Thermodynamical behaviour of the Variable Chaplygin gas
aa r X i v : . [ g r- q c ] F e b Thermodynamical behaviour of the VariableChaplygin gas
D. Panigrahi Abstract
The thermodynamical behaviour of the Variable Chaplygin gas (VCG)model is studied, using an equation of state like P = − Bρ , where B = B V − n .Here B is a positive universal constant, n is also a constant and V is the vol-ume of the fluid. From the consideration of thermodynamic stability, it isseen that only if the values of n are allowed to be negative, then (cid:0) ∂P∂V (cid:1) S < c V shows positive expression. Using the best fit value of n = − . et al [9] gives that the fluid is thermodynamically stablethrough out the evolution. The effective equation of state for the special caseof, n = 0 goes to ΛCDM model. Again for n < flip occurs for the value of n < T →
0, the thermal equa-tion of state reduces to (cid:0) − n (cid:1) which is identical with the equation of statefor the case of large volume. KEYWORDS : cosmology; chaplygin gas; thermodynamicsPACS : 05.70.Ce;98.80.Es;98.80.-k
1. Introduction
Recent observational evidences suggest that the present universe is accelerating [1,2]. A Chaplygin type of gas cosmology [3] is one of the plausible explanations ofrecent phenomena, which is a new matter field to simulate dark energy. This type ofequation of state(EOS) is not applicable in the case of primordial universe. This wasdiscussed in the several articles [4–8]. Such equation of state leads to a componentwhich behaves as dust at early stage and as cosmological constant (Λ) at later stage.The form of the equation of state (EoS) of matter is the following, P = − Bρ (1)Here P corresponds to the pressure of the fluid and ρ is the energy density of thatfluid and B is a constant. Recently a variable Chaplygin gas (VCG) model was Sree Chaitanya College, Habra 743268, India and also
Relativity and Cosmology ResearchCenter, Jadavpur University, Kolkata - 700032, India , e-mail: [email protected] B to dependon scale factor of our metric chosen. Now we have taken the above relation as B = B V − n where V is the volume of the fluid. For n = 0 the VCG equation ofstate reduces to the original Chaplygin gas equation of state. The value of n may bepositive or negative. Guo et al [9] showed that the best fit value of n = − . gold sample of 157 SNe Ia data . Later in another article [10] they constrainedon VCG and determine the best fit value of n = 0 . +1 . − . using gold sample of 157SNe Ia data and X-ray gas mass fraction in 26 galaxy cluster [11]. This resultfavors a phantom-like Chaplygin gas model which allows for the possibility of thedark energy density increasing with time. Relevant to mention that recently thereare some indications that a strongly negative equation of state, w ≤ −
1, may givea good fit [12–14] with observations. But in another work [15], we have seen thatthe value of n lie in the interval ( − . , .
6) [
WMAP 1st Peak + SNe Ia(3 σ )] and( − . , .
8) [
WMAP 3rd Peak + SNe Ia(3 σ )].Recently Santos et al [16] have studied the thermodynamical stability in gener-alised Chaplygin gas model. In the present work we investigate thermodynamicalbehaviour of the variable Chaplygin gas (VCG) by introducing the integrability con-dition equation (3) and the temperature of equation (16). All thermal quantitiesare derived as functions of temperature and volume. In this case, we show that thethird law of thermodynamics is satisfied with the Chaplygin gas. Furthermore, wefind a new general equation of state, describing the Chaplygin gas as function ofeither volume or temperature explicitly. For the variable Chaplygin gas we expectto have similar behaviours as the Chaplygin gas did show. Consequently, we confirmthat Chaplygin gas could show a unified picture of dark matter and energy whichcools down through the universe expansion. Returning to the stability criterion ofthe Chaplygin gas we find that the value of n should be negative. Interestingly Guo et al [9] showed that the best fit value of n = − . n may be positive or negative. Fromthe thermodynamical stability considerations we can constrain the value of n andfound that n should always have negative value. The paper is organised as follows:in section 2 we build up the thermodynamical formalism of the VCG model anddiscuss the thermodynamical behaviour of this model. Finally in section 3 we givea brief discussion.
2. Formalism
Before proceeding further we define the uniform density of the fluid filling the uni-verse as ρ = UV (2)where U and V are the internal energy and volume filled by the fluid respectively.Now the energy U and pressure P of Variable Chaplygin gas may be taken as afunction of its entropy S and volume V . From general thermodynamics [17], one2as the following relationship (cid:18) ∂U∂V (cid:19) S = − P (3)With the help of equations (1) - (3) we get (cid:18) ∂U∂V (cid:19) S = B V − n VU (4)Integrating we get U = " B V − n − n + c = (cid:18) B V − n N (cid:19) V (cid:26) (cid:16) ǫV (cid:17) N (cid:27) (5)the parameter c is the integration constant which may be a universal constant ora function of entropy S only; c = c ( S ) and B = B ( S ). The term N = − n and ǫ = h Nc B i N which has the dimension of volume. Now the energy density ρ of theVCG reduces to the following form ρ = V − n (cid:20) B − n + cV − − n (cid:21) = (cid:18) B V − n N (cid:19) (cid:26) (cid:16) ǫV (cid:17) N (cid:27) (6)Now we want to discuss the thermodynamical behaviour of this model. (a) Pressure : The pressure P of the VCG is also determined as a function of entropy S andvolume V in the following form P = − B V − n h B − n + cV − − n i = − (cid:20) N B V − n (cid:21) (cid:20) (cid:16) ǫV (cid:17) N (cid:21) − (7)For n = 0 , the above results reduce to CG model [3] and its thermodynamicalbehaviour was discussed earlier by Santos et al [16].It is seen from the fig-1 that for n ≤
0, the pressure goes more and more negativewith volume. We get P = 0 at V = 0 for any value of n . It also follows from thefig-1 that as n becomes more and more negative the pressure falls sharply. (b) Equation of state: Now from the equations (6) and (7) we get the effective equation of state as W = Pρ = − N (cid:0) ǫV (cid:1) N (8)3 = n = - - - - - ® P ® Figure 1:
The nature of variations of P and V for different values of n are shown.This figure shows that P goes more and more negative with V .(Taking B = 1 , c = 1 ). n = - n =
00 1 2 3 4 - - - - ® w ® Figure 2:
The variations of W and V for different values of n are shown. This figureshows that a quiescence phenomenon for n = 0 and phantom-like phenomenon for n < .(Taking B = 1 , c = 1 ). i) For small volume, V ≪ ǫ , i.e. , ǫV ≫ W →
0, therefore P ≈
0. we get dust dominated universe and the EoS isindependent of n .ii) For large volume, V ≫ ǫ , i.e. , ǫV ≪ W ≈ − n n < W is always greater than −
1. So this isnot ΛCDM, but for n = 0, this will be ΛCDM. Influence of n is prominent in thiscase. From equation (9) it follows that for positive values of n , the value of W willbe 0 > W > −
1. So we get a quiescence phenomenon and the big rip is avoided.However, in what follows we shall presently see that to preserve the thermodynamicstability of VCG n should be negative. For n < W < −
1, the phantom-like model. It is seen from the fig-2 that W is more negative for n = − .
4. In anearlier work [18] the present author studied modified Chaplygin gas in higher di-mensional space time and showed that in the presence of extra dimension the modelbecame phantom like, but when the extra dimension is absent our results seem to be4CDM. One may see that the experimental results favour like VCG model [9,10,15]. (c) Deceleration parameter:
Now we calculate the deceleration constant. q = 12 + 32 Pρ = 12 − N (cid:0) ǫV (cid:1) N (10) n = - n = - - - - ® q ® Figure 3:
The variations of q and V for different values of n are shown. Early flip isshown for n = 0 . (Taking B = 1 , c = 1 ). i) For small volume, V ≪ ǫ , i.e. , ǫV ≫ q ≈ i.e ., q is positive,universe decelerates for small V .ii) For large volume, V ≫ ǫ , i.e. , ǫV ≪ q ≈ − n Thus we see that initially, i.e. , when volume is very small there is no effect of n on q . q is positive, universe decelerates. From fig-3 it follows that as volumeincreases q goes to zero first and then universe accelerates. For flip to occur the flipvolume ( V f ) is in the following form V f = ǫ (cid:20) − n (cid:21) N (11)A little analysis of the equation (11) shows that for V f to have real value n < flip . This is in accord with the findings of the observa-tional result [9]. (d) Stability: To verify the thermodynamic stability conditions of a fluid along its evolution, itis necessary (i) to determine if the pressure reduces through an adiabatic expansion (cid:0) ∂P∂V (cid:1) S < c V > ∂P∂V (cid:19) S = P V " (6 − n ) ( −
11 + (cid:0) ǫV (cid:1) N ) − n (12) n = - n =
00 2 4 6 8 - - - - - - - ® d P d V ® Figure 4:
The variations of dPdV and V for different values of n are shown. The natureof evolution of graphs are quite different for n = 0 and n = − . but they give dPdV < throughout the evolution. (Taking B = 1 , c = 1 ). In an earlier work Sethi et al [15] showed that the range of n lies in the interval( − . , .
6) [WMAP 1st Peak + SNe Ia(3 σ )] and ( − . , .
8) [WMAP 3rd Peak +SNe Ia(3 σ )]. But from equation (12) we see that for n ≤ (cid:0) ∂P∂V (cid:1) S < n is notcompatible in VCG model. It may be concluded that to get thermodynamical stableevolution the positive value of n should be discarded. One may mention that thenature of evolution of graphs are quite different initially for n = 0 and n = − . dPdV < n .Now we should also verify if the thermal capacity at constant volume c V is alwayspositive. First, we determine the temperature T of the Variable Chaplygin gas asa function of its volume V and its entropy S . The temperature T of this fluid isdetermined from the relation T = (cid:0) ∂U∂S (cid:1) V . Using this relation of the temperatureand with the help of equation (5) we get the expression of T as follows T = 12 (cid:20) V N N dB dS + dcdS (cid:21) (cid:20) B V N N + c (cid:21) − (13)If c and B are also assumed to be universal constants, then dcdS = 0 and dB dS = 0,the fluid, in such condition, remains at zero temperature for any value of its vol-ume and pressure. Therefore, to check the thermodynamic stability of the variableChaplygin gas whose temperature varies during its expansion, it is necessary to as-sume that the derivatives in equation (13) are not simultaneously zero. We haveno apriori knowledge of the functional dependance of B and c on S . From phys-ical considerations, however, we know that this function must be such as to givepositive temperature and cooling along an adiabatic expansion, and we choose that (cid:0) ∂c∂S (cid:1) >
0. 6ow from dimensional analysis, we observe that [ c ] = [ U ] which implies [ c ] =[ U ] = [ T ][ S ]. Thus c = 1 β S (14)Here β − is a universal constant with the dimension of the inverse of the temperature: β − = τ . Differentiating equation (14) we get dcdS = 2 τ S (15)In order to have positive temperatures and cooling along an adiabatic expansion,we must impose for mathematical simplicity dB dS = 0, which makes the constant B a universal constant. In that case equation (13) reduces to T ≈ dcdS (cid:18) B V N N + c (cid:19) − = τ S (cid:18) B V N N + τ S (cid:19) − (16)After straight forward calculations we get the expression of entropy as S = (cid:18) B N (cid:19) V N Tτ (cid:18) − T τ (cid:19) − (17)For positive and finite entropy 0 < T < τ . Evidently at T = 0, S = 0 implying thatthe third law of thermodynamics is satisfied.The thermal capacity at constant volume can be written as c V = T (cid:18) ∂S∂T (cid:19) V = (cid:18) B N (cid:19) V N Tτ (cid:0) − T τ (cid:1) (18)Since, 0 < T < τ , c V > n . (e) Thermal equation of State: Since P = P ( T, V ), using equations (5), (14) and (17) we get the internal energyas a function of both V and T as follows U = (cid:18) B N (cid:19) V N (cid:18) − T τ (cid:19) − (19)Now with the help of equations (1), (2) and (19) the pressure will be P = − (cid:18) B N (cid:19) V − n (cid:18) − T τ (cid:19) (20)which is also a function of both V and T . For T = τ , P = 0, the universe behaveslike a dust-like or a pressureless universe, as the Chaplygin gas equation of statecan not explain the primordial universe. Unlike the work of Santos et al [16] we donot get de Sitter like universe due to the presence of the term V − n in equation (20)for the case of T →
0. Again we have seen that the isobaric curve for the VCG do7ot coincide with its isotherms in the diagram of thermodynamic states. Now usingequation (2) and (19) we further get, ρ = (cid:18) B N (cid:19) V − n (cid:18) − T τ (cid:19) − (21)We find exactly similar expressions of ρ with the help of equations (1) and (20).From equations (20) and (21) we get the thermal equation of state parameter ω = Pρ = (cid:16) − n (cid:17) (cid:18) − T τ (cid:19) (22)This thermal equation of state parameter is an explicit function of temperature onlyand it is also depends on n . As volume increases temperature falls during adiabaticexpansions. In our case, for T →
0, the equation (22) yields ω = − n which isidentical with the equation (9) as it is the case of large volume. Again as T → τ (the maximum temperature), ω → (cid:18) ∂U∂V (cid:19) T = T (cid:18) ∂P∂T (cid:19) V − P (23)Using equations (19) and (20) we find the relation (23) is also satisfied.We can also express the maximum temperature τ as a function of the initialconditions of the expansion. If we consider that the initial conditions at V = V are ρ = ρ , P = P and T = T , then we can get from equation (5) as c = (cid:18) ρ − B N V − n (cid:19) V (24)With the help of equations (6), (7) and (24), we obtained the energy density ρ andthe pressure P as a function of the volume V as ρ = V − n ρ " B N ρ + (cid:18) − B N ρ V − n (cid:19) (cid:18) V V (cid:19) V n (25)and P = − B (cid:16) B o ρ (cid:17) V − n h B Nρ + (cid:16) − B Nρ V − n (cid:17) (cid:0) V V (cid:1) V n i (26)Now the equations (20), (25) and (26) can be written as function of the reducedparameters ε , v , p , κ , and t such that ε = ρρ , v = VV , p = PB ,κ = 2 B N ρ , t = TT , τ ∗ = τT (27)8he equations (20), (25) and (26) can be written in the reduced units respectivelyas p = − (cid:18) N (cid:19) V − n (cid:18) − t τ ∗ (cid:19) (28) ε = V − n (cid:20) κ + (cid:0) − κV − n (cid:1) V n v (cid:21) (29) p = − κ (cid:0) N V − n (cid:1) h κ + (cid:0) − κV − n (cid:1) V n v i (30)At P = P , V = V and T = T , we have t = 1 and v = 1, and we get fromequations (28) and (30) p = − κ (cid:18) N (cid:19) V − n = − (cid:18) N (cid:19) V − n (cid:18) − τ ∗ (cid:19) (31)hence κ and τ ∗ can be determine as follows κ = V n (cid:18) − τ ∗ (cid:19) (32)and τ ∗ = 1 (cid:16) − κV − n (cid:17) (33)Interestingly, we have seen that τ ∗ is depend on both κ , V and n also. For n = 0, allthe above equations reduce to the equations of Santos et al [16]. At present epoch, κ = B Nρ , therefore, ρ = (cid:0) B Nκ (cid:1) . If we considered that the temperature τ = 10 K(temperature of Planck era) and T = 2 .
7K (the temperature of the present epoch),the ratio, τ ∗ = τT = 3 . × . So the ratio κ will be, κ = V n (cid:20) − . × ) (cid:21) ≈ V n (34)Again from equation (21), for the case of present epoch when temperature T isvery small ( i.e., T → ρ ≈ B N V n ! ≈ (cid:18) B N κ (cid:19) (35)The same result can be obtained from equation (6) for large volume.Thus, consideration from equation (14), at the present epoch, the energy density ρ of the universe filled with the VCG must be very close to (cid:0) B Nκ (cid:1) .9 . Discussion We have studied thermodynamical behaviour of VCG model. We consider the valueof n = − . et al [9]. In an earlier literature we have seenthat the value of n may be negative or positive [15]. For large volume, when n > n = 0 goes toΛCDM model. Again for n < n = − .
4, the pressure goes more and more negativeas volume increases (fig-1).ii) The effective equation of state is shown in equation (8). At large volume wehave seen that for n = 0, gives ΛCDM model. Influence of n is prominent for suffi-ciently large volume. For n < W < flip to occur the value of n < (cid:0) ∂P∂V (cid:1) S < n , (cid:0) ∂P∂V (cid:1) S < n must be negative. Interestingly, this result is in agreement with theobservational result found earlier by Guo et al [9]. Due to this reason we have taken n = − . c V is also determined and it is seen that c V is alwayspositive irrespective of the value of n . So both the conditions of thermodynamicstability of the fluid are studied which shows that the fluid is thermodynamicallystable through out the evolution process.(v) The expression of entropy is derived and shown in equation (17). In thisequation it is seen that at T = 0, S = 0 implying that the third law of thermody-namics is satisfied.(vi) Finally the thermal equation of state is discussed in this work where it isseen that the volume is not explicitly present in the expression (22). This thermalequation of state parameter is an explicit function of temperature only. As volumeincreases temperature falls during adiabatic expansions. In this case, for T → ω = − n which is identical with the equation (9) as itis the case of large volume. Again as T → τ , ω → τ is expressed as a function of the initial conditions10f the expression. Acknowledgment :
DP acknowledges the financial support of UGC, ERO fora MRP ( No- F-PSW- 165/13-14) and also CERN, Geneva, Switzerland for a shortvisit. The author thanks the referee for valuable comments and suggestions.
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