I argue that, in the chaotic version of string cosmology proposed recently,
classical and quantum effects generate, at the time of exit to radiation, the
correct amount of entropy to saturate a Hubble (or holography) entropy bound
(HEB) and to identify, within our own Universe, the arrow of time. Demanding
that the HEB be fulfilled at all times forces a crucial "branch change" to
occur, and the so-called string phase to end at a critical value of the
effective Planck mass, in agreement with previous conjectures.Comment: Latex file, 8 pages, 1 Figur

Veneziano, G.

Physics Letters B

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arXiv

G. Veneziano

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Pre-bangian origin of our entropy and time arrow

I argue that, in the chaotic version of string cosmology proposed recently, classical and quantum effects generate, at the time of exit to radiation, the correct amount of entropy to saturate a Hubble (or holography) entropy bound (HEB) and to identify, within our own Universe, the arrow of time. Demanding that the HEB be fulfilled at all times forces a crucial "branch change" to occur, and the so-called string phase to end at a critical value of the effective Planck mass, in agreement with previous conjectures.I argue that, in the chaotic version of string cosmology proposed recently, classical and quantum effects generate, at the time of exit to radiation, the correct amount of entropy to saturate a Hubble (or holography) entropy bound (HEB) and to identify, within our own Universe, the arrow of time. Demanding that the HEB be fulfilled at all times forces a crucial "branch change" to occur, and the so-called string phase to end at a critical value of the effective Planck mass, in agreement with previous conjectures

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CERN-TH/99-32hep-th/9902126Pre-bangian origin of our entropy and time arrowG. VenezianoTheory Division, CERN, CH-1211 Geneva 23, SwitzerlandAbstractI argue that, in the chaotic version of string cosmology proposed recently, classical andquantum eects generate, at the time of exit to radiation, the correct amount of entropy tosaturate a Hubble (or holography) entropy bound (HEB) and to identify, within our ownUniverse, the arrow of time. Demanding that the HEB be fullled at all times forces acrucial \branch change" to occur, and the so-called string phase to end at a critical value ofthe eective Planck mass, in agreement with previous conjectures.CERN-TH/99-32February 1999The origin of the present entropy of our Universe, S0, is one of the deepest cosmologicalmysteries. The 2.7 K cosmic microwave background (CMB), if it indeed lls our observableUniverse uniformly, contributes a gigantic 1090 to S0. However, as repeatedly emphasizedby many people, most notably by Roger Penrose [1], such an amount falls very short ofwhat entropy could have been expected to be, even if we go back to the Planckian era, i.e.to t = tP  10−43 s after the big bang. Since entropy can only grow, the entropy of ourUniverse at t = tP , SP , must be smaller than S0; yet, on the basis of the energy content andof the size of the Universe RP at t  tP  lP /c, we might have expected 1SP  EP RP /ch  ρP R4P  (RP/lP )4  10120 . (1)The fact that the entropy of our Universe must have been at least 30 orders of magnitudesmaller than the value in (1) would nicely \explain" our arrow of time, by identifying thebeginning of the Universe with this state of incredibly small entropy near the Planck time [1].In order to solve the problem, Penrose [1] invokes, without much justication, a new \Weyl-curvature hypothesis". The expected value given in Eq. (1) coincides with the so-calledBekenstein entropy bound (BEB) [2], which states that, for any physical system of energyE and physical size R, entropy cannot exceed SBB = ER/ch. This bound is saturated by ablack hole of mass E and size equal to its Schwarzschild radius R = GE. If the newly bornUniverse were a single black hole its Schwarzschild radius would have been much larger thanRP , and an even higher entropy, O(10180), would have resulted. What could have made theinitial entropy much smaller than SBB is instead the possibility that the Universe, right afterthe big bang, was already in a very ordered, homogeneous state. But this is just restating thepuzzle in terms of the usual homogeneity problem of standard (non-inflationary) cosmology[3].The way the two problems are related can be made explicit by introducing a strongerbound on entropy, which, unlike Bekenstein’s general bound, should apply to the specialcase of (fairly) homogeneous cosmological situations. We shall call it the \Hubble entropybound" and formulate it as follows: Consider a suciently homogeneous Universe in whicha (local) Hubble expansion (or contraction) rate can be dened, in the sinchronous gauge,as:H  1/6 ∂t(log g) , g  det (gij) , (2)with H varying little (percentage-wise) over distances O(H−1). In this case H−1, the so-called Hubble radius, is known to correspond to the scale of causal connection, i.e. to thescale within which microphysics can act. In such a context it is hard to imagine that a blackhole larger than H−1 can form, since, otherwise, dierent parts of its horizon would be unableto hold together. Thus, the largest entropy we may conceive is the one corresponding tohaving just one black hole per Hubble volume H−3. Using the Bekenstein{Hawking formula1Throughout this paper we will stress functional dependences while ignoring numerical factors.1for the entropy of a black hole leads to our proposal of a \Hubble entropy bound" (HEB):S < SHB  l−2P∑iH−2i  nHSH , (3)where the sum runs over each Hubble-size region. The last estimate in (3) assumes a fairlyconstant H throughout space, and denes nH as the number of Hubble-size regions, eachone carrying maximal entropy SH = l−2P H−2. This bound appears to be related, at least insome cases, to the one recently proposed by Fischler and Susskind [4] on the basis of theso-called holography principle. We may thus take SHB to stand for Hubble or Holographyentropy bound according to taste. If one applies the HEB to the initial Universe, one nds[4] that, unlike the BEB, it is practically saturated. It is widely fullled thereafter [4].This letter aims at explaining why saturation of the HEB naturally takes place at the bigbang in the context of pre-big bang (PBB) cosmology [5]. Before proceeding, we note that,in ordinary inflation, the entropy problem is solved [3] by invoking a non-adiabatic reheatingprocess occurring after inflation. Since inflation has already made the Universe homogeneous,after thermal equilibrium is reached entropy is given by its standard thermodynamic relationto temperature (here the reheating temperature) asSRH  T 3RHR3RH  S0 . (4)One thus naturally obtains the correct value. However, unlike what will be shown to be thecase in the PBB scenario, SRH fails to saturate the HEB since:SHB(t = trh) = l−2P HRHR3RH = H−1RHT4RHR3RH = (TRH/HRH) SRH  SRH . (5)In order to discuss various forms of entropy in the PBB scenario, let us recall some basicideas, which have emerged from recent studies of the latter (see [6] for a review). It nowlooks quite certain that generic {though suciently weak{ initial conditions lead to a formof stochastic PBB, which, in the Einstein-frame metric, can be seen as a sort of chaoticgravitational collapse [7, 8]. Black holes of dierent sizes form but, for an observer insideeach horizon measuring distances with a stringy meter 2, this is experienced as a pre-big banginflationary cosmology in which the t = 0 (hopefully fake) big bang singularity is identied[8] with the (hopefully equally fake) black hole singularity at r = 0.We are thus led to identifying our observable Universe as what became of a portion ofspace that was originally inside a large enough black hole. In general, if we want to achievea very flat and homogeneous Universe, we should better identify our present Universe withjust a tiny piece of the collapsing/inflating region. For the purpose of this note, however,this would only complicate the equations without adding new physical information. Thisis why, hereafter, we shall identify our present Universe with the whole interior of a singleinitial black hole.2Note that, while we shall work in the string frame throughout, the same results would also follow in theEinstein frame.2SBBSBBSBBSHBSHBSHBSqfSqfti ts(HR)r(HR)ttr = tex teq t0 tS0ScollStot = Scoll+SqfScoll101201090(Hin p,in)-2 -2Figure 1: At the beginning of the DDI era (t = ti) the entropy of the just-formed black hole,Scoll, coincides with both the BEB, SBB, and the HEB, SHB, while the entropy in quantumfluctuations, Sqf , is completely negligible. At the beginning of the string phase, t = ts, bothScoll and SHB still have their common initial value. SBB and Sqf have grown considerably,but the latter is still negligible if the string coupling is still small at t = ts. During thestring phase, Sqf catches up with Scoll rst, and with SHB later, i.e. when the energy in thequantum fluctuations becomes critical and exit to radiation is expected (t = tr). Finally,during the radiation and matter-dominated phases, SHB grows towards SBB, while our ownentropy Stot lags far behind and increases only slowly as the result of dissipative phenomenaand growth of inhomogeneities.It is helpful to follow the evolution of various entropies with the help of Fig.1. At timet = ti, corresponding to the rst appearance of a horizon, we can use the Bekenstein{Hawkingformula to argue thatScoll  (Rin/lP,in)2  (HinlP,in)−2 = SBB = SHB , (6)where we have used the fact [8] that the initial size of the black-hole horizon determinesalso the initial value of the Hubble parameter. Thus, at the onset of collapse/inflation,there is no hierarchy between the two bounds and entropy is as large as allowed by them.Furthermore, since the collapsing region is large in string (and a fortiori in Planck) units,eq. (6) corresponds to a large number. Incidentally, this number is also close to the numberof quanta needed for the collapse to occur [8]. We have also assumed the initial quantumstate to be the ground state. Because of the small initial coupling and curvature, quantumfluctuations around it are very small [9], initially, and contribute a negligible amount Sqf tothe total entropy.3After a short transient phase, dilaton-driven inflation (DDI) should follow and last untilts, the time at which a string-scale curvature O(Ms) is reached. We expect the process notto generate further entropy (unless more energy flows into the black hole, but this wouldonly increase its total comoving volume), but what happens to the two bounds? This isthe crucial observation: while the HEB also stays constant, the BEB grows, causing a largediscrepancy between the two at the end of the DDI phase. In order to show this, let us recallone of the equations of string cosmology [5], the conservation law:∂t(e−φpgH)= ∂t((pgH3) (e−φH−2))= ∂t (nHSH) = 0 . (7)Comparing with (3), we recognize that (7) simply expresses the time independence of theHEB during the DDI phase. While at the beginning of the DDI phase nH = 1, and the wholeentropy is in a single Hubble volume, as DDI proceeds the same total amount of entropybecomes equally shared among very many Hubble volumes until, eventually, each one ofthem contributes a relatively small number. By contrast, it is easy to see that the BEB isincreasing fast during the DDI phase since, using (7),SBB MR  ρR4  H2e−φpgR = const. (HR) , (8)and both R and H grow during DDI. Also, having assumed that the string coupling is stillsmall at the end of DDI, we can easily argue that the entropy in quantum fluctuationsremains at a negligible level during that phase.Something interesting happens if we now consider the string phase 3, characterized by aconstant H and _φ. It is easy to nd that, if _φ > 3H , the HEB starts to decrease while for_φ < 3H it increases. Clearly, the rst alternative leads to a contradiction with the HEB,since Scoll cannot decrease. We are thus led to the amusing result that the HEB demands_φ  _φ−3H  0 during the string phase as opposed to the _φ > 0 condition that characterizesthe DDI phase. Thus, the HEB implies a \branch change" occurring between the DDI andthe string phase, a well known necessary condition for achieving a graceful exit [11]. Thecondition _φ < 0 for the string phase also follows from (apparently independent) argumentsbased on the study of late-time attractors [12, 13].When will the nal exit to the FRW phase occur? It has been assumed [14] that it doeswhen the energy in the quantum fluctuations becomes critical, i.e. whenρqf  Neff H4max = e−φexitM2s H2max , (9)where Neff is the eective number of particle species produced. Taking Hmax  Ms, xesthe value of the dilaton at exit, eφexit  1/Neff . Using known results on entropy productiondue to the cosmological squeezing of vacuum fluctuations [15], we nd:Sqf(ex)  Neff H3maxV  e−φexitM3s V (l2s/l2P)exitV l−3s  SHB(ex) , (10)3We concentrate here on the standard PBB scenario [5] and not on its variant [10] in which the DDIphase flows directly into a low-energy M-theory phase.4i.e. saturation of the HEB by Sqf . Unless exit occurs at this point, the HEB will be violatedat later times.We thus arrive, generically, at the situation shown in Fig.1. At t = tex  tr, the entropyin the quantum fluctuations has catched up with (and possibly overcome) that of the classicalcollapse and has become equal to the HEB, SHB  (HR)3  1090. By then, SBB is a factorHR larger, which is precisely the factor 1030 that we are running after. From there on,the story is simple: our entropy remains, to date, roughly constant and around 1090, whileSHB keeps increasing {with somewhat dierent rates{ during the radiation and the matter-dominated epochs. SBB always remains a factor HR above SHB, but this factor, originallyhuge, shrinks to unity today, by denition.In conclusion, the entropy and arrow-of-time problems are neatly solved, in PBB cosmol-ogy, by the identication of our observable Universe with (part of) the interior of an originalblack hole. As such, its initial entropy saturates both the HEB and the BEB and is largebecause of the assumed large size (in string or Planck units) of the initial black hole. Fromthere on, there is a natural mechanism to provide saturation of the HEB at the beginningof the radiation-dominated phase, i.e. when the BEB lies some thirty orders of magnitudehigher. This is precisely what is needed to account for the initial entropy of our Universe,and to unambiguously identify its time arrow.We do not wish to conceal the fact that our choice of the initial size of the collaps-ing/inflating region can be objected to, along the lines of Refs. [16], as representing a hugeamount of ne-tuning. Our answer to this objection has been expressed elsewhere [7]: theclassical collapse/inflation process is a scale-free problem in General Relativity; as such, itshould lead to a flattish distribution of horizon sizes, extending from the string length tovery large scales, including those appropriate for giving birth to our Universe. No otherdimensionless ratio is tuned to a particularly large or small value as evidentiated in Fig.1by the three upper curves all originating from the same point at t = ti. Finally, we wishto stress again that the entropy considerations discussed in this note appear to provide newgeneral arguments supporting previous conjectures on the way pre-big bang inflation shouldmake a graceful exit into standard, post-big bang FRW cosmology.Useful discussions with M. Bowick, R. Brustein, M. Gasperini, A. Ghosh, F. Larsen, R.Madden and E. Martinec are gratefully acknowledged.Note added: After completion of this work I became aware of a very recent paper [17]which reaches similar conclusions on the role of H−1 in cosmological entropy bounds.5References[1] See, for instance, R. Penrose, The Emperor’s new mind (Oxford University Press, NewYork, 1989), Chapter 7.[2] J.D. Bekenstein, Phys. Rev. D23 (1981) 287; D49 (1994) 1912, and references therein.[3] E.W. Kolb and M.S. Turner, The early Universe (Addison-Wesley, Redwood City, CA,1990); A.D. Linde, Particle physics and inflationary cosmology (Harwood, New York,1990).[4] W. Fischler and L. Susskind, Holography and cosmology, hep-th/9806039; see alsoD. Bak and S.-J. Rey, Holographic principle and string cosmology, hep-th/9811008;A. K. Biswas, J. Maharana and R.K. Pradhan, The holography principle and pre-bigbang cosmology, hep-th/9811051; S.K. Rama and T. Sarkar, Holographic principle duringinflation and a lower bound on density fluctuations, hep-th/9812043.[5] G. Veneziano, Phys. Lett. B265 (1991) 287; K.A. Meissner and G. Veneziano, Phys.Lett. . B267 (1991) 33; Mod. Phys. Lett. A6 (1991) 3397; M. Gasperini and G.Veneziano, Astropart. Phys. 1 (1993) 317; Mod. Phys. Lett. A8 (1993) 3701; Phys.Rev. D50 (1994) 2519. An updated collection of papers on the pre-big bang scenariois available at http://www.to.infn.it/~gasperin/.[6] G. Veneziano, Inflating, warming up, and probing the pre-bangian Universe, CERN-TH/99-22, hep-th/9902097.[7] G. Veneziano, Phys. Lett. B406 (1997) 297; A. Buonanno, K.A. Meissner,C. Ungarelli and G. Veneziano, Phys. Rev. D57 (1998) 2543, and references therein.[8] A. Buonanno, T. Damour and G. Veneziano, Pre-big bang bubbles from the gravitationalinstability of generic string vacua, hep-th/9806230.[9] A. Ghosh, G. Pollifrone and G. Veneziano, Phys. Lett. B440 (1998) 20.[10] M. Maggiore and A. Riotto, D-branes and cosmology, hep-th/9811089;see also T. Banks, W. Fishler and L. Motl, Duality versus singularities, hep-th/9811194.[11] R. Brustein and G. Veneziano, Phys. Lett. B329 (1994) 429; N. Kaloper, R. Maddenand K.A. Olive, Nucl. Phys.B452 (1995) 677, Phys. Lett. B371 (1996) 34; R. Easther,K. Maeda and D. Wands, Phys. Rev. D53 (1996) 4247.[12] M. Gasperini, M. Maggiore and G. Veneziano, Nucl. Phys. B494 (1997) 315.[13] R. Brustein and R. Madden, Phys. Lett. B410 (1997) 110, Phys. Rev. D57 (1998) 712.6[14] G. Veneziano, in Eective theories and fundamental interactions, Erice, 1996, A. Zichichied., (World Scientic, Singapore, 1997) p. 300; A. Buonanno, K.A. Meissner, C. Un-garelli and G. Veneziano, JHEP01, 004 (1998).[15] M. Gasperini and M. Giovannini, Phys. Lett. B301 (1993) 334, Class. Quant. Grav. 10(1993) L133; R. Brandenberger, V. Mukhanov and T. Prokopec, Phys. Rev. Lett. 69(1992) 3606, Phys. Rev. D48 (1993) 2443.[16] M. Turner and E. Weinberg, Phys. Rev. D56 (1997) 4604; N. Kaloper, A. Linde and R.Bousso, Phys. Rev. D59 (1999) 043508.[17] R. Easther and D.A. Lowe, Holography, cosmology and the second law of thermodynam-ics, hep-th/9902088.7

Pre-bangian origin of our entropy and time arrow

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