Holographic Foam Cosmology: From the Late to the Early Universe
aa r X i v : . [ g r- q c ] F e b Holographic Foam Cosmology: From the Late to theEarly Universe
Y. Jack Ng Institute of Field Physics, Department of Physics and Astronomy,University of North Carolina, Chapel Hill, NC 27599, U.S.A.
Abstract
Quantum fluctuations endow spacetime with a foamy texture. The degree offoaminess is dictated by blackhole physics to be of the holographic type. Appliedto cosmology, the holographic foam model predicts the existence of dark energy withcritical energy density in the current (late) universe, the quanta of which obey infi-nite statistics. Furthermore we use the deep similarities between turbulence and thespacetime foam phase of strong quantum gravity to argue that the early universe wasin a turbulent regime when it underwent a brief cosmic inflation with a “graceful”transition to a laminar regime. In this scenario, both the late and the early cosmicaccelerations have their origins in spacetime foam.
Keywords: spacetime foam, holography, dark energy, cosmic inflation, infinite statis-tics, turbulence [email protected] Introduction
There are two cosmic accelerations that we are aware of: a brief inflationary acceleration [1]in the early universe and the present (“late” universe”) acceleration [2] that is attributedto dark energy. Normally they are treated independently and separately; but there arealso some works [3, 4] that consider both regimes of accelerated expansions. We think itis conceptually necessary and aesthetically pleasing to trace both cosmic accelerations toa common cause in fundamental theory. Following Wheeler [5], we believe that space iscomposed of an ever-changing geometry and topology called spacetime foam and that thefoaminess is due to quantum fluctuations of spacetime. We argue for a scenario in whichspacetime foam is the origin of both cosmic accelerations.The outline of this Letter is as follows. We begin with a brief review of the holographicspacetime foam model. In section 2 we use the quantum uncertainty principle coupled withblack-hole physics to show that spacetime is indeed foamy and the degree of foaminess isconsistent with the holographic principle; we further argue that there necessarily exists adark sector in the universe. In section 3 we apply the holographic spacetime foam to cosmol-ogy (with the corresponding cosmology called holographic foam cosmology (HFC) [6, 7]) andargue for the existence of dark energy with critical energy density in the present universeand its quanta obey infinite statistics. In section 4 we use the deep similarities between thephysics of turbulence and the universal geometric properties of the holographic spacetimefoam to heuristically argue that the early universe was in a turbulent phase during whichthe universe underwent a brief cosmic inflationary acceleration. Section 5 contains our con-cluding remarks.We will use the subscript “P” to denote Planck units (with, e.g., l P ≡ ( ~ G/c ) / ∼ − cm being the Planck length.) And for simplicity, ~ , c , and the Boltzmann constant k B are often put equal to unity. One manifestation of spacetime fluctuations is in the induced uncertainties in any distancemeasurement. Consider the following gedanken experiment [8] to measure the distance l between a clock at one point and a mirror at another. By sending a light signal from theclock to the mirror in a timing experiment, we can determine the distance. The quantumuncertainty in the positions of the clock and the mirror introduces an inaccuracy δl . Let usconcentrate on the clock (of mass m ). If it has a linear spread δl when the light signal leavesthe clock, then its position spread grows to δl + ~ l ( mcδl ) − when the light signal returns2o the clock, with the minimum uncertainty at δl = ( ~ l/mc ) / . Hence one concludesthat δl & ~ lmc . One can supplement this requirement with a limit from general relativity[9],viz., δl must be larger than the Schwarzschild radius Gm/c of the clock, yielding δl & Gmc , the product of which with the bound from quantum fluctuations finally gives [9, 10] δl & ( ll P ) / = l P (cid:18) ll P (cid:19) / . (1)This bound on δl can also be derived by the following method which provides additionalvaluable insights. Consider a spherical volume of radius l over the amount of time T = 2 l/c it takes light to cross the volume. One way to map out the geometry of this spacetime region[11] is to fill the space with clocks, exchanging signals with other clocks and measuring thesignals’ times of arrival. This process of mapping the geometry is a sort of computation;hence the total number of operations is bounded by the Margolus-Levitin theorem[12], whichstipulates that the rate of operations for any computer cannot exceed the amount of energy E that is available for computation divided by π ~ /
2. To avoid collapsing the region into ablack hole, the total mass M of clocks must be less than lc / G , corresponding to the upperbound on energy density ρ ∼ l/Gl = ( ll P ) − . (2)Together, these two limits imply that the total number of operations that can occur in aspatial volume of radius l for a time period 2 l/c is no greater than ∼ ( l/l P ) . (Here andhenceforth we set c = 1 = ~ .) To maximize spatial resolution, each clock must tick only onceduring the entire time period. And if we regard the operations partitioning the spacetimevolume into “cells”, then on the average each cell occupies a spatial volume no less than ∼ l / ( l /l P ) = ll P , yielding an average separation between neighboring cells no less than l / l / P . This spatial separation is interpreted as the average minimum uncertainty in themeasurement of a distance l , that is, δl & l / l / P , in agreement with the result Eq.(1) ob-tained above.We can now heuristically derive the holographic principle. Since, on the average, eachcell occupies a spatial volume of ( δl ) . ll P , a spatial region of size l can contain no morethan l / ( ll P ) = ( l/l P ) cells. Thus this spacetime foam model corresponds to the case ofmaximum number of bits of information l /l P in a spatial region of size l , that is allowed bythe holographic principle [13, 14]. Accordingly, we will refer to this spacetime foam model(corresponding to δl & l / l / P ) as the holographic spacetime foam model. Henceforth we will neglect multiplicative constants of order unity. From Spacetime Foam to Dark Energy
As a corollary to the above discussion, we can now give a heuristic argument [11, 6, 15] onwhy the Universe cannot contain ordinary matter only. Start by assuming the Universe (ofsize l ) has only ordinary matter. According to the statistical mechanics for ordinary matterat temperature T , energy scales as E ∼ l T and entropy goes as S ∼ l T . Black holephysics can be invoked to require E . lG = ll P . Then it follows that the entropy S andhence also the number of bits I (or the number of degrees of freedom) on ordinary matterare bounded by . ( l/l P ) / . We can repeat verbatim the argument given in section 2 toconclude that, if only ordinary matter exists, δl & (cid:16) l ( l/l P ) / (cid:17) / = l / l / P which is muchgreater than l / l / P , the result found above from our analysis of the the gedanken experimentand implied by the holographic principle. Thus, there must be other kinds of matter/energywith which the Universe can map out its spacetime geometry to a finer spatial accuracythan is possible with the use of only conventional ordinary matter. We conclude that a darksector necessarily exists in the Universe!The above discussion leads to the prediction of dark energy. To see that, let us nowgeneralize this discussion for a static spacetime region with low spatial curvature to the caseof the recent/present universe by substituting l by 1 /H , where H is the Hubble parameter.[6, 15] Eq.(2) yields the cosmic energy density ρ ∼ (cid:16) Hl P (cid:17) ∼ ( R H l P ) − ∼ − M P . Next,recall that we have also shown that the Universe contains I ∼ ( R H /l P ) bits of information( ∼ for the current epoch).[6] Hence the average energy carried by each of these bitsor quanta is ρR H /I ∼ R − H . These long-wavelength bits or “particles” (quanta of spacetimefoam) carry negligible kinetic energy. (Note: Such long-wavelength quanta can hardly becalled particles. We will simply call them “particles” in quotation marks.) Since pressure(energy density) is given by kinetic energy minus (plus) potential energy, a negligible kineticenergy means that the pressure of the unconventional energy is roughly equal to minus itsenergy density, leading to accelerating cosmic expansion, in agreement with observation [2].This scenario is very similar to that of quintessence [16], but it has its origin in the holo-graphic spacetime foam. [7, 17]How do these long-wavelength quanta differ from ordinary particles? Consider N ∼ ( R H /l P ) such “particles” in volume V ∼ R H at T ∼ R − H , the average energy carriedby each “particle”. If these “particles” obey Boltzmann statistics, the partition function Z N = ( N !) − ( V /λ ) N gives the entropy of the system S = N [ ln ( V /N λ ) + 5 / λ ∼ T − ∼ R H . But then V ∼ λ , so S becomes negative unless N ∼ Alternatively one can interpret these quanta as constituents of dark energy, contributing a more or lessuniformly distributed cosmic energy density and hence acting as a dynamical effective cosmological constantΛ ∼ H . N inside the log in S , i.e, the Gibbs factor ( N !) − in Z N , must be absent. (This means that the N “particles”are distinguishable!) Then the entropy is positive: S = N [ ln ( V /λ ) + 3 / ∼ N . Now,the only known consistent statistics in greater than 2 space dimensions without the Gibbsfactor is the quantum Boltzmann statistics, also known as infinite statistics. [18, 19] Thuswe conclude that the “particles” constituting dark energy obey infinite statistics, ratherthan the familiar Fermi or Bose statistics. [15, 20] For completeness, let us list some of theproperties of infinite statistics [18, 19]. A Fock realization of infinite statistics is given by a k a † l = δ k,l . It is known that particles obeying infinite statistics are distinguishable, andimportantly their theories are non-local. [21, 19] (To be more precise, the fields associatedwith infinite statistics are not local, neither in the sense that their observables commute atspacelike separation nor in the sense that their observables are pointlike functionals of thefields.) Their quanta are extended (consistent with what we show above for dark energy).The number operator and Hamiltonian, etc., are both nonlocal and nonpolynomial in thefield operators. This property of non-locality will be useful later in the discussion of theearly Universe. But we should note that TCP theorem and cluster decomposition still hold;and quantum field theories with infinite statistics remain unitary. [19] So far we have applied HFC to the present and recent cosmic eras (with ρ ∼ − M P ). Butwhat about the early universe (with ρ ∼ − M P )? Actually the discussion in the preced-ing section has already given us some helpful hints [6] especially with respect to inflation[1] in the early universe. For example: (1) The flatness problem is largely solved because,according to HFC, the cosmic energy is of critical density. (2) It is quite possible that HFCprovides sufficient density perturbation as the model contains the essence of a k-essencemodel. Nevertheless one important aspect of the early Universe appears to be missing inHFC. It is connected to the expectation (supported by Wheeler’s insight [23]) that, due toquantum fluctuations, spacetime, when probed at very small scales, as is the case for theearly universe, will appear very complicated —- something akin in complicity to a chaoticturbulent froth. So, was spacetime turbulent in the early universe? To this question wenow sketch a positive response.First let us show the deep similarities between the problem of quantum gravity and tur- Furthermore, the horizon problem may also be solved since spacetime foam physics is essentially quantumblackhole physics and thus is closely related to wormhole physics which can be used [22] to solve the horizonproblem. √− g ∂ a ( √− gg ab ∂ b ϕ ) = 0 , where the effective space time metrichas the canonical ADM form ds = ρ c [ c dt − δ ij ( dx i − v i dt )( dx j − v j dt )], with c being thesound velocity. In this expression for the metric, it is apparent that the velocity of the fluid v i plays the role of the shift vector N i in the canonical Dirac/ADM treatment of Einsteingravity: ds = N dt − h ij ( dx i + N i dt )( dx j + N j dt ). Hence in the fluid dynamics context, N i → v i , and a fluctuation of v i would imply a fluctuation of the shift vector (and hence afluctuation of the spacetime metric) and vice versa.Next, let us note that, in fully developed turbulence in three spatial dimensions, Kol-mogorov scaling specifies the behavior of n -point correlation functions of the fluid velocity.The scaling [26] follows from the assumption of constant energy flux, v t ∼ ε , where v standsfor the velocity field of the flow, and the single length scale ℓ is given as ℓ ∼ v · t . Thisimplies that v ∼ ( ε ℓ ) / , consistent with the experimentally observed two-point func-tion h v i ( ℓ ) v j (0) i ∼ ( ε ℓ ) / δ ij . Now recall our discussion above on distance fluctuations δℓ ∼ ℓ / ℓ / P , and define the velocity as v ∼ δℓt c , since the natural characteristic time scaleis t c ∼ ℓ P c , then it follows that v ∼ c (cid:0) ℓℓ P (cid:1) / . It is now obvious that a Kolmogorov-likescaling [26] in turbulence has been obtained. This interpretation of the Kolmogorov scalingin the quantum gravitational setting implies that the quantum fluctuation phase of strongquantum gravity in the early Universe could be governed by turbulence.The discussion above is for the case of very large Reynolds number Re = Lv/ν , where v is the velocity field, L is a characteristic scale and ν is the kinematic viscosity which is givenby the product of the mean free path ˜ l and an effective velocity factor ˜ v . But was Re actuallylarge enough in the early Universe to set off turbulence? For the purpose of comparisonand illustration, let us recall that, in conventional cosmology at time, say, 10 − sec., vL in the numerator of Re is roughly given by the product of v ∼ − c and L ∼ l P . [27]For HFC close to Planckian time, the denominator of Re is given by ν ∼ cl P since, in thatregime, momentum transport could only be due to Planckian dynamics. For the discussionto follow, let us note that, relatively speaking, the effective velocity factor ˜ v in ν is not thatdifferent from the v factor in the numerator of Re . The onset of turbulence was due to thesmallness of the length scale (which plays the role of an effective mean free path) in thedenominator of Re , viz., ˜ l = l P << L ∼ l P that was mainly responsible for yielding alarge Re in the (very) early Universe.There remains one crucial hurdle to overcome. If, as we suggest, turbulence in the earlyUniverse was related to inflation, we have to confront the “graceful” exit problem: How to6et a small enough Re to transit to the laminar phase and to end inflation in the process? Itis here the nonlocality property enjoyed by the quanta of spacetime foam (due to the factthat they obey infinite statistics) came to the rescue since the length scale ˜ l in ν wouldnaturally and eventually extend to the order of L yielding a small enough Re to suppressturbulence and to naturally end inflation. We have sketched a scenario in which both the late and the early cosmic accelerations havea common origin and can be traced to spacetime foam. The case for dark energy in the cur-rent/recent (“late”) universe was proposed before [6, 15], while the case for cosmic inflationin the early Univerese is the main focus of this Letter. One attractive feature of our presentproposal is that the scheme is very economical, involving no arbitrary or fine-tuned parame-ters. It is also natural in that inflation was inevitable as turbulence set off by the Planckiandynamics was inevitable in the early Universe. The scheme also provides a rationale for whyinflation lasted only briefly (say, ∼ − sec.) as the turbulent phase was quickly termi-nated due to the nonlocal (extended) property of the quanta of spacetime foam. Of courseit is important to check if this scenario is supported by more quantitative arguments andcalculations. In passing we should also mention that it will be of great interest to see ifour scenario for inflation discussed above can mitigate or at least amelioriate some of thecriticism [28] against the inflation paradigm.We conclude with two observations the first of which involves the crucial question ofwhether our approach yields enough e-folds of inflation to solve the myriad of cosmologicalproblems. The following heuristic argument would seem to say probably there were. Let usfollow the folklore: at the end of inflation, the energy stored in the quanta of spacetime foamwould be converted into hot ordinary particles (as well as dark matter). Since the GrandUnification era is around ∼ − sec., which, as an order-of-magnitude estimate, should alsomark the end of inflation, giving enough (say &
65) e-folds of inflation. This argument maybe strengthened, if, as has been proposed [29], quantum gravity can actually be the originof (ordinary-) particle statistics, and that infinite statistics (the statistics obeyed by quantaof spacetime foam) is the underlying statistics. In that case, ordinary particles that obeyBose or Fermi statistics are actually some sort of collective degrees of freedom of “particles”of infinite statistics. (See Ref. [30] for a discussion of such a construction.) Thus, arguably,at the end of inflation, quanta of spacetime foam could be converted into ordinary particlesas required. Compare with the case of dark energy discussed in Section 3.
7e end this paper with our second observation. Our aesthetically pleasing scenario canbe compared to a recent model [3] of quintessential inflation based on the assumption thatthe slow roll parameter has a Lorentzian form as a function of the number of e-folds. Itsform corresponds to the vacuum energy both in the inflationary (with ρ ∼ − M P ) and thedark energy (with ρ ∼ − M P ) epochs which are treated symmetrically. In this modelthe inflationary scale is exponentially amplified while the dark energy scale is suppressed,producing a curious cosmological see-saw mechanism. In the present work, the two cosmicaccelerations are also attributed to a single mechanism; but they are related by some sort ofturbulent-laminar duality (or chaotic-smooth complementarity). It would be interesting tosee if our scheme can be approximated by an effective theory in which a similar ansatz forthe slow roll parameter naturally arises as in the see-saw model [3]. But a full investigationmay require a non-perturbative treatment of quantum gravity, involving a truly non-localfield theory of “particles” obeying infinite statistics. Handling turbulence in such a contextof quantum gravity may also prove to be challenging.
Acknowledgments
I thank David Benisty and Eduardo Guendelman for a useful correspondence. I amgrateful to the Bahnson Fund and the Kenan Professors Research Fund of the University ofNorth Carolina at Chapel Hill for partial financial support. We note that when the see-saw model is realized in the context of a single scalar field, the extractedpotential of the scalar field is fairly complicated. Can this complication be a reflection of non-perturbativequantum gravity involving non-local quanta of spacetime foam? eferenceseferences