aa r X i v : . [ h e p - t h ] J u l Emergence of a Cyclic Universe from the Hagedorn Soup
Tirthabir Biswas e-mail: [email protected] of Physics,Institute for Gravitation and the Cosmos,The Pennsylvania State University,104 Davey Lab, University Park, PA,16802, U.S.A (Dated: April 21, 2019)One of the challenges of constructing a successful cyclic universe scenario is to be able to in-corporate the second law of thermodynamics which typically leads to Tolman’s problem of evershrinking cycles. In this paper we construct a non-singular toy model where as the cycles shrinkin the past they also spend more and more time in the entropy conserving Hagedorn phase. Thusin such a scenario the entropy asymptotes to a finite non-zero constant in the infinite past. Theuniverse “emerges” from a small (string size) geodesically complete quasi-periodic space-time. Thisparadigm also naturally addresses some of the classic puzzles of Big Bang cosmology, such as thelargeness, horizon and flatness problems.
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Cyclic models of universe offer a promising resolution to the Big Bang singularity problem [1, 2, 3, 4, 5].The hope is that rather than having to deal with a “beginning of time” when the universe starts from a singularity,time can be made “eternal” in either direction (past and future). However, there are three main reasons why it stillremains an extreamely challenging task to construct a theoretically consistent and phenomenologically viable model: (i)
In General Relativity cyclic universes involve violation of various energy conditions at the “bounce” [6] wherecontraction gives way to expansion and typically this requires invoking problematic matter sources, such as ghosts.( ii)
We are confronted with the second law of thermodynamics according to which the total entropy in the universecan only increase monotonically. As first pointed out by Tolman [1], this immediately tells us that the evolution can atmost be “quasi-periodic” where the length of the cycles monotonically increase with the increase in entropy from cycleto cycle. Moreover, the problem of the “beginning of time” comes back to haunt us, as the cycles become vanishinglysmall at a finite proper time in the past [1]. (iii)
One of the biggest successes of inflationary cosmology is, that itnot only explains why the universe is so large, flat and homogeneous today, but also provides a causal mechanism toproduce small inhomogeneities which we observe in CMB and LSS. Thus any seriously competing alternative to thestandard inflationary paradigm should be able to reproduce these basic successes of inflation.The main purpose of this paper is to address in a theoretically consistent (ghost and singularity free) framework (i) ,Tolman’s problem of ever-shrinking cycles ( ii ) by taking recourse to a crucial feature of string theory, the existenceof the Hagedorn phase where all the different string states (massless and massive) are in thermal equilibrium andtherefore entropy is conserved. This allows us to construct an “emerging cyclic universe” where as one goes back incycles (time) and the cycles become shorter and on an average hotter, it spends more and more time in the Hagedornphase where no entropy is produced. Thus the universe asymptotes to a constant entropy state with almost-periodiccontractions and expansions. One of the key difference between the approaches of this paper to previous ones is thatwe incorporate non-singular bounces where the Hubble rate remains bounded. In a singular big crunch/bang scenariothe interaction rates which can potentially maintain thermal equilibrium among the different species, cannot keepup with the diverging Hubble rate. Thus most previous literature assumes that such transitions will produce largeamounts of entropy which eventually leads to Tolman’s problem.In the model we propose, entropy is produced afterwards when the different species in the Hagedorn phase fall out ofthermal equilibrium below some critical temperature, say T p , and energy starts to flow from hotter to colder species.Consequently, we find a class of solutions where the length of the cycles increase monotonically with time. Eachof these cycles spend some amount of time in both the Hagedorn and non-Hagedornic phase, and as a result someamount of entropy is always produced. However, as one goes back in the past the universe spends less and less timein the entropy producing non-Hagedornic phase, the entropy produced in a given cycle therefore goes to zero, and theuniverse asymptotes towards a periodic evolution.Apart from ( i ) and ( ii ), this new cosmological scenario naturally explains the flatness, largeness and horizon problems.The universe can “start” out small ( ∼ string length) with a low entropy and with curvature density around the stringscale and yet through the course of entropy production in the infinite cycles in the past, the evolution eventuallyproduces cycles with a large entropy where curvature remains negligible for a long time. Through the course ofthese infinite number of cycles there is also obviously enough time to establish causal connection. The only a priori assumption that we have to make is that of spatial homogeneity. Thus an important open question is whether the“initial cycles” can drive one to homogeneity? Finally, we do not attempt to address ( iii ) in this paper, but we pointto mechanisms that have been invoked [3, 7] to produce nearly scale-invariant spectrum without inflation, and thatencouraging progress have been made to include these in the framework of non-singular bouncing cosmologies [8, 9]. Hagedorn Physics, Casimir Energy and Bounces:
We start by considering the Hagedorn phase of matter on a space-time which includes our observed three dimensional universe, along with appropriately compactified extra dimensions.For us, the most crucial property of the Hagedorn phase is, that in this phase all the different string states are inthermal equilibrium with each other and the entropy remains conserved. In our analysis we will only care about the“universal” leading order behaviour of entropy [10, 11] S = β H E + O ( V T H /E ) ⇒ ρ hag = T H S/V (1)where S and E are the entropy and energy of the Hagedorn phase, V is the spatial volume of the universe, and T H is the Hagedorn temperature.As mentioned before, finding bounces with consistent physics is challenging, but the existence of negative Casimirenergies offers an interesting possibility. As a simple illustrative example, in this paper we will consider Casimir2nergies coming from minimally coupled free scalar fields in a closed universe setting. It was shown in [12] that inthis case the Casimir energy ∼ − a − ( t ). The Hubble equation therefore reads as H = T H M p (cid:20) Sa − Ω c a − Ω k a (cid:21) (2)where we have defined a dimensionless “scale factor” a ≡ T H V / and Ω c , Ω k are dimensionless ∼ O (1) constantsassociated with the Casimir energy [12], and spatial curvature respectively.Now, if both the bounce and the turnaround occurs within the Hagedorn phase then we end up having an “eternallyperiodic cyclic universe” given as a function of the conformal time, τ : a ( τ ) = S − √ S − c Ω k cos ντ k with ν ≡ r Ω k T H M p (3)Phenomenologically this is rather uninteresting, as the temperature cannot fall below T p which is expected to beclose to the string scale. We are therefore interested in the quasi-periodic evolution where cycles can grow with theproduction of entropy, once out of the Hagedorn phase. Ordinary matter, Curvature and Turnarounds:
From simple thermodynamic considerations it follows that once thedifferent species fall out of equilibrium and there is exchange of energy (from the hotter to the colder species), entropyin generated. We want to incorporate such entropy production in a toy model setting. For this purpose we willconsider a two species model with radiation, ρ r , and non-relativistic matter, ρ m , and assume that the entire string gascan be clubbed into two of these categories near the transition from Hagedorn phase to radiation. Thus the picture is,that both these species are in thermal equilibrium in the Hagedorn phase, but after the transition temperature, thetwo species fall out of equilibrium and energy starts to flow from the dust-like matter to radiative degrees of freedom.Before we model this energy exchange, it is instructive to look at the usual solution of a closed universe with ordinarymatter and radiation. The Hubble equation and it’s solution is of the same form as (2) and (3) respectively with thesubstitution, S → Ω m and Ω c → Ω c − Ω r .To complete the story one needs to relate the quantities in the two halves of the cycle. To keep the calculations simple,we are going to assume that one can ignore Ω c as compared to Ω r which is always possible provided Ω c ≪ Ω k [13].Now, unfortunately we do not have access to the complicated transitionary dynamics from the Hagedorn to theradiation dominated epoch as the corrections in (1) are not under control, but by our explicit construction the totalentropy should be conserved till the transition point, leading to S = b r Ω / r a p + b m Ω m a p (4)where the constants b r , b m depend on the number of radiative and non-relativistic species one has, and their properties.We also know that at the point of phase transition the matter and radiation were still just in thermal equilibrium, i.e. had the same temperature T m = T H b m = T H / r b r a = T r (5)Finally, we make a phenomenological ansatz about the relative energy densities of matter and radiation at thetransition epoch µ ≡ ρ m /ρ r = Ω m a p / Ω r (6)Ideally, this ratio should be calculable if we understood the transition from Hagedorn to radiation+matter phase, butphenomenological considerations (matter-radiation equality ∼ ev ) already suggests µ to be very small ∼ − .We can now determine all the dynamical quantities involved in the second half in terms of the entropy of the Hagedornphase (which varies from cycle to cycle), and phenomenological constants, b m , b r and µ [13]. Under the simplifying In string theory massless moduli scalar fields are ubiquitous, but typically they also couple to the Ricci scalar. Such couplings maymodify the form of the casimir energy. Also, these scalars can be dynamic in the early universe which will modify the evolution, butthese questions can hopefully be answered through a more elaborate study in the future. b m ∼ µ, Ω c / Ω k ≪ r = (cid:18) Sb r (cid:19) / , Ω m = 34 µS and a p = 4 S / b / r (7) Entropy Production:
Having understood the behaviour of our universe in a “non-interacting” entropy-preservingsetting, let us now try to understand how energy exchange between ordinary matter and radiation effects the dynamics.The energy exchange process can be captured via phenomenological equations of the form [4]˙ ρ r + 4 Hρ r = T H s and ˙ ρ m + 3 Hρ m = − T H s (8)which preserves conservation of stress-energy tensor, but which “typically” breaks the time reversal symmetry pro-viding an “arrow of time”. s characterizes the energy flow and one expects it to depend on the densities and the scalefactor. It is easy to compute the net entropy production in such models:˙ S = ˙ S r + ˙ S m = a s b r T H ρ / r − b m ! = a s (cid:18) T H T r − T H T m (cid:19) (9)We will assume s >
0. Consistency with 2 nd law of thermodynamics then implies that the quantity within bracketsmust be positive. This is nothing but the condition that the temperature of the non-relativistic gas be greater thanthe radiative gas, so that energy flows from the hotter non-relativistic species to colder radiation in accordance with1 st law of thermodynamics. Since in our picture the two species have the same temperature T p , at the transition point,where after T r decreases, while T m stays fixed, (9) is consistent with both the 1 st and 2 nd law of thermodynamics.Let us now see why we can realize the emergent cyclic universe. It is sufficient to look at small cycles which (we claim)asymptotically approaches the periodic evolution (3). In these cycles matter density is always negligible as comparedto radiation, so that the expression for the turnaround point reduces to a max ≈ p Ω r / Ω k ≈ S / / ( p Ω k b / r ) (10)We notice that a max ∼ S / , while a p ∼ S / . In other words as we go back in the past and the entropy decreases, a p catches up with a max , and the universe spends less and less time in this entropy-generating phase. In turn, less andless entropy is produced, and in fact, the entropy approaches a constant given by a max = a p ⇒ lim n →−∞ S n = S cr ≡ (4 / Ω / k b r (11)It is not difficult to estimate the entropy growth for “small” cycles defined by ντ p ≪
1, where τ p is the conformaltime corresponding to the transition point a p = a ( τ p ). Under the technically simplifying assumption s ≪ M p /T H ,one finds by solving (9), the increase in entropy in a given cycle to be given by [13]∆ S = 2 a max s cr ν τ p a p ≈ dδSdn where s cr ≡ s ( a p ( S cr )) (12)where “ n ” labels the number of the cycle and δS ≡ S − S cr . Since, a max , a p and τ p are all known functions of theentropy, this differential equation can be solved. As n → −∞ we find the leading order result: S ≈ S cr (cid:20) C n (cid:21) where C = 8 (cid:18) (cid:19) / s cr νb r (13)We now explicitly see that, S → S cr at the past infinity.We note in passing that the divergence of entropy at n = 0 is only an artifact of approximating s by a constant inderiving (12), which breaks down once the cycles become large. For large cycles the growth of entropy with n woulddepend on the modeling of s (see [13] for an illustrative example), but on physical grounds we expect S n to diverge(if at all) only as n → ∞ . Conclusions:
We have proposed a new non-singular cyclic cosmology which addresses Tolman’s problem of evershrinking cycles as well as some of the standard puzzles of Big Bang cosmology, such as the horizon, entropy, largeness,and flatness problems. The success of the scenario largely rests on the crucial assumption of the existence of a thermal4tringy Hagedorn phase near the bounce. One expects this to hold if the string interaction rate Γ str ≫ H max . Basedon naive estimates of Γ str [14] one can check that this would be true as long as e ψ > T H / M p [13], where ψ isthe dimensionally reduced dilaton [14]. This seems a rather reasonable bound to satisfy, however one would like toperform a more careful investigation in the future.Likewise, to make this scenario more concrete we need to include moduli dynamics/stabilization (which should alsoexplain why the observed three dimensions became large as compared to the extra dimensions, see [15] for a pos-sible relevant mechanism), sources of stringy Casimir energies and mechanisms for generating near-scale invariantperturbations, but at the least the scenario looks promising. [1] R.C.Tolman, Relativity,Thermodynamics andCosmology,(OxfordU.Press,Clarendon Press,1934); “On the Problem of theEntropy of the Universe as a Whole”, Phys. Rev. , 1639 (1931)[2] H. Bondi and T. Gold, Mon. Not. Roy. Astron. Soc. , 252 (1948). F. Hoyle, MNRAS, , 372 (1948); F. Hoyle andJ. V. Narlikar, Proc. Roy. Soc. A 278 , 465 (1964); J. V. Narlikar, G. Burbidge and R. G. Vishwakarma, arXiv:0801.2965[astro-ph].[3] J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok, “The ekpyrotic universe: Colliding branes and the origin of thehot big bang,” Phys. Rev. D , 123522 (2001) [arXiv:hep-th/0103239]; P. J. Steinhardt and N. Turok, Phys. Rev. D ,126003 (2002) [arXiv:hep-th/0111098]; Science , 1436 (2002).[4] J. D. Barrow, D. Kimberly and J. Magueijo, Class. Quant. Grav. , 4289 (2004) [arXiv:astro-ph/0406369]; T. Clifton andJ. D. Barrow, Phys. Rev. D , 043515 (2007) [arXiv:gr-qc/0701070].[5] M. Novello and S. E. P. Bergliaffa, arXiv:0802.1634 [astro-ph].[6] C. Molina-Paris and M. Visser, Phys. Lett. B , 90 (1999) [arXiv:gr-qc/9810023].[7] A. Nayeri, R. H. Brandenberger and C. Vafa, Phys. Rev. Lett. , 021302 (2006) [arXiv:hep-th/0511140];[8] S. Alexander, T. Biswas and R. H. Brandenberger, arXiv:0707.4679 [hep-th].[9] T. Biswas, A. Mazumdar and W. Siegel, JCAP , 009 (2006) [arXiv:hep-th/0508194]; T. Biswas, R. Brandenberger,A. Mazumdar and W. Siegel, arXiv:hep-th/0610274.[10] N. Deo, S. Jain, O. Narayan and C. I. Tan, Phys. Rev. D , 3641 (1992); N. Deo, S. Jain and C. I. Tan, Phys. Rev. D , 2626 (1989); N. Deo, S. Jain and C. I. Tan, Phys. Lett. B , 125 (1989).[11] A. A. Tseytlin and C. Vafa, Nucl. Phys. B , 443 (1992) [arXiv:hep-th/9109048].[12] C. A. R. Herdeiro and M. Sampaio, Class. Quant. Grav. , 473 (2006) [arXiv:hep-th/0510052].[13] T. Biswas and S. Alexander, arXiv:0812.3182 [hep-th].[14] R. Danos, A. R. Frey and A. Mazumdar, Phys. Rev. D , 106010 (2004) [arXiv:hep-th/0409162].[15] R. H. Brandenberger and C. Vafa, Nucl. Phys. B , 391 (1989)., 391 (1989).