Constraining Entropic Cosmology
aa r X i v : . [ a s t r o - ph . C O ] F e b Preprint typeset in JHEP style - HYPER VERSION
Constraining Entropic Cosmology
Tomi S. Koivisto
Institute for Theoretical Physics and the Spinoza Institute, Utrecht University,Leuvenlaan 4, Postbus 80.195, 3508 TD Utrecht, The NetherlandsE-mail: [email protected]
David F. Mota
Institute of Theoretical Astrophysics, University of Oslo, 0315 Oslo, NorwayE-mail: [email protected]
Miguel Zumalac´arregui
Institute of Cosmos Sciences (ICC-IEEC), University of BarcelonaMarti i Franques 1, E-08028 Barcelona, SpainE-mail: [email protected]
Abstract:
It has been recently proposed that the interpretation of gravity as an emer-gent, entropic phenomenon might have nontrivial implications to cosmology. Here severalsuch approaches are investigated and the underlying assumptions that must be made inorder to constrain them by the BBN, SneIa, BAO and CMB data are clarified. Presentmodels of inflation or dark energy are ruled out by the data. Constraints are derived onphenomenological parameterizations of modified Friedmann equations and some featuresof entropic scenarios regarding the growth of perturbations, the no-go theorem for entropicinflation and the possible violation of the Bekenstein bound for the entropy of the Universeare discussed and clarified.
Keywords: cosmology: miscellaneous; cosmology: observations; cosmology: theory;. ontents
1. Introduction 12. Modifications from surface terms 3
3. Modifications from quantum corrections to the entropy-area law 8
4. Dark energy from generalized entropy-area law? 9
5. General comments on entropic cosmologies 12
6. Conclusions 18
1. Introduction
The notion of gravity as an emergent force has been contemplated for a long time [1]. Thederivation of gravitational field equations from thermodynamics by reference [2] supportsthis, and has lead to further substantial hints of evidence for the idea [3]. Recently, theproposal was put forward that gravity is a thermodynamic phenomenon emerging fromthe holographic principle [4]. It was argued that the Newton’s law of gravitation canbe understood as an entropic force caused by the change of information holographicallystored on a screen when material bodies are moving with respect to the screen. This isdescribed by the first law of thermodynamics, F ∆ x = T ∆ S , connecting the force F andthe displacement ∆ x to the temperature T of the screen and the change of its entropy,∆ S . T can be then identified with the Unruh temperature without referring to a horizon.Postulating ∆ S = 2 πm ∆ x , m being a particle mass, Newton’s second law follows. More to– 1 –he point, assuming equipartition of the energy [5] given by the enclosed mass, Newtoniangravitation emerges. See reference [6] for subtly different viewpoints. It has also beenargued that the entropic scenario fails to reproduce the quantum states of gravitationallytrapped neutrons [7], which experimentally match the predictions of Newtonian gravity. Ifthis is confirmed the entropic scenario would be ruled out.Cosmology has been also considered in this framework [8, 9]. As is well known, theFriedman equation can be deduced from semi-Newtonian physics. Thus it ensues from theabove arguments as shown by reference [10]. Reference [4] has also inspired modificationsto the cosmic expansion laws. The purpose of the present paper is to uncover implicationsof such modifications. Two approaches are investigated . One set of corrections to theFriedman equations is motivated by the possible connection of the surface terms in thegravitational action to the holographic entropy. Reference [16] noted that (at the presentlevel of the formulation) this is equivalent to introducing sources to the continuity equations,previously considered in the trans-Planckian context. Thus non-conservation of energy isimplied. In another approach, the derivation of the Friedman equation as an entropic forcefrom basic thermodynamic principles is generalized by taking into account loop correctionsto the entropy-area law [17]. The result is corroborated by its accordance with previousconsiderations [18], and is consistent with energy conservation though yet lacks a covariantformulation. While it might seem preliminary to investigate in detail the predictions ofthese models whose foundations, at the present stage, are rather heuristic, we believe itis useful to explore the generic consequences such models may have. Knowing about thepossible form of viable extensions to our standard Friedmannian picture along the lines ofreference [19] can shed light on the way towards more rigorous derivation of the effectiveentropic cosmology, and on the prospects of eventually testing the above cited ideas bycosmological observations.An encouraging result in this respect is that the viable cosmologies in a realisations ofboth the quite different approaches we focus on, possess an identical expansion rate (in thesimplest but relevant setting of a universe filled by a single fluid dominated cosmology).It is also interesting that at high curvatures this expansion rate reduces to a constant.Thus not a big bang singularity, but instead inflation is found in the past. We presenta simple argument why this inflation avoids the no-go theorem formulated in reference[20], which indeed is valid for material sources violating the strong energy condition. Wealso find that the higher curvature corrections, motivated by quantum corrections, arenot viable as they lack a consistent low-energy limit. Furthermore, we impose boundson the unknown parameters of the models by considering the scale of inflation, big bangnucleosynthesis (BBN) and from the effects of a modified behaviour of dark matter in thepost-recombination universe.Only phenomenologically motivated, but an interesting case is a monomial correctionto the area-entropy law. Such can result in acceleration without dark energy with a goodfit to the data. We perform a full Markov Chain Monte Carlo (MCMC) likelihood anal- Other approaches include considering two holographic screens [11] or Debye modifications of the tem-perature dependence [12]. Cosmological implications of the doublescreen model [13] and of the Debye model[14, 15] have also been considered. – 2 –sis exploiting astronomical data from baryon acoustic oscillations [21], supernovae [22]and cosmic microwave background [23]. A slightly closed universe turns then out to bepreferred by the data, unlike in the standard ΛCDM model. We consider also the evo-lution perturbations, which is determined uniquely if the Jebsen-Birkhoff law is valid. Acharacteristic feature is then the growth of gravitational potentials in conjunction withthe modified growth of overdensities. We also show that the visible universe bounded bythe last scattering surface is less entropic than a black hole the enclosed volume wouldform. This consistency check proposed in reference [24] can also be employed to constrainentropic cosmologies.The surface term approach is discussed in section 2. In section 3 we consider theimplications of a specific form of area-entropy law motivated by quantum gravity, andin section 4 we explore a phenomenological power-law parametrization of this law. Theperturbation evolution and constraints from more theoretical considerations are discussedin section 5. Each of these sections can be read independently of the others. Finally, theresults we obtained are concisely summarized in section 6.
2. Modifications from surface terms
Easson, Frampton and Smoot recently argued that extra terms should be added to theacceleration equation for the scale factor. This was discussed from various points of view,in particular it was conjectured that the additional terms can stem from the usually ne-glected surface terms in the gravitational action. Present acceleration of the universe [25]and inflation [26] were proposed to be explained by the presence of these terms withoutintroducing new fields. This is obviously an exciting prospect warranting closer inspection.Slightly different versions of the acceleration equation were introduced in both of theabove mentioned two papers. The following parametrization of the two Friedman equationsencompass all those versions and their combinations : H = 8 πG ρ + α H + α ˙ H + 8 πGα H , (2.1)˙ H + H = − πG w ) ρ + β H + β ˙ H + 8 πGβ H . The six coefficients α i , β i are dimensionless for all i = 1 , ,
3. The extrinsic curvature atthe surface was argued to result in something like α = β = 3 / π and α = β = 3 / π and quantum corrections in nonzero β [26, 25]. The equations imply that dHdN = H (cid:0) πGc H − c (cid:1) , (2.2)where N = log( a ) is the e-folding time and c ≡ (3 − α )(1 + w ) − β − α (1 + 3 w ) − β , (2.3) c = α (1 + 3 w ) + 2 β − α (1 + 3 w ) − β . (2.4) We will not adress the case in which the two equations degenerate to one by a particular choice of theparameters. – 3 –ote that c is proportional to the lower order corrections, and c is proportional to thehigher order contributions ∼ H . The information lost by having only one differentialequation in (2.2) should be compensated by imposing boundary conditions from (2.1) toits solutions. In the general case of multiple fluids, the model does not uniquely determinehow to the EoS (equation of state) w evolves. The reason is that the two equations (2.1)result in only one (non)conservation equation for the total density, and we have no uniqueprescription how the relative densities behave if the total density consists of a mixture offluids. From the viewpoint of reference [16], the source terms for the individual fluids areundetermined. However, the most relevant special case of a single-fluid dominated universeallows an exact solution where these ambiguities are absent. In the case that w is a constant, Eq.(2.2) can be easily solved: H ( a ) = c πG (cid:20) c (cid:16) aa (cid:17) c + c (cid:21) , (2.5) c = 132 π G (1 + w ) ρ . (2.6)We chose the integration constant c in such a way that when the entropic correctionsvanish, we recover the standard Hubble law. It is now clear that though there are sixunknown factors in (2.1) we cannot derive from first principles, the cosmological impli-cations are rather unambiguous (some degeneracies are broken for evolving w as we alsoexplicitly see below), and can be encoded in the two numbers c and c . Thus it is bothfeasible and meaningful to constrain them, despite our considerable ignorance of the morefundamental starting point. From the form (2.5) it is also transparent that as a →
0, wehave de Sitter solution: so the model indeed predicts inflation. As the scale factor grows,(nearly) standard evolution is recovered: so it is also simple to see the present versions ofthe model don’t provide dark energy. At early times, w = 1 /
3, we can obtain constraintsfrom BBN, and from the inflationary scale by estimating the amplitude of fluctuations.Both the scaling modification c and the constant term c can be bounded. At late times, w = 0, we can obtain constraints at least from the modified scaling law for dust, c . Beforethis let us however consider obtaining the present acceleration. As we need acceleration at late times, we need a Λ-term to accelerate the universe. Then(2.2) generalizes to dHdN = H (cid:0) πGc H − c (cid:1) + c Λ H , (2.7) c = 1 + w − α (1 + 3 w ) + 2 β . (2.8)This equation is solved by H = c πGc + p πGc c Λ − c πGc tanh (cid:18)q πGc c Λ − c ( N − N ) (cid:19) (2.9)– 4 –ence, the form of the Friedman equation is quite completely different from the usual. Thisis due to the nonlinearity of stemming from the presence the higher curvature corrections.At low curvatures, their effect doesn’t disappear as the most naive expectation wouldbe. In fact, the limit c → ΛThus we saw that the higher curvature corrections together are not compatible with acosmological constant in a viable cosmology. Let us therefore consider the case that thehigher curvature corrections given by c vanish, but allow a Λ term to accelerate theuniverse. Then the behavior of the Hubble rate is just what is expected from Eq.(2.5).Now (2.7) is solved by H ( a ) = 8 πG ρ a − c + 13 − w β Λ . (2.10)The integration constant ρ corresponds to the renormalised energy density at a = 1. Simi-larly, the cosmological constant is slightly ”dressed”. The conclusion is that the observableeffect to the expansion is the modified scaling of matter density. Below we consider thecosmological bounds on such scaling. First we consider the constraints from early universe. During radiation domination, Eq.(2.5) becomes H = 8 πGρ c a c − + 32 πG c ρ . (2.11)From this we see that the variation effective Newton’s constant is δG eff /G ≈ ( 12 c − − c log a ) − π G c c ρ . (2.12)This variation can be bounded by requiring successful BBN. For instance, reference [27]derived that δG eff = 0 . +0 . − . . The radiation energy density is given by ρ = g ∗ π T , (2.13)where we use for the number of effective relativistic degrees of freedom g ∗ at the nucle-osynthesis temperature T ∼ g ∗ = 10 .
75. Plugging in the numbers, weobtain − . · − < − c < . · − . (2.14) − · < c < · . (2.15)– 5 –ecause of the tremendous hierarchy between the Planck and the BBN scale there is avery poor constraint on the high curvature corrections c . This can be also written as2 √ πGc / < . GeV − by restoring the dimensions.If inflation is considered to be driven by the entropic corrections, we can estimatetheir magnitude from the amplitude of perturbations observed in CMB. Amplitude of thespectrum of quantum fluctuations of massless fields is expected to be given by the ratio h δφδφ i = 8 πGH ǫ ∼ − , (2.16)where ǫ is the slow-roll parameter and the right hand side is determined from observa-tions. The spectral index as determined from observations gives ǫ ∼ O (0 . πGH = c /c , and we know that c must be of order one, successful generation of ob-served fluctuations from entropic inflation suggest that c ∼ . To obtain inflation atthe GUT scale for example, we have to consider a very large amplification of the Planck-scale suppressed effect of c ∼ M P /M GUT . This estimate is much more tentative than theother we impose, since it depends on the physics of inflation, that are not well establishedeven in standard cosmology. Reference [28] discussed holographic view of inflation and theinterpretation of quantum fluctuations as thermal fluctuations on the screen.
The modified scaling law of dark matter can be used to impose tight bounds from the lateuniverse. This has been explored in reference [29], who derived constraints on the EoS fordark matter, taking into account experimental data both on the background and on theperturbations. Adopting the prescription where the Newton frame sound speed vanishes we can translate the result into our case as: − . · − < c − < . · − . (2.17)for 99.7% C.L. bounds. Reference [29] took into account the full CMB and LSS data.However, as we cannot deduce the perturbation evolution in these models unambiguosly,it is useful to consider constraints ensuing solely from background expansion. It turns outthat by including the latest data on SNeIa, BAO and CMB, the reached precision is onlyslightly lower. The result is shown in Fig. 1 and corresponds to the bounds − . · − < c − < . · − . (2.18)at 99.7% C.L. As proposed in reference [26], a more complete version of the model couldalso be constrained by the precision data on the equivalence principle. Ref.[29] uses slightly nonstandard nomenclature for the sound speeds. Usually ˙ p/ ˙ ρ is referred to as theadiabatic sound speed. The gauge-dependent quantity δp/δρ , when evaluated in the rest frame of the fluid,gives the sound speed according to the usual definition, see e.g. [30, 31, 32]. – 6 – L / L m δ c MCMC
Figure 1:
Constraints on the modified equation of state for CDM and baryons scaling law by usingSNe, BAO and CMB distance priors. Radiation has been assumed to follow the usual scaling. Onlyvery slight deviations from the usual scaling law are allowed.
ΛAs mentioned above, one can also consider the case that matter continuity equation is notviolated. Then the Λ-term must be responsible for the non-conservation in a consistentsystem. The Hubble law can be derived analogously to the above cases and one readilyfinds that it now has the form H = 8 πGρ (cid:16) α − β )3(1+ w ) (cid:17) + Λ a α − β ) . (2.19)So the Λ-term acquires a dynamical behavior. In case α − β < Now the BBN constraint for the effective gravitational constant gives − . < β − α < . . (2.20)From WMAP7 measurements on the equation of state of dark energy, combined with othercosmological data [34], we obtain an even tighter bound, − . < β − α < . . (2.21)– 7 –his is in qualitative agreement with reference [33], where the entropic corrections werebounded with the CMB acoustic scale.
3. Modifications from quantum corrections to the entropy-area law
There is evidence from string theory and from loop quantum gravity that the two leadingquantum corrections to the area entropy-law are proportional to the logarithm and theinverse of the area [35, 36].Reference [17] derived the Friedman equation from an underlying entropic force takinginto account quantum corrections to the entropy formula. We slightly generalise his endresult Eq.(25) by allowing multiple fluids (labeled i ) with constant equations of state w i and a cosmological constant: H + ka = X i (cid:20) πG − β (1 + 3 w i ) G w i ) (cid:18) H + ka (cid:19) − γ (1 + 3 w i ) G π (5 + 3 w i ) (cid:18) H + ka (cid:19) ρ i (3.1)+ Λ3 (cid:20) − βπ G ( H + ka ) log a + γ π G ( H + ka ) (cid:21) . Again there is a problem recovering usual evolution at low curvature if we include the highcurvature correction proportional to γ . This can be seen easily. Defining the shorthandnotations S β ≡ − βG X i w i w i ρ i − β G Λ3 π log a , (3.2) S γ ≡ − γG π X i w i w i ρ i + γ G Λ12 π , (3.3)the solution(s) for the Hubble rate may be written as H + ka = 1 − S β GS γ ± p − πG ρS γ GS γ , (3.4)where the total matter density is denoted by ρ = P ρ i . It is obvious that at the limit wherethe corrections tend to zero, we do not recover standard cosmological evolution. Thus thehigher order corrections here suffer from a similar graceful exit problem as we encounteredin the previous section.Therefore we set γ = 0 and consider only effect of the leading logarithm correction tothe entropy, proportional to β . The solution for the Hubble rate can then be written as,neglecting the cosmological constant, H + ka = 8 πG ρ " βG X i (cid:18) w i w i (cid:19) ρ i − . (3.5)– 8 –hus, the corrections occur near the Planck scale. If β is large enough this can supportinflation since the RHS tends to a constant when matter is relativistic and w i = 1 / i . Again, we can constrain this from the effective G at BBN. It is interestingto note that the form of the entropic Friedman equation assumes the same form as inthe previous section, where the derivation was quite different, the underlying physicalassumptions leading to a (non-)conservation and apparently different form of the forcelaw. The only difference between (2.5) and (3.5) is that in the case where the matterconservation laws are modified, (2.1) can describe also a slightly modified scaling of thematter density. This may be regarded as support for the robustness of the prediction ofthe expansion law in entropic cosmologies. From Eq.(3.5), the effective variation of the Newton’s constant is now given by δG N /G = 11 + βG ρ BBN − ≈ − βG ρ . (3.6)Using Eq.(2.13) for the radiation density at nucleosynthesis and proceeding analogously tosection 2.3.1, we find that the BBN bound on the magnitude constant β is | β | . . · ,translating to | p πGβ | < . /GeV . (3.7)Again it is clear that BBN is not efficient to constrain the corrections. Furthermore, sincethe scale of inflation is below the Planck scale, we have to consider very large values of β .From considerations of loop quantum gravity and string theory however, the natural valuefor β is of order one. Considering such values, inflation takes place at the Planck scale -where we cannot trust the perturbatively entropy-area law, which can be expected to holdonly at the limit of large horizon size. In fact, e.g. [37] has argued that the description ofspacetime as a differential manifold may be justified only asymptotically at macroscopiclength scales.
4. Dark energy from generalized entropy-area law?
In the following we consider the possibility of infrared modifications to the large-scalebehavior of gravity. Such can ensue from corrections to the S ∼ A relation that growfaster than A . Among such is the volume correction that scales as ∼ A / . Interestingly,corrections of this type imply, within the entropic interpretation of gravity, a modifiedNewton’s law which may explain the galactic rotation curves without resorting to darkmatter. This has been shown by reference [38]. We can then, effectively, generate MOND[39, 40] at the galactic scales. This motivates us to study whether we may generate modifiedgravity at the largest scales in such a way that we would avoid the introduction of darkenergy field or a cosmological constant.For this purpose, we consider the area-entropy law of the form S = A ℓ P + s ( A ) , (4.1)– 9 –here the function s ( A ) represents the quantum corrections and ℓ P = G ~ /c . We assumethat A = QN , where Q is a constant to be determined, and that the entropy changes byone fundamental unit (corresponding to unit change in the number of bits on the screenwith radius R ) when ∆ r = η ~ / ( mc ), r being the comoving radial coordinate. Then thefirst law of thermodynamics together with the equipartition of energy leads to the modifiedNewtonian law of gravitation F = − Q c M m πk B ~ ηR (cid:18) ℓ P + ∂s∂A (cid:19) = − GM mR (cid:18) ℓ P ∂s∂A (cid:19) , (4.2)where in the second equality we made the identification Q = 8 πk B ηℓ P . Let us assume thepower-law correction s ( A ) = 4 πσn (cid:18) arℓ P (cid:19) n ∼ A n . (4.3)This type of parametrization for entropic gravity effects has been recently consideredby other authors [42]. Taking into account that in the cosmological context the activegravitational mass is given by the Tolman-Komar mass (5.30), and that the R = ar =1 / p H + k/a , we obtain the Friedman equation H + ka = 8 πG X i " σ (1 + 3 w i )1 + 3 w i − n ℓ P ( H + ka ) ! n − ρ i . (4.4)Not surprisingly, the possible infrared corrections, n >
1, are precisely those which couldbe significant in cosmology at late times. The nonperturbative form of s ( A ) is of courseis unknown, but the volume correction is known to be given by n = 3 /
2, so n > w and thecorrections dominate over the standard term in (4.4), the expansion is described by theeffective EoS w eff = 1 + wn − . (4.5)Thus a matter dominated universe accelerates given n >
3. With larger n , the effective EoSis more negative, but phantom expansion can be achieved only when w is itself negative.The exact evolution of w eff including the effects of possible spatial curvature is shown inFigure 2. In order to obtain the bounds on the parameters arising from the modified Friedman equa-tion, a suitably modified version of CMBeasy [43] was employed together with a MCMCcode, taking into account astronomical data from baryon acoustic oscillations [21], super-novae [22] and cosmic microwave background [23]. The results are displayed in Figure 3 In Ref.[41] it was instead assumed that the number of bits is directly proportional to entropy, which is notcompatible with our assumption A = QN . From the former assumption follows instead F = − GMm/ ( R + ℓ P s ( A ) /π ). – 10 – w e ff ln(a) Ω m = 0.2 Ω m = 0.3 Ω m = 0.4 Figure 2:
Effective equation of state for the generalized entropy-area law. The lines correspond toan open, flat and closed universe with Ω S = 0 . n = 2 .
5. Equation (4.5) would give w = − . and Table 1. Due to the geometric nature of the modifications, a possible curvature of thespatial sections was allowed. This revealed a preference towards slightly closed universes,which might be due to the appearance of k in the r.h.s. of (4.4). Relatively lower values of n are favoured with respect to higher ones because higher values reproduce a total equationof state which is too close to −
1. Note also the existence of degeneracies in the individualdatasets, which are broken by the combined constraints.
Figure 3:
Bounds on parameters for the generalized area-entropy law using Sne (blue), CMB(orange), BAO (green) and the three combined datasets (gray). Note that closed universes (Ω m +Ω S >
1) are slightly preferred in this model. Entropic corrections gave χ S = 533 .
28, slightly higherthan the value obtained for a similar MCMC for ΛCDM with χ = 532 . – 11 –ll BAO CMB SNe h . ± . − . ± . − n . ± . > . − > . S . ± . − . ± .
12 0 . ± . m . ± .
02 0 . ± .
05 0 . ± .
13 0 . ± . k . ± . − − . ± . − . ± . Table 1:
Maximum likelihood values and 1 sigma error bars from the constraints of Section 4.1.
Table 2 shows the results of model comparison with ΛCDM including the Bayesian andthe Akaike criteria given by − L max + p log d and − L max + 2 p respectively, with p the number of free parameters and d the number of experimental data points. Eventhougha cosmological constant is favoured in all cases, the values of χ are very similar and mostof the difference in these cases is due to the additional parameter n in the entropy-correctedmodel. S ( A ) Λ χ χ /d.o.f Table 2:
Model comparison according to different criteria.
5. General comments on entropic cosmologies
Reference [44] have shown that by assuming the Jebsen-Birkhoff theorem [45] it is possibleto deduce the evolution of the spherical overdensities in a dust-filled universe given thebackground evolution. This is equivalent to a brane-motivated set-up where an effectiveenergy density (in our case with fractional density Ω s ) evolves adiabatically with cold darkmatter as shown in Ref. [46]. It is not clear to us whether the entropic gravity obeys theJebsen-Birkhoff theorem. Though this seems in line with the equipartition principle, in thepresence of corrections to the S ∼ A law the case is more nontrivial. Therefore we didn’tinclude the constraints from perturbations into the likelihood analysis above. However, inthe spirit of reference [44] (where the approach was developed to study DGP and relatedmodels) we take this as a first approximation to gain insight into the clustering of matterin entropic cosmology. The perturbation evolution equation can be derived by tracking thesurface of a star in a Schwarzchild metric embedded in the background of FRW where theexpansion is given by some gravity theory deviating from Einstein’s GR.– 12 –onsider the curved background ds = − dt + a ( t ) (cid:18) dλ − Kλ + λ d Ω (cid:19) . (5.1)We want to match this with the Schwarzchild-like metric ds = − g ( r ) dT + g rr ( r ) dr + r d Ω . (5.2)For this purpose consider the coordinate transformation r = r ( t, λ ), T = T ( t, λ ) whichallows to rewrite (5.2) in the form ds = − N ( t, λ ) dt + a ( t ) dλ − Kλ + r d Ω . (5.3)This implies the following conditions are satisfied: − g ˙ T + g rr ˙ r = − N , (5.4) − g ˙ T T ′ + g rr ˙ rr ′ = 0 , (5.5) − g ( T ′ ) + g rr ( r ′ ) = a − Kλ . (5.6)Here prime indicates derivative wrt λ , and an overdot derivative wrt t . We consider aspherical object whose interior expands like (5.1), and match the boundary at λ = λ ∗ smoothly with the exterior metric (5.2). This requires r ( t, λ ∗ ) = a ( t ) λ ∗ , (5.7) r ′ ( t, λ ∗ ) = a ( t ) , (5.8) N ( t, λ ∗ ) = 1 , (5.9) N ′ ( t, λ ∗ ) = 0 . (5.10)With some algebra employing equations (5.4-5.9) we infer that g rr ( r ( t, λ ∗ )) = 11 − λ ∗ ( ˙ a + K ) . (5.11)One obtains directly from (5.7) that the radius at the boundary λ ∗ satisfies¨ r = r ( H + ˙ H ) . (5.12)Since the top-hat overdensity δ ( t ) contained within radius r in the background density ρ M is (1 + δ ) ∼ / ( ρ M r ), we can recast (5.12) into an evolution equation for δ . At linearorder, we obtain ¨ δ + 2 H ˙ δ = H + ¨ HH ! δ . (5.13)In contrast, when the same background expansion is due to a smooth dark energy compo-nent, the growth of perturbations is governed by¨ δ + 2 H ˙ δ = 4 πGρ M δ . (5.14)– 13 –he difference is thus only the source term in the RHS of Eq.(5.13) due to the clumpinessof the effective fluid, whereas with smooth dark energy only the matter density acts asa gravitational source in the RHS of (5.14). This also determines the behaviour of themetric perturbations, which can be probed by various observations, in particular weaklensing and ISW and its correlations. In the longitudinal gauge the scalar perturbationsare parameterized by ds = − (1 + 2Ψ) dt + a ( t )(1 + 2Φ) (cid:20) dλ − Kλ + dλ d Ω (cid:21) . (5.15)Now we know that in an overdense region the line element can be written as ds = − d ˜ t + ˜ a (˜ t ) " d ˜ λ − ˜ K ˜ λ + d ˜ λ d Ω , (5.16)where the curvature ˜ K is associated with the overdensity and ˜ a = (1 + δ ) a . Analogouslyto Eq.(5.11), one may now infer that g − rr = 1 − ˙ r − ˜ λ ∗ ˜ K . (5.17)It follows that ˜ K − K = − (cid:16) ˙ Hδ + H ˙ δ (cid:17) . (5.18)What remains to do is a coordinate transformation that brings (5.16) into the form (5.15).With the top-hat profile for the overdensity, we can then identify the coefficients Φ and Ψ.Follwing reference [44] one may then find that ∇ a Φ = ˙
Hδ , (5.19) ∇ a Ψ = − H + ¨ HH ! δ . (5.20)This shows that the entropic corrections forces the gravitational potential unequal andthereby producing effective anisotropic stress. This is an interesting prediction as it allowsto distinguish the possible entropic origin of acceleration from for instance scalar fieldmodels.As an example we consider the power-law parametrization (4.3). The growth rate f ofthe perturbations can be defined as f ≡ d log δd log a . (5.21)Asymptotically, when the the effective EoS is given by (4.5), the two solutions to the evo-lution equation (5.13) correspond to growth rate f = 3 / (2 n ) and f = 3 /n −
2. The firstone is the growing solution and thus dominates at late times if n > /
4. As n increases,the decay of perturbations becomes less rapid, and when n → ∞ , the background is deSitter and the matter density is frozen. In dark energy cosmologies described by (5.14), the– 14 – | H ’ + H ’’ / H | , π G ρ m t [Mpc] n = 2 n = 4 n = 6 n = 10 Λ δ en t r o / δ Λ t [Mpc] n = 2 n = 4 n = 6 n = 10 0.4 0.6 0.8 1 1.2 1.4 0.1 1 d l og ( δ ) / d l og ( a ) z n = 2 n = 4 n = 6 n = 10 Λ -2e-06-1e-06 0 1e-06 2e-06 3e-06 4e-06 5e-06 6e-06 7e-06 8e-06 0.1 1 k / a ( Φ - Ψ ) en t r o [ M p c - ] z n = 2 n = 4 n = 6 n = 10 Figure 4:
Growth of structure in entropic cosmologies.
Top left:
Coefficient of δ in the r.h.sof (5.13) (thick lines) and (5.14) (thin lines) for some values of n . Top right:
Time evolution of δ normalized to the ΛCDM value. Bottom left:
Evolution of the growth factor (5.21) obtainedusing the modified equation (5.13) (thick lines) and the usual one (5.14) (thin lines). Data pointscorrespond to Table 1 in Ref. [47].
Bottom right:
Anisotropic stress caused by the modifiedgravitational potentials (5.19,5.20). All lines correspond to flat universes with Ω m = 0 .
3, Ω S = 0 . solution with growing δ is absent and the transition behaviour, which is relevant to obser-vations, is different. The growing solution corresponds asymptotically to Φ , Ψ ∼ a p , where p = 2 − / (2 n ) if 0 < n < / p = 0 if n > /
4. Thus we expect that the gravitationalpotentials will tend asymptotically to a constant value if the present acceleration stemsfrom entropic nature of gravity. This feature may produce the observational signatures ofthese models which could be used to distinguish them from dark energy models with thesame background expansion. The growing solution to Eq.(5.14) corresponds to f = 3 /n − p = − / (2 n ) when n > /
4. Thus in the case of smooth dark energy field, the gravi-tational potentials as well as the overdensities decay faster and vanish asymptotically. Therelative amplification of the of gravitational potentials we find in the entropic cosmologiescould be detected for instance in negative LSS-ISW correlations. Of course the presentuniverse is only approaching the exact solutions mentioned above.Numerical solutions for the full evolution of perturbations are plotted in Figure 4. Ta-ble 3 shows the numerical values of the growth factor (5.21) at a = 1 and 50, together with– 15 – a = 1 a = 50 a → ∞ Table 3:
Numerical and asymptotic values of the growth factor f . the limit previously described. Discrepancies occur for the cases with low n , which have lessnegative e.o.s. and need longer time to reach the limit. The growth of inhomogeneities canbe enhanced or damped depending on the value of n . The weakening of gravity responsiblefor the acceleration is reflected on smaller scales in the lower value of the r.h.s. factor of(5.13) responsible for gravitational instability. However, that term can become larger than4 πGρ m around the beginning of the acceleration due to the variation of H through its timederivatives . This effect can easily overcome the weakening of gravity if n is large enough(more pronounced transition), leading to higher values of δ . The enhancement effect is alsoresponsible for the resulting anisotropic stress due to the same factors appearing in (5.13)and in (5.20). Reference [24] raised an interesting point concerning a possible violation of the Bekensteinbound by the entropy contained in the visible universe. As is well known, it can be reasonedthat the entropy S of a region cannot exceed the entropy S BH of a black hole of the mass M contained in the region, S ≤ S BH = 4 πR S ℓ P , (5.22) R S = 2 GM being the Schwarzchild radius. When the considered region is the whole visibleuniverse, it is natural to take the radius to be given by d ∗ , the comoving distance to thesurface of last scattering. In holographic cosmology, the entropy (neglecting corrections fornow) is then just S V U = 4 πd ∗ ℓ P . (5.23)Now the mass (can be the Tolman-Komar) contained within the sphere of the radius d ∗ is M = 4 π d ∗ ρ M . (5.24)As reference [24] showed, it is quite convenient to write these results in terms of the CMBshift parameter R [48] R ≡ p Ω M H d ∗ . (5.25) Ref. [44] argue that slower growth of inhomogeneities follow from the acceleration condition. However,they consider a Friedman equation which is locally H ∼ ρ n which is inequivalent to (4.4). – 16 –sing Eq.(5.24) and then Eq.(5.25) we obtain that R S = 2 GM = 8 πGρ M d ∗ = H Ω M d ∗ = R d ∗ . (5.26)This enables us to write immediately the ratio S V U S BH = R − . (5.27)The observational constraint on the shift parameter is R = 1 . ± .
018 [34]. The factthat the visible universe respects the entropy bound is a consequence of it being confinedwithin its own Schwarzchild radius. This is a nontrivial consistency check that the entropiccosmology passes. One can also check that this continues to hold when the quantumcorrections are taken into account. For simplicity restricting to the flat case, we obtainthen S V U S BH = 1 R n Ω M + (1 − n )Ω s ( d ∗ /H ) n − n Ω M + (1 − n )Ω s ( R d ∗ /H ) n − . (5.28)As n increases, the ratio gets smaller and the bound is fulfilled with a wider margin, ascan be seen in Figure 5. S V U / S B H n Ω S = 0.5 Ω S = 0.7 Ω S = 0.8 Figure 5:
The ratio of the entropy of the visible universe and of the black hole of the same massin the model where the the monomial correction of power n accelerates the universe. The differentlines correspond to flat universes with different values of Ω S . A further interesting observation is that these inflationary models may avoid the no-gotheorem prohibiting inflation in entropic force law scenarios derived by reference [20]. Thetheorem is based on the observation that the active gravitational mass becomes negativein accelerating cosmology, which implies negative temperature on the holographic screen.The active mass is considered to be the Tolman-Komar mass, M = 2 Z Σ (cid:18) T ab − T (cid:19) n a ξ b dV , (5.29)– 17 –here Σ encloses the considered volume, n a is normal to it and ξ b is a time-like Killingvector. In the cases considered here, the Einstein equations are not satisfied, and thus M can be positive even if the universe accelerates. Equating the two vectors with thefour-velocity of matter, one obtains in FRW M = 4 π ρ + 3 p ) a r . (5.30)In our examples we have radiation dominated universe where inflation is driven by entropiccorrections, and clearly the Tolman-Komar mass (5.30) is positive.
6. Conclusions
In this paper we performed some elementary considerations of the possible consequences ofmodifications to the Friedman equations that have been suggested to describe the effectsof holographic entropy on cosmology.We found that the higher order curvature corrections, motivated by quantum correc-tions, lead to a graceful exit problem and thus can be, at least in the simplest scenarios,excluded. It was also observed that in two quite different approaches, the entropic cor-rections lead to a similar expansion law that predicts inflation in the early universe. Thisinflationary period has a natural transition to a radiation dominated universe. In the sur-face term approach of section 2, we identified the parameter combinations (2.3) that canbe constrained by observations. There c quantifies the lower order and c the higher ordercontributions. We obtained: − . · − < c − < . · − , p | πGc | < .
02 1 /GeV . from late and early universe constraints, respectively. In an alternative prescription retain-ing the cosmological matter conservation laws, previously introduced and bounded [33], anestimate for the parameters can be given as β − α = 0 . ± . . (6.1)In the quantum corrected approach discussed in section 3, the leading logarithm correctionto the entropy-area relation was shown to be constrained by BBN in a similar way, andthe following inverse correction was the cause of the graceful exit problem.In addition, we studied a phenomenological power-law correction to the entropy for-mula. It was found that such can generate accelerating expansion in the late universe.Combining the available data to bound on the power of the correction, we obtained n = 3 . ± . . (6.2)We may go significantly further if the evolution of spherical metrics can be argued todepend only on the amount of enclosed matter. Then the features of linearized structureis encaptured by the three equations (5.13,5.19,5.20).– 18 –e also pointed out that the inflation realized by the correction terms may avoid theno-go theorem prohibiting inflation driven by material sources in entropic cosmologies. Inthis light, we may claim to have not only have verified that the entropic corrections can drive inflation and the present acceleration of the universe but that they must be responsiblefor it, if the entropic emergence proposal in reference [4] is true. Coupled with the hintsin reference [38] that quantum corrections to entropy may also eliminate the need for darkmatter, this would suggest a drastic reinterpretation of cosmological observations withinan entropic paradigm. acknowledgements TK is supported by the Academy of Finland and the Yggdrasil grant of the Research Coun-cil of Norway. DFM thanks the Research Council of Norway FRINAT grant 197251/V30and the Abel extraordinary chair UCM-EEA-ABEL-03-2010. DFM is also partially sup-ported by the projects CERN/FP/109381/2009 and PTDC/FIS/102742/2008. MZ isfunded by MICINN (Spain) through the project AYA2006-05369 and the grant BES-2008-009090.
References [1] M. Visser,
Sakharov’s induced gravity: A modern perspective , Mod. Phys. Lett.
A17 (2002)977–992, [ gr-qc/0204062 ].[2] T. Jacobson,
Thermodynamics of space-time: The Einstein equation of state , Phys. Rev. Lett. (1995) 1260–1263, [ gr-qc/9504004 ].[3] T. Padmanabhan, Thermodynamical Aspects of Gravity: New insights , Rept. Prog. Phys. (2010) 046901, [ arXiv:0911.5004 ].[4] E. P. Verlinde, On the Origin of Gravity and the Laws of Newton , arXiv:1001.0785 .[5] T. Padmanabhan, Equipartition of energy in the horizon degrees of freedom and theemergence of gravity , arXiv:0912.3165 .[6] J. Makea, Notes Concerning ’On the Origin of Gravity and the Laws of Newton’ by E.Verlinde (arXiv:1001.0785) , arXiv:1001.3808 .[7] A. Kobakhidze, Gravity is not an entropic force , arXiv:1009.5414 .[8] T. Padmanabhan, Why Does the Universe Expand ? , arXiv:1001.3380 .[9] M. Li and Y. Wang, Quantum UV/IR Relations and Holographic Dark Energy from EntropicForce , Phys. Lett.
B687 (2010) 243–247, [ arXiv:1001.4466 ].[10] R.-G. Cai, L.-M. Cao, and N. Ohta,
Friedmann Equations from Entropic Force , Phys. Rev.
D81 (2010) 061501, [ arXiv:1001.3470 ].[11] Y.-F. Cai, J. Liu, and H. Li,
Entropic cosmology: a unified model of inflation and late- timeacceleration , Phys. Lett. B (2010) 213–219, [ arXiv:1003.4526 ].[12] C. Gao,
Modified Entropic Force , Phys. Rev.
D81 (2010) 087306, [ arXiv:1001.4585 ]. – 19 –
13] Y.-F. Cai and E. N. Saridakis,
Inflation in Entropic Cosmology: Primordial Perturbationsand non-Gaussianities , arXiv:1011.1245 .[14] H. Wei, Cosmological Constraints on the Modified Entropic Force Model , Phys. Lett.
B692 (2010) 167–175, [ arXiv:1005.1445 ].[15] S.-W. Wei, Y.-X. Liu, and Y.-Q. Wang,
Friedmann equation of FRW universe in deformedHorava- Lifshitz gravity from entropic force , arXiv:1001.5238 .[16] U. H. Danielsson, Entropic dark energy and sourced Friedmann equations , arXiv:1003.0668 .[17] A. Sheykhi, Entropic Corrections to Friedmann Equations , arXiv:1004.0627 .[18] R.-G. Cai, L.-M. Cao, and Y.-P. Hu, Hawking Radiation of Apparent Horizon in a FRWUniverse , Class. Quant. Grav. (2009) 155018, [ arXiv:0809.1554 ].[19] G. Dvali and M. S. Turner, Dark energy as a modification of the Friedmann equation , astro-ph/0301510 .[20] M. Li and Y. Pang, A No-go Theorem Prohibiting Inflation in the Entropic Force Scenario , Phys. Rev.
D82 (2010) 027501, [ arXiv:1004.0877 ].[21] W. J. Percival et al. , Baryon Acoustic Oscillations in the Sloan Digital Sky Survey DataRelease 7 Galaxy Sample , Mon. Not. Roy. Astron. Soc. (2010) 2148–2168,[ arXiv:0907.1660 ].[22] R. Amanullah et al. , Spectra and Light Curves of Six Type Ia Supernovae at 0.511 ¡ z ¡ 1.12and the Union2 Compilation , arXiv:1004.1711 .[23] E. Komatsu et al. , Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP)Observations: Cosmological Interpretation , arXiv:1001.4538 .[24] P. H. Frampton, Holographic Principle and the Surface of Last Scatter , arXiv:1005.2294 .[25] D. A. Easson, P. H. Frampton, and G. F. Smoot, Entropic Inflation , arXiv:1003.1528 .[26] D. A. Easson, P. H. Frampton, and G. F. Smoot, Entropic Accelerating Universe , arXiv:1002.4278 .[27] C. Bambi, M. Giannotti, and F. L. Villante, The response of primordial abundances to ageneral modification of G N and/or of the early universe expansion rate , Phys. Rev.
D71 (2005) 123524, [ astro-ph/0503502 ].[28] Y. Wang,
Towards a Holographic Description of Inflation and Generation of Fluctuationsfrom Thermodynamics , arXiv:1001.4786 .[29] C. M. Muller, Cosmological bounds on the equation of state of dark matter , Phys. Rev.
D71 (2005) 047302, [ astro-ph/0410621 ].[30] R. Bean and O. Dore,
Probing dark energy perturbations: the dark energy equation of stateand speed of sound as measured by WMAP , Phys. Rev.
D69 (2004) 083503,[ astro-ph/0307100 ].[31] T. Koivisto and D. F. Mota,
Dark Energy Anisotropic Stress and Large Scale StructureFormation , Phys. Rev.
D73 (2006) 083502, [ astro-ph/0512135 ].[32] D. F. Mota, J. R. Kristiansen, T. Koivisto, and N. E. Groeneboom,
Constraining DarkEnergy Anisotropic Stress , Mon. Not. Roy. Astron. Soc. (2007) 793–800,[ arXiv:0708.0830 ]. – 20 –
33] R. Casadio and A. Gruppuso,
CMB acoustic scale in the entropic accelerating universe , arXiv:1005.0790 .[34] E. Komatsu et al. , Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP)Observations: Cosmological Interpretation , arXiv:1001.4538 .[35] A. W. Peet, TASI lectures on black holes in string theory , hep-th/0008241 .[36] S. Carlip, Black Hole Entropy and the Problem of Universality , arXiv:0807.4192 .[37] P. Nicolini, Entropic force, noncommutative gravity and un-gravity , arXiv:1005.2996 .[38] L. Modesto and A. Randono, Entropic corrections to Newton’s law , arXiv:1003.1998 .[39] M. Milgrom, A Modification of the Newtonian dynamics as a possible alternative to thehidden mass hypothesis , Astrophys. J. (1983) 365–370.[40] F. Bourliot, P. G. Ferreira, D. F. Mota, and C. Skordis,
The cosmological behavior ofBekenstein’s modified theory of gravity , Phys. Rev.
D75 (2007) 063508, [ astro-ph/0611255 ].[41] Y. Zhang, Y.-g. Gong, and Z.-H. Zhu,
Modified gravity emerging from thermodynamics andholographic principle , arXiv:1001.4677 .[42] A. Sheykhi and S. H. Hendi, Power-Law Entropic Corrections to Newton’s Law andFriedmann Equations From Entropic Force , arXiv:1011.0676 .[43] M. Doran, CMBEASY:: an Object Oriented Code for the Cosmic Microwave Background , JCAP (2005) 011, [ astro-ph/0302138 ].[44] A. Lue, R. Scoccimarro, and G. Starkman,
Differentiating between Modified Gravity and DarkEnergy , Phys. Rev.
D69 (2004) 044005, [ astro-ph/0307034 ].[45] N. Voje Johansen and F. Ravndal,
On the discovery of Birkhoff ’s theorem , Gen. Rel. Grav. (2006) 537–540, [ physics/0508163 ].[46] T. Koivisto, H. Kurki-Suonio, and F. Ravndal, The CMB spectrum in Cardassian models , Phys. Rev.
D71 (2005) 064027, [ astro-ph/0409163 ].[47] S. Nesseris and L. Perivolaropoulos,
Testing LCDM with the Growth Function δ ( a ) : CurrentConstraints , Phys. Rev.
D77 (2008) 023504, [ arXiv:0710.1092 ].[48] J. R. Bond, G. Efstathiou, and M. Tegmark,
Forecasting Cosmic Parameter Errors fromMicrowave Background Anisotropy Experiments , Mon. Not. Roy. Astron. Soc. (1997)L33–L41, [ astro-ph/9702100 ].].