On the thermodynamic stability of the modified Chaplygin gas

Physics Letters B 646 (2007) 215–221 www.elsevier.com/locate/physletb

On the thermodynamic stability of the modified Chaplygin gas F.C. Santos a , M.L. Bedran b,1 , V. Soares a,∗ a Instituto de Física da Universidade Federal do Rio de Janeiro, Bloco A – C.T. Cidade Universitária, 21945-970 Rio de Janeiro, Brazil b Departamento de Física da Universidade Federal de Juiz de Fora, ICE – Campus Universitário, 36036-330 Juiz de Fora, Brazil

Received 10 November 2006; received in revised form 23 January 2007; accepted 27 January 2007 Available online 3 February 2007 Editor: N. Glover

Abstract This work discusses the thermodynamic stability of an exotic fluid known as modified Chaplygin gas [MCG]. In the literature, one considers such a fluid as a perfect one which obeys the adiabatic equation of state P = Bρ − A/Aρ α , where P stands for the pressure and ρ is the energy density of the fluid; the parameters A and B are positive constants, and α  0. Extending the analysis presented in [F.C. Santos, M.L. Bedran, V. Soares, On the thermodynamic stability of the generalized Chaplygin gas, Phys. Lett. B 636 (2006) 86–90] to the MCG, it is remarked that if the energy density of the Chaplygin fluid in its generalized form (B = 0) or modified form (B = 0) depends on volume only, the temperature of the fluid remains zero at any pressure or volume it may attain. One sets up a scenario to determine the corresponding thermal equation of state of the MCG and it reveals that the MCG only presents thermodynamic stability during any expansion process if its thermal equation of state depends on temperature only, P = P (T ). This scenario also establishes physical constraints on the parameters B and α of this equation. Moreover, the modified Chaplygin gas may cool down through any thermodynamic process without facing any critical point or phase transition. © 2007 Elsevier B.V. All rights reserved. PACS: 98.80.Es; 05.70.Ce; 95.30.Tg; 98.80.-k Keywords: Cosmology; Equation of state; Chaplygin gas; Thermodynamic stability

1. Introduction In the last decade, a growing number of observational data of redshift and luminosity-distance relations of type Ia supernovae have supported the conclusion that the expansion rate of our present universe is augmenting [1–4]. As required by general relativity, such a dynamics demands that most of the energy density of the universe consists of a certain fluid with an effective negative pressure. Several mechanisms have been proposed during the last years in order to clarify the physical nature of this fluid and, among them, the single-component fluid known as Chaplygin gas has attracted large interest [5–17]. Such an exotic fluid is considered a perfect one which, in its more general form, is named modified Chaplygin gas (MCG)

* Corresponding author.

E-mail address: [email protected] (V. Soares). 1 On leave from Universidade Federal do Rio de Janeiro, Brazil.

0370-2693/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2007.01.038

and obeys the following caloric equation of state (EoS): A . (1) ρα In Eq. (1), P corresponds to the pressure of the fluid and ρ to its energy density: P = Bρ −

U (2) , V where U and V are the internal energy and volume filled by the fluid, respectively. The parameter α is a universal constant: α  0. The parameters A and B are also positive and considered as universal constants. This equation of state, with α = 1 and B = 0, has been first considered to describe lifting forces on a plane wing in aerodynamic process [14]; its generalization for α > 0 and B = 0—known as generalized Chaplygin gas, or GCG for short—was originally proposed in Refs. [7,15]. Inasmuch as GCG acts as a fluid with essentially constant negative pressure at low energy densities it also presents some drawbacks. Among its disadvantages, it behaves as an almost

ρ=

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pressureless fluid at high energy densities. To interpolate states between standard fluids at high pressures at high energy densities and constant negative pressure at low energy densities, the extended form from the GCG (B = 0) to the MCG (B = 0) equation of state must be considered [13]. Then, for B = 1/3, for instance, the energy density of the MCG smoothly changes from that of radiation era at early times to an almost constant value at late times, a period in which the fluid acts analogously as a cosmological constant, giving rise to accelerated expansion [13,16]. Moreover, the sound speed of the GCG is negligible at early times and approaches the speed of light in the late-time limit. A value of the sound speed near to the speed of light has implications that apparently disfavor the generalized Chaplygin gas as a useful model of the cosmic medium. Particularly, it should be connected with fluctuations of the medium on small (subhorizon) scales. The fact that these fluctuations are not observed has led some authors [17] to the conclusion that the generalized Chaplygin gas model should be ruled out as a competitive candidate. Albeit there have been several papers discussing various aspects of the behavior of the GCG [10,11,18–22] and MCG [22, 23] to reconcile the standard cosmological model with observations, there has not been, to our knowledge, a full analysis of the physical constraints that can be placed on such a fluid from a general thermodynamic point of view. In a previous paper, we have addressed this question with respect to the GCG [24]. In particular, we have shown that the GCG behaves as a stable fluid with a thermal equation of state that is temperature dependent only. The purpose of the present Letter is to show that a similar scenario can also be set up for the MCG. In this Letter, we determine the corresponding thermal equation of state of the MCG and we analyze its temperature evolution as well as its thermodynamic stability considering both caloric and thermal equations of state. To attain this objective, we organized the work as follows. In Section 2 we consider the Chaplygin fluid as a single system and calculate its internal energy U as a function of its natural variables, the entropy S and volume V . Its adiabatic equation of state P is also determined as a function of the natural variables of U . From these results, in Section 3, we determine the thermal EoS, investigate the behavior of the temperature of the gas and discuss its thermodynamic stability. In Section 4 we discuss the behavior of a universe filled with the MCG and the possible physical constraints on the values of the parameters of the MCG. We summarize our conclusions in the last section. 2. Caloric EoS of the modified Chaplygin gas In this section we determine the energy density U and pressure P of MCG as a function of its entropy S and volume V . From general thermodynamics [25], one has the following relationship: 

∂U ∂V

 = −P . S

(3)

After combining Eqs. (1)–(3), one finds the following differential equation   Vα U ∂U =A α −B , (4) ∂V S U V whose solution gives the internal energy U of the MCG in its natural variables, S and V :   1/(α+1) A b U= (5) . V (α+1) + B(α+1) B +1 V The parameter b is the integration constant of Eq. (4), which may be a universal constant or a function of entropy S only: b = b(S). It is interesting to remark that even in the case when A and B are not universal constants but functions of entropy only, A = A(S) and B = B(S), Eq. (5) remains valid. In the present work we always consider these parameters as universal constants. For further discussion, it is convenient to rewrite Eq. (5) in the form:  R 1(α+1) 1(α+1)   δ A V 1+ , U= (6) B +1 V where R = (B + 1)(α + 1) > 1 and δ represents the ratio   (B + 1) 1/R , δ= b A

(7)

which has dimension of volume. Therefore, the energy density ρ of the MCG is determined from Eqs. (2) and (6):  1/(α+1) A ρ= + V −R b B +1  1/(α+1)   R 1/(α+1) A δ = , (8) 1+ B +1 V and one remarks that it reduces as the volume filled by the gas expands adiabatically. The pressure P of the MCG is also determined as a function of entropy S and volume V , with the help of Eqs. (1) and (8):  1  α+1 (B + 1) A P =− α B +1 [1 + (δ/V )R ] α+1    R  B δ × 1− (9) 1+ . B +1 V One observes from the equation above that for B = 0 (GCG) the pressure has only negative values and for B = 0 (MCG), the pressure may have positive or negative values, depending if the volume V is lower or greater than a certain characteristic value V ∗ , defined as: V ∗ = δB 1/R .

(10) V∗

Then, when the fluid occupies a volume V equal to its corresponding pressure is zero. At small volumes (V  δ), Eq. (9) suggests that the energy density of the MCG behaves like  1/(α+1)  (B+1) A δ ρ≈ (11) B +1 V

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and is very high (one remarks that Eq. (11) also imposes that b > 0). In this case, the pressure P of the Chaplygin gas behaves like 

A P ≈B B +1

1/(α+1) 

δ V

(B+1) .

(12)

For large volumes (V  δ), the energy density behaves as a mixing of two fluids:  ρ≈

A B +1

1/(α+1)  1+

 R  δ 1 . α+1 V

(13)

In the equation above one has a term with constant energy density and another one whose density depends on volume. The pressure in this case is like  P ≈−

A B +1

1/(α+1) .

(14)

Thus, the modified Chaplygin gas has extremely high and positive pressures at small volumes, is essentially pressureless at volumes near V ∗ , and has practically constant negative pressure at large volumes. It is interesting to see that when the gas has high energy densities, its equation of state reduces to: P ≈ Bρ.

(15)

For low energy densities, its equation of state becomes: P ≈ −ρ.

(16)

The pressure, in both limits, is proportional to the energy density but the coefficient of proportionality is positive at low volumes and negative at large ones. These results are in agreement with those obtained in Refs. [7,15], when B = 0 (GCG), and [13,16], B = 0 (MCG), in the framework of Friedman– Robertson–Walker cosmological model. Assuming the volume occupied by the fluid is related to the cosmological expansion factor as: V ∝ a 3 , and B = 1/3, the energy density ρ and the pressure at small a are approximated by ρ ≈ b1/(α+1) and P ≈ ρ/3, respectively, which corresponds to a universe dominated by radiation. At cosmological factor a = a ∗ ∝ (V ∗ )1/3 the pressure is zero. For large values of the cosmological factor a it follows that: ρ ≈ (3A/4)1/(α+1) and P ≈ −(3A/4)1/(α+1) which, in turn, corresponds to a universe with a cosmological constant [3A/4]1/(α+1) (i.e., a de Sitter universe). These results show that the MCG is able to evolve from radiation like dominated era to a dust dominated period and finally to a de Sitter universe. However, to verify the thermodynamic stability conditions of a fluid along its evolution, it is necessary to determine if the pressure is reduced through an adiabatic expansion [25]:   ∂P < 0. (17) ∂V S After deriving Eq. (9) with respect to the volume and rearranging the terms one finds that Eq. (17) is satisfied only if:

  R − 2α−1 α+1 δ R 1+ − (α + 1)V V    R  δ × α+B α+1+ < 0. V

217

(18)

Therefore, the simultaneous conditions α = 0 and B = 0 must be discarded because it will place a severe constraint on the stability of this fluid: in such a case, (∂P /∂V )S = 0, and the pressure will remain the same through any adiabatic change of volume. For α > 0 and B  0 this derivative is always negative at any value of the volume V . Then, if one considers Eq. (17) only as the criterion of stability, we must conclude that the Chaplygin fluid is not prohibited to attain all pressures defined by Eq. (9). However, to discuss the stability of the fluid extensively, it is not enough to consider Eq. (17) only. It is also necessary to determine if the pressure reduces or remains constant as the fluid expands at constant temperature T , in the same region where Eq. (17) is negative [25]. Thus, one must also verify if   ∂P  0. (19) ∂V T In addition, it is also necessary to verify if the thermal capacity at constant volume cV is always positive [25]:       ∂S ∂U ∂ρ = =V > 0. cV = T (20) ∂T V ∂T V ∂T V Finally, one must consider which constraints these stability conditions impose on the possible values of the parameters B and α. To discuss these questions it is necessary, at first, to determine how the temperature T of the fluid depends on volume V and entropy S. This is considered in the next section. 3. Thermal EoS of the modified Chaplygin gas In this section we determine initially the temperature T of the MCG as a function of its volume V and its entropy S, and we establish under certain conditions its thermal equation of state. From general thermodynamics [25], the temperature T of a fluid is determined from the following equation:   ∂U . T= (21) ∂S V Replacing Eq. (5) into Eq. (21), one obtains the temperature of MCG: −α/(α+1)   V −(R−1) db A −R T= (22) +V b . (α + 1) B +1 dS If b is also assumed to be a universal constant, then db =0 (23) dS and the fluid, in such a condition, remains at zero temperature for any value of its volume and pressure. Therefore, to discuss extensively the thermodynamic stability of the modified Chaplygin gas whose temperature varies during its expansion, it is necessary to assume that the derivative in Eq. (22) is not zero.

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Empirically we have no direct knowledge of what this function should be. Theoretically, however, we know that this function must be such as to give positive temperatures and cooling along an adiabatic expansion, and we choose that db > 0. (24) dS Unfortunately, the condition in Eq. (24) is not sufficient for the analytical determination of the temperature T of the modified Chaplygin fluid. Therefore, a unique thermal equation of state P = P (T , V ) for the MCG, from Eq. (5), is not possible without knowledge of b = b(S). However, we remark that for small volumes the MCG equation of state reduces to p ∼ Bρ. Under these conditions, in astrophysical applications, this fluid must mimic radiation or ultrarelativistic particles in thermal equilibrium. In such a case, one must have ρ = ρ(T ) and P = P (T ) [26]. With the help of the above considerations, it is possible to set up a scenario where the thermal equation of state of the MCG is assumed to be a function of the temperature only, P = P (T ), and b = b(S) as well as ρ = ρ(T ) are completely determined by changing from the variables (S, V ) to (P , T ), in a similar way as it has been done for GCG in Ref. [24]. To make such a transformation one needs to know the Jacobian [25]:         ∂P ∂P ∂T ∂T ∂(P , T ) − . = (25) ∂(S, V ) ∂S V ∂V S ∂V S ∂S V This Jacobian can also be written as       T ∂P ∂T ∂(P , T ) ∂P =− , =− ∂(S, V ) ∂S V ∂V T cV ∂V T

(26)

where the thermal capacity at constant volume cV , from the condition in Eq. (20), is supposed to be positive. Since one assumes that the pressure is a function of the temperature only, then (∂P /∂V )T = 0. In such a case, the Jacobian in Eq. (25) must satisfy the identity:         ∂P ∂P ∂T ∂T (27) − = 0. ∂S V ∂V S ∂V S ∂S V The requirement in Eq. (27) allows the determination of b = b (S) when the pressure depends on the temperature alone. After performing these derivatives with the help of Eqs. (1), (8), and (21), one finds after a lengthy manipulation that     b αb V −R − − b b α + 1 ρ α+1    −R  αb 1 V + 1− (28) − b = 0, α + 1 ρ α+1 R where the symbols b and b represent the first and second derivatives of b with respect to the entropy S: db d 2b . and b = dS dS 2 Then, Eq. (28) is satisfied when:   R − 1 b b − = 0. b R b

b =

(29)

(30)

Solving the differential equation in Eq. (30) yields   R−1 ln b − ln b = ln λ, R

(31)

with λ = const > 0. Therefore, when the pressure P is a function of the temperature T only, the integration constant b in the solution for the internal energy U in Eq. (5) must satisfy the relationship: db = λb(R−1)/R , (32) dS whose solution relates the entropy S to the constant b as: R 1/R b . λ Choosing the constant λ as

S=

λ=

(α + 1) , β

(33)

(34)

one determines the following expression for the entropy as a function of b: (B + 1)(α + 1) 1/R = β(B + 1)b1/R . b S= (35) λ When the entropy S is constant the parameter b is also constant, as expected. Considering Eqs. (22), (32), and (34), one finds the following equation for the temperature T of the Chaplygin gas:  −α/(α+1)   −R (R−1)/R A −R βT = (36) V b +V b B +1 or B/R   R −α/(α+1)  R−1  δ δ A . 1+ βT = (37) B +1 V V Finally, combining Eqs. (7), (8), and (36), one obtains the temperature T as a function of the energy density ρ only:  (R−1)/R A βT = ρ B/(B+1) 1 − (38) . (B + 1)ρ α+1 It is not possible to solve the equation above explicitly for any arbitrary value of R and then write the energy density as a function of the temperature. Nonetheless, one observes that Eqs. (1) and (38) together express the pressure in function of the temperature in parametric form. In such a scenario, the thermal equation of state of the MCG is then dependent on temperature only. That means the isobars in the phase diagram of thermodynamic states for the MCG are coincident with the isotherms. One determines the thermal capacity at constant volume cV of the MCG using the general thermodynamic relationship, Eq. (20), together with Eq. (38):   ∂ρ cV = βV ∂(βT ) V 1/R  (B + 1)ρ α+1 α+1 A = βV − . (39) ρ B +1 αA + Bρ α+1 Since R > 1 and ρ > [A/(B + 1)]1/(α+1) , then one has the thermal capacity at constant volume always positive and finite at

F.C. Santos et al. / Physics Letters B 646 (2007) 215–221

any temperature the MCG may attain. It is interesting to observe that when the temperature goes to zero the energy density goes to ρ ≈ [A/(B + 1)]1/(α+1) and the thermal capacity goes to zero, as expected from the third law of the thermodynamics. Therefore, in the scenario defined by Eq. (32), the MCG has its thermal capacity at constant volume equal to a positive value and the possible thermodynamic states of the MCG always satisfy the condition of stability cV > 0. One also observes from Eqs. (1) and (38) that all derivatives (∂ n P /∂V n )T , n = 1, 2, 3, . . . , are equal to zero for any volume V . (Constant temperature T implies constant ρ and then constant P .) Therefore, in the scenario built up from the condition defined by Eq. (32), the modified Chaplygin fluid has its thermal equation of state depending on temperature only and presents stable behavior through any expansion or contraction process. Thus, the MCG is a perfect fluid that cools down during an adiabatic expansion. It can also attain the negative and essentially constant pressures suggested by Eq. (9) without breaking its thermodynamic stability and without facing any critical point in the process. 4. Physical constraints on the parameters of the EoS Let us consider the thermodynamic states of the MCG in the limiting cases. At small volumes (V  δ), its energy density is very high, as well as its pressure: 1/(α+1)  R/(α+1)  δ A , P ∼ Bρ. ρ≈ (40) B +1 V Thus, for small volumes, one determines the temperature from Eqs. (38) and (40):   R/(α+1) (R−1)/R V B/(B+1) ∼ ρ B/(B+1) . (41) 1− βT ≈ ρ δ After combining Eqs. (35), (40), (7), and (41), one also estimates the entropy in this same limit:  1/R   A δ 1/B 1/(B+1) ∼ρ ∼ (βT ) B +1 V 1 S ∼ (42) . (B + 1)β V Therefore, in the limit of small volumes the entropy of the MCG is S ∼ (B + 1)β(βT )1/B V .

(43)

Then, one observes that, in this limit, P V ∼ BU

(44)

and, from Eqs. (41) and (43), the internal energy, volume, temperature and entropy are related as  B+1 S . U V B = ρV B+1 ∼ (βT )(B+1)/B V B+1 ∼ (B + 1)β (45) From the relations above one observes that at the limit of small volumes the MCG behaves as an ideal quantum gas [27]. General thermodynamics only is not able to fix a unique value to

219

the parameter B and one must rely on the experiments. As remarked in the previous section, however, since the energy density depends on temperature only, a particular theoretical choice of astrophysical interest is B = 1/3. In such a case, the equation of state of the MCG is reduced to 1 P ∼ ρ(T ), (46) 3 which mimics the pressure at given temperature T not only for the energy density of radiation (ργ ), but also for the energy density of a fluid of ultrarelativistic electron–positron pairs (ρe ) or for the energy density of a gas of massless neutrinos (ρν ) [26]: U ∼ β 4T 4V ,

(47)

depending on how the parameter β is fitted: 7 7 βe4 = σB , or βν4 = σB , (48) 8 16 where σB is the Stefan–Boltzmann constant. The theory of relativity imposes constraints on the maximum value of the speed of sound in the fluid: the propagation speed cS of small adiabatic perturbations in the MCG must be lower than the speed of light c:   cS2 ∂P = < 1. (49) ∂ρ S c2

βγ4 = σB ,

Thus, one must add the following restriction to the equation of state of the MCG:   ∂P < 1, 0< (50) ∂ρ S which imposes that: 0
α(B + 1) < 1. 1 + V −R b(B + 1)/A

(51)

This result establishes some constraints on the values of the constants B and R: for small volumes, Eq. (51) leads to 0 < B < 1 and, for large volumes, one has 1 < R < 2. Therefore, the possible values of α are also constrained: (i)

if B ∼ 0,

(ii)

if B = 1/3, then 0 < α < 1/2;

(iii)

if B ∼ 1,

then 0 < α < 1; then α ∼ 0.

(52)

In order to discuss these constraints it is convenient to write the pressure P , volume V , and temperature T in reduced units: P , [A/(B + 1)]1/(α+1) βT τ= , [A/(B + 1)]B/R

π=

φ=

V , δ (53)

as well as the energy density ρ as ε=

ρ . [A/(B + 1)]1/(α+1)

(54)

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F.C. Santos et al. / Physics Letters B 646 (2007) 215–221

The reduced caloric EoS for the MCG is then written as    (B + 1) B

−R π =− , 1+φ 1− B +1 [1 + φ −R ]α/(α+1) and the reduced energy density is given as 1/(α+1)

. ε = 1 + φ −R

temperature T , so that: (55)

(56)

The reduced temperature is then   φ 1−R 1 (R−1)/R B/(B+1) =ε . τ= 1 − α+1 ε [1 + φ −R ]α/(α+1)

(57)

Replacing the constraints from condition (i) in Eq. (52) into Eqs. (55)–(57), yields: 1

π =−

[1 + φ −(α+1) ]α/(α+1) 1/(α+1)

, ε = 1 + φ −(α+1) and

 τ = 1−

1 ε α+1

, (58)

α/(α+1) ,

(59)

which gives the thermal EoS for the GCG in the following form: α/(α+1)

. π = − 1 − τ (α+1)/α (60) The properties of this solution have already been discussed in Ref. [24]. In summary, in such conditions the fluid always presents negative pressures. Moreover, if one assumes that at present epoch the energy density of the universe filled with the GCG is slightly greater than unit, ε ≈ 1, then this universe would have at an early epoch (φ → 0, ε → ∞) an essentially constant temperature τ ≈ 1. Then, this temperature remains the same up to a recent past when it begins to rapidly drop and goes to very low values at the present epoch. In the future (φ → ∞, ε → 1) the temperature goes to zero (τ → 0). The condition (ii) in Eq. (52), combined with Eqs. (55)–(57), gives the reduced pressure in the form    1

4/3 −4(α+1)/3 , 1− 1+φ π =− 4 [1 + φ −4(α+1)/3 ]α/(α+1) (61) the reduced energy density as

1/(α+1) ε = 1 + φ −4(α+1)/3 (62) , and the reduced temperature as   1 (4α+1)/[4(α+1)] . τ = ε 1/4 1 − α+1 ε

1 ε ≈ τ 4. π ≈ ε, (65) 3 The pressure of the MCG is then positive and behaves in a similar way as the radiation pressure. When this universe expands from small volumes to volumes of the order of φ ≈ φ ∗ = (1/3)3/[4(α+1)] (Eq. (10) in reduced units) then in this region the MCG behaves as a pressureless fluid: ε(φ ∗ ) ∼ 41/(α+1) ,

π ∼ 0,

and the temperature of the MCG is then given by  (4α+1)/[4(α+1)] 3 τ∼ ε 1/4 . 4

(66)

(67)

Consequently, at this stage, if it continues to expand, its energy density goes down and then cools to lower temperatures: the universe behaves then as a dust-like one. If the expansion continues, the pressure becomes negative, the volume becomes very large and the energy density of the fluid goes to ε ∼ 1.

(68)

Then, at this stage, its pressure becomes essentially a negative constant and its temperature goes to zero: π ∼ −1,

τ ∼ 0,

(69)

that means at very low temperatures the universe filled with MCG behaves as a de Sitter universe. Thus, in the evolution from a radiation-like to a de Sitter cosmological model, the temperature of the universe filled with the MCG cools down as expected. The condition (iii) of Eq. (52), combined with Eqs. (55)– (57), gives the following expression for the reduced pressure and the reduced energy density:   

 1

−2 , ε ∼ 1 + φ −2 . π ∼ −2 1 − 1 + φ (70) 2 The reduced temperature is then τ = φ −1

(71)

or

  1 1/2 . τ ∼ ε 1/2 1 − ε

(72)

Eq. (72) can be solved for ε: (63)

Eq. (1), in reduced units, 4 1 π = ε− α (64) 3 3ε and Eq. (63) give the thermal EoS for the MCG in the parametric form. Therefore, if it is assumed again that at the present epoch the energy density of the universe filled with the MCG is slightly greater than unity, ε ≈ 1, the behavior of the energy density at an early epoch is similar to that of a blackbody at

ε ∼ τ 2 + 1,

(73)

and the replacement of Eq. (73) into Eq. (1), in reduced units, leads to π ∼ ε − 2,

(74)

which gives the thermal EoS for the MCG: π ∼ τ 2 − 1.

(75)

Therefore, if one assumes again that at the present epoch the energy density of the universe filled with the MCG is of the order

F.C. Santos et al. / Physics Letters B 646 (2007) 215–221

of unity, ε ≈ 1, the pressure of the universe at an early epoch is then positive and its energy density goes with the square of its temperature: π ∼ ε,

ε ∼ τ 2.

(76)

When this universe expands from small volumes to volumes of the order of φ ≈ φ ∗ ∼ 1 (Eq. (10) in reduced units) then in this region the MCG behaves as a pressureless fluid: ε ∼ 2,

π ∼ 0,

(77)

and the temperature of the MCG is then 1 τ ∼ √ ε 1/2 . 2

(78)

Consequently, at this stage of its expansion the fluid becomes essentially pressureless and cools to lower temperatures: the universe behaves then as a dust-like one. If the fluid continues the expansion process, the pressure becomes negative, the volume becomes very large and its energy density goes to ε ∼ 1.

(79)

Then, its pressure becomes essentially a negative constant and its temperature goes to zero: π ∼ −1,

τ ∼ 0.

(80)

Therefore, at very low temperatures the universe filled with MCG under the condition (iii) also behaves as a de Sitter universe. 5. Conclusions Using general thermodynamics, we have shown in this Letter under which conditions the modified Chaplygin gas behaves as a thermodynamically stable fluid. We have also determined its thermal equation of state and shown that it must be dependent on temperature only to fulfill all conditions of thermodynamic stability during any expansion process. In such a scenario, when the fluid occupies small volumes it behaves as an ideal quantum gas. The speed of light constrains the maximum value of the speed of sound in the MCG and therefore imposes boundaries in the parameters B and α in the equation of state. It is determined that the parameter B must be in the range 0 < B < 1 and, if one assumes that the present energy density is slightly higher than its limit value, all constraints predict that an expanding universe filled with MCG must cool down from high temperatures at an early epoch, to low temperatures at the present epoch and to even lower temperatures later. Specifically, the MCG reduces to GCG at the limit B ∼ 0 and this finding constrains the value of α to the range 0 < α < 1. Moreover, one also recovers the same previous results of the thermodynamic analysis of the

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GCG: during any process the temperature of the GCG remains in the range 0 < T < T ∗ , where T ∗ is the maximum finite temperature the GCG may sustain when it fills small volumes. For B = 1/3 the fluid presents high pressures and analogous behavior as the radiation or ultrarelativistic fluids when it occupies small volumes. It behaves essentially as a pressureless gas as the fluid continues expanding and it has constant negative pressure when it occupies large volumes, in accordance with the results obtained in the framework of Friedman–Robertson– Walker cosmology. In such a case, however, the value B = 1/3 constrains the parameter α to values in the range 0 < α < 1/2. One also finds that the limit B ∼ 1 constrains α to extremely low values (α ∼ 0). However, the energy density of the MCG at small volumes, in this limit, is not radiation-like but depends on the square of its temperature. The scenario set up in this work also shows that there is no critical point in the diagram of thermodynamic states of the modified Chaplygin gas. Therefore, this fluid does not sustain any phase transition under any thermodynamic process. References [1] S. Perlmutter, et al., Bull. Am. Astron. Soc. 29 (1997) 1351. [2] S. Perlmutter, et al., Astrophys. J. 483 (1997) 565; S. Perlmutter, et al., Astrophys. J. 517 (1999) 565. [3] A.G. Riess, et al., Astrophys. J. 607 (2004) 665. [4] M. Hamuy, et al., astro-ph/9609059. [5] J.D. Barrow, Phys. Lett. B 235 (1990) 40. [6] M. Hassaïne, et al., Lett. Math. Phys. 57 (2001) 33. [7] A. Kamenshchik, U. Moschella, V. Pasquier, Phys. Lett. B 511 (2001) 265. [8] J.C. Fabris, S.V.B. Gonçalves, P.E. de Souza, Gen. Relativ. Gravit. 34 (2002) 53–63. [9] N. Bili´c, G.B. Tupper, R.D. Viollier, Phys. Lett. B 535 (2002) 17. [10] A. Dev, J.S. Alcaniz, J. Deepak, Phys. Rev. D 67 (2003) 023515. [11] G.M. Kremer, Gen. Relativ. Gravit. 35 (2003) 1459. [12] P.F. González-Díaz, Phys. Lett. B 562 (2003) 1. [13] H.B. Benaoum, hep-th/0205140. [14] S. Chaplygin, Sci. Mem. Moscow Univ. Math. 21 (1904) 1. [15] M.C. Bento, O. Bertolami, A.A. Sen, Phys. Rev. D 66 (2002) 043507. [16] U. Debnath, A. Banerjee, S. Chakraborty, Class. Quantum Grav. 21 (2004) 5609. [17] H.B. Sandvik, M. Tegmark, M. Zaldariaga, I. Waga, Phys. Rev. D 69 (2004) 123524. [18] R. Bean, O. Doré, Phys. Rev. D 68 (2003) 023515. [19] T. Multamäki, M. Manera, E. Gaztañaga, Phys. Rev. D 69 (2004) 023004. [20] L. Amendola, F. Finelli, C. Burigana, D. Carturan, J. Cosmol. Astropart. Phys. 7 (2003) 005. [21] J.S. Alcaniz, J. Deepak, A. Dev, Phys. Rev. D 67 (2003) 043514. [22] L.P. Chimento, Phys. Rev. D 69 (2004) 1235171. [23] L.P. Chimento, R. Lazkoz, Phys. Lett. B 615 (2005) 146. [24] F.C. Santos, M.L. Bedran, V. Soares, Phys. Lett. B 636 (2006) 86. [25] L.D. Landau, E.M. Lifschitz, Statistical Physics, third ed., Course of Theoretical Physics, vol. 5, Butterworth–Heinemann, London, 1984. [26] S. Weinberg, Gravitation and Cosmology, Wiley, NY, 1972. [27] P. Landsberg, Thermodynamics and Statistical Mechanics, Dover, NY, 1990, p. 62.