertheless, an infinitely oscillatory universe is a very attractive alternative to theBig Bang. One new ingredient in the cosmic make-up is the dark energy discovered onlyin 1998 and so it natural to ask whether this can avoid the difficulties with entropy.Some work has been started to exploit the dark energy in allowing cyclicity possiblywithout the need for inflation in [4–8]. Another new ingredient is the use of branes anda fourth spatial dimension as in [9–12] which examined consequences for cosmology. TheBig Rip and replacement of dark energy by modified gravity were explored in [13, 14].If the dark energy has a super-negative equation of state, ωΛ = pΛ/ρΛ < −1, it leadsto a Big Rip [15] at a finite future time where there exist extraordinary conditions withregard to density and causality as one approaches the Rip. In the present article we explorewhether these exceptional conditions can assist in providing an infinitely cyclic model.We consider a model where, as we approach the Rip, expansion stops due to a branecontribution just short of the Big Rip and there is a turnaround at time t = tT whenthe scale factor is deflated to a very tiny fraction (f) of itself and only one causal patchis retained, while the other 1/f 3 patches contract independently to separate universes.Turnaround takes place an extremely short time (< 10−27s) before the Big Rip wouldhave occurred, at a time when the universe is fractionated into many independent causalpatches [14].We discuss contraction which occurs with a very much smaller universe than in expan-sion and with almost vanishing entropy because it is assumed empty of dust, matter andblack holes.A bounce takes place a short time before a would-be Big Bang. After the bounce,entropy is injected by inflation [16], where is assumed that an inflaton field is excited.Inflation is thus be a part of the present model which is one distinction from the work of[5–8]. For cyclicity of the entropy, S(t) = S(t+τ) to be consistent with thermodynamics itis necessary that the deflationary decrease by f 3 compensate the entropy increase acquiredduring expansion including the increase during inflation.A possible shortcoming of the proposal could have been the persistence of spacetimesingularities in cyclic cosmologies [17], but to our understanding for the model we outlinethis problem is avoided, provided that the time average of the Hubble parameter duringexpansion is equal in magnitude and opposite in sign to its average during contraction.This model is published because it gives renewed hope for the infinitely oscillatoryuniverse saught in [1–3]. Time will tell whether the present model is consistent, but atpresent we see no fatal flaw.1Friedmann equation for expansion phase. Let the period of the Universe be designated byτ and the bounce take place at t = 0 and turnaround at t = tT . Thus the expansion phaseis for times 0 < t < tT and the contraction phase corresponds to times tT < t < τ . Weemploy the following Friedmann equation for the expansion period 0 < t < tT :(ȧ(t)a(t))2=8πG3[((ρΛ)0a(t)3(ωΛ+1)+(ρm)0a(t)3+(ρr)0a(t)4)−ρtotal(t)2ρc](1)where the scale factor is normalized to a(t0) = 1 at the present time t = t0 ≃ 14Gy. Toexplain the notation, (ρi)0 denotes the value of the density ρi at time t = t0. The first twoterms are the dark energy and total matter (dark plus luminous) satisfyingΩΛ =8πG(ρΛ)03H20= 0.72 and Ωm =8πG(ρm)03H20= 0.28 (2)where H0 = ȧ(t0)/a(t0). The third term in the Friedmann equation is the radiation densitywhich is now Ωr = 1.3 × 10−4. The final term ∼ ρtotal(t)2 is derivable from a brane set-up [9,10,12]; we use a negative sign arising from negative brane tension (a negative sign canarise also from a second timelike dimension but that gives difficulties with closed timelikepaths). ρtotal = Σi=Λ,m,rρi. As the turnaround is approached, the only significant termsin Eq.(1) are the first (where ωΛ < −1) and the last. As the bounce is approached, theonly important terms in Eq.(1) are the third and the last. (We shall later argue that thesecond term, for matter, is absent during contraction.) In particular, the final term of Eq.(1), ∼ ρtotal(t)2, arising from the brane set-up is insignificant for almost the entire cyclebut becomes dominant as one approaches t → tT for the turnaround and again for t → τapproaching the bounce.Turnaround. Let us assume for algebraic simplicity ωΛ = −4/3 = constant. This value isalready almost excluded by WMAP3 [18] but to begin we are aiming only at consistency ofinfinite cyclicity. More realistic values may be discussed elsewhere. With the value ωΛ =−4/3 we learn from [13] that the time to the Big Rip is (trip − t0) = 11Gy(−ωΛ − 1)−1 =33Gy which is, as we shall discuss, within 10−27 second or less, when turnaround occurs att = tT . So if we adopt t0 = 14Gy then tT = t0+(trip−t0) ∼ (14+33)Gy = 47Gy. From theanalysis in [13–15] the time when a system becomes gravitationally unbound correspondsapproximately to the time when the dark energy density matches the mean density of thebound system. For an object like the Earth or a hydrogen atom water density ρH2O is apractical unit.With this in mind, for the simple case of ω = −4/3 we see from Eq.(1) that thedark energy density grows proportional to the scale factor ρΛ(t) ∝ a(t) and so given thatthe dark energy at present is ρΛ ∼ 10−29g/cm3 it follows that ρΛ(tH2O) = ρH2O whena(tH2O) ∼ 1029. We can estimate the time tH2O by taking on the RHS of the Friedmannequation only dark energy(ȧa)2= H20ΩΛa−β with β = 3(1 + ω). When we specialize toω = −4/3 it follows thata(tH2O)(a(t0) = 1)=((trip − t0)(trip − tH2O))2(3)2so that (trip − tH2O) = 33Gy × 10−14.5 ≃ 103.5s ∼ 1 hour. [The value is sensitive to ω] Itis instructive to consider approach to the Rip a more general critical density ρc = ηρH2Oand to compute the time (trip − tη) such that ρΛ(tη) = ρc = ηρH2O. We then find, usinga(tη) = 1029η, that(trip − tη) = (trip − t0)10−14.5η−1 ≃ η−1hours (4)which is the required result. We shall see η > 1031 so the time in (4) is < 10−27s.To discuss the turnaround analytically we keep only the first and last terms, the onlysignificant ones, on the RHS of Eq.(1) which becomes for the special case ω = 4/3(ȧa)2= α1a− α2a2 (5)in whichα1 =8πG3(ρΛ)0 α2 =8πG3(ρΛ)20ρc(6)Writing a = z2 and z = (α1/α2)1/2sinθ givesdt =2√α2α1dθsin2θ=2√α2α1d(−cotθ) (7)Integration then gives for the scale factora(t) =(α1α2)sin2θ =ρc(ρΛ)0[11 +(tT−tC)2](8)where C = −(3/2πGρc)1/2. At turnaround t = tT , a(tT ) = [ρC/(ρλ)0] = (a(t))max. At thepresent time t = t0, a(t0) = 1 and sin2θ0 = [(ρΛ)0/ρC ] ≪ 1, increasing during subsequentexpansion to θT = π/4.A key ingredient in our model is that at turnaround t = tT our universe deflates dra-matically with efffective scale factor a(tT ) shrinking before contraction to â(tT ) = fa(tT )where f < 10−28. This jettisoning of almost all, a fraction (1 − f), of the accumulatedentropy is permitted by the exceptional causal structure of the universe. We shall seelater that the parameter η at turnaround lies in the range η = 1031 to η = 1087 whichimplies a dark energy density at turnaround (Planckian density of ρΛ ∼ 10104ρH2O canbe avoided) such that, according to the Big Rip analysis of [13, 14], all known, and yetunknown smaller, bound systems have become unbound and the constituents causally dis-connected. Recall that the density of a hydrogen atom is approximately ρH2O and we arereaching a dark energy density of from 31 to 87 orders of magnitude higher.According to these estimates, at t = tT the universe has already fragmented into anastronomical number (1/f 3) of causal patches, each of which independently contracts as aseparate universe leading to an infinite multiverse. The entropy at t = tT is thus divided3between these new contracting universes and our universe retains only a fraction f 3. Sinceour model universe has cycled an infinite number of times, the number of parallel universesis infinite.Friedmann equation for contraction phase. The contraction phase for our universe occursfor the period tT < t < τ . The scale factor for the contraction phase will be denoted byâ(t) while we use always the same linear time t subject to the periodicity t + τ ≡ t. Atthe turnaround we retain a fraction f 3 of the entropy with â(tT ) = fa(tT ) and for thecontraction phase the Friedmann equation is(˙̂a(t)â(t))2=8πG3[((ρ̂Λ)0â(t)3(ωΛ+1)+(ρ̂r)0â(t)4)−ρ̂total(t)2ρ̂c](9)where we have definedρ̂i(t) =(ρi)0f3(ωi+1)â(t)3(ωi+1)=(ρ̂i)0â(t)3(ωi+1)(10)but in contrast to Eq.(1) we have set ρ̂m = 0 because our hypothesis is that the causal patchretained in the model contains only dark energy and radiation but no matter including noblack holes. This is necessary because during a contracting phase dust or matter wouldclump, even more readily than during expansion, and inevitably interfere with cyclicity.Perhaps more importantly, presence of dust or matter would require that our universego in reverse through several phase transitions (recombination, QCD and electroweak toname a few) which would violate the second law of thermodynamics. We thus require thatour universe comes back empty! Any tiny entropy associated with radiation is constantduring adiabatic contraction.The contraction of our universe will proceed from one of the 1/f 3 causal patchesfollowing Eq.(9) until the radiation balances the brane tension at the bounce.Bounce. At the bounce, the contraction scale is given, using ρc = ηρH2O, from Eq. (1) asa(τ)4 =((ρr)0ηρH2O)(11)Now the model’s bounce at t = τ must be before the electroweak transition at tEW =10−10s when a(tEW ) = 10−15, and after the Planck scale when a(tP lanck) = 10−32 in orderto accommodate the well established weak transition and to avoid uncertainties associatedwith quantum gravity. With this in mind, here are three illustrative values (A, B, C) forthe bounce temperature TB:• (A) At a GUT scale TB = 1017GeV, a(tB) = 10−30.• (B) At an intermediate scale TB = 1010GeV, a(tB) = 10−23.• (C) At a weak scale TB = 103GeV, a(tB) = 10−16.4From Eq.(11) and Eq.(4) for these three cases one finds• (A) η = 1087 and (trip − tT ) = 10−87hr.• (A) η = 1059 and (trip − tT ) = 10−59hr.• (A) η = 1031 and (trip − tT ) = 10−31hr.Immediately after the bounce, we assume that an inflaton field is excited and there isconventional inflation with enhancement E = a(τ + δ)/â(τ). Successful inflation requiresE > 1028. Consistency requires therefore f < E−1 to allow for the entropy accrued duringexpansion after inflation. The fraction of entropy jettisoned from our universe at deflationis thus extremely close to one, being less than one and more than (1− 10−28)3.Entropy. Consider first the present epoch t = t0. The contributions of the radiation tothe entropy density s follows the relations =2π245g∗T3 (12)Photons contribute g∗ = 2he present CMB temperature is T = 2.73K ≡ 0.235meV ∼1.191(mm)−1. Substitution in Eq.(12) gives a present radiation entropy density sγ(t0) =1.48(mm)−3. Using a volume estimate V = (4π/3)R3 with R = 0Gly ≃ ×1029mm gives atotal radiation entropy Sγ ∼ 6.3 × 1087. Including neutrinos increase g∗ in Eq.(12) fromg∗ = 2 to g∗ = 3.36 = 2 + 6 × (7/8) × (4/11)4/3. This increases Sγ = 6.3 × 1087 toSγ+ν ∼ ×1088.This total entropy is interpretable as exp(1088) degrees of freedom, or in informationtheory [19] to a number I of qubits where 2I = eS so that I = S/(ln2 = 0.693) ∼ 1088.This is well below the holographic bound which is dictated by the area in terms of Planckunits 10−64mm2 which gives Sholog(t0) = 4π(1029mm)2/(10−32mm)2 ∼ 10123 about 1035times bigger. In [19] it is suggested that at least some of this difference may come fromsupermassive black holes. The entropy contribution from the baryons is smaller than Sγby some ten orders of magnitude, so like that of the dark matter, is negligible.What is the entropy of the dark energy? If it is perfectly homogeneous and non-interacting it has zero temperature and entropy. Finally, the 4th term in Eq.(1) corre-sponding to the brane term is neglible, as we have already estimated. The conclusion isthat at present Stotal(t0) ∼ 1088.Now consider the entropy approoaching turnaround at t = tT . We have estimatedthat a(tT ) = 1029η and representative values for η = ρc/ρH2O are 1031, 1059 and 1087.The temperature Tγ of the radiation scales as Tγ ∝ a(t)−1 so using the entropy densityof Eq.(12) a comoving 3-volume ∝ a(t)3 will contain the same total radiation entropySγ(tT ) = Sγ(t0) as at present; this is simply the usual adiabatic expansion. The expansionfrom t = 0 to tT is not purely adiabatic because irreversible processes take place. Thefirst is inflation which increases entropy by > 1084. There are phase transitions such as5the electroweak transition at tEW ∼ 100ps, the QCD phase transition at tQCD ∼ 100µs,and recombination at trec ∼ 1013s. Further irreversible processes occur during duringstellar evolution. Although the expansion of the radiation, the dominant contributor tothe entropy, is adiabatic, the entropy of matter increases in accord with the second lawof thermodynamics. In our model, the entropy of the matter increases between t = 0and tT ∼ 47Gy. Setting the entropy of the dark energy to zero and the radiation asadiabatic, the matter part represented by ρm will cause the entropy to rise from S(t = 0)to S(tT ) = S(t = 0)+∆S where ∆S causes the contradiction plaguing previous oscillatorymodel universes [1–3].Our main point is that in order for entropy to be cyclic, the entropy which was enhancedby a huge factor E3 > 1084 at inflation must be reduced dramatically at some point duringthe cycle so that S(t) = S(t + τ) becomes possible. Since it increases during expansionand contraction, the only logical possibility is the decrease at turnaround as accomplishedby our causal patch idea. The second law of thermodynamics continues to obtain forother causal patches, each with practically vanishing entropy at turnaround, but these arepermanently removed from our universe contracting instead into separate universes.For contraction tT < t < τ we are assuming the universe during contraction is empty ofmatter until the bounce so its entropy is vanishingly small. Immediately after the bounceinflation arises from an inflaton field, assumed to be excited. We find the counterpoiseof inflation at the bounce and deflation at turnaround an appealing aspect of the presentmodel.Conclusion. The standard cosmology based on a Big Bang augmented by an inflation-ary era is impressively consistent with the detailed data from WMAP3 [18] when darkenergy, most conservatively a cosmological constant, is included. Our objections to thisstandard model are more aesthetic than motivated directly by observations. The firstobjection is the nature of the initial singularity and the initial conditions. A second ob-jection, not of concern to all colleagues, is that the predicted fate of the universe is aninfinitely long expansion. We have outlined here a cyclic cosmology resting on phantomdark energy where these objections are ameliorated: the classical density and temperaturenever become infinite and future expansion is truncated. Also, our proposal of deflationnaturally leads to a multiverse picture, somewhat reminiscent of that predicted in eternalinflation, though here the proliferation of universes must be infinite and originates at theopposite end of a cyclic cosmology, at its maximum rather than at its minimum size.We publish our infinitely cyclic model mainly in the hope that it will stimulate a moredetailed and compelling formulation.AcknowledgementsWe thank Alex Vilenkin and Robert Wald for useful discussions. This work was sup-ported in part by the U.S. Department of Energy under Grant No. DE-FG02-06ER41418.6References[1] A. Friedmann, Z. Phys. 10, 377 (1922).[2] R.C. Tolman, Phys. Rev. 38, 1758 (1931).[3] R.C. Tolman, Relativity, Thermodynamics and Cosmology. Oxford University Press(1934).[4] J. Khoury, B.A. Ovrut, P.J. Steinhardt and N. Turok. Phys. Rev. D64, 123522(2001). hep-th/0103238.[5] P.J. Steinhardt and N. Turok, Science 296, 1436 (2002).[6] P.J. Steinhardt and N. Turok, Phys. Rev. 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Deflation at Turnaround for Oscillatory Cosmology

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