Study of Thermodynamic Quantities in Generalized Gravity Theories
Surajit Chattopadhyay, Ujjal Debnath, Samarpita Bhattacharya
aa r X i v : . [ g r- q c ] A ug Study of Thermodynamic Quantities in Generalized GravityTheories
Surajit Chattopadhyay ∗ , Ujjal Debnath † and Samarpita Bhattacharya ‡ Department of Computer Application (Mathematics Section),Pailan College of Management and Technology, Bengal Pailan Park, Kolkata-700 104, India. Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-711 103, India. (Dated: November 8, 2018)In this work, we have studied the thermodynamic quantities like temperature of the universe, heatcapacity and squared speed of sound in generalized gravity theories like Brans-Dicke, Hoˇrava-Lifshitzand f ( R ) gravities. We have considered the universe filled with dark matter and dark energy. Alsowe have considered the equation of state parameters for open, closed and flat models. We haveobserved that in all cases the equation of state behaves like quintessence. The temperature andheat capacity of the universe are found to decrease with the expansion of the universe in all cases.In Brans-Dicke and f ( R ) gravity theories the squared speed of sound is found to exhibit increasingbehavior for open, closed and flat models and in Hoˇrava-Lifshitz gravity theory it is found to exhibitdecreasing behavior for open and closed models with the evolution of the universe. However, for flatuniverse, the squared speed of sound remains constant in Hoˇrava-Lifshitz gravity. I. INTRODUCTION
Recently, it has become well known that the universe has not only undergone the period of early-timeaccelerated expansion (inflation), but also is currently in the so-called late-time accelerating epoch (dark energyera). The unified description of inflation and dark energy is achieved by modifying the gravitational action atthe very early Universe as well as at the very late times [1, 2]. A number of viable modified gravity theorieshas been suggested [3, 4, 5, 6, 7]. In reference [8], the connection between modified gravity and M-string theorywas indicated. The modified gravity gives the qualitative answers to the number of fundamental questionsabout dark energy. Indeed, the origin of dark energy may be explained by some sub-leading gravitationalterms which become relevant with the decrease of the curvature (at late times). Moreover, there are manyproposals to consider the gravitational terms relevant at high curvature (perhaps, due to quantum gravityeffects) as the source of the early-time inflation. Hence, there appears the possibility to unify and to explainboth: the inflation and late-time acceleration as the modified gravity effects [8]. Reviews on modified gravityare available in the references like [9] and [10]. Among the recent attempts to construct a consistent theory ofquantum gravity, much attention has been paid to the quite remarkable Hoˇrava-Lifshitz quantum gravity [11].An extensively studied generalization of general relativity involves modifying the Einstein-Hilbert Lagrangianin the simplest possible way, replacing R −
2Λ by a more general function f ( R ) [10, 11, 12]. Recently themodified Hoˇrava-Lifshitz f ( R ) gravity has been proposed in ref.[1]. Discussions on Hoˇrava-Lifshitz gravity havebeen made in references [13, 14]. The basic idea of Hoˇrava-Lifshitz gravity is to modify the UV behavior ofthe general theory so that the theory is perturbatively renormalizable [14]. However this modification is onlypossible on condition when we abandon Lorentz symmetry in the high energy regime [14]. In reference [15],the very interesting physical implications of Hoˇrava-Lifshitz gravity are summarized as: (i) the novel solutionsubclasses, (ii) the gravitational wave production, the perturbation spectrum, (iii) the matter bounce, (iv) thedark energy phenomenology, (iv) the astrophysical phenomenology, and (v) the observational constraints onthe theory. Recently, scalar-tensor theories have received renewed interest. The Brans-Dicke theory [16] is thesimplest example of a scalar-tensor theory of gravity. In Brans-Dicke theory, Newton’s constant becomes afunction of space and time, and a new parameter ω is introduced. General relativity is recovered in the limit ω → ∞ [17]. Interacting dark energy [18, 19] and holographic dark energy [20, 21, 22, 23, 24] models havebeen considered in Brans-Dicke theory. Brans-Dicke scalar field as chameleon field has been considered in thereferences [25] and [26].A profound connection between gravity and thermodynamics was first established by Jacobson [27], who first ∗ surajit − † [email protected] , [email protected] ‡ samarpita − [email protected] showed that the Einstein gravity can be derived from the first law of thermodynamics in the Rindler spacetime.Thermodynamic aspects of the cosmological horizons have been reviewed in [28] and [29]. Investigating thegeneralized second law (GSL) of thermodynamics in gravity has gained immense interest in recent years. Aplethora of papers have studied the thermodynamics in Einstein gravity theory [30, 31, 32, 33, 34, 35]. As themodified theory of gravity was argued to be a possible candidate to explain the accelerated expansion of ouruniverse by various authors [36, 37, 38], thus it is interesting to examine the GSL in the extended gravity theories[39, 40, 41, 42]. Thermodynamics has been studied in the brane world scenario [43, 44, 45, 46], Hoˇrava-Lifshitzgravity [47, 48, 49], Brans-Dicke gravity [50, 51, 52] and in f ( R ) gravity [53, 54, 55]. Extending the study of [15],two of the authors of the present paper, examined the validity of the GSL in various cosmological horizons ofa universe governed by the Hoˇrava-Lifshitz gravity and the GSL was proved to be valid in different horizons [56].In the present work, we have studied the thermodynamic quantities of the universe in generalized gravitytheories like Brans-Dicke, Hoˇrava-Lifshitz and f ( R ) gravities. Instead of investigating the validity of the lawsof thermodynamics, we have tried to investigate how the thermodynamic quantities like heat capacity ( C v ),temperature T and squared speed of sound v s behave during the evolution of the universe governed by thesaid gravity theories. In addition to this, the equation of state parameters have also been studied for all of thesaid gravity theories. Organization of the rest of the paper is as follows: In section II, we have discussed thethermodynamic quantities. In sections IIIA, IIIB and IIIC we have discussed the thermodynamic quantitiesunder Brans-Dicke, Hoˇrava-Lifshitz and f ( R ) gravity theories respectively. Finally, in section IV, we havediscussed the results. II.
GENERAL DESCRIPTION OF THERMODYNAMIC QUANTITIES
The Einstein field equations for homogeneous, isotropic FRW universe are given by [57] (choosing c = 1) H + ka = 8 πG ρ (1)and ˙ H − ka = − πG ( ρ + p ) (2)where H (= ˙ aa ) is the Hubble parameter and k = 0 , − , +1 denote the curvature index for flat, open andclosed universe respectively. Here, ρ and p denote the energy density and pressure of the universe. The energymomentum tensor T µν is conserved by virtue of the Bianchi identities, leading to the continuity equation [57]˙ ρ + 3 H ( ρ + p ) = 0 (3)where p is the isotropic pressure and ρ is the energy density of the fluid defined by ρ = UV (4)Here, U is the internal energy and V is the volume of the universe.We consider the FRW universe treated as a thermodynamical system. Then from Gibb’s equation of thermo-dynamics, we have [33] T dS = d ( ρV ) + pdV = d (( ρ + p ) V ) − V dp (5)where S is the entropy, T is the temperature and V is the volume of the universe. The integrability conditionof thermodynamic system is given by [58] ∂ S∂T ∂V = ∂ S∂V ∂T (6)which leads to the relation between pressure, energy density and temperature as dp = ρ + pT dT (7)From (5) and (7), we get dS = d (cid:18) ( ρ + p ) VT (cid:19) (8)and integrating, we can obtain the expression of the entropy as (except for an additive constant) S = ( ρ + p ) VT (9)However, for adiabatic process entropy is constant and consequently, the equation (5) becomes d [( ρ + p )] = V dp (10)Relation (9) can also be obtained using (7) into (10). Hence for adiabatic process equation (9) may beconsidered as the temperature defining equation as T = ( ρ + p ) VS (11)The square speed of sound and heat capacity are defined by v s = ∂p∂ρ (12)and C V = V ∂ρ∂T (when entropy S is constant = S , say) (13)These thermodynamic quantities would be investigated for their evolution with the expansion of the universein the subsequent sections. III.
THERMODYNAMIC QUANTITIES IN GENERALIZED GRAVITY THEORIES A. Brans-Dicke Theory
The Jordan-Fierz-Brans-Dicke theory (heretofore, we will call it Brans-Dicke (BD) theory for simplicity) isthe simplest example of a scalar-tensor theory of gravity. A brief introduction of the BD theory has beenpresented in the previous section. The Lagrangian density for the Brans-Dicke theory is L = √− g (cid:20) − φR + ωφ g µν ∂ µ φ + L m (cid:21) (14)where φ is the Brans-Dicke field, and L m is the Lagrangian density for the matter fields. The self-interactingBD theory is described by the Jordan-Brans-Dicke (JBD) action (choosing c = 1) as: S = Z d x √− g π (cid:20) φR − ω ( φ ) φ φ ,α φ, α − V ( φ ) + 16 π L m (cid:21) (15)where V ( φ ) is the self-interacting potential for the BD scalar field φ and ω ( φ ) is modified version of the BDcoupling parameter which is a function of φ . In this theory φ plays the role of the gravitational constant G .This action also matches with the low energy string theory action for ω = −
1. The matter content of theUniverse is composed of matter fluid, so the energy-momentum tensor is given by T mµν = ( ρ + p ) u µ u ν + p g µν (16)where u µ is the four velocity vector of the matter fluid satisfying u µ u ν = − ρ, p are respectively energydensity and isotropic pressure.From the Lagrangian density we obtain the field equations G µν = 8 πφ T mµν + ω ( φ ) φ (cid:20) φ ,µ φ ,ν − g µν φ ,α φ ,α (cid:21) + 1 φ [ φ ,µ ; ν − g µν φ ] − V ( φ )2 φ g µν (17)and φ = 8 πT ω ( φ ) −
13 + 2 ω ( φ ) (cid:20) V ( φ ) − φ dV ( φ ) dφ (cid:21) − dω ( φ ) dφ ω ( φ ) φ, µ φ ,µ (18)where T = T mµν g µν . Equation (17) can also be written as G µν = 8 π ˜ T µν = 8 πφ (cid:18) T mµν + 18 π T φµν (cid:19) (19)where ˜ T µν can be treated as effective energy momentum tensor. The line element for Friedman-Robertson-Walker space-time is given by ds = − dt + a ( t ) (cid:20) dr − kr + r ( dθ + sin θdφ ) (cid:21) (20)where, a ( t ) is the scale factor and k (= 0 , − , +1) is the curvature index describe the flat, open and closedmodel of the universe.We are considering the universe filled with dark energy (with energy density ρ D ) and dark matter (withenergy density ρ m ). As we are not considering interacting situation, the conservation equations are separatelysatisfied for dark matter and dark energy. Thus˙ ρ D + 3 H ( ρ D + p D ) = 0 (21)and ˙ ρ m + 3 H (1 + w m ) ρ m = 0 (22)Solving the conservation equation for dark matter, we get the density and pressure of dark matter as ρ m = ρ m (1 + z ) w m ) and p m = ρ m (1 + w m )(1 + z ) w m ) . Defining ρ = ρ m φ and p = p m φ the Einstein’s fieldequations can be written as - - - - - - w t o t a l Fig.1Fig. 1 shows the EOS parameter w total = p + p D ρ + ρ D for k = − k = 1 (the green line) and k = 0 (the blueline) for Brans-Dicke model where dark energy and dark matter satisfy the conservation equation separately. We havetaken α = 3, B = 0 . ρ m = 0 . w m = 0 . H + ka = 8 π ρ + ρ D ) (23)˙ H − ka = − π ( ρ + p + ρ D + p D ) (24)where ρ D = ω π ˙ φ φ − π H ˙ φφ + V ( φ )16 πφ (25)and p D = ω π ˙ φ φ + H π ˙ φφ + 18 π ¨ φφ − V ( φ )16 πφ (26)To find the thermal quantities we use the choices of φ and V as φ = φ a α , V = V φ − wm ) α (27)Using EOS for dark energy p D = wρ D and the solution of (21) ρ D = ρ D a − w ) in the field equations weget H = kAa − + Ba − α ((1+ ω ) α − α + Ca − α − w ) (28)˙ H = kA a − + B a − α ((1+ ω ) α − α + C a − α − w ) (29) - C v - T Fig.2 Fig.3 - - v s Fig.4Fig. 2 shows the heat capacity C v for k = − k = 1 (the dotted line) and k = 0 (the broken line) forBrans-Dicke model where dark energy and dark matter satisfy the conservation equation separately. We have taken α = 3, B = 0 . ρ m = 0 . w m = 0 . T with evolution of the universe for k = − k = 1 (the green line) and k = 0 (the blue line) for Brans-Dicke model. We have taken α = 3, B = 0 . ρ m = 0 . w m = 0 . v s with evolution of the universe for k = − k = 1 (the greenline) and k = 0 (the blue line) for Brans-Dicke model. We have taken α = 3, B = 0 . ρ m = 0 . w m = 0 . where, A = α ((1+ ω ) α − − (2+ α ) ; C = − π (1+ w ) ρ D φ [(2+ α )( − α − w ))+2 α ((1+ ω ) α − A = α (1 − α (1+ ω )) A α ; B = α (1 − α (1+ ω )) B α ; C = α (1 − α (1+ ω )) φ C − π (1+ w ) ρ (2+ α ) φ A = 6 φ + AB /B ; B = 6 ρ D (1 + α − ωα / B ; C = B C/B − πρ D (30)Equation of state parameter w total = p + p D ρ + ρ D is computed for flat, open as well as closed universes and areplotted against redshift z in figure 1 and it is found that in all of the above cases the EOS parameters arestaying above −
1, which indicates quintessence-like behavior. The behaviour of the EOS parameters furtherindicate that the energy density is increasing with evolution of the universe irrespective of it curvature.We replace p and ρ by ( p + p D ) and ( ρ + ρ D ) in equation (11) we get the temperature T , which is used inequation (13) to get the heat capacity C v . Heat capacity C v is computed for open, closed and flat universesand are plotted in figure 2. This figure shows that for all of the three universes, the heat capacity is decreasingwith increase in the redshift. This means that the heat capacity is increasing with evolution of the universe - - - - - - - - - w t o t a l Fig.5Fig. 5 shows the behavior of the the equation of state parameter w total with evolution of the universe for k = − k = 1 (the green line) and k = 0 (the blue line) for Hoˇrava-Lifshitz gravity. We have taken λ = 1 . µ = 1 . w m = 0 .
03 and ρ m = 0 . irrespective of its curvature. Also, we present the temperature T against redshift z for all of the curvatures.We find that the temperature is decreasing with evolution of the universe. B. Ho ˇ rava-Lifshitz Gravity Thermodynamics in cosmology has been extensively studied either in Einsteins theory of gravity or in modifiedtheories of gravity. In this section, we shall generalize such studies to the Hoˇrava-Lifshitz (HL) Cosmology. Anexhaustive review of HL cosmology is available in [59]. We briefly review the scenario where the cosmologicalevolution is governed by HL gravity. The dynamical variables are the lapse and shift functions, N and N i respectively, and the spatial metric g ij . In terms of these fields the full metric is written as [60] ds = − N dt + g ij ( dx i + N i dt )( dx j + N j dt ) (31)where indices are raised and lowered using g ij . The scaling transformation of the coordinates reads: t → l t and x i → lx i .The action of the HL gravity is given by [60] I = dt R dtd x ( L + L + L m ) L = √ gN h κ ( K ij K ij − λK ) + κ µ (Λ R − )8(1 − λ ) i L = √ gN h κ µ (1 − λ )32(1 − λ ) R − κ ω ( C ij − µω R ij )( C ij − µω R ij ) i (32)where, κ , λ , µ , ω and Λ are constant parameters, and C ij is Cotton tensor (conserved and traceless, vanishingfor conformally flat metrics). The first two terms in L are the kinetic terms, others in ( L + L ) give the potentialof the theory in the so-called “detailed-balance” form, and L m stands for the Lagrangian of other matter field.Comparing the action to that of the general relativity, one can see that the speed of light and the cosmological Newtons constant are - - T - - C v ]Fig.6 Fig.7Fig. 6 shows the behavior of temperature T with evolution of the universe for k = − k = 1 (the greenline) and k = 0 (the blue line) for Hoˇrava-Lifshitz gravity. We have taken λ = 1 . µ = 1 . w m = 0 .
03 and ρ m = 0 . C v with evolution of the universe for k = − k = 1 (the greenline) and k = 0 (the blue line) for Hoˇrava-Lifshitz gravity. We have taken λ = 1 . µ = 1 . w m = 0 .
03 and ρ m = 0 . - - v s Fig.8Fig. 8 shows the squared speed of sound v s with evolution of the universe for k = − k = 1 (the greenline) and k = 0 (the blue line) for Hoˇrava-Lifshitz gravity. We have taken λ = 1 . µ = 1 . w m = 0 .
03 and ρ m = 0 . c = κ µ r Λ1 − λ , G c = κ c π (3 λ −
1) (33)It may be noted that when λ = 1, L reduces to the usual Lagrangian of Einsteins general relativity. Thus,when λ = 1, the general relativity is approximately recovered at large distances.As we are considering dark energy with dark matter the conservation equations are given by (21) and (22).The field equations are H + ka = 8 πG c ρ m + ρ D ) (34)and ˙ H + 32 H + k a = − πG c ( p m + p D ) (35)where ρ D ≡ κ µ k λ − a + 3 κ µ Λ λ − ≡ πG c (cid:18) k Λ a + 3Λ (cid:19) (36)and p D ≡ κ µ k λ − a − κ µ Λ λ − ≡ πG c (cid:18) k Λ a − (cid:19) (37)Using the solution for the conservation equation for dark matter given in (22) we get the total energy densityas a function of redshift ( z = a −
1) as ρ ( z ) = ρ m + ρ D = ρ m (1 + z ) w m ) + 116 πG c (cid:18) k (1 + z ) Λ + 3Λ (cid:19) (38)Similarly, the total pressure as a function of redshift z is p ( z ) = p m + p D = ρ m w m (1 + z ) w m ) + 116 πG c (cid:18) k (1 + z ) Λ − (cid:19) (39)The squared speed of sound is given as a function of z by v s ( z ) = k (1 + z ) κ µ + 6(1 + z ) w m ( − λ ) w m ρ m (1 + w m )3 { k (1 + z ) κ µ + 2(1 + z ) w m ( − λ )(1 + w m ) ρ m } (40)and the heat capacity becomes C v ( z ) = 3 S { k (1 + z ) κ µ + 2(1 + z ) w m ( − λ )(1 + w m ) ρ m } k (1 + z ) κ µ + 6(1 + z ) w m ( − λ ) w m ρ m (1 + w m ) (41)The thermodynamic quantities expressed above are now plotted against redshift to see their behavior withthe evolution of the universe. In figure 5, where we have plotted the equation of state parameter for the Hoˇrava-Lifshitz gravity, we see that the behavior is like Brans-Dicke theory. It is staying above −
1. However, at lowerredshifts the equation of state parameter is tending to −
1. However, it never crosses −
1. Like Brans-Dicke,this behavior remains the same for flat, open and closed universes. In figure 6 we find that in the case of flat,open and closed universes, the temperature T is decreasing with the evolution of the universe. From figure 6we see that the heat capacity C v is increasing as we are approaching towards the lower redshifts. From figure7 we understand that for open and closed universes, the squared speed of sound v s decreases with the evolu-tion of the universe. However, for flat universe, the v s remains constant throughout the evolution of the universe. C. f ( R ) Gravity
Motivated by astrophysical data which indicate that the expansion of the universe is accelerating, the modifiedtheory of gravity (or f ( R ) gravity) which can explain the present acceleration without introducing dark energy,has received intense attention. Extensive review of f ( R ) gravity is available in [61]. The action of f ( R ) gravityis given by [62] S = Z d x √− g (cid:20) f ( R )2 κ + L matter (cid:21) (42)0where g is the determinant of the metric tensor g µν , L matter is the matter Lagrangian and κ = 8 πG . The f ( R )is a non-linear function of the Ricci curvature R that incorporates corrections to the Einstein-Hilbert actionwhich is instead described by a linear function f ( R ). The gravitational field equations in this theory are H + ka = κ f ′ ( R ) ( ρ + ρ c ) (43)˙ H − ka = − κ f ′ ( R ) ( ρ + p + ρ c + p c ) (44)where ρ c and p c can be regarded as the energy density and pressure generated due to the difference of f ( R )gravity from general relativity given by [61] (choosing G = 1) ρ c = 18 πf ′ (cid:20) − f − Rf ′ − Hf ′′ ˙ R (cid:21) (45) p c = 18 πf ′ (cid:20) f − Rf ′ f ′′ ¨ R + f ′′′ ¨ R + 6 f ′′ ˙ R (cid:21) (46)where, the scalar tensor R = − (cid:16) ˙ H + 2 H + ka (cid:17) .As we are considering both dark matter and dark energy, the Friedman equations take the form H + ka = 8 π ρ total (47)˙ H − ka = − π ( ρ total + p total ) (48)where, ρ total = ρ + ρ c , p total = p + p c (49)with ρ = ρ m f ′ and p = p m f ′ . As there is no interaction, like the previous two cases the dark energy and darkmatter satisfy the conservation equation separately. Therefore, we have the density and pressure of dark matteras ρ m = ρ m (1 + z ) w m ) and p m = ρ m (1 + w m )(1 + z ) w m ) . In the present section, while considering the f ( R ) gravity, we have illustrated with a solution, f ( R ) = βR + αR m , R = Aa n , m > , α > , β > n ( m −
1) = 1.Using the above form of f ( R ) in (44) and (45) we have computed the temperature T , squared speed of sound v s and heat capacity C v as functions of the redshift z as follows: ρ = (1 + z ) w m ) ρ m m ( A (1 + z ) n ) − m α + β (51) p = w m (1 + z ) w m ) ρ m m ( A (1 + z ) n ) − m α + β (52)1 - - C v - T Fig.9 Fig.10 - v s Fig.11Fig. 9 shows the plot of heat capacity C v against redshift z in f ( R ) gravity. We see that C v is increasing with theevolution of the universe. Here α = 2 . , β = 2 . , w m = 0 . , m = 2 . , ρ m = 0 .
23 and the red, green andblue lines correspond to k = − , , T against redshift z . We find that the temperature is decreasing with theevolution of the universe in f ( R ) gravity α = 12 . , β = 10 . , w m = 0 . , m = 12 . , ρ m = 0 .
23 and the red, greenand blue lines correspond to k = − , , v s against redshift z in f ( R ) gravity. We find that v s is increasingwith the evolution of the universe. Here, α = 12 . , β = 10 . , w m = 0 . , m = 1 . , ρ m = 0 .
23 and the red, green andblue lines correspond to k = − , , ρ c = α ( − m ) A m (1 + z ) mn π ( mA − m (1 + z ) n ( − m ) α + β ) mn (2 m − z ) − n (cid:16) C − k (1+ z ) + A (1+ z ) − n n − (cid:17) A (53) p c = A (1+ z ) n mπ (cid:8) − A + 6( − m ) mn (1 + n ) (1 + z ) − n ++6 mn (1 + z ) − n ( − n + z + nz ) q C − k (1+ z ) + A (1+ z ) − n − n ) o (54)Using (51), (52), (53) and (54) we get temperature T , squared speed of sound v s and heat capacity C v asfunctions of redshift z in the following forms T = ρ + p + ρ c + p c (1 + z ) S (55)2 v s = ξ ( z ) ξ ( z ) (56)where ξ ( z ) = (cid:2) m ( A (1 + z ) n ) − m α + β ) × An (1+ z ) − n ( − A +6( − m ) mn (1+ n ) (1+ z ) − n +6 mn (1+ z ) − n ( − n + z + nz ) q C − k (1+ z )2 + A (1+ z ) − n − n ) )24 mπ + A mπ × − − m ) m ( − n ) n (1 + n ) (1 + z ) n + mn (1+ z ) n ( − n + z + nz ) (cid:0) k (1+ z )3 + A (1+ z ) − n (cid:1)q C − k (1+ z )2 + A (1+ z ) − n − n ) +6 mn (1 + n )(1 + z ) n q C − k (1+ z ) + A (1+ z ) − n − n ) − m ( − n ) n (1 + z ) n ( − n + z + nz ) × q C − k (1+ z ) + A (1+ z ) − n − n ) (cid:17) + w m (1+ w m )(1+ z ) n +3 wm ρ m m ( A (1+ z ) n ) − m α + β − A ( − m ) mnw m (1+ z ) n +3 wm ( A (1+ z ) n ) m αρ m ( m ( A (1+ z ) n ) m α + A (1+ z ) n β ) oi (57)and ξ ( z ) = π (1+ z ) − ( − m ) mn ( A (1 + z ) n ) m − m ) mn (1+ z ) − n (cid:16) C − k (1+ z )2 + A (1+ z ) − n − n (cid:17) A α ×{ ( − m )( A (1 + z ) n ) − m α + m ( A (1 + z ) n ) − m α + β } (cid:3) + − m )( − m ) mn (1+ z ) − n ( A (1+ z ) n ) m (cid:16) − − n ) C + k (4 − n + n z )2 − An (1+ z ) n ( − n )(1+ z )4 (cid:17) α ( m ( A (1+ z ) n ) − m α + β ) A π − − m ) mn (1 + z ) w m ( A (1 + z ) n ) − m αρ m + 48(1 + w m )(1 + z ) w m ( m ( A (1 + z ) n ) − m α + β ) ρ m (58)and C v = ζ ( z ) ζ ( z ) (59)where ζ ( z ) = ( − m ) mn (1+ z ) − n ( A (1+ z ) n ) − m n A +2( − m ) mn (1+ z ) − n (cid:16) C − k (1+ z )2 + A (1+ z ) − n − n (cid:17)o Aπ ( m ( A (1+ z ) n ) − m α + β ) α × n α ( − m )( A (1+ z ) n ) − m ( m ( A (1+ z ) n ) − m α + β ) o ++ ( − m )( A (1+ z ) n ) m n − m ) mn (1+ z ) − n (cid:0) k (1+ z )3 + A (1+ z ) − n (cid:1) +2( − m ) m (4 − n ) n (1+ z ) − n (cid:16) C − k (1+ z )2 + A (1+ z ) − n − n (cid:17)o α π ( m ( A (1+ z ) n ) − m α + β ) A − A ( − m ) mn (1+ z ) − n +3(1+ wm ) ( A (1+ z ) n ) − m αρ m ( m ( A (1+ z ) n ) − m α + β ) + w m )(1+ z ) − wm ) ρ m m ( A (1+ z ) n ) − m α + β (60)3 - - - - - w t o t a l Fig.12Fig. 12 shows the evolution of the equation of state parameter w total with the evolution of the universe in f ( R ) gravity.We find that for k = − , , w total > −
1. The indicates quintessence era.Here, α = 10 . , β = 10 . , w m = 0 . , m = 12 . , ρ m = 0 .
23 and the red, green and blue lines correspond to k = − , , and ζ ( z ) = − S (1+ z ) A (1+ z ) n (cid:16) − A +6( − m ) mn (1+ n ) (1+ z ) − n +6 mn (1+ z ) − n ( − n + z + nz ) q C − k (1+ z )2 + A (1+ z ) − n − n ) (cid:17) mπ ++ ( − m )( A (1+ z ) n ) m (cid:16) A +2( − m ) mn (1+ z ) − n (cid:16) C − k (1+ z )2 + A (1+ z ) − n − n (cid:17)(cid:17) α πA ( m ( A (1+ z ) n ) − m α + β ) + (1+ z ) wm ) ρ m m ( A (1+ z ) n ) − m α + β + w m (1+ z ) wm ) ρm m ( A (1+ z ) n ) − m α + β + S (1+ z ) × A (1+ z ) n mπ − − m ) m ( − n ) n (1 + n ) (1 + z ) − n + mn (1+ z ) − n ( − n + z + nz ) (cid:0) k (1+ z )3 + A (1+ z ) − n (cid:1)q C − k (1+ z )2 + A (1+ z ) − n − n ) +6 mn (1 + z ) − n q C − k (1+ z ) + A (1+ z ) − n − n ) ((1 + n )(1 + z ) − ( − n )( − n + z + nz )) (cid:17) + An (1+ z ) − n (cid:16) − A +6( − m ) mn (1+ n ) (1+ z ) − n +6 mn (1+ z ) − n ( − n + z + nz ) q C − k (1+ z )2 + A (1+ z ) − n − n ) (cid:17) mπ − ( − m ) mn ( A (1+ z ) n ) m α n A +2( − m ) mn (1+ z ) − n (cid:16) C − k (1+ z )2 + A (1+ z ) − n − n (cid:17)o A (1+ z ) π ( m ( A (1+ z ) n ) − m α + β ) (cid:16) α ( − m )( A (1+ z ) n ) − m ( m ( A (1+ z ) n ) − m α + β ) + 1 (cid:17) ( − m )( − m ) mn (1+ z ) − n ( A (1+ z ) n ) m (cid:16) − − n ) C + k (4 − n + n z )2 − An (1+ z ) n ( − n )(1+ z )4 (cid:17) α A π ( m ( A (1+ z ) n ) − m α + β ) + w m )(1+ z ) wm ρ m m ( A (1+ z ) n ) − m α + β + w m (1+ w m )(1+ z ) wm ρ m m ( A (1+ z ) n ) − m α + β − A ( − m ) mn (1+ z ) n +3 wm ( A (1+ z ) n ) m αρ m ( m ( A (1+ z ) n ) m α + A (1+ z ) n β ) (1 + w m ) i (61)The thermodynamic quantities expressed above are now plotted against redshift z to see their behaviorwith the evolution of the universe. We proper choice of the parameters we plot all of the quantities infigures 8, 9, and 10 respectively. In figure 8 we find the increasing behavior of the heat capacity with theevolution of the universe. This behavior remains the same irrespective of the curvature of the universe.From figure 9 we see that as the universe in evolving, the temperature T is decreasing. Here also weget the same behavior for open, closed and flat universes. It may be interpreted that the temperature ofthe universe decreases as it expands under f ( R ) gravity. In figure 8 we plot the squared speed of sound v s . Like the temperature, v s is decreasing with the expansion of the universe under f ( R ) gravity. The4choices of the parameters are mentioned in the figure captions. Behavior of the equation of state parameter w total is observed in figure 11. Throughout the evolution of the universe w total > −
1. This indicatesquintessence like behavior of the equation of state parameter. Therefore, we see that in f ( R ) gravity,where we are considering the coexistence of dark energy and dark matter with very small pressure withoutinteraction, the equation of state parameter behaves like quintessence era. This holds true for flat, closedas well as open universes. However, it also discerned that w total is gradually increasing in the negative direction. IV.
DISCUSSIONS
In the present work, we have considered modified gravities as Brans-Dicke, Hoˇrava-Lifshitz and f ( R )gravities. Various thermodynamic quantities like temperature, heat capacity and squared speed of sound havebeen investigated for all the gravity theories. In each case we have considered that the universe is filled withdark matter and dark energy which are not interacting. Prior to evaluating the thermodynamic quantitieswe have studied the behaviors of the equation of state parameters. In figure 1 we see that in the case ofBrans-Dicke gravity theory, the equation of state parameter is staying above − −
1. However, at lower redshifts the equation of state parameter is tending to −
1. However, it never crosses −
1. Like Brans-Dicke, this behavior remains the same for flat, open and closed universes. Similar behaviorof the equation of state parameter is discernible in f ( R ) gravity also. The evolution of the equation of stateparameter for f ( R ) gravity has been presented in figure 12. From the figures 3, 6 and 10 we see that thetemperature T of the universe is decreasing with evolution of the universe in Brans-Dicke, Hoˇrava-Lifshitz and f ( R ) gravities respectively. This behavior remains the same in open, closed and flat universes. From figures2, 7 and 9 we find that the heat capacity C v of the universe increases with the evolution of the universe.Moreover, in all of the cases C v remains at the positive level throughout the evolution of the universe. We havealso investigated the squared speed of sound v s in all of the cases. From figure 4 we see that for closed universe v s starts decreasing from redshift − .
2. However, up to z = − . v s has an increasing behavior throughout the evolution of the universe. From figure 8we see that in Hoˇrava-Lifshitz gravity v s has a decaying behavior throughout the evolution of the universe inthe case of open and closed universes. However, for flat ( k = 0) universe, the squared speed of sound remainsconstant throughout the evolution of the universe. From figure 11 we understand that v s increases throughoutthe evolution of the universe irrespective of the curvature of the universe. Acknowledgement:
The authors wish to sincerely acknowledge the warm hospitality provided by Inter-University Centre forAstronomy and Astrophysics (IUCAA), Pune, India, where part of the work was carried out during a scientificvisit in January, 2011.
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