Note on Dark Energy and Cosmic Transit in a scale-invariance cosmology
aa r X i v : . [ g r- q c ] J a n Note on Dark Energy and Cosmic Transit in a scale-invariancecosmology
Nasr Ahmed , and Tarek M. Kamel Mathematics Department, Faculty of Science, Taibah University, Saudi Arabia Astronomy Department, National Research Institute of Astronomy and Geophysics, Helwan, Cairo, Egypt Abstract
In general, the laws of physics are not invariant under a change of scale. To find outwhether the ’scale-invariance hypothesis’ corresponds to nature or not, a careful exami-nation to its implications is required. As a consequence, the scale-invariance cosmologicalmodels need to be carefully checked with many tests in order to confirm or disconfirmthem. In this paper, three different toy models have been introduced in the framework ofa scale-invariance cosmology to examine dark energy and cosmic transit. Although cosmictransit exists in the three models, the pressure stays always negative during cosmic evo-lution. In addition, there is always a singularity in the evolution of the equation of stateparameter which is not suitable for a complete investigation of dark energy evolution. Theundesirable features of the parameters have been discussed, and a comparison with othercosmological contexts has been done.
PACS: 04.50.-h, 98.80.-k, 65.40.gdKeywords: Modified gravity, cosmology, dark energy.
The discovery of the accelerated expansion of the universe has been considered as one of the mostchallenging problems in modern physics [1, 2, 3]. In order to find a satisfactory explanation,several proposals have been suggested among them is the dark energy (DE). Dark energy isassumed to be an exotic form of energy with negative pressure which acts as a repulsive gravityand, consequently, pushes the the universe to expand faster. Some dynamical scalar field modelsfor DE have been introduced including quintessence [4], Chaplygin gas [5], phantom energy [6],k-essence [7], tachyon [8], holographic [9, 10] and ghost condensate [11, 12]. Another approachto explain this cosmic acceleration is by modifying the geometrical part of Einstein-Hilbertaction [24]. It has been shown that such ’modified gravity theories’ can explain the galacticrotation curves without assuming the existence of dark energy [18, 19, 20]. Examples of thisapproach include f ( R ) gravity [25] in which the Lagrangian is a function of the Ricci scalar R ,Gauss-Bonnet gravity [26], f ( T ) gravity [27] where T is the torsion scalar, and f ( R, T ) gravity[28], where T is the trace of the energy momentum tensor ( see [35] for a detailed review ).In f ( T ) gravity, the torsion scalar T is used in the Einstein-Hilbert action instead of the Ricciscalar R . An advantage of f ( T ) gravity is that it has second-order field equations while f ( R ) [email protected] R in the action isreplaced by the the Gauss-Bonnet term G = R − R µν R µν + R µνρδ R µνρδ , i.e. a function f ( G )can be used.The equation of state parameter ω is the division of cosmic pressure and energy density, ω = pρ . Investigating the evolution of this parameter is very essential to understand the nature ofdark energy. This parameter is equal to 0 for dust, 1 / − − ω ≤ − − ≤ ω ≤ ω = − ω of a single perfectfluid in Friedmann-Robertson-Walker geometry to cross the − ω = 1 is the largest possible value consistent with causality, it is assumed tohappen for some exotic type of matter called Zel’dovich fluid where the speed of sound is equalto the speed of light [14]. A matter fluid with ω = pρ ≫ ω Λ = −
1, dynamical models of darkenergy are still possible especially those with ω across − • The main aim of the current work is to test the viability of the recently suggested ’scale-invariance cosmology’ [30] by investigating the cosmic transit and dark energy assumptionin its framework. • In order to do this, we have introduced three models based on three different ansatze.These ansatze have been used before in several cosmological studies because of theirconsistency with observations, represented mainly in the evolution of both the decelerationand jerk parameters. They all lead to a deceleration-acceleration cosmic transit and a flatLambda CDM universe at late time. • Dark energy is usually defined as an exotic form of energy with negative pressure. If thedeceleration-acceleration cosmic transit exists in a certain model, the evolution of cosmic2ressure in that model should reveal a positive-to-negative sign flipping in correspondingto the positive-to-negative sign flipping of the deceleration parameter. Also, the evolutionof the EoS parameter shouldn’t reveal any improper behaviour or singularities. • The behaviors of cosmic pressure p and EoS parameter ω in the three models have beenfound to be disappointing. The pressure is always negative with cosmic time which doesn’thelp to explain the cosmic transit, and the EoS parameter always shows a singularity whichmeans that it is not possible to get a complete description for the evolution of dark energy. • Because of such incomplete and undesirable behaviors of p and ω in the three models withthree different empirical forms of the scale factor, we conclude that the recently suggestedscale-invariance cosmology in [30] might not be able to provide a complete description fordark energy and cosmic evolution. • The three ansatze used in this work provide excellent behavior of both p and ω in theframework of other cosmological contexts and different gravity theories. In the last section,we have mentioned some examples of such theories where the same ansatze lead to apositive-to-negative sign flipping in cosmic pressure, and a complete evolution of EoSparameter with no singularities.The current work represents an attempt to test the scale-invariance cosmology throughsome ansatze which are shown to be observationally consistent and also have been successfulin other cosmological contexts. Although the present work brings doubts to the viability ofthe scale-invariance cosmological models, more empirical forms of the scale factor still need tobe utilized to make a final conclusion. Also, a future modified version of the scale-invariancecosmology may avoid the bad features we have got here.The rest of the paper is organized as follows: In section 2, we give a review to therecently suggested scale-invariant gravity and the associated scale-invariance cosmology. Insection 3, we investigate three different solutions to the spatially flat cosmological equationsand analyze the behavior of different parameters. In section 4, we compare the obtained resultswith the results of other gravity theories where the same ansatze have been used. The finalconclusion is included in section 5. Scale-invariance means that the equations don’t change under the line element transformation ds ′ = λ ( t ) ds . where ds ′ is the line element of general relativity and ds is the line element ofa more general space [33]. Although some modified scale-invariance gravity theories have beenintroduced to explain the cosmic acceleration [29, 30, 32], the ability of the scale-invariancecosmological models to provide a complete description to cosmic evolution still not confirmed.The Robertson-Walker metric is written as ds = dt − a ( t ) (cid:20) dr − κr + r ( dθ + sin θdφ ) (cid:21) (1)3here r , θ , φ are comoving spatial coordinates, a ( t ) is the cosmic scale factor, t is time, κ is either 0, − e g µν = e Q g µν (2)have been applied to the Robertson-Walker metric where Q is the quantum potential with theform Q = 32 (cid:18) q − (cid:19) H m (3) q = ¨ RR ˙ R is the deceleration parameter, H = ˙ RR is the Hubble parameter and m is the particle’smass. The modified Einstein field equation now becomes [29, 30] e R µν − e g µν e R = 8 πG e T µν + Λ e g µν (4)where e R µν is the Ricci tensor with respect to the modified metric e g µν , e R is the Ricci scalar, e T µν is the energy-momentum tensor, and Λ is the effective cosmological constant. The energy-momentum tensor e T µν = e T ( M ) µν + e T ( Q ) µν where e T ( M ) µν is related to the matter contribution, and e T ( Q ) µν arises from the energy density of the quantum potential. The solution of (4) gives the followingmodified Friedmann equations˙ R R = 8 πG ρ + Λ λ − λ ˙ RλR + ˙ λ λ − κR , (5)¨ RR = − πG ρ + 3 p ) + Λ λ − ˙ λ ˙ RλR − ˙ λ λ − ¨ λλ . where λ = e Q , for λ = 1 we get the ordinary FRW equations. We consider only the flat casewhich is consistent with recent observations [34]. While in [32] a universe with Λ = 0 has beenconsidered, we will use a very small positive value for the cosmological constant as suggestedby observations. We are going to try three different solutions, each of them satisfies two observational conditions:1- The deceleration-acceleration cosmic transit which means there is a sign flipping from positiveto negative in the evolution of the deceleration parameter.2- Since we are considering a flat universe ( κ = 0), the jerk parameter must have the asymptoticvalue j = 1 at late-time. We recall that flat ΛCDM models have j = 1 [36]. The following ansatz gives a good agreement with observations for 0 < n < a ( t ) = A sinh n ( bt ) (6)4he main motivation behind using the ansatz (6) is its consistency with observations, and ithas been used in constructing several cosmological models in different gravity theories [37, 38,39, 41, 42, 57, 58, 59, 60, 62]. Since the deceleration-acceleration cosmic transit is supportedby recent observations [1], a specific form of the scale factor a ( t ) that leads to a sign flippingin the evolution of the deceleration parameter q from positive to negative can be utilized. Suchhyperbolic scale factor ansatz has appeared in many cosmological contexts such as Bianchicosmology [37], and the cosmological models in the framework of Chern-Simons modified gravity[39, 40]. In [42], A. Sen introduced a Quintessence model using the same hyperbolic ansatz wherethe consistency of this ansatz with observations was the main motivation behind using it. It hasalso been shown that this hyperbolic form can also provide a unified description for cosmologicalevolution up to the late-time future [41]. The deceleration and jerk parameters are respectivelygiven as q ( t ) = − ¨ aa ˙ a = − cosh ( bt ) + n cosh ( bt ) , (7) j = ... aaH = 1 + 2 n − n cosh ( bt ) , (8)The sign flipping of q ( t ) is shown in Fig.1(a) for n = where − ≤ q ( t ) ≤
1. The present valueof q ( t ) is supposed to be around − .
55 [43]. De-Sitter expansion occurs at q = −
1, power-lawexpansion happens for − < q <
0, and a super-exponential expansion occurs for q < −
1. Using(7), the cosmic transit is supposed to occur at q = 0 ( or ¨ a = 0). Here we have, t q =0 = 12 b ln(3 + 2 √ , (9)which gives t ≈ .
88 for b = 1. The parameter j ( t ) provides a convenient method to describemodels close to ΛCDM [44, 45]. This parameter has the asymptotic value j = 1 at late-time forthe current flat model (figure 1(d)). For the most important papers that have shed the light on q and j constraints and on further terms beyond them see [46, 47, 48, 49, 50, 51, 52, 53, 54, 55].In this case, we get 5 ( t ) = 16144 sinh ( bt ) π (cid:0) ( − b + 512Λ) coth( bt ) + ( − b + 1152 b − bt ) (10)+ ( − b − b − b + 1536Λ) coth( bt ) + 144 b + 384 b − (cid:1) .ρ ( t ) = 12048 sinh( bt ) π (cid:0) (192 b − bt ) + ( − b − b + 768Λ) cosh( bt ) (11)+ (27 b + 144 b + 192 b − bt ) + 256Λ (cid:1) .ω ( t ) = 13 (cid:18)(cid:18) − (cid:18) b −
43 Λ (cid:19) cosh( bt ) + ( − b + 1152 b − bt ) (12)+ ( − b − b − b + 1536Λ) cosh( bt ) + 384 b − b (cid:1)(cid:1) ÷ (cid:18) (cid:18) b −
43 Λ (cid:19) cosh( bt ) + ( − b − b + 768Λ) cosh( bt ) + ( − b + 144 b + 192 b − bt ) + 256Λ (cid:1) .ω ( z ) = (cid:18) − A (1 + (1 + z ) A )(1 + z ) b − z ) A m (cid:18)
13 + (1 + z ) A (cid:19) (13) × b − b m + 512Λ m (cid:1) ÷ (cid:0) A (1 + (1 + z ) A )(1 + z ) b − A × (1 + (1 + z ) A )(1 + z ) m b + (576 + 576(1 + z ) A ) m b − m (cid:1) The last equation expresses the EoS parameter ω in terms of the redshift z . Bycalculating the limit of this equation as z tends to 0 (at the current epoch) we find that ω ( z ) = − z → ω ( z ) = − ρ ( t ) (2(d)) and afuture quintessence-dominated flat universe (2(g)). But the cosmic pressure p ( t ) doesn’t changethe sign from positive to negative in corresponding to the cosmic transit. According to the darkenergy assumption in which the negative pressure acts as a repulsive gravity, p ( t ) should bepositive in the early-time decelerating epoch and negative in the late-time accelerating epoch. Another interesting ansatz that has been proved to be consistent with CMB observations is theso called logamediate inflation scenario [63] where the scale factor expands as a ( t ) = e B ln( t ) α .Its consistency with observations is the main motivation behind using it in the current work.In this case, we get the jerk and deceleration parameters as j ( t ) = 1 B α (cid:2) ( − α + 3) ln( t ) − α +1 + 2 ln( t ) − α +2 − Bα ln( t ) − α +1 (14)+ ( α + 3 α + 2) ln( t ) − α + (3 Bα − Bα ) ln( t ) − α + B α (cid:3) .q ( t ) = ( − α + 1) ln( t ) − α − Bα + ln( t ) − α +1 Bα (15)Figures (1(b)) and (1(e)) show the behavior of q and j of this solution. The deceleration6arameter changes sign from negative (at the very early time) to positive, and then becomesnegative again at late times which provides a more general description of cosmic expansion thanthe first solution. That means we have both an acceleration-deceleration cosmic transit whichis expected to happen in the very early times, and a deceleration-acceleration cosmic transitwhich is supposed to happen at late times according to observations. (see [64] for a descriptionof a similar scale factor). The pressure, energy density and Eos parameter can be expressed as p ( t ) = 1384 1 πm t (cid:18) α ( α − (cid:18)(cid:18) m − (cid:19) α − t m + 1712 (cid:19) B ln( t ) − α (16) − A α ( α − ( α − ln( t ) ( − α ) − B α ( α − α − ) ln( t ) − α +162 B α ( α − α − ln( t ) − α + (324 t m − α ( α − B × ln( t ) − α − (862 t m − α ( α − B ln( t ) α − − B α ( α − × ln( t ) − α + 648 α ( α − B (cid:18) −
54 + α (cid:19) ln( t ) − α − α (cid:18) m t − t m + 98 (cid:19) B ln( t ) α − + 486 B α ( α −
1) ln( t ) − α +( − t m + 324) B α ln( t ) α − − A ln( t ) − α α +36 t (cid:18) Bα ( α − α − α −
3) ln( t ) − α − Bα ( α − (cid:18) t m − (cid:19) ln( t ) α − − Bα ( α − α −
2) ln( n ) α − + (cid:18) t m − (cid:19) αB ln( t ) α − + 89 m t λ (cid:19) m (cid:19) .ρ ( t ) = 1128 1 πm t (cid:18) − α ( α − B (cid:18)(cid:18) t m − (cid:19) α − t m + 518 (cid:19) × (17)ln( t ) − α + 27 B α ( α − ( α − ln( t ) − α + 162 B α ( α − × ( α − ln( t ) − α − B α ( α − α − ln( t ) − α − α × ( α − B (cid:18) t m − (cid:19) ln( t ) − α + 216 α ( α − B × (cid:18) t m − (cid:19) ln( t ) α − + 243 B α ( α − ln( t ) − α − α × ( α − B (cid:18) −
54 + α (cid:19) ln( t ) − α + 48 B α (cid:18) t m − (cid:19) × ln( t ) α − − B α ( α −
1) ln( t ) − α + 216 B α (cid:18) t m − (cid:19) × ln( t ) α − − λm t + 243 B ln( t ) − α α (cid:1) . ( t ) = (cid:0) − B α ( α − α − ln( t ) α − α B ( α −
1) ln( t ) α (18) − B α (cid:18) t − (cid:19) ( α −
1) ln( t ) α + 243 ln( t ) α α B + 324 × B α ( t − t + 1) ln( t ) α − B α ( α − (cid:18)(cid:18) t − (cid:19) α − t + 1712 (cid:19) ln( t ) α +2 + 864 B (cid:18) t − (cid:19) α ( α −
1) ln( t ) α +3 +96 B α (cid:18) t − t + 98 (cid:19) ln( t ) α +4 + 27 B α ( α − ( α − ) × ln( t ) α + 162 B α ( α − α − ln( t ) α + 243 α B ( α − × ln( t ) α + 216 Bαt ( α − α −
2) ln( t ) α − Bαt ( α − × ( α − α −
3) ln( t ) α +2 + 48 Bα ( α − (cid:18) t − (cid:19) t ln( t ) α +4 − Bα (cid:18) t − (cid:19) t ln( t ) α +5 − t ln ( t ) − B α ( α − × (cid:18) −
54 + α (cid:19) ln( t ) α +1 (cid:19) ÷ (cid:18) B (cid:18)(cid:18) t − (cid:19) α − t + 518 (cid:19) α × ( α −
1) ln( t ) α +2 + 486 B α ( α − α − ln( t ) α + 648 B α × (cid:18) t − (cid:19) ( α −
1) ln( t ) α − B α ( α − (cid:18) t − (cid:19) ln( t ) α +3 ×− A α (cid:18) t − (cid:19) ln( t ) α +4 + 1944 B α ( α − (cid:18) −
54 + α (cid:19) × ln( t ) α +1 + 1458 α B ( α −
1) ln( t ) α − B α (cid:18) t − (cid:19) ln( t ) α −
729 ln( t ) α α B − B α ( α − ( α − ln( t ) α − B × α ( α − α − ln( t ) α − α B ( α − ln( t ) α + 48Λ t ln( t ) (cid:1) Figure (2) shows a physically acceptable behavior of the energy density ρ ( t ). We cansee that, as in the first solution, the cosmic pressure p ( t ) doesn’t change sign from positive tonegative during its evolution in corresponding to the deceleration-acceleration cosmic transit.Because cosmic transit already exists in the model (Figur 1(b)), this behavior of p ( t ) whichis always negative doesn’t help to explain this cosmic transit. In addition to this incompletebehavior of cosmic pressure, the evolution of the EoS parameter ω ( t ) also shows a singularityat early-time. For late-times, ω ( t ) has the asymptotic value ≈ − which is exactly the sameasymptotic value of the first solution. So, both solutions predict a future quintessence-dominateduniverse and also suffer from a divergence at some point.8 .3 Hybrid solution. Another ansatz which leads to a deceleration-acceleration cosmic transit with the jerk parametertends to the flat Λ
CDM model for late-time is the hybrid scale factor [65, 66]. This consistencywith observations is the main motivation behind using it. The hybrid ansatz is a mixture ofboth power-law and exponential-law cosmologies, the scale factor expands as [65, 66]: a ( t ) = a t α e βt , (19)where a > α ≥ β ≥ β = 0 we get the power-law cosmology, andfor β = 0 we obtain the exponential-law cosmology. New cosmologies can be investigated for α > β >
0. In this case we get the deceleration and jerk parameters as q ( t ) = − ¨ aa ˙ a = α ( βt + α ) − j ( t ) = α + (3 β t − α + (3 β t − β t + 2) α + β t ( β t + α ) (21)The cosmic transit occurs at t = √ α − α β which restricts α in the range 0 < α < p ( t ) = 1384 1 m πt (cid:0) − α + ( − m t − βt + 324) α + ( − m t (22) − m t β −
108 + ( − β + 540 m ) t + 324 βt ) α − m × (cid:18)
98 + m t β + (cid:18) − m + 2716 β (cid:19) t − βt (cid:19) t α − m (cid:18) β −
13 Λ (cid:19) t (cid:19) ρ ( t ) = 1128 m πt (cid:0) α + (216 m t + 486 βt − α + (48 m t + 432 m t β (23)+ 108 + (243 β − m ) t − βt ) α + 96 m (cid:18) m t + 94 βt − (cid:19) × βt α + 48 m (cid:18) β −
13 Λ (cid:19) t (cid:19) ω ( t ) = (cid:0) − α + ( − m t − βt + 324) α + ( − m t − m t β (24) −
108 + ( − β + 540 m ) t + 324 βt ) α − m (cid:18)
98 + m t β ++ (cid:18) − m + 2716 β (cid:19) t − βt (cid:19) t α − m (cid:18) β −
13 Λ (cid:19) t (cid:19) ÷ (729 × α + (648 m t + 1458 βt − α + (324 + 144 m t m t β + (729 β − m ) t − βt ) α + 288 m (cid:18) m t + 94 βt − (cid:19) βt α + 144 m (cid:18) β −
13 Λ (cid:19) t (cid:19) p ( t ), ρ ( t ) and ω ( t ). Figure (2) shows a physically acceptable behavior of ρ ( t ), a singularity inthe evolution of the EoS parameter ω ( t ), and no sign flipping of p ( t ) from positive to negativein corresponding to the deceleration-acceleration cosmic transit. (a) q (b) q (c) q (d) j (e) j (f) j Figure 1: The deceleration and jerk parameters for the hyperbolic, logamediate inflation andhybrid solutions labelled as q , q , q , j , j and j respectively. We have used these three ansatzemainly because of their consistency with observations. The cosmic deceleration-accelerationtransit is allowed in all of them where there is a sign flipping of q ( t ) from positive to negative.In addition, the behavior of the jerk parameter j ( t ) at the current epoch represents anothersupport for using those three ansatze. Since flat ΛCDM models have j = 1, the plots showthat the jerk parameter tends to the flat Λ CDM model for late-time in the three models. Herewe have adopted the following numerical values: n = , A = 0 . b = 63, α = 5, B = 1, and α = β = 0 .
1. 10 a) p (b) p (c) p (d) ρ (e) ρ (f) ρ (g) ω (h) ω (i) ω Figure 2: The pressure, energy density and EoS parameter for the three solutions. We havea physically accepted behavior of the energy density where ρ → + ∞ as t →
0. However, forthe three different ansatze, the scale-invariance cosmology fails to provide a complete behaviorof p ( t ) and ω ( t ). Since dark energy has a negative pressure effect, there should be a positive-to-negative sign flipping in cosmic pressure in corresponding to the positive-to-negative signflipping of the deceleration parameter (Figure 1). We also observe a singularity in the behaviorof ω ( t ) which means that a complete description for the dark energy evolution can’t be obtained.While the three forms of a ( t ) lead to a complete description of p ( t ) and ω ( t ) in other theories,the failure to predict a complete behavior of p ( t ) and ω ( t ) using the same forms in the scale-invariance cosmology puts the cosmological viability of the theory under dispute. Here we haveadopted the following numerical values: n = , A = 0 . b = 63, α = 5, B = 1, α = β = 0 . m = 1, and Λ = 0 . Comparison with other theories
In the previous section, we have studied three different toy models in the framework of a scale-invariance cosmology where p ( t ) and ω ( t ) always show undesirable and incomplete behavior.This result makes the ability of the scale-invariance cosmology to provide a complete descriptionfor dark energy evolution uncertain. As we have mentioned in the introduction section, the threeempirical forms of a ( t ) used in the current study have been utilized in other modified gravitytheories by many researchers where better behaviors of both p and ω have been obtained. Thepositive-to-negative sign flipping of cosmic pressure helps to explain the cosmic transition fromdeceleration to acceleration. In this section, we give some examples of those works where thesame forms of a ( t ) have been used in different cosmological contexts.In the framework of the entropy-corrected cosmology [56], a stable flat universe hasbeen reached using the hyperbolic ansatz in [57] and using the hybrid ansatz in [62]. In both ofthese models, the cosmic pressure shows a sign flipping from positive to negative in correspondingto the positive-to-negative sign flipping of the deceleration parameter. Also, there is no animproper behavior in the evolution of the equation of state parameter in both of them asindicated in figure (3). The sign flipping in the evolution of cosmic pressure has also obtainedin the framework of Swiss-cheese brane-worlds by utilizing the hybrid ansatz [58]. Using thehyperbolic ansatz in universal extra-dimensional cosmology also leads to a sign flipping of cosmicpressure [59]. It has also been shown that this change of sign in the evolution of cosmic pressureexists in the context of Chern-Simons modified gravity upon using both the hyperbolic and thelogamediate inflation forms [60]. It also appears in the framework of cyclic cosmology [61]. We have have investigated the cosmic transit and dark energy assumption in a scale-invariancecosmological framework using three different toy models. In order to explain the cosmic transitaccording to the dark energy assumption, the pressure should be positive in the early-timedecelerating epoch and negative in the late-time accelerating epoch. Although cosmic transitexists in the three models, the pressure stays always negative during cosmic evolution. Anotherbad feature is the evolution of the equation of state parameter which always shows a singularityat specific time. It is interesting to note that a future quintessence-dominated universe withdifferent asymptotic values is predicted in the three models. Finally, We have compared theresults we obtained in the framework of the scale-invariance cosmology with other cosmologicalmodels in different gravity theories where the same ansatze have been utilized. While the samethree forms of a ( t ) lead to a complete description of p and ω in other theories, the failure topredict a complete behavior of p and ω using the same forms in the scale-invariance cosmologyputs the cosmological viability of the theory in doubt.12 a) p (b) ω (c) p (d) ω (e) p (f) ω Figure 3: The behavior of p ( t ), ρ ( t ) and ω ( t ) in other cosmological frameworks where the sameforms of a ( t ) have been utilized. (a) and (b) show the behavior of p ( t ) and ω ( t ) for the hyperbolicsolution in the entropy-corrected cosmology where − ≤ ω ( t ) ≤ and the solid lines representthe flat case. (c) and (d) show the behavior of p ( t ) and ω ( t ) for the hybrid solution in theentropy-corrected cosmology where ω ( t ) varies in the same range. (e) and (f) show the behaviorof p ( t ) and ω ( t ) for the hybrid solution in the Swiss-cheese brane-world cosmology. The detailedcalculations and numerical values adopted in these plots have been given in [57], [62] and [58]. Acknowledgment
We are so grateful to the reviewer for his many valuable suggestions and comments that signif-icantly improved the paper.