aa r X i v : . [ g r- q c ] D ec Holography constrains quantum bounce
D.H. Coule
Institute of Cosmology and Gravitation,Dennis Sciama Building, Burnaby Rd, University of Portsmouth,Portsmouth PO1 3FX, UK
Abstract
Recent work in quantum loop cosmology suggests the universe un-dergoes a bounce when evolving from a previously collapsing phase.However, with matter sources that obey the strong-energy conditioni.e. non-inflationary, the scenario appears to strongly violate the holo-graphic bound on entropy S ≤ A/ A now represents the cross-sectional area of the bounce.We also give a simple argument why any inflationary phase afterthe bounce is unlikely due to the prior dissipation of any kinetic dom-inated scalar field with large frequency m ∼ − over semi-infinitecollapsing time scales. Rather entropy increase should be expectedand any subsequent inflationary phase would have entailed a violationof the generalized second law before the bounce occurs.PACS numbers: 04.20, 98.80 Bp
1n some recent approaches to quantum gravity the usual singularity inbig bang cosmology is possibly replaced by a bounce due to the influence ofquantum gravitational effects. As an approximation to this phenomena thestandard Friedmann equation for a FRW model e.g.[1,2] H + ka = ρ (1)can become modified such that H + ka = ρ − ρ ρ c (2)where ρ c represents the critical energy scale, typically expected to be aroundthe Planck scale. This behaviour occurs in loop quantum gravity [3-6] al-though in that case the curvature dependence is actually slightly more in-volved, but for our purposes this simplified equation will first prove adequate.A similar description might be obtained with brane models with an extratime dimension [7] although this is probably observationally discounted [8].Note that a suggested [9] single negative tension brane is not suitable: itdiffers from eq.(2) by an overall minus sign on the R.H.S. since startingwith a positive 5-dimensional Planck mass the negative tension causes the4-dimensional Newton’s constant to become negative cf.[10].The holography bound [11] on entropy S ≤ A/
4, where A is the corre-sponding area measured in Planck units appears saturated for black holes butwith usual matter there is a stronger restriction S ≤ A / - see e.g.[12] forextensive reviews. By choosing a closed radiation dominated FRW universewith corresponding maximum size a max ∼ A / this stronger bound is justsaturated and a max ∼ S / r , where S r represents the corresponding entropyof the radiation present, ignoring some numerical factors that correspond tothe number of spin states cf.[13-15].By starting at the maximum size of a closed radiation dominated uni-verse we then know the amount of entropy initially present, By letting themodel collapse we can find the corresponding minimum radius a min and thecorresponding allowed entropy at the bounce S b . Since we know the amountof entropy, and assuming adiabatic behaviour so that the entropy remainsconstant, we can check whether the holography bound S ≤ A/ We use Planck units throughout and set numeric factors like 8 π/ k/a onthe LHS of eq.(2). Also since the matter behaviour is expected to display itsusual form we take ρ = α/a . Then a max ∼ α and assuming ρ c ≃
1, we find S b ≃ a min ∼ α / . Then since α / > α / the holography bound is stronglyviolated i.e. S r > S b . We typically have in mind starting with a largeclassical universe with α ∼ , so the starting entropy of the radiation is10 but only a value of 10 should be possible if the holographic bound isstill valid across the constricted bounce. This result apparently contradictsthose obtained previously in ref.[28]. But there they erroneously claim that α (their K o constant of eq.(2.4), (3.8) and (3.10) is some “integration con-stant without physical significance” but rather this constant determines themaximum size and hence entropy content of a closed FRW model.To correctly include curvature k = 1 in loop quantum cosmology actuallyinvolves a more complicated equation so that eq.(2) becomes [5,16] H = (cid:18) ρ − a (cid:19) − ρρ c + 1 ρ c a ! (3)For the case of a massless scalar field the corresponding values are a max ∼ p φ and a min ∼ p / φ [5]. The parameter p φ is here analogous to the previous α . The corresponding holography bound is now violated, but less severely,since now the corresponding values of entropy are p / φ > p / φ .A more careful analysis of eq.(3), together with other relevant equationspresented in ref.[5,16], confirms the results and shows that the apparent vio-lation becomes larger as the strong-energy condition boundary is approachedand that the bound could become satisfied for ultra-stiff equations of statei.e w > /
3, where for a perfect fluid p = wρ . Such equations of state havepreviously been used during the collapsing stage of the Ekpyrotic scenario[17] with certain possible advantages [18].The use of a closed model was not strictly relevant for this argument sinceone could start with any finite physical size and derive analogous quantities.The crucial point is that the universe is being constricted into a finite sizeduring the bounce regardless of the underlying geometry k . This in turnsimplifies the application of holography bound compared to various compli-cations that can occur in usual FRW models [12].3 Figure 1: Sketch of the universe “threading the needle”: bouncing at a sizebelow the holographic bound of area AThe bound might also be satisfied if the critical density ρ c is vastly de-creased, so the repulsive gravitational effect can intervene sooner, but this isprobably anyway inconsistent with other phenomena. There might also bescope to increase the bounce size a min by employing different quantizationprocedures that introduce further dependence on the lattice size employed.Although so-called polymer quantization appears to then actually reduce thebounce size [19]. It is worrying, though, that the procedures have alreadymeant to have agreed on the standard Bekenstein-Hawking black hole result S = A/ ρ c value, see e.g.[20].If black holes are also present, and during the collapsing phase theymight be expected to more easily form and congeal cf.[15], the bound willbe much harder to satisfy since the entropy can potentially increase towards S ∼ a max ∼ α ∼ : essentially the matter during the bounce is be-ing more compressed than in the usual black hole case, such densely en-tropic states have previously been dubbed monsters [21]. One can see thispotential problem by simply relating the mass of the observable universe ∼ kg to the Planck density ∼ kgm − so the universe could be appar-ently squashed into a volume 10 − m if one could work with Planck densitymatter: yet the Schwarzschild radius of a supermassive black hole is alone ∼ m so we never remotely deal with Planck energy densities with largeblack holes: the average density of a large black hole can be small!Another way to evade this entropy problem ( and this might be the causeof the discrepancy with ref.[28] ) would be to have only Planck sized quan-tities during the collapsing phase i.e. α ∼ p φ ∼ / ∼ − is preserved if the bound is to be satisfied -so only a limited knowledge of the collapsing phase would still be possiblecf.[23].Whether this “eye of the needle” constriction -see Fig.(1)- is too severeand the holography bound should instead be respected; or else other processesintervene to reduce the entropy and so violate the generalized second law,will have to be resolved before such a (monster?) quantum bounce can becountenanced.Let us consider further why the possibility of inflation after the bounce isdifficult to envision. The model is envisioned to display the following stagesof first scalar field ‘fast-roll’ kinetic, or strictly speaking the oscillatory be-haviour around the potential’s minimum (Osc) with effectively a p = 0 equa-tion of state [1,2], then potential V ( φ ) domination and finally matter pro-duction, where matter represents the eventual production of standard mattercomponents. The model is therefore claimed to have the following stages ofdevelopmentOsc → Potential → Osc → MatterFirstly there is an argument using the canonical measure [24] that sug-gests inflation, driven for example by a massive scalar field V ( φ ) = 1 / m φ ,is likely if applied at the bounce point [25]. It is known that applying suchan argument at Planck energy densities is generally conducive to obtain-ing inflationary behaviour [26]. But this argument is akin to the universesuddenly commencing its existence at the bounce itself. The canonical mea-sure is only valid for non-dissipative Hamiltonian systems so it can only beapplied in certain situations. The analysis should rather take into accountthe previous mostly classical evolution of the universe collapsing towards thebounce having started presumably at time t = −∞ . Incidentally in [24] theydivide out by a claimed irrelevant “gauge” a divergence related to an arbi-trary fixing of the initial fiducial cell. This requires further justification assetting various parameters just to agree with quantum gravity phenomenacan simply transfer the issue into why the Planck units take the values theydo. This “gauge” factor is instrumental in potentially solving the flatness5roblem when curvature k = 0 is present [26]: various matter componentsare affected differently by scaling the scale factor. By removing this factorthey remove any possibility of resolving the flatness problem in the modelsconsidered since it already exists in the pre-bouncing phase. For exampleone might wonder why a curvature dominated Milne model (k=-1) does notjust perpetually expand from time t = −∞ ?Here is where a difficulty with setting up the scalar field arises: if wesimply set the field at the minimum of the potential it will typically oscillate about the minimum with a high frequency m ∼ − and as the universecontracts the amplitude of the field φ will increase as typically ∼ a − / e.g.[1,2,27]. But now whatever the couplings are that typically reheat after anyinflationary phase, where again the field oscillates with frequency m , willinterfere and dissipate this high frequency oscillation into particle creationas the universe slowly contracts: so either the couplings are mysteriouslyabsent and the universe would always be dominated by the scalar field andany subsequent inflationary phase could not reheat by the usual oscillatingscalar field mechanism; or the reheating effectively occurs during the finalstages of collapse preventing any large and coherent scalar field being presentand no subsequent inflation caused by V ( φ ) would result. This problemhas been overlooked cf.[3] because “the reheating has simply been assumedto occur after rolling into the minimum but not when it already was inthe minimum” so introducing an asymmetry that is unwarranted. Theremight be convoluted ways of escaping this dilemma but at first sight is seemsthere will be entropy production as the universe collapses due to the highfrequency oscillations of φ together with the presence of any remotely non-zero couplings to other matter, essentially because the mass m has to set thescale of perturbations during the inflationary phase but is also responsible forhigh frequencies oscillations when H < m : ∼ − in Planck units is vastlyabove, for example, electroweak scales. This then suggests the universe willbecome simply dominated by standard non-inflationary matter as the bounceapproaches and so does not allow any inflationary low entropic conditions tonaturally occur.The scalar field development would therefore be more realistically of theform We ignore the further issue that one might expect the oscillations about the minimumto be out of phase beyond some initial coherent length scale , but this would entail using aninhomogeneous metric. Arbitrary small initial amplitudes for φ would contradict quantumuncertainty conditions. → Matter × → Potential → Osc → Matterwith a bottleneck developing before potential domination can occur whichwould apparently require violation of the generalized second law to proceedin the required manner. One naturally expects entropy to increase from leftto right in this development and the entropy S of a potential driven inflation-ary phase is comparatively small S ≃ /V ( φ ) [1,2]. In ref.[25] they obtaineda limit, on their variable F B , which corresponds to entropy S ∼ /F B ≤ so being very restrictive on the pre-bouncing phase. Even if matter was notformed, because of extremely weak coupling, it is known that climbing thepotential is unstable e.g.[28] and the scalar field will instead become kineticenergy dominated so the next most realistic scenario would beOsc → Kinetic energy → Osc → MatterIn neither case is it realistic to obtain that the potential would come todominate and so drive a subsequent stage of inflation.In conclusion, we have outlined problems for the bouncing scenario eitherwith just standard matter or when a scalar field is included to hopefully drivean extra inflationary phase. Either large initial entropy or growth during thesemi-eternal collapsing phase, due to quantum dissipative effects , will tendto cause violation of the holography bound during any possible bounce - soin apparent contradiction with the claims in [29]. Any subsequent usefulinflation caused by any generated homogeneous scalar field is not compatiblewith placing the field simply at the minimum of a potential with, howeverslight, couplings to standard matter.We finally note that there is some similarity with bouncing models occur-ring in Horava-Lifshitz gravity to those considered here-see eg.[30]. It wouldbe of interest to see if they also suffer from related concerns. Even if the effects are mainly occurring just seconds before the bounce one gener-ally needs reheating fairly rapidly, so the couplings cannot be excessively weak, to becompatible with nucleosynthesis constraints [1,2]. eferences
1. A.D. Linde, “Particle Physics and Inflationary cosmology” (HarwoodPress) 1990.2. E.W. Kolb and M.S. Turner, “The Early Universe” (Addison-Wesley:New York) 1990.3. P. Singh, K. Vandersloot and G.V. Vereshchagin, Phys. Rev. D 74(2006) p. 043510.4. P. Singh, K. Vandersloot and G.V. Vereshchagin, Phys. Rev. Lett. 96(2006) p.1413015. A. Ashtekar, T. Pawlowski, P. Singh and K. Vandersloot, Phys. Rev.D 75 (2007) p.024035.6. A. Ashtekar, A. Corichi and P. Singh, Phys. Rev. D 77 (2008) p.024046.7. Y. Shtanov and V. Sahni, Phys. Lett. B 557 (2003) p.1.M.G. Brown, K. Freese and W.H. Kinney, JCAP 3 (2008) p.002.8. G. Dvali, G. Gabadadze and G. Senjanovic, hep-ph/9910207. see also
I. Quiros, arXiv:0706.24009. L. Baum and P.H. Frampton, Phys. Rev. Lett. 98 (2007) p.071301.10. C. Barcelo and M. Visser, Phys. Lett. B 482 (2000) p.183.11. G.’t Hooft, gr-qc/9310026.L. Susskind, J. Math. Phys. 36 (1995) p.6377.12. D. Bigatti and L. Susskind, hep-th/0002044.R. Bousso, Rev. Mod. Phys. 74 (2002) p.825.13. J.D. Barrow, New. Astron. 4 (1999) p.333.14. J.D. Barrow and M.P. Dabrowski, Mon. Not. R. Astron. Soc. 275(1995) p.850.15. R. Durrer and J. Laukenmann, Class. Quant. Grav. 13 (1996) p.1069.86. L. Parisi, M. Bruni, R. Maartens and K. Vandersloot, Class. Quant.Grav. Grav. 24 (2007) p.6243.17. P.J. Steinhardt and N. Turok, Phys. Rev. D 65 (2002) p.126003.18. J.K. Erickson, D.H. Wesley, P.J. Steinhardt and N. Turok, Phys. Rev.D 69 (2004) p.063514.19. A. Corichi, T. Vukasinac and J.A. Zapata, Phys. Rev. D 76 (2007)p.0440163.20. A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, Phys. Rev. Lett. 80(1998) p.904.21. R.D. Sorkin, R.M. Wald and Z.J. Zhang, Gen. Rel. Grav. 13 (1981)p.1127.S.D.H. Hsu and D. Reeb, Phys. Lett. B 658 (2008) p.244. see alsosee also