Thermodynamics of the Variable Modified Chaplygin gas
aa r X i v : . [ g r- q c ] M a r Thermodynamics of the Variable ModifiedChaplygin gas
D. Panigrahi and S. Chatterjee Abstract
A cosmological model with a new variant of Chaplygin gas obeying anequation of state(EoS), P = Aρ − Bρ α where B = B a n is investigated inthe context of its thermodynamical behaviour. Here B and n are constantsand a is the scale factor. We show that the equation of state of this ‘Vari-able Modified Chaplygin gas’ (VMCG) can describe the current acceleratedexpansion of the universe. Following standard thermodynamical criteria wemainly discuss the classical thermodynamical stability of the model and findthat the new parameter, n introduced in VMCG plays a crucial role in deter-mining the stability considerations and should always be negative. We furtherobserve that although the earlier model of Lu explains many of the currentobservational findings of different probes it fails the desirable tests of ther-modynamical stability. We also note that for n <
KEYWORDS : cosmology;Chaplygin gas;thermodynamicsPACS : 98.80.-k,98.80.Es,95.30.Tg,05.70.Ce
1. Introduction
Following the high redshift supernovae data in the last decade [1, 2] we know thatwhen interpreted within the framework of the standard FRW type of universe (ho-mogeneous and isotropic) we are left with the only alternative that the universeis now going through an accelerated expansion with baryonic matter contributingonly 5% of the total budget. Later data from CMBR studies [2] further corrobo-rate this conclusion which has led a vast chunk of cosmology community ( [3] andreferences therein) to embark on a quest to explain the cause of the acceleration.In fact the studies on accelerated expansion and its possible interpretations fromdifferent angles have been reigning the research paradigm for the last few decades.The teething problem now confronting researchers in this field is the identificationof the mechanism that triggered the late inflation. But as they are already discussed Sree Chaitanya College, Habra 743268, India and also
Relativity and Cosmology ResearchCentre, Jadavpur University, Kolkata - 700032, India , e-mail: [email protected] Relativity and Cosmology Research Centre, Jadavpur University, Kolkata - 700032, India,e-mail : chat [email protected] e.g. quintessence) [5], phan-tom [6], holographic models [7], string theory landscape [8], Born-Infeld quantumcondensate [9], modified gravity approaches [10], inhomogeneous spacetime [11],higher dimensional space time [12] etc.While the above mentioned alternatives to address the observed acceleration ofthe current phase have both positive and negative aspects a number of papers havecome up taking into account the Chaplygin gas [13–15] as a new form of matterfield to simulate unified dark energy and dark matter model. It is presumed thatfor gravitational attraction, the dark matter component is responsible for galaxystructure formation while dark energy provides necessary repulsive force for currentaccelerated expansion.Motivated by the desire to explain away the observational fallouts better andbetter the form of the equation of state (in short, EoS) of matter is later generalisedin stages first through the addition of an arbitrary constant with an exponent overthe mass density, generally referred to as generalised Chaplygin gas (GCG) [3].Barring the serious disqualification e.g. , it violates the time honoured principle ofenergy conditions, it is fairly successful to interpret the observational results comingout of gravitational lensing or recent CMBR and SNe data in varied cosmic probesvia the fine tuning of the value of the newly introduced arbitrary constant. The formof EoS is again modified through the addition of an ordinary mater field, which istermed in the literature as modified Chaplygin gas (MCG) [16–18], claiming an evenbetter match with observational results.It may be appropriate at this stage to call attention to a recent work by Guo andZhang [19,20] where a very generalised form of the Chaplygin gas relation is invoked,assuming the constant B to depend on the scale factor a . Taking , e.g. , B = B a − n ,it is shown that for a very large value of the scale factor the model interpolatesbetween a dust-dominated phase and a quiessence phase (i.e., dark energy with aconstant equation of state ) [12, 14] given by W = − n .As mentioned earlier, although different variants of dark energy models as wellas Chaplygin type of gas models are brought in the cosmological arena as alsoits associated success to explain the observations coming out of different cosmicprobes it has not escaped our notice that scant attention has been paid so farto address the important issue if all the so called perfect gas models are atleastthermodynamically stable. Otherwise they would lose the claim to be treated asa physically realistic system. Following this there has been of late a resurgenceof interests among workers to address their queries to this aspect of the problem.Recently Santos et al have studied the thermodynamical stability in generalised [21]and modified Chaplygin [22] gas model on the basis of standard prescription [23]where both (i) (cid:0) ∂P∂V (cid:1) S < (cid:0) ∂P∂V (cid:1) T < c V > P ), effectiveequation of state ( W ), deceleration parameter ( q ) etc. on the basis of thermody-namics. Later many thermal quantities are derived as functions of temperature orvolume. In this case, we also show as consistency check that the third law of ther-modynamics is satisfied by the new form of the Chaplygin gas. For the generalisedChaplygin gas we expect to have almost similar behaviour as the Chaplygin gasequations did show. Further, we see that the Chaplygin gas which shows a unifiedpicture of dark matter and energy cools down through the adiabatic expansion ofthe universe without any critical point. Returning to the stability criterion of theChaplygin gas we find a striking difference from the analysis of Santos et al. Inour case of variable modified Chaplygin gas (VMCG) [24] we find that the stabilitydepends critically on the new parameter n , as introduced by Guo et al. We alsonotice that stability decreases with the increase of magnitude of n , which apparentlydisfavors the formalism of Guo-Zhang [19]. In the section on acoustic velocity weinterestingly note that unlike the previous Chaplygin gas models where the squaredsound velocity is always positive definite, the velocity here depends on the value ofthe newly introduced parameter n . But as noted earlier the stability criteria dictatesthat n should be negative which, however, makes the squared velocity also negativeat the late stage of evolution. This means that the perfect fluid model for Guo’svariant of the chaplygin gas is classically unstable and is in line with the resultsobtained earlier by Myung [25] while dealing with the holographic interpretationfor Chaplygin gas and tachyon. The paper ends with a short discussion.
2. Formalism
The line element corresponding to spatially flat FRW spacetime is given by ds = dt − a ( t ) (cid:0) dr + r dθ + r sin θdφ (cid:1) , (1)where a ( t ) is the scale factor. In this work we consider the following equation ofstate P = Aρ − Bρ α . (2) A and B are positive constants. As discussed in the introduction we have taken B = B V − n where n is an arbitrary constant and B an absolute constant. Here P corresponds to the pressure and ρ the energy density of that fluid such that ρ = UV , (3)where U and V are the internal energy and volume filled by the fluid respectively.We try to find out the energy density U and pressure P of Variable ModifiedChaplygin gas as a function of its entropy S and volume V . From general thermo-dynamics, one has the following relationship (cid:18) ∂U∂V (cid:19) S = − P. (4)3ith the help of the above equations we get (cid:18) ∂U∂V (cid:19) S = B V − n V α U α − A UV , (5)which, on integration yields U = " B (1 + α ) V α ) − n A + 1)(1 + α ) − n + cV A (1+ α ) α . (6)The parameter c is the integration constant, which may be a universal constant ora function of entropy S only. Now we rewrite the above equation in the followingform U = (cid:20) B (1 + α ) V − n N (cid:21) α V (cid:20) (cid:16) ǫV (cid:17) N (cid:21) α , (7)where N = A +1)(1+ α ) − n > A + 1)(1 + α ) > n for real U and ǫ = (cid:20) A + 1)(1 + α ) − n B (1 + α ) c (cid:21) N = (cid:20) N cB (1 + α ) (cid:21) N , (8)which has a dimension of volume. Now the energy density ρ of the VMCG comesout to be ρ = (cid:20) B (1 + α ) V − n N + V − N − n c (cid:21) α (9a)= (cid:20) B (1 + α ) V − n N (cid:21) α (cid:20) (cid:16) ǫV (cid:17) N (cid:21) α . (9b)From what has been discussed above we like to obtain the expression of relevantphysical quantities and investigate their behaviour. (a) Pressure : Using equations (2) and (9b) the pressure P of the VMCG may also be deter-mined as a function of entropy S and volume V in the following form P = − (cid:20) B (1 + α ) V − n N (cid:21) α (cid:18) N α (cid:19) h − A (1+ α ) N n (cid:0) ǫV (cid:1) N oih (cid:0) ǫV (cid:1) N i α α . (10)The equation (10) gives a very general expression of pressure. Under special condi-tions 4) For n = 0 & A = 0, the equation (10) reduces to P = − ( B ) α (cid:26) (cid:16) cB V (cid:17) α (cid:27) α α , (11)which is, as expected, similar to generalised Chaplygin gas (GCG) model [21].ii) For n = 0 & A = 0, we get the modified Chaplygin gas (MCG) model withpressure given by [22] P = − (cid:18) B A (cid:19) α (cid:26) A − A (cid:20) n (1+ A ) cB V o (1+ A )(1+ α ) (cid:21)(cid:27)(cid:20) n (1+ A ) cB V o (1+ A )(1+ α ) (cid:21) α α . (12)iii) For A = 0, but n = 0, the model represents the Variable Chaplygin gas(VCGC) [26] and equation (10) becomes P = − (cid:0) B V − n (cid:1) α N (1 + α ) n (cid:0) ǫV (cid:1) N o α α . (13)Again if we put α = 1 in equation (13), we get P = − (cid:20) NB V − n n ( ǫV ) N o (cid:21) which isidentical with our previous work [27] when we consider the Variable Chaplygin gas(VCG) model.The pressure may have positive or negative values, depending on the magnitudeof both A & n and also on volume V (fig - 1). For P = 0, let V is denoted by V c ,which is given by V c = ǫ (cid:20) A (1 + α )3(1 + α ) − n (cid:21) N , (14)which restricts n as n < α ). Initially, i.e. , V < V c , P is positive, which indi-cates a radiation dominated universe. For V = V c , P = 0 and V > V c , P is negativepointing to a state of accelerating universe. This is an interesting result showing that V c introduces a new scale in the analysis, beyond which a dust dominated universeenters the acceleration era. With expansion an initially decelerating universe tendsto reverse its motion and prepares to accelerate when its volume crosses a criticalvalue designated by V c . It is also to be noted that for physically realistic values ofconstants both V c and ǫ are of the same order of magnitude i.e. ǫ also signifies avolume scale beyond which accelerating era commences. So V ≫ ǫ represents a verylarge volume and V ≪ ǫ the reverse. In what follows we will see that this statementhas significant cosmological implications.5 = - 2A = 0A = 1A = 1/30 5 10 15 20 - - - ® P ® (a) The graphs clearly show that pres-sure P can be both positive and nega-tive depending on the value of A . For A = 0, P is always negative, i.e. Chap-lygin type gas with n which is also inaccord with the equation (2). A > 0n = 2n = 0n = - 20 2 4 6 8 10 - - ® P ® (b) It show that pressure P is positivefor small value of V and negative forlarge V when A >
0. As the value of n tends towards negative, P becomesmore negative. A = 0, n = 00.0 0.5 1.0 1.5 2.0 - - - - - ® P ® (c) For A = 0, n = 0, P is always nega-tive, chaplygin type case. Figure 1:
The variation of P and V for different values of A and n . Here we havetaken B = 1 , α = 1 & c = 1 for constant S . (b) Caloric EoS: Now using the expression (9b) and (10) we get the caloric equation of stateparameter W = Pρ = A − (cid:18) N α (cid:19)
11 + (cid:0) ǫV (cid:1) N . (15)As the last EoS is very involved in nature it is very difficult to extract muchphysics out of it. So we look forward to its extremal cases as:1. For small volume, V ≪ ǫ , we get from equation (15) that P ≈ Aρ. (16)This is a barotropic equation of state. In this case no influence of n on smallvolume.2. For large volume, V ≫ ǫ , the equation (15) reduces to W ≈ − n α ) . (17)Equation (14) shows that n < α ). So there are three possibilities for W depending on the signature of n as (i) n > W > −
1, here the caloric equation6 = 2n = 0n = 20 10 20 30 40 50 - - - ® w ® Figure 2:
The variation of W and V for different values of n . The values of constantsare taken as A = , B = 1 , α = 1 , c = 1 . of state results in a quiescence type and big rip is avoided in this case, (ii) n = 0, W = − i.e. , we get ΛCDM. (iii) n < W < − n is prominentin this case.Initially, i.e. , when volume is sufficiently small, W >
0. As V increases to V c , W tends to zero. When V = V c = ǫ h A (1+ α )3(1+ α ) − n i N , W = 0. Again V increases and W becomes negative. We have seen from fig-2 that n = 0, W = − i.e. , ΛCDMmodel which is currently fashionable. But for n > > W > − V . Similar result was shown earlier in higher dimensional case also [12]. We donot discuss much about n > n ≤ (c) Deceleration parameter: Now using equation (15) we calculate the deceleration parameter of the VMCGfluid as q = 12 + 32 Pρ = 12 + 32 ( A − (cid:18) N α (cid:19)
11 + (cid:0) ǫV (cid:1) N ) , (18)For mathematical simplicity here also we discuss the extreme cases.1. For small volume, V ≪ ǫ , it gives q ≈
12 + 32 A, (19) i.e. , q is positive, universe decelerates for small V .2. For large volume, V ≫ ǫ , the equation (18) reduces to q ≈ − n α ) . (20)Initially, i.e. , when volume is very small there is no effect of n on q , here q is positive,universe decelerates. But for large volume, q is negative and this depends on thevalue of n also. For flip in velocity to occur the flip volume ( V f ) becomes7 = 2n = 0n = - 20 10 20 30 40 50 - - - ® q ® Figure 3:
The variation of q and V for different values of n . We have considered here A = , B = 1 , α = 1 & c = 1 . V f = ǫ (cid:20) (1 + 3 A )(1 + α )2(1 + α ) − n (cid:21) N . (21)Therefore n < α ), which interestingly does not violate our previous restrictionon n . A little analysis of the above equation shows that for V f to have real value n < α ), otherwise there will be no flip . This also follows from the fig - 3 whereonly n < flip (for α = 1). Alternatively the inequality n < α ) maybe treated as our acceleration condition. Thus V < V f , we get deceleration; and V > V f , acceleration occur.It has not also escaped our notice that we are here getting two scales ( V c and V f )for volume where apparently acceleration flip occurs. Again we know that in a FRWcosmology for flip to occur pressure should not only be negative but its magnitudeshould be less than ρ ( i.e. , ρ + 3 P < V c < V f , which also follows from their expressions (14) and (21). In this connectionit may also be mentioned that with V = V f , ρ + 3 P = 0 as expected. (d) Velocity of Sound: Let us consider v s be the velocity of sound, then using equation (10) we can write v s = (cid:18) ∂P∂ρ (cid:19) S = A + N α (1 + α ) n (cid:0) ǫV (cid:1) N o − nNn + (3 N + n ) (cid:0) ǫV (cid:1) N . (22)Since sound speed should be 0 < (cid:16) ∂P∂ρ (cid:17) S <
1, now our analysis shows ,1. For small volume, i.e. , at early universe, the equation (22) leads to 0 < A < A = , the radiation dominated universe.2. For large volume, the equation (22) reduces to v s = − n α ) . (23)8he equation (23) does not depend on A , it depends on n and α only. In whatfollows we shall see that from the thermodynsmical stability condition the value of n <
0, leading to a phantom universe [28]. Moreover the equation (23) gives animaginary speed of sound for α >
0, leading to a perturbative cosmology. One neednot be too sceptic about it because it favours structure formation [29].It may not be out of place to draw some correspondence to a recent work byY.S. Myung [25] where a comparison is made between holographic dark energy,Chaplygin gas, and tachyon model with constant potential. We know that theirsquared speeds are crucially important to determine the stability of perturbations.They found that the squared speed for holographic dark energy is always negativewhen imposing the future event horizon as the IR cutoff, while those for originalChaplygin gas and tachyon are always non-negative. This is in sharp contrast to ourvariant (VMCG) of the Chaplygin gas model where we observe that depending onthe signature of n the squared velocity may be both positive or negative. However asdiscussed earlier a non negative value of n is clearly ruled out from thermodynamicstability considerations. This points to the fact that the perfect fluid model forVMCG dark energy model is classically unstable like the holographic interpretationfor Chaplygin gas and tachyon and hence problematic in the long run. (e) Thermodynamical Stability: To verify the thermodynamic stability conditions of a fluid along its evolution,it is necessary (a) to determine if the pressure reduces both for an adiabatic andisothermal expansion [23] (cid:0) ∂P∂V (cid:1) S < (cid:0) ∂P∂V (cid:1) T < (b) also to examine if thethermal capacity at constant volume, c V > (cid:18) ∂P∂V (cid:19) S = P V (1 + α ) A (1 + α ) n n h (cid:0) ǫV (cid:1) N i + 3 N (cid:0) ǫV (cid:1) N o + N (cid:26) Nα ( ǫV ) N h ( ǫV ) N i − n (cid:27) N − A (1 + α ) h (cid:0) ǫV (cid:1) N i . (24)Now we have to examine the negativity of (cid:0) ∂P∂V (cid:1) S . As this expression (24) is soinvolved we can not make any conclusion considering this equation as a whole. Ouranalyses naturally split into two parts :1. Firstly, we have considered small volume, V ≪ ǫ , where equation (24) gives (cid:0) ∂P∂V (cid:1) S ≈ − (1 + A ) PV . We get from previous analysis that at the early stageof evolution P ≈ Aρ , therefore, in this case (cid:0) ∂P∂V (cid:1) S ≈ − A (1 + A ) ρV which isindependent of n but very much depends on A and at this stage of evolution (cid:0) ∂P∂V (cid:1) S < V ≫ ǫ , the equation (24) reduces to (cid:0) ∂P∂V (cid:1) S ≈ − nP V (1+ α ) .Since P is negative at the late stage of evolution, so n must be negative tomake (cid:0) ∂P∂V (cid:1) S <
0. We have seen that the dependence of n is prominent at thelater case. From fig - 4 we get the similar type of conclusion.Now we discuss some special cases to constrain the parameters used here.9 = 2n = 0 n = - 20 5 10 15 20 - - - - - ® d P d V ® Figure 4:
The variation of (cid:0) ∂P∂V (cid:1) S and V for different values of n . The nature of graphsshows that for n ≤ , (cid:0) ∂P∂V (cid:1) S < throughout the evolution unlike n > where (cid:0) ∂P∂V (cid:1) S < only at the early stage. We have taken A = 1 , B = 1 , α = 1 , c = 1 . (i) The simultaneous conditions n = 0, α = 0 and A = 0 must be discardedbecause it will place a severe restriction on the stability of this fluid, in such a case (cid:0) ∂P∂V (cid:1) S = 0 and the pressure will remain the same through any adiabatic changeof volume. However B here behaves like a Cosmological Constant. We get thede-Sitter type of metric.(ii) Again for α = 0, A = 0 and n = 0, (cid:0) ∂P∂V (cid:1) S = n B V − (1+ n ) , i.e. , for n < (cid:0) ∂P∂V (cid:1) S < et al [21]where simultaneously α = 0, A = 0 is not possible. Relevant to mention that, n ≤ A = 0, n = 0 and α = 0, the equation (24) reduces to Santos’s work[21]. In this case (cid:0) ∂P∂V (cid:1) S will be (cid:18) ∂P∂V (cid:19) S = α PV (cid:16) ǫV (cid:17) α (cid:26) (cid:16) ǫV (cid:17) α (cid:27) − , (25)which gives (cid:0) ∂P∂V (cid:1) S < α >
0. It also agrees with the work of Santos et al [21]in this field.(iv) when α = 0, n = 0 and A = 0, (cid:0) ∂P∂V (cid:1) S = − AB V (cid:0) ǫV (cid:1) , i.e. , (cid:0) ∂P∂V (cid:1) S < A > B > A = 0 but α = 0 & n = 0 the equation (24) reduces to (cid:18) ∂P∂V (cid:19) S = PV (cid:0) ǫV (cid:1) N − A A n (cid:0) ǫV (cid:1) N o " A + α (1 + α )1 + (cid:0) ǫV (cid:1) N . (26)This equation (26) is identical with the another work of Santos et al [22].(vi) Again when A = 0 but α = 0 & n = 0 the equation (24) gives (cid:18) ∂P∂V (cid:19) S = P V (1 + α ) " N α (cid:0) ǫV (cid:1) N (cid:0) ǫV (cid:1) N − n . (27)10his is the case of Variable Generalised Chaplygin gas (VGCG) model [26]. A = 0, Α = 1.53, n = 0.755 10 15 20 - - - - - ® d P d V ® Figure 5:
The variation of (cid:0) ∂P∂V (cid:1) S and V is shown for A = 0 , α = 1 . , n = 0 . andalso B = 1 , c = 1 . It is clear from equation (27) that for α > n should be negative for (cid:0) ∂P∂V (cid:1) S < n = 0 .
75 and α = 1 . A = 0 but α = 1 & n = 0, where we have seen that (cid:0) ∂P∂V (cid:1) S < n < A , B , α and n . As it is notpossible to constrain all the parameters simultaneously from equation (24), we havetried it step by step. Now we want to concentrate on the signature of n ratherthan the other parameters because we are dealing with VMCG model where n hasa special role throughout the evolution, particularly at late time. Now n = 0 wasdiscussed earlier by several authors [22]. It is seen from the equation (24) that forpositive values of A , B and α , it will be (cid:0) ∂P∂V (cid:1) S < n < n at small volume (cid:0) ∂P∂V (cid:1) S < i.e. ,at the late universe where the influence of n is significant, for n > (cid:0) ∂P∂V (cid:1) S > n >
0. Therefore, inthe context of thermodynamical stability, we have to conclude that the value of n should be negative for the VMCG model.Now we have to examine if (cid:0) ∂P∂V (cid:1) T ≤ n < c v where c v = T (cid:0) ∂S∂T (cid:1) V = (cid:0) ∂U∂T (cid:1) V = V (cid:0) ∂ρ∂T (cid:1) V . Now we determine the temperature T ofthe Variable modified Chaplygin gas as a function of it volume V and its entropy S .The temperature T of this fluid is determined from the relation T = (cid:0) ∂U∂S (cid:1) V . Usingthe above relation of the temperature and with the help of equation (6) we get theexpression of T as 11 = V − N − n α ) α (cid:20) B (1 + α ) N + V − N c (cid:21) − α α (cid:18) ∂c∂S (cid:19) V (28a)= V − N − n α ρ − α (cid:18) ∂c∂S (cid:19) V . (28b)If c is also assumed to be a universal constant, then dcdS = 0 and the fluid, insuch a condition, remains at zero temperature for any value of its volume and pres-sure. Therefore, to discuss extensively the thermodynamic stability of the variablemodified Chaplygin gas whose temperature varies during its expansion, it is neces-sary to assume that the derivative of equation (28) is not zero implying (cid:0) ∂c∂S (cid:1) = 0.We have no a priori knowledge of the functional dependence of c . From physicalconsiderations, however, we know that this function must be such as to give posi-tive temperature and cooling along an adiabatic expansion and so we choose that (cid:0) ∂c∂S (cid:1) > U ] = ( [ c ][ V ] A (1+ α ) ) α . (29)Since [ U ] = [ T ] [ S ], we can write[ c ] = [ T ] α [ S ] α [ V ] A (1+ α ) . (30)It is difficult to get an analytic solution of c from equation (30), so as a trial case,we take an empirical expression of c and then to check if the resulting expressionssatisfy standard relations of thermodynamics. But as c is a function of entropy only,the expression of c will be c = (cid:0) τ v A (cid:1) α S α , (31)where τ and v are constants having the dimensions of time and volume respectively.Now dcdS = (1 + α ) (cid:0) τ v A (cid:1) α S α . (32)Using equation (28) and (32), we get the expression of temperature T = V − N − n α ) (cid:20) B (1 + α ) N + V − N c (cid:21) − α α (cid:0) τ v A (cid:1) α S α (33a)= V − N − n (cid:0) τ v A (cid:1) α ρ − α S α (33b)= τ v A V A ( −
11 + (cid:0) ǫV (cid:1) N ) α α , (33c)and from equation (33a), the entropy is 12 = h B (1+ α ) N V N i α (cid:0) Tτ α (cid:1) α (cid:0) Vv α ) (cid:1) Aα (cid:26) − (cid:0) Tτ (cid:1) αα (cid:0) Vv (cid:1) A (1+ α ) α (cid:27) α , (34)It follows from equation (34) that for positive and finite entropy one should have0 < T V A < τ v A , but individually 0 < T < τ and v < V < ∞ , i.e. , τ represents themaximum temperature whereas v represents the minimum volume. It is shown fromthe equation (33c) that as volume of the VMCG increases, temperature decreases.It is also proved from the equation (33c) that when T → V → ∞ and when T → τ , V → v . Thus we can apparently avoid the initial singularity.Evidently at T = 0, S = 0 which implies that the third law of thermodynamicsis satisfied in this case.Now using equation (34) we get the expression of thermal heat capacity as c V = T (cid:18) ∂S∂T (cid:19) V = h B (1+ α ) N V N i α (cid:0) Tτ α (cid:1) α (cid:0) Vv α (cid:1) Aα α (cid:26) − (cid:0) Tτ (cid:1) αα (cid:0) Vv (cid:1) A (1+ α ) α (cid:27) α α . (35)Since 0 < T V A < τ v A and α > c V > n . This ensures the positivity of α . It is interesting to note that whenthe temperature goes to zero c V goes to zero as expected from the third law ofthermodynamics.If we put A = 0 and α = 1, i.e. Variable Chaplygin gas model, we get theidentical expression of c V of our previous work [27]. Again for A = 0 and n = 0,the equation (35) reduces to the work of Santos et al [21].To end the section a final remark may be in order. While positivity of specificheat is strongly desirable vis-a-vis when dealing with special relativity, in a recentcommunication Luongo et al [30] argued that in a FRW type of model like theone we are discussing a negative specific heat at constant volume and a vanishinglysmall specific heat at constant pressure ( c P ) are compatible with observational data.In fact they have derived the most general cosmological model which is agreeablewith the c V < c P ∼ (f ) Thermal EoS: Since P = P ( V, T ), using (6), (31) and (34) we get the internal energy as afunction of both V and T as follows: U = V B (1+ α ) N V − n − (cid:0) Tτ (cid:1) αα (cid:0) Vv (cid:1) A (1+ α ) α α . (36)Now using (2), (3) and (36) the Pressure is13 = ρ " A − N α ( − (cid:18) Tτ (cid:19) αα (cid:18) Vv (cid:19) A (1+ α ) α ) . (37)We have seen from equation (37) that for A = 0 and α = 1, the solution reducesto our previous work [27]. It is to be seen that for A = 0 and n = 0 the abovesolution goes to an earlier work of Santos et al [21].Now the thermal EoS parameter is given by ω = A − N α ( − (cid:18) Tτ (cid:19) αα (cid:18) Vv (cid:19) A (1+ α ) α ) . (38)Thus thermal EoS parameter is, in general, a function of both temperature andvolume. If we consider early stage of the universe when at very high temperatureand small volume, i.e. , at T → τ and also V → v , we get from equation (38) that ω ≈ A , i.e. , P ≈ Aρ . This is same as equation (16).Figure 6: The variations of (cid:0) ∂P∂V (cid:1) T vs V are shown. The graphs clearly show that (cid:0) ∂P∂V (cid:1) T < throughout the evolution only for negative values of n . Here we have taken A = 0 . , α = 0 . , B = 1 , τ = 1 and v = 1 . Secondly we consider for large volume, i.e. , very low temperature , i.e. , T →
0, weget from equation (38) that ω ≈ − n α ) , which is identical with equation (21) asis customary with the existing literature in this field [21,27]. Thus thermodynamicalstate represented by equations (16) and (17) are essentially same at both early andthe late stage of the universe.Now from equation (37) we have to examine if (cid:0) ∂P∂V (cid:1) T ≤
0, and a lengthy butstraight forward calculation shows that only for negative value of n , this conditionsatisfies. It is very difficult to find out (cid:0) ∂P∂V (cid:1) T from the equation (37) in a compactform and to get requisite inferences from it. So we have taken recourse to graphicalmethod instead. From fig - 6, we clearly infer that (cid:0) ∂P∂V (cid:1) T < n < (cid:0) ∂P∂V (cid:1) T < (cid:0) ∂P∂V (cid:1) S < i.e. , both isothermal and adiabatic situations needto be addressed. In the process we have found that (cid:0) ∂P∂V (cid:1) S and (cid:0) ∂P∂V (cid:1) T are negativefor n <
0. Relevant to mention that in a previous work by Santos et al [22] it14as assumed (cid:0) ∂P∂V (cid:1) T = 0 for simplicity, but in our case we have not made such asimplistic approach. In fact we have used a different type of approach to find outthe expression of c = c ( S ). As we have not made this assumption our analysis ismore general in nature and our formulations would not reduce to there case when n = 0.As pointed out in the previous section we here take up a standard relation ofthermodynamics (e.g. the first internal energy equation) [31] as a trial case to seeif the equation (31) is satisfied. (cid:18) ∂U∂V (cid:19) T = T (cid:18) ∂P∂T (cid:19) V − P. (39)Using equations (36) and (37) we verify the relation (39) which proves the cor-rectness of our approach. (f ) Pressure-Volume relation: It is very difficult to arrive at a ( P ∼ V ) relation in a general way. So one hasto take recourse to extremal conditions.1. For small volume, V ≪ ǫ , which gives P ∼ Aρ . In this case energy densityand also pressure are very high. Using (33c) the temperature will be T ≈ τ v A V A . (40)At the early stage of the universe V → v (minimum volume) implies that T → τ (maximum temperature) So the temperature is high enough at this stage. Usingequations (33c) and (40) we get ρ ≈ S τ v A V A , (41)therefore, U V A = ρV A +1 = Sτ v A . (42)We know at this stage that P ∼ Aρ = A UV , i.e. , P V ∼ AU , which gives usingequation (42) P V A = Sτ v A . Since the entropy remains constant in an adiabaticprocess, this relation leads to P V A = Constant. So it is observed that for smallvolume, i.e. , at high temperature the VMCG behaves as a fluid of γ (= c P c V ) = 1 + A .We can also rewrite the EoS as P = ( γ − ρ . Since early universe is radiationdominated i.e. , A = , the value of γ = 1 + A = , the pressure is related tothe volume as P V = constant. Thus the VMCG behaves like a photon gas. Theequation of adiabatic photon gas coincides with extreme relativistic electron gas.2. For large volume, V >> ǫ ,Due to low density at this stage, entropy density is sufficiently small at the lowtemperature. We know from equation (17) that P ≈ (cid:26) − n α ) (cid:27) ρ. (43)Again, from equation (9b), in this case we get15 ≈ (cid:26) B (1 + α ) V − n N (cid:27) α . (44)Now using equations (43) and (44), we get P V n α ) = (cid:26) − n α ) (cid:27) (cid:26) B (1 + α ) N (cid:27) α . (45)Equations (43), (44) and (45) can be obtained from the equations (38), (36) and(37) respectively at T →
0. For an adiabatic system, the above equation looks like
P V γ = k , where k is a constant. Since we know that VMCG is thermodynamicallystable for n < k <
0. In this case, P is also negative. At thelate stage of evolution, i.e. , at low temperature, the VMCG behaves like a fluid of γ = n α ) . Interestingly, it is to be noted that the value of this γ depends on both n and α at this stage of evolution. It further follows from equation (33c) that T → ǫV <<
1. Since any system near T = 0 is in states very close to its ground state,quantum mechanics is essential to the understanding of its properties. Indeed, thedegree of randomness at these low temperatures is so small that discrete quantumeffects can be observed on a macroscopic scale.
3. Discussion
Following the discovery of late acceleration of the universe there is a proliferationof varied dark energy models as its possible rescuers. While many of them sig-nificantly explain the observational findings coming out of different cosmic probesserious considerations have not been directed so far to the query whether the modelsare thermodynamically viable, for example if they obey the time honoured stabil-ity criteria. We have here considered a very general type of exotic fluid, termed‘Variable Modified Chaplygin gas’ and studied its cosmological implications, mainlyits thermodynamical stability. Regarding the cosmological dynamics we have comeacross two characteristic volumes of the fluid - V c and V f representing critical volumeand flip volume respectively. The former refers to the case where pressure changesits sign while the latter gives the volume when the acceleration flip occurs. Fromphysical consideration one should get V c < V f - which also matches with our analy-sis. As discussed at the end of section 2( c ), for flip to occur in FRW cosmology onlya negative pressure is not the necessary and sufficient condition. The magnitude ofpressure should be also less than ρ , which also follows from our analysis.Although the exhaustive analysis of the latest cosmological observations providesa definite clue of the existence of dark energy in the universe but it is difficult todistinguish between the merits of various forms of dark energy at present. For thestability criteria we have followed the standard prescription : (cid:0) ∂P∂V (cid:1) S < (cid:0) ∂P∂V (cid:1) T < c V >
0. This, however, dictates that the new parameter, n introduced VMCGshould be negative definite. This contrasts sharply with an earlier contention ofLu’s [26] where to explain the observational results they have to choose n as posi-tive definite. This is pathological because it makes the system thermodynamicallyunstable. 16gain this model shows that at early stage, the EoS becomes P = Aρ , where 0 ≤ A ≤
1. But at late stage, it reduces to equation (43). From the thermodynamicalstability conditions, we find that n <
0, which favours a phantom like evolution andbig rip is thus unavoidable. So far as phantom model is concerned, it is found tobe compatible with SNe Ia observations and CMB anisotropy measurements [28].The most important conclusion coming out of our analysis is that this model coversboth big bang and big rip in the whole evolution process.It is to be noted that at T = 0 the entropy of VMCG vanishes as in conformitywith the third law of thermodynamics. We have studied both the thermal and thecaloric EoS which shows that both 0 < T < τ and v < V < ∞ where τ is amaximum temperature and v is a minimum volume attainable. Here τ and v arecanonical in the sense that as T → τ , V → v at the early stage, i.e. , τ representsthe maximum temperature that our VMCG model can sustain for the small finitevolume, v .We have also discussed ( P ∼ V ) relation and at early stage ( A = ) it is shownthat for an isentropic system VMCG behaves like a photon gas, as it was at thetime of radiation dominated era. It is also shown that for large volume, i.e. , atlow temperature, entropy is sufficiently low which agrees well with the currentlyavailable low energy density of the universe.Finally it may not be out of place to point out that as the field equations arevery involved in nature we have to adopt an ansatz to determine the function ofintegration c = c ( S ). However to justify it we have checked an important relation(39) (first internal energy equation) and have found it to be correct. So our ansatz is essentially viable. Moreover, this model is very general in the sense that many ofearlier works in this field may be obtained as a special case.To end a final remark may be in order. We have here concentrated on thecosmological and thermodynamical behaviour of the VMCG model mainly on atheoretical premise. However, as a future exercise one should try to constrain thevalue of the parameters associated with both thermal and caloric EoS in the lightof observational values. Acknowledgment :
One of us(SC) acknowledges the financial support ofUGC, New Delhi for a MRP award and acknowledges CERN for a short visit. DPacknowledges the financial support of UGC, ERO for a MRP (No- F-PSW- 165/13-14) and also acknowledges IRC,NBU for short visit. The authors wish to thank theanonymous referee for valuable comments and suggestions.
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