aa r X i v : . [ h e p - t h ] N ov Thermal non-Gaussianitity in holographic cosmology
Yi Ling ∗ and Jian-Pin Wu † Center for Relativistic Astrophysics and High Energy Physics,Department of Physics, Nanchang University, 330031, China
Abstract
Recently it has been shown that the thermal holographic fluctuations can give rise to an almostscale invariant spectrum of metric perturbations since in this scenario the energy is proportionalto the area of the boundary rather than the volume. Here we calculate the non-Gaussianity of thespectrum of cosmological fluctuations in holographic phase, which can imprint on the radiationdominated universe by an abrupt transition. We find that if the matter is phantom-like, the non-Gaussianity f equilNL can reach O (1) or even be larger than O (1). Especially in the limit ω → − / k if we neglect the variationin T during the transition (fixed temperature). ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION Inflation is an extremely successful model in solving the horizon and flatness problemsin the standard hot big bang cosmology [1]. Furthermore inflation shows us that the CMBanisotropy [2, 3, 4] and the observed structures in the universe [5, 6, 7, 8, 9] are results ofthe quantum fluctuations occurring in the very early stage of the universe. Its predictionof a nearly scale invariant spectrum has been confirmed very well in the experiments [10].Nevertheless, the possibility that primordial thermal fluctuations might seed the structureof our universe is also an intriguing alternative to quantum fluctuations [11, 12, 13, 14].Unfortunately the spectral index of thermal fluctuations is either too red ( n s = 0) or tooblue ( n s = 4), failing to generate a nearly scale invariant spectrum [15, 16, 17]. Therefore,thermal scenarios call for new physics in order to produce nearly scale invariant fluctuations.For example, a thermal scenario in which the effect of new physics to change the equationof state of thermal matter, can produce nearly scale invariant spectrum. This happensin non-commutative inflation [12, 18, 19], and also in LQC [16, 20]. Another happens inthe Hagedorn phase for certain types of string gas [21] or in holographic phase [17], inwhich the energy becomes strongly non-extensive, specifically proportional to the area. Inaddition, postulating a mildly sub-extensive contribution to the energy density in Near-Milneuniverse [15], the scale-invariance spectrum can also be obtained. In this paper, we focus onholographic thermal fluctuations and investigate the holographic thermal non-Gaussianity.This scale invariance suggests that the primordial perturbation arising from a thermalholographic phase may be able to seed the structure of observable universe, and lead to theanisotropy observed in CMB as well. Thus it is interesting and significant to ask what is thedistinct feature of this holographic primordial perturbations and which is testable in comingobservations. There are at least two possible observables, accessible in the near future,that have the potential to rule out or support large classes of models: non-Gaussianityand primordial gravitational waves. In Ref. [22], Y. S. Piao has investigated the tensorperturbation. He finds that the tensor perturbation amplitude has a moderate ratio, whichmay be tested in coming observations. For comparison, we can also see Ref. [23] for tensorperturbation from string gas cosmology. Here we focus on the non-Gaussianity in holographiccosmology.In the WMAP convention, the degree of non-Gaussianity is parameterized by f NL , which2an be written as ζ = ζ g + 35 f NL (cid:0) ζ g − h ζ g i (cid:1) , (1)where ζ is the primordial curvature perturbation and ζ g is its linear Gaussian part. Theintroduced factor 3 / f NL should be defined by thethree-point function h ζ k ζ k ζ k i and has different shapes. We usually focus on two cases.One is the local, squeezed limit k ≪ k ≃ k , denoted by f local NL . Another is the non-local,equilateral limit k ≃ k ≃ k , labled by f equil NL [25](for brief, we can also refer to [26]).Recently, the WMAP 5-year data shows that at 95% confidence level, the primordialnon-Gaussianity parameters for the local and equilateral models are in the region − 111 and − < f equil NL < | f NL | < et al. find that thermal fluctuations can potentially producelarge non-Gaussianity. They develop a method to calculate the non-Gaussianity of thefluctuations with thermal origin and apply it to study the non-Gaussianity in string gascosmology [34]. In this paper, we intend to investigate the thermal non-Gaussianity in theholographic cosmology.The outline of this paper is the following. We first present a brief review on the holo-graphic thermal fluctuations in section II. Then we use thermal correlation functions tocalculate the power spectrum and the non-Gaussianity in section III. Finally, we discuss theconditions which may lead to a large non-Gaussianity in this scenario, in comparison withthose in string gas cosmology in section IV. 3 I. BRIEF REVIEW ON THE HOLOGRAPHIC COSMOLOGY A holographic version of thermal cosmology was originally proposed by J. Magueijo etal. (for more details, see Ref. [17]). They postulate that the very early universe underwenta phase transition from a high temperature, holographic phase (Phase I) to a usual lowtemperature phase of standard cosmology (Phase II). In holographic phase, the universeis disordered such that there is no classical metric [35], but its thermodynamics may bedescribed by making use of the holographic principle [36]. The spacetime geometry emergedonly during the phase transition such that in Phase II the universe may be described interms of Einstein gravity theory. The basic idea in holographic cosmology is that owing tothe existence of an early holographic phase, in which a specific heat scales as the area ofthe boundary, the thermal fluctuations which arise in holographic phase can finally imprinta scale invariant spectrum on Phase II through the phase transition [17].More explicitly, in holographic phase it is conjectured that h E i = bM pl T h A i , (2)where b is some dimensionless constant, M pl is the Planck mass, T is the temperature, and h A i is the expectation value of the area of the boundary. Thus, at fixed area, using therelation c A = ( ∂ h E i ∂T ) h A i , the specific heat can be expressed as c A = b h A i ~ G , (3)where ~ is the Plank constant and G is the Newton constant. Therefore, the specific heatat fixed area is proportional to the area which leads to scale-invariant spectrum.When the temperature falls to a critical temperature T c , the phase transition beginsand classical metric emerges. As the concept of length is created, the area and volumecan be expressed as h A i = A = 4 πR , and V ( R ) = πR as usual, where R is the thermalcorrelation length. For more precisely depicting the phase transition process, the dependenceof R on T was introduced [17], R ( T ) l = ( T c T c − T ) γ , (4)where the relation is valid for T ≤ T c , l is the smallest scale but not zero, which is determinedby quantum geometry, the critical exponent γ is introduced to parameterize the speed ofthe phase transition. 4rom the above relation, we can see that when T = T c , the correlation length R isdivergent and T < T c , R is created, the transition happens and enters into the usual radiationphase of standard cosmology.In holographic cosmology, the holographic phase has a nearly divergent length, which isrequired to assure all interesting modes observed today are in causal contact before tran-sition. This solves the horizon problem. In the next section, we will mainly discuss theholographic thermal non-Gaussianity. III. THE HOLOGRAPHIC THERMAL NON-GAUSSIANITY In this section, our purpose is to discuss the non-Gaussianity of thermal fluctuations inholographic cosmology. Before discussing the non-Gaussianity, a key question which has tobe addressed is on what scale the initial conditions should be imposed. We note that inRef. [33], they hold the thermal horizon R as a parameter during the calculation. Theyfind that if thermal horizon is smaller than Horizon scale at the horizon crossing, thermalfluctuations can lead to a large non-Gaussianity. In Ref. [19], the thermal correlation length R = T − is adopted, which is a lower bound and so will lead to larger non-Gaussianity.In our work, we will adopt the Hubble scale R = H − , beyond which causality prohibitslocal causal interactions [37]. Therefore we will calculate at R = H − , i.e. k = a/R = aH .This means that when T = T c , R ( T ) is infinite, H ≃ 0. Next, we will firstly calculatethe 2-point correlations, 3-point correlations and the power spectrum. Then we give thenon-Gaussianity of holographic thermal fluctuations.Fluctuations in a thermal ensemble can be determined from the thermodynamic partitionfunction Z = X r e − βE r , (5)where the summation runs over all states, E r is the energy of the state, and β = T − . Let U represents the total energy inside region R . Then the average energy of the system is givenby h U i = h E i = P r E r e − βE r P r e − βE r = − d log Zdβ , (6)The 2-point correlation function for the energy fluctuations δU ≡ U − h U i is given by h δU i = d log Zdβ = − dUdβ = T c h A i , (7)5imilarly, the 3-point correlation function can be expressed as h δU i = − d log Zdβ = d Udβ = T (2 c h A i + T c ′h A i ) . (8)where prime denotes the derivative with respect to the temperature T .Since the energy density perturbations δρ = δE/V . Using (3) and (7), the 2-pointcorrelation functions of δρ is given as h δρ i = h δU i V = T c h A i R = 4 πbT ~ GR . (9)Now we calculate the 3-point correlation function of δρ . We note that c ′ A = 0. Using (3)and (8), the 3-point correlation function of δρ can be expressed as h δρ i = h δU i V = T (2 c h A i + T c ′h A i ) R = 8 πbT ~ GR . (10)Performing the fourier transformation, the density fluctuations δρ k in momentum spacecan be related the fluctuation δρ in position space by δρ k = k − δρ . (11)In longitudinal gauge (see [38, 39, 40]), and in the absence of anisotropic matter stress,the metric takes the form ds = a ( η )[ − dη (1 − dx ] , (12)where Φ represents the fluctuations in the metric. Since during the phase transition, R ( T )is infinite, H ≃ 0, we can have k ≫ H , which means that the perturbations are deep in thehorizon. Thus the Eq. (12) of metric perturbation may be reduced to the Poisson equation k Φ k = 4 πGa δρ k . (13)Using the horizon crossing condition k = a/R = aH , the Poisson equation can be ex-pressed as Φ kL = 4 πGδρ k R . (14)Using (9), (11) and (14) , the 2-point correlation for Φ k is given h Φ k i = (4 π ) GbT ~ k − . (15)6imilarly, the 3-point function for Φ k is expressed as h Φ k i = 2(4 π ) G bT ~ R k − . (16)If we neglect the variation in T during the transition (fixed temperature), the powerspectrum for Φ k can be written as P Φ ≡ k π h Φ k i = 32 πb T c T P l . (17)where T P l = q ~ G is Planck temperature. Owing to the energy proportional to the area, thatis a scale-invariant spectrum but not the white noise.Next, we will calculate the holographic thermal non-Gaussianity. Note that Φ perturbsCMB through the so-called Sachs-Wolfe effect [41]. However, it is useful to introduce asecond variable, ζ , which is the primordial curvature perturbation on comoving hypersurfaces[9, 42]. Then the non-Gaussianity estimator f equilNL can be calculated theoretically by f equilNL = 518 k − h ζ k ih ζ k ih ζ k i . (18)The variables Φ and ζ are related by ζ = Φ − H ˙ H ( ˙Φ + H Φ) . (19)The variable ζ remaining nearly constant at super-horizon scales for adiabatic fluctuationsbut Φ not [38]. However, if the equation of state is constant, then Φ also remains constantat super-horizon. Therefore the relation (19) reduces to [43], ζ = 5 + 3 ω ω Φ . (20)We also note that the primordial curvature variable ζ is independent of ω , but the variableΦ, which perturbs the CMB, changes as ω changes. For more detailed discussion on thevariables ζ and Φ, we can refer to [19, 38, 44].Therefore, combining Eqs. (15), (16), (18) and (20), the non-Gaussianity estimator f equilNL can be expressed as f equilNL = 5(1 + ω ) ~ ω ) π bT c R = 5(1 + ω ) ~ H ω ) π bT c . (21)7sing the condition that modes exit the Hubble radius k = a c H and k = a H , andconsidering the relationship between scale factor and temperature a c a = T T c in radiation-dominated era, we can estimate the non-Gaussianity estimator f equilNL to be f equilNL = 5(1 + ω ) ~ ω ) π b H T kk = 5(1 + ω ) ~ ω ) π b × − kk , (22)where H , T , and k represent today’s values, a c is the scale factor during the phase tran-sition and we have neglected the variation in a c .From the above relation, we find that if ω → − 1, the non-Gaussianity will be suppressedas in usual inflationary phase. However, if the matter is phantom-like, the non-Gaussianity f equilNL can reach O (1) or larger than O (1) by fine tuning of the equation of state ω . Especiallyin the limit ω → − / 3, the non-Gaussianity can be very large. Therefore, the thermalholographic non-Gaussianity depends on what kind of matter in holographic phase (PhaseI), in which these fluctuations propagate to the primordial curvature fluctuation ζ or thepotential Φ when the classical metric emerges in Phase II, so that the large thermal non-Gaussianity occurs in holographic phase imprints on Phase II. Also, we must point out thatwhen ω < − 1, the non-Gaussianity estimator f equil NL < 0. Therefore the large non-Gaussianityin holographic cosmology will be negative.Moreover, the non-Gaussianity estimator f equilNL depends linearly on the mode k . Thisis similar with string gas cosmology [34], where the non-Gaussianity estimator f equilNL alsodepends linearly on the mode k , but very different from what happens in inflationary mod-els, where the non-Gaussianities are almost scale invariant. As pointed out in Ref. [34],depending linearly on the mode k means that modes reentering the Hubble radius earlierhave a larger non-Gaussianity. Therefore, if non-Gaussianity with an amplitude growinglinearly with k were to be detected, then thermal holographic cosmology with phantom-likematter should be desirable.In addition, we must point out that the non-Gaussianity estimator f equilNL also dependssensitively on the parameter b . If the parameter b is small, then the non-Gaussianity couldbe large. However, it will be confronted with the same problem as in string gas cosmology,in which if the string scale is decreased to the TeV scale, in order to obtain a power spectrumwith reasonable amplitude, a fine-tuning on the temperature T of the thermal string gasshould be required [34]. In this one would also require a fine-tuning on the phase transitiontemperature T c . 8ow, in fact, if we depict more precisely the phase transition by the relation (4), thespectrum in phase II cannot be exactly scale-invariant and the non-Gaussianity estimator f equilNL cannot be exactly depends linearly on the mode k . In this case, the spectrum can beexpressed as more precisely P Φ = 32 πb T c T P l [1 − ( l R ) γ ] = 32 πb T c T P l [1 − ( l T c H T kk ) γ ] . (23)Similarly, the non-Gaussianity estimator f equilNL takes the form f equilNL = 5(1 + ω ) ~ ω ) π b H T kk [1 − ( l T c H T kk ) /γ ] − ≃ ω ) ~ ω ) π b H T kk [1+( l T c H T kk ) /γ ] . (24)We can see that the non-Gaussianity estimator f equilNL depends on k in a more complicatedmanner, which is very different from the string gas cosmology. IV. CONCLUSION AND DISCUSSION In this paper, we have calculated the non-Gaussianity parameter f equilNL for thermal fluc-tuations in holographic cosmology with the use of the strategy developed in [33]. Sincethe energy is proportional to the area, the thermal holographic non-Gaussianity with anamplitude growing linearly with k is obtained if we neglect the variation in T during thetransition (fixed temperature). Furthermore, if we assume the dependence of R on T as (4)during the transition, then we find that the non-Gaussianity estimator f equilNL depends on k in a more complicated manner.Furthermore, the thermal holographic non-Gaussianity depends on what kind of matterin Phase I. If ω → − 1, the non-Gaussianity will be suppressed as in usual inflationaryphase. However, if the matter is phantom-like, the non-Gaussianity f equilNL can reach O (1)or larger than O (1) by fine tuning of the equation of state ω . Especially in the limit ω → − / 3, the non-Gaussianity can be very large. In fact, from the expression (18) of thenon-Gaussianity estimator f equilNL and the relation (20), we can immediately see that whenthe ω → − / ζ → 0, so the non-Gaussianity estimator f equilNL → ∞ . If ω → − ζ → ∞ ,the non-Gaussianity estimator f equilNL → 0. In single field slow roll inflationary model, inorder to obtain the scale-invariant spectrum, near deSitter spacetime ( ω → − 1) is required,then ζ → ∞ , the non-Gaussianity estimator f equilNL → 0. Therefore, the non-Gaussianity issuppressed in single field slow roll inflationary model. However, in holographic cosmology,9he scale-invariant spectrum is obtained by the specific heat scaling as area in holographicphase. Since the holographic phase transition is determined by geometry rather than thematter content [17], the large non-Gaussianity can be achieved by an appropriate choice of ω . We also note that the phase transition in string gas cosmology depends on the mattercontent (in the Hagedorn phase ω = 0 and in the radiation dominated universe ω = 1 / ω in thermal holographic cosmology. Next, we will also consider the thermal non-Gaussianityin semiclassical loop cosmology and Near-Milne universe [45]. 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