Energy Density Bounds in Cubic Quasi-Topological Cosmology
EEnergy Density Bounds in Cubic Quasi-Topological Cosmology
U. Camara dS ∗ , A.A. Lima ∗ and G.M.Sotkov ∗ ∗ Departamento de F´ısica - CCEUniversidade Federal do Esp´ırito Santo29075-900, Vit´oria - ES, Brazil
ABSTRACT
We investigate the thermodynamical and causal consistency of cosmological models of thecubic Quasi-Topological Gravity (QTG) in four dimensions, as well as their phenomenologicalconsequences. Specific restrictions on the maximal values of the matter densities are derived byrequiring the apparent horizon’s entropy to be a non-negative, non-decreasing function of time.The QTG counterpart of the Einstein-Hilbert (EH) gravity model of linear equation of stateis studied in detail. An important feature of this particular QTG cosmological model is thenew early-time acceleration period of the evolution of the Universe, together with the standardlate-time acceleration present in the original EH model. The QTG correction to the causaldiamond’s volume is also calculated.KEYWORDS: Cubic Quasi-Topological Gravity, Effective EoS, Entropic bounds, CausalEntropic Principle. e-mail: [email protected] e-mail: andrealves.fi[email protected] e-mail: [email protected], [email protected] a r X i v : . [ g r- q c ] A p r ontents Regions containing extremely dense matter are a common feature of all the big-bang inflationarycosmological space-times. The consistent description of such high energy states in the evolutionof the Universe requires certain “higher curvature” extensions of Einstein-Hilbert (EH) gravity,involving powers (of the traces) of the Riemann tensor, believed to take into account the shortdistance quantum effects. Such corrections to the EH action are known to arise as counter-terms in the perturbative quantization both of matter in curved spaces [1] and of pure EHgravity as well [2, 3]. The main problem with “higher curvature” gravity theories concerns thepresence of higher derivatives of the metric in the equations of motion, which in general lead tocausal and unitarity inconsistencies [2, 3] when considered out of the framework of superstringtheory. Nevertheless, the extensive studies of such models, in particular the so called modified f ( R )-gravities, have found interesting applications in different areas of modern cosmology (see,e.g., [4–6] and references therein).The present paper is devoted to the investigation of the effects caused by the higher curva-ture terms in a particular cosmological model in four dimensions, based on the simplest “mostphysical” extended gravity, given by the following cubic action for Quasi-Topological Gravity [7]: S GBL = (cid:82) √− g d x κ (cid:110) R − λL (cid:2) R − R αβ R αβ + 2 R αβµν R αβµν (cid:3) + µL (cid:104) R + 18 R αβγδ R γδµν R µν αβ − R αβγδ R γµβν R δν αµ − R αβγδ R αγ R βδ + 8 R αβ R βγ R γα (cid:3) + L matter (cid:9) , (1)where κ ≡ πG = 4 l ; l Pl is the Planck length; λ , µ are dimensionless “gravitational coupling”constants, while L is a new length scale, which can be chosen as L = l Pl by an appropriateredefinition of λ and µ . We include the cosmological constant Λ > L matter .The remarkable feature of this cubic extension of the EH action, is that the correspondingequations of motion for all conformally flat metrics, as for example those of domain walls andof flat Friedmann-Robertson-Walker (FRW) space-times:d s = − d t + a ( t ) d x i d x i , (2)are of second order [8], while arbitrary metrics yield, in general, equations of motion of fourthorder . This fact, together with the introduction of an appropriate superpotential and the The particular combination of quadratic terms represents the d = 4 Gauss-Bonnet topological invariant, anddoes not contribute to the dynamics. space-times.The most important and universal new property of these higher curvature cosmologicalmodels is that, differently from the EH case, the entropy prescribed to the apparent horizons [9] s ( t ) = π κ H (cid:0) − λL H + 3 µL H (cid:1) (3)with H ≡ ˙ a/a denoting the Hubble factor, is not automatically positive definite and increasing .As a consequence, the requirement of the thermodynamical consistency of these models: s ( t ) ≥ s/ d t >
0, introduces certain restrictions on the available maximal values (cid:37) max of the matterdensities (cid:37) = κ H ≤ (cid:37) max and certain minimal scales L min (related to (cid:37) max ) up to which wecan have a physically meaningful description of the Universe evolution within the framework ofthe cubic QTG cosmologies.In order to exemplify the effects caused by the higher curvature terms, we choose a par-ticularly simple and yet quite rich cosmological model, representing a QTG extension of thefollowing EH model of a linear equation of state : p /(cid:37) = w − w /(cid:37) . (4)The matter content is that of a barotropic fluid with constant equation of state parameter w ,together with a dark energy “fluid” representing the cosmological constant Λ = κ w / w ).This EH cosmological model has been widely studied [10–13], with a particularly remarkableresult by Bousso et al. [10], who deduce the value of the cosmological constant for a universedominated by dust through most of its history ( w = 0). The new QTG features established inthe present paper are: The presence of a new (early-time) acceleration period; changes in theduration of the acceleration and deceleration periods, as well as of the future and past eventhorizon radii; and finally certain very small corrections to the volume of the causal diamond (tobe compared with the EH one [10]). Consider an universe filled with a barotropic fluid with energy density (cid:37) and pressure p ,components of a ‘bare’ energy-momentum matter-tensor T (0) µν . For the ansatz (2), the QTGequations of motion derived from (1) are the modified Friedmann equations: κ (cid:37) = 6 H (cid:0) − µL H (cid:1) , (5a) κ ( (cid:37) + p ) = − H (cid:0) − µL H (cid:1) ; (5b)˙ (cid:37) + 3 H ( p + (cid:37) ) = 0 . (5c)They reduce to the usual EH-Friedmann equations when µ = 0; otherwise the contributions fromthe QTG terms may be regarded as separating a “gravitational energy momentum tensor” T QTG µν at the right-hand side of the Einstein equations, thus composing an effective energy momentumtensor T eff µν , viz. G µν = T eff µν ; T eff µν = T (0) µν + T QTG µν , where G µν is the Einstein tensor. Notice that the effective energy-momentum tensor has allthe properties of a perfect fluid tensor, i.e. its components, T eff µν = Diag ( − (cid:37), p, p, p ) satisfy the2sual, EH Friedmann equations: (cid:37) = κ H , (cid:37) + p = − κ ˙ H , (6)as well as the continuity equation, ˙ (cid:37) + 3 H ( p + (cid:37) ) = 0 , (7)which is a simple consequence of the Bianchi identities. Because of its direct connection to theHubble function, the cosmological observations (of distances and red shifts) should perceive notthe bare energy density (cid:37) , but rather the effective one, (cid:37) , related to the former via Eq.(5a): (cid:37) = (cid:37) (cid:16) − µL κ (cid:37) (cid:17) . (8)Although such a hydrodynamical interpretation of T eff µν is rather formal, we further assume thatthis “effective fluid” obeys the weak energy condition, i.e. p + (cid:37) ≥
0, what assures that ˙ H ≤ C ≡ − µL H = 1 − µL κ (cid:37) (9)must be positive (or vanishing), as can be seen from Eq.(5b). While this condition holds auto-matically for µ ≤
0, for positive values of µ the sign of C will depend on the value of the energydensity; it vanishes for (cid:37) = L κ (cid:112) /µ and it is indeed negative for greater values of (cid:37) . Sincenear a singularity the value of (cid:37) grows without bounds, for µ > nonsingular universe.This would be the case, for example, in bounce-like models beginning and ending at de Sitterspaces with (asymptotic) densities (cid:37) dS ≤ L κ (cid:112) /µ . We leave the study of such spaces for amore thorough discussion in [14], and focus for the remainder of this letter in the singular cases,thus considering only µ ≤ Killing event horizons are known to posses thermodynamical properties: a temperature T relatedto the surface gravity, and an entropy s which in EH gravity is given by the Bekenstein-Hawkingformula s = A/ A being the area of the horizon [15–17]. Then theEinstein equations can be rewritten as a Clausius relation [18], d E = T d s , for the flux of energyd E across the horizon .The lack of time-like Killing vectors in non-stationary space-times – as for example typicalFRW spaces – represents an obstacle in the definition of a surface gravity for the correspond-ing dynamical apparent horizons. A possible consistent generalization of the “Thermodynam-ics/Gravity” correspondence can nevertheless be achieved by using the Kodama vector to definethe horizon’s Kodama-Hayward temperature T KH [25]. As it was recently shown by Cai etal. [21,22], the corresponding Friedmann equations, for EH gravity and for certain modified the-ories as well, turn out to be again equivalent to the Clausius relation d E MS = T KH d s KW , with E MS being the Misner-Sharp energy and s KW an appropriately defined Kodama-Wald entropy,which for the cubic QTG cosmologies is given by Eq.(3). Notice that for de Sitter space-times, Remarkably, this equivalence remains valid in a variety of “higher curvature” gravitational theories [19–22]with the horizon’s entropy given then by the Wald formula [23, 24]. The proof of the equivalence between the QTG modified Friedmann equations (5) and the above generalizationof the Clausius relation is given in our forthcoming paper [26]. H is constant and the apparent horizon coincides with the Killing event horizon, ourformula (3) reduces to the known static QTG Wald entropy [9].The consistent interpretation of s QTG ( t ), given by Eq.(3), as an entropy for the apparenthorizon in the considered QTG cosmological models requires that it must be a non-negative andnon-decreasing function of time. The later is always true if both effective and bare fluids dosatisfy the weak energy condition. Then as a consequence we have that ˙ s = − G κ C ( (cid:37) ) (cid:37) ˙ (cid:37) ≥ s QTG ( t ) ≥ λ and µ , they turn out to introduce certain upper boundson the energy density (cid:37) . Gauss-Bonnet Gravity:
The topological nature of the GB term in the action renders thedynamics (i.e. the equations of motion) of GB gravity, for which µ = 0 and λ (cid:54) = 0, identical tothat of EH gravity. But the horizon entropy is not simply proportional to H − , in fact s ( t ) = G (cid:0) H − λL (cid:1) . Thus if λ < λ > H ( t ) –or, equivalently, of (cid:37) – past which the entropy becomes negative . Therefore the assumption ofpositivity of entropy places as an upper boundary, (cid:37) GB , on the possible values of the energydensity: 0 ≤ (cid:37) ≤ (cid:37) GB ; (cid:37) GB ≡ κ L λ (10) Quasi-Topological Gravity:
Similar considerations applied to the QTG model, for the nega-tive values of the coupling µ < finite interval:0 ≤ (cid:37) ≤ (cid:37) QTG ≡ κ L µ (cid:16) λ − (cid:112) λ − µ (cid:17) . (11)In a singular universe, it is then inevitable that at some instant the (divergent) effective energydensity violates the entropic threshold of Eqs.(10) or (11) for a finite value of (cid:37) . Hence in theconsidered GB gravity (with λ >
0) and QTG models of µ <
0, the cosmological singularitylies in a region of space-time which is already unphysical, for the apparent horizon entropy isnegative.
In order to describe the changes in the evolution of the homogeneous and isotropic Universecaused by the cubic QTG terms, we next address the problem concerning the constructionof analytic solutions of the modified Friedmann equations (5) in the particular example of thematter stress-energy tensor T (0) µν , whose components are related by the following linear barotropicequation of state [10–13]: p /(cid:37) = ω ( (cid:37) ) ; ω ( (cid:37) ) = w − w /(cid:37) . (12)The constant w , if positive, represents an energy density which contributes to the pressure p independently of the variable energy density (cid:37) – thus when the former dominates (i.e. w /(cid:37) (cid:29) p = − w , resultingin a de Sitter geometry. In this sense, w may be identified with the cosmological constant and412) is an example of a simple quintessence model. The constant w , which may be seen as the“matter EoS parameter” (as opposed to the “cosmological constant parameter” w ) is equal tothe velocity of sound in the fluid: v = ∂p /∂(cid:37) = w (in Plank units, c = 1 = (cid:126) ). The causalitycondition w < doesnot have a “phantom” phase , then we have to impose | w | < H ( t ) obtained by combining Eqs.(5) with the EoS (12):(1 − µL H ) ˙ H = − H (1 − µL H )(1 + w ) + κ w , (13)the integration of which yields t ( H ) = − w ) 1 L µ (cid:40) υ (cid:104) arc tg (cid:16) H − ξζ (cid:17) + arc tg (cid:16) H + ξζ (cid:17)(cid:105) − χ log (cid:16) ( H − ξ ) + ζ ( H + ξ ) + ζ (cid:17) + αH Λ arc cth ( H/H Λ ) − πυ (cid:41) . (14)Here H is the only real root of the cubic equation (8) when written in terms of H = κ (cid:37)/ H , with H ≡ κ (cid:37) / h and its complex conjugate ¯ h , are denoted by ξ and ζ , while χ and υ arethe real and imaginary parts of α/ ≡ − µL h h ( h − H )( h − ¯ h ) . Notice that we have chosen one specificsingular solution H ( t → → ∞ of Eq.(13), representing big-bang space-times with singularityat t = 0. Depending on the initial conditions imposed on H ( t ) (or equivalently on (cid:37) ( t )) onecan construct other non-singular bounce-like solutions (both for QTG or EH models), which arehowever out of the scope of the problems discussed in the present paper.The function t ( H ) (14) is not trivially invertible; nevertheless it allows to describe the prop-erties of the different periods of acceleration and deceleration of the QTG corrected Universeevolution, that can be obtained directly from Eqs.(13), (5b) and (6). We next recall that onecan also use the deceleration parameter q ≡ − ¨ a a/ ˙ a (for ˙ a (cid:54) = 0), written in a suggestive form: q = (1 + 3 p/(cid:37) ) , (15)in order to determine whether the universe undergoes accelerated ( q <
0) or decelerated ( q > p/(cid:37) ≡ ω eff between the components of the effective energy-momentumtensor plays the role of an ‘ effective equation of state ’. Its explicit form : ω eff ( (cid:37) ) = − w ) (cid:0) − µL κ (cid:37) / (cid:1) (cid:37) − w (cid:37) (1 − µL κ (cid:37) /
12) (16)is derived by substituting Eq.(12) into Eqs.(5b) and (6). As expected, for µ = 0 we get ω eff = ω . Since q ≷ ω eff ≷ − /
3, it is convenient to consider only Eq.(16). Observe that asone approaches the initial singularity and (cid:37) diverges, we can take the zeroth order limit of1 / ( µL κ (cid:37) ) (cid:28) ω eff ≈ (1 + ω ). Therefore, even if ω is in the“most decelerated range” possible, viz. ω (cid:46)
1, in the considered QTG cosmology we have anaccelerated phase: ω eff (cid:46) − /
3. Thus, the addition of the cubic QTG terms (1) to the EHaction results in a new acceleration period at the beginning of the universe. Changes in the EoS due to the higher curvature terms arising within the context of f ( R )-modified gravityhave been studied in Refs. [27, 28]. iii) (ii) (i) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:144) (cid:144) Ω e ff Ρ (cid:144) Ρ Pl Figure 1: Evolution of the equation of state parameter for µ = − : the solid line correspondsto the effective EoS of QTG as a function of the effective energy density, Eq.(16); the dashedblack line corresponds to the EH case, ω ( (cid:37) ); the red lines are the entropy densities of theapparent cosmological horizon, Eq.(3), for (i) λ = − . × ; (ii) λ = − . × ; (iii) λ = 7 . × .This effect may be seen in Fig.1, where ω ( (cid:37) ), given by Eq.(12), is shown as the dashed line,while the continuous black line depicts the effective equation of state (16). Here w = 0, and w is fixed by the observed value of the cosmological constant (cf. Eq.(20)). One can clearly seethat as the energy density increases the plot of ω eff sinks beneath the line of − /
3, indicating thenew period of accelerated expansion. For smaller densities, however, there is very little differencebetween the EH plot and the QTG one. Although some of the indispensable features of inflation– slow-roll, for example – are not present , we shall freely nominate this early acceleration periodas an “inflationary”. The duration of such a “rustic inflation” evidently depends on the valueof µ . By imposing ω eff ( (cid:37) acc ) = − /
3, we get a cubic equation: ( w − κ µL (cid:37) − w +13 (cid:37) acc + w = 0 , (17)whose positive real roots give the threshold of the two acceleration periods now present in thedynamics – the initial one due to QTG and the final due to the cosmological constant. Thenumber of distinct real solutions depends on the sign of the discriminant of (17) and, for µ < w > − /
3, it determines a critical value | µ acc | = 16(3 w + 1) − w ) κ w L for which the discriminant vanishes. Thus if | µ | < | µ acc | there are two positive real roots for(17), corresponding to two distinct periods of acceleration. But for large enough values ofthe gravitational coupling, namely | µ | ≥ | µ acc | , there is no positive real root to (17) and as aconsequence the initial QTG acceleration period lasts for such a long time that it merges withthe final one – the universe is then never decelerated.It is worthwhile to remark here that in the QTG counterpart (16) of the EH linear EoS, for µ < v = ∂p/∂(cid:37) also satisfies the causality conditions − < v <
1, under the same restrictions on the parameters w and w . On the other hand, for µ >
0, wesee that v → ∞ as (cid:37) → L κ (cid:112) /µ . This non-causal behaviour of the effective fluid near thepoints where C ( (cid:37) ) = 0 is yet another reason to consider here only the case of negative valuesof µ . 6lthough the form of t ( H ) given by Eq.(14) does not allow to analytically determine theexact form of the Hubble function H ( t ), we may invert it in a first order approximation. Thisis possible when the dimensionless quantity | µ | L H is much smaller than unity. For L = l Pl ,such an approximation is valid during most of the universe history, since H ( t ) is typically of acosmological order (greater than 1 Mpc ∼ × l Pl ). In what follows, we shall refer to thisapproximation as “first order in | µ | ”: H ( t ; µ ) = H ( t ) + µ H ( t ) + · · · ,a ( t, µ ) = a ( t ) [1 + µ A ( t ) + · · · ] , with the scale factor parametrized as a ( t ) = e A ( t ) . In zeroth order we have, naturally, the EHsolution: H ( t ) = κ (cid:113) w w ) cth [( t − t ) /τ ] , (18) a ( t ) = ˜ a sh δ [( t − t ) /τ ] , δ = w ) . (19)Here 1 /τ = κ (cid:112) w ) w /
8, and ˜ a is a normalization constant. At early times a ( t ) ∼ ˜ a ( t/τ ) δ the observed behaviour is typical of cosmologies with constant EoS: p /(cid:37) ≈ w . Later,for t/τ (cid:29) = κ w / w ) (20)dominates the EoS and the universe enters a final, accelerated, asymptotically de Sitter phase,with an asymptotically constant energy density (cid:37) Λ = w / (1 + w ). Then a ( t ) ∼ exp (cid:26) (cid:113) Λ ( t − t ) (cid:27) . (21)In particular, by choosing w = 0, and thus δ = 2 /
3, we see that (12) describes fairly well ourobserved Universe, neglecting inflation and the radiation dominated era: we begin at t = 0 witha dust-filled space-time, which ends at a de Sitter space with Λ presenting observed value [10]Λ ≈ . × − × l − , (22)if we choose w accordingly, using Eq.(20).The first order corrections can be easily calculated from Eq.(13): H ( t ) = ˜ H sh (( t − t ) /τ ) (cid:110) t − t τ − sh (cid:16) t − t ) τ (cid:17) −− cth (cid:0) t − t τ (cid:1) (cid:2) sh − (cid:0) t − t τ (cid:1) + (cid:3) (cid:111) ; (23) A ( t ) = ˜ H τ (cid:110) cth (cid:0) tτ (cid:1) (cid:2) (cid:0) tτ (cid:1) − tτ (cid:3) + sh − (cid:0) tτ (cid:1) (cid:111) , (24)where ˜ H = − κ w τ L / w ) . The constant t = 4 πυ/ w ) µL assures that thesingularity is placed in t = 0, and as µ → t →
0. From the scale factor and the Hubblefunction, one can determine all other relevant quantities, and in particular the effective energydensity (cid:37) ( t ), plotted in Fig.2.As one approaches the initial singularity at t = 0, Eq.(14) shows that H diverges. Eventuallywe then have H (cid:38) /l Pl and the first order approximation is bound to fail – indeed, H ( t ) divergesmore rapidly than H ( t ) as t →
0. This can be clearly seen in Fig.2: the black continuous line7 (cid:45) t Pl (cid:64) (cid:180) t Pl (cid:68) Ρ e ff (cid:64) (cid:45) (cid:180) Ρ P l (cid:68) Figure 2: Time evolution of the energy density, for the same parameters of Fig.1. The blackcontinuous curve gives the exact function (cid:37) ( t ), obtained from graphical inversion of t ( (cid:37) ) givenby Eq.(14); the black dashed line gives the first order approximation; the dotted gray linedepicts the EH density (cid:37) ( t ); the horizontal gray line marks the final energy density of the earlyacceleration period.shows the exact function (cid:37) ( t ), obtained from graphical inversion of t ( (cid:37) ) given by Eq.(14); theblack dashed line shows the first order approximation H ( t ) + µH ( t ). It is clear that the firstorder approximation is only valid valid for times greater than an instant t ∗ , when the curve hasa maximum, but for t > t ∗ it is in good agreement with the exact solution. To smallest order in | µ | , we have t ∗ ≈ (cid:16) w ) / + π (cid:17) √ w ) L | µ | / . (25)Now, approximating to first order Eq.(17) we find that the initial period ends when the effectiveenergy density decreases to the value (cid:37) initial ≈ (cid:113) w − w l − ( L | µ | ) − / , (26)which according to Eq.(14) happens at the instant t acc ≈ ( √ π ) √ w ) L | µ | / . (27)This shows that t ∗ is slightly posterior to t acc , hence an approximation to only first order is notsufficient to describe the initial acceleration period created by QTG.As we have shown above, if | µ | (cid:29) | µ acc | we may have an eternally accelerated expansion. Suchlarge values of | µ | , therefore, do not correspond reasonably to the observed universe. One mightthus pose the question: What are the restrictions on the values of the gravitational couplings λ and µ , which guarantee the physical consistency of the quasi-topological effects? Regarding theinitial acceleration period as an inflation era, we may assume that it would occur in the rangeof energies (cid:37) Pl (cid:38) (cid:37) (cid:38) − × (cid:37) Pl (see [29]), thus we must have µ acc such that the root of (17)lies within this bound. The upper bound of this interval, viz. (cid:37) acc ∼ − × (cid:37) Pl , is indeedthe case depicted in Figs.1 and 2, what serves to demonstrate the validity of the first orderapproximation. Another phenomenological restriction stems from the fact the the apparenthorizon entropy should not be vanishing for too small values of (cid:37) . In the GB case, there is noinitial acceleration period, and this entropic restriction is in fact the only condition we have on8 . It is quite evident from Eq.(10) that one may choose λ to get (cid:37) GB as large as one needs –e.g. for λ = 3 / (cid:37) GB = (cid:37) Pl . In QTG instead, as it may be easily seen from Eq.(11),for each given value of µ determining the end of the acceleration period, we can choose λ inorder to place (cid:37) QTG : (i) before or (iii) after (cid:37) initial , or even to have (ii) (cid:37) QTG = (cid:37) initial . This isexemplified in Fig.1, where the red lines represent the values of (cid:37) for which s ( (cid:37) ) = 0. Note thatin cases (ii) and (iii) the whole initial acceleration period is rendered “unphysical” on accountof the negative horizon entropy there. Although the more significant effects of QTG take place when the curvature, as well as the energydensity, are big enough – namely, at early times – it turns out that the whole evolution of theuniverse is modified by the QTG terms. Clearly, at later times, as the curvature diminishes,the cubic and quadratic terms in (1) become more and more negligible and the first orderapproximation made in the last section is then justified.An example of such changes is given by the fact that the cosmological constant, whichcharacterizes the geometry of the asymptotically dS spaces in the limit t → ∞ , is not equalto the bare one, Λ , defined by the matter Lagrangian. Indeed, in QTG one observes the effective cosmological constant Λ eff , related to the Hubble function H ( t ). To first order in µ ,one may determine Λ eff by the approximation (23) for H ( t ), or else by inverting directly theexact equation Λ = Λ eff (cid:0) − µL Λ / (cid:1) , (28)obtained from Eq.(8): Λ eff ≈ Λ + L µ Λ / vacuum. Notice that for the very small value of the observed cosmological constant, both Λ and Λ eff are practically equal.Another feature of asymptotically de Sitter space-times is the presence of a future eventhorizon. Its comoving radius is given by the integral r f ( t ) = (cid:82) ∞ t d t/a ( t ) and may be calculatedto first order in µ by using the results (19) and (24). At zeroth order (i.e. in the EH case) thisyields r (0) f ( t ) = τδ ˜ a δ ( t/τ ) 2 F (cid:104) δ , (cid:0) δ (cid:1) ; δ ; ( t/τ ) (cid:105) , (29)while the first order QTG correction is given by: r (1) f ( t ) = ˜ H τ a (cid:40) /δ ch δ ( t/τ ) 2 F (cid:104) δ , δ ; δ ; ( t/τ ) (cid:105) ++ / (4+ δ )[ch( t/τ )] (4+ δ ) F (cid:104) δ , δ ; δ ; ( t/τ ) (cid:105) − t/ ( δτ )sh δ ( t/τ ) −− /δ ch δ ( t/τ ) 2 F (cid:104) δ , δ ; δ ; ( t/τ ) (cid:105) (cid:41) . (30)The function r f ( t ) describes the past light cone of the (infinite) future of the comoving observerat the origin. The tip of this cone is placed at { r f = 0 , t = ∞} . Then as t → ∞ the physicalradius , l f ( t ) = a ( t ) r f ( t ) becomes equal to the (constant) de Sitter radius 1 /H . The first order9pproximation to l f ( t ) can be easily obtained with the help of Eq.(24): for example its valuetoday, at t = 13 . l f ≈ (cid:0) . × + µ × . × − (cid:1) Mpc .Alternatively, the comoving radius r p ( t ) = (cid:82) t ˜ t d t/a ( t ) describes the future light cone for anobserver at the origin, starting from its tip at { r p = 0 , t = ˜ t } . In a singular universe, if thistip is placed at the beginning of time, ˜ t = 0, then r p ( t ) is the particle horizon. In practice, thechoice of ˜ t determines only a constant, for we may write r p ( t ) = − (cid:90) ∞ t d t/a ( t ) + (cid:90) ∞ ˜ t d t/a = − r f ( t ) + r f (˜ t ) . (31)Thus for example, in the EH case with ˜ t = 0, Eq.(29) gives r (0) p ( t ) = − τ/δ ˜ a ch δ ( t/τ ) 2 F (cid:104) δ , δ ; δ ; ( t/τ ) (cid:105) + τ a B (cid:0) δ , − δ (cid:1) , (32)and in general Eqs.(29) and (30) determine also the first order correction for r p ( t ), for a given ˜ t .Recall that the first order approximation for the scale factor is only valid for t (cid:38) t ∗ , with t ∗ denoting the instant where the approximation fails, Eq.(25). Therefore in QTG we have the“technical” impossibility of placing the tip of the light cone on the initial singularity – the earlierwe may place it is at ˜ t = t ∗ . Due to the fact that t ∗ (at our first approximation) is slightlyposterior to t acc , we are then technically prevented from describing the increase of the particlehorizon during this early “inflationary” period. Consequently, we cannot determine its numberof e -foldings, and whether it solves the usual problems of non-inflationary cosmology – such as,for example, the horizon problem.The knowledge of the above expressions for the radii r p ( t ) and r f ( t ) provides an analyticdescription of the causal diamond [30, 31] and its comoving volume V c . d . ( t ), in the QTG cosmo-logical model under investigation. The former is defined as the intersection of the causal pastof the “point” { r f = 0 , t = ∞} and the causal future of { r p = 0 , t = ˜ t } . At each instant t thevolume is given by V c . d . = πr ( t ) with r = r p on the upper and r = r f on the lower half ofit. It is easily seen that V c . d . has a single maximum at t = t edge , when r p ( t edge ) = r f ( t edge ) [31].With the aid of Eq.(31), the instant t edge is determined from the equation r f ( t edge ) = r f (˜ t ),which is exact, i.e. independent from the first order approximation. If r f ( t ) is a continuousfunction, then we conclude that a first order correction to ˜ t must imply a first order change in t edge . Therefore for small t ∗ = ˜ t in QTG, the edge of the causal diamond occurs at an instantdiffering not more than at first order from the corresponding value in EH.In EH gravity with linear EoS (12) and w = 0, describing an early universe dominated bydust , Bousso et al. [10] have calculated V c . d . ( t ) and used it to predict the order of magnitude ofthe observed cosmological constant. Their analysis, based on the rather universal (phenomeno-logical) evaluation of the entropy production rate in our universe, demonstrates that only whenΛ ∼ − × l − the causal entropic principle (CEP) , requiring maximal entropic productionwithin the corresponding causal diamond volume, is fulfilled. According to their arguments themain production of bulk entropy occurs during the matter-dominated era. Therefore one canperform a similar analysis for the QTG extension of this linear EoS model, by imposing the CEP conditions for the causal diamond whose inferior tip is placed at ˜ t = t ∗ , thus respectingthe restrictions of the considered first approximation. Surely, even if the tips of the causal di-amonds in QTG and in EH gravity were placed at the same point – say, by replacing t = 0with t = t ∗ in the EH case as well –, still they would not be identical due to the changes in In which not only the initial inflationary period, but also the radiation-dominated era of the concordancemodel are absent. eff ≈ Λ + L µ Λ / ∼ − × l − .Throughout this discussion, we have not taken into account the entropic bounds derived inSect.3. Regardless of our technical restrictions for placing the inferior tip of the causal diamond,in the case of µ <
0, we cannot place it before the instant t QTG when the apparent horizon’sentropy vanishes. The same is true in the GB case, for λ >
0. We are however assuming that t QTG is very small, in particular that t QTG (cid:46) t ∗ . This can always be set for an appropriate valueof λ . Either way, the entropic restrictions do not allow the inferior tip of the causal diamond tobe placed on the singularity.Let us list in conclusion a few open problems concerning the considered (linear EoS) cos-mological model of the cubic Quasi-Topological gravity: (a) The stability conditions for theasymptotic dS cosmological QTG solutions, which requires the calculation of the spectrum ofthe corresponding linear fluctuations at least in the probe approximation; (b) The analysis ofthe properties of QTG models with more realistic matter content by considering EoS or equiv-alently (string inspired) matter superpotentials [32] giving rise to early-time inflation with a desired slow-roll behaviour , which is under investigation [14]. It is worthwhile to also mentionthat the methods and some of the results of the present paper seem to have a straightforward“cosmological” application to the recently constructed “higher curvature” quartic QTG exten-sion [33] of the EH gravity, as well as to the case of spatially curved FRW solutions of theconsidered four dimensional, cubic QTG model. Acknowledgments . We are grateful to C.P. Constantinidis for his collaboration in the initialstage of this work and for the discussions.
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