Generalized Second Law of Thermodynamics in Quintom Dominated Universe

Physics Letters B 641 (2006) 130–133 www.elsevier.com/locate/physletb

Generalized second law of thermodynamics in quintom dominated universe

M.R. Setare

Physics Department, Kurdistan University, Sanandej, Iran Received 8 June 2006; received in revised form 17 August 2006; accepted 17 August 2006 Available online 30 August 2006 Editor: N. Glover

Abstract In this Letter we will investigate the validity of the generalized second law of thermodynamics for the quintom model of dark energy. Reviewing brieﬂy the quintom scenario of dark energy, we will study the conditions of validity of the generalized second law of thermodynamics in three cases: quintessence dominated, phantom dominated and transition from quintessence to phantom will be discussed. © 2006 Elsevier B.V. All rights reserved.

1. Introduction ing to study their cosmological implication. Recently there are many relevant studies on phantom energy [12]. The analysis of One of the most important problems of cosmology, is the the properties of dark energy from recent observations mildly problem of so-called dark energy (DE). The type Ia supernova favor models with w crossing −1 in the near past. But, nei- observations suggests that the universe is dominated by dark ther quintessence nor phantom can fulfill this transition. In the energy with negative pressure which provides the dynamical quintessence model, the equation of state w = p/ρ is always mechanism of the accelerating expansion of the universe [1–3]. in the range −1 w 1forV(φ)>0. Meanwhile for the The strength of this acceleration is presently matter of debate, phantom which has the opposite sign of the kinetic term com- mainly because it depends on the theoretical model implied pared with the quintessence in the Lagrangian, one always has when interpreting the data. Most of these models are based on w −1. Neither the quintessence nor the phantom alone can dynamics of a scalar or multi-scalar fields (e.g. quintessence fulfill the transition from w>−1tow<−1 and vice versa. [4,5] and quintom model of dark energy, respectively). Primary Although for k-essence [13] one can have both w −1 and scalar field candidate for dark energy was quintessence sce- w<−1, it has been lately considered by Refs. [14,15] that it nario, a fluid with the parameter of the equation of state lying is very difficult for k-essence to get w across −1 during evolv- − −1 in the range, 1 −1to would violate not only the strong energy condition ρ + 3P>0, w<−1, or vice versa, namely from w<−1tow>−1isalso but the dominate energy condition ρ + P>0, as well. Fluids one realization of quintom, as can be seen clearly in [18].We of such characteristic dubbed phantom fluid [7,9]. In spite of must mention that there are another possibilities in model build- the fact that the field theory of phantom fields encounter the ing regarding quintom, see [19] for a dark energy model which problem of stability which one could try to bypass by assuming includes higher derivative operators in the Lagrangian with a them to be effective fields [10,11], it is nevertheless interest- single scalar field which gives rise to an equation of state larger than −1 in the past and less than −1 at the present time. One an- E-mail address: [email protected] (M.R. Setare). other is the scalar field model with non-minimal coupling to the

0370-2693/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2006.08.039 M.R. Setare / Physics Letters B 641 (2006) 130–133 131 gravity [20]. Another model is a single scalar field minimally So, the equation of state can be written as coupled to the gravity but with a non-minimal kinetic term, in −φ˙2 +˙σ 2 − 2V(φ,σ) this model a dimensionless function of temperature f(T)is in w = . (5) − ˙2 +˙2 + front of the kinetic term. During the evolution of the universe φ σ 2V(φ,σ) when f(T)change sign from positive to negative, the possibil- From the equation of state, it is seen that for σ>˙ φ˙, w −1 ity of quintessence-phantom transition appears [16]. and for σ<˙ φ˙, we will have, w<−1. Alike [17], we consider In 1973, Bekenstein [21] assumed that there is a relation be- a potential with no direct coupling between two scalar fields tween the event of horizon and the thermodynamics of a black − − = + = λφ φ + λσ σ hole, so that the event of horizon of the black hole is a measure V(φ,σ) Vφ(φ) Vσ (σ ) Vφ0e Vσ 0e , (6) of the entropy of it. This idea has been generalized to horizons where the λφ and λσ , are two dimensionless positive numbers of cosmological models, so that each horizon corresponds to characterizing the slope of the potential for φ and σ respec- an entropy. Thus the second law of thermodynamics was mod- tively. So, the evolution equation for two scalar fields in FRW ified in the way that in generalized form, the sum of all time model will have the following form derivative of entropies related to horizons plus time derivative dVφ(φ) of normal entropy must be positive, i.e. the sum of entropies φ¨ + 3H φ˙ − = 0, (7) dφ must be increasing function of time. In [23], the validity of gen- dV (σ ) eralized second law (GSL) for the cosmological models which σ¨ + 3H σ˙ + σ = 0, (8) departs slightly from de Sitter space is investigated. However, dσ it is only natural to associate an entropy to the horizon area where, H is the Hubble parameter. as it measures our lack of knowledge about what is going on beyond it. In this Letter we show that the sum of normal en- 3. Generalized second law and quintom model of dark tropy and the horizon entropy in phantom dominated universe energy is non-decreasing function of time. Also, the transition from quintessence to phantom dominated universe is considered and To study the GSL through the universe which is dominated the conditions of the validity of GSL in transition is studied. by quintom scenario, we deduce the expression for normal en- Also for quintom model of dark energy [16], we study the GSL tropy using the first law of thermodynamics. in quintom dominated universe and conclude the same results TdS= dE + PdV= (P + ρ)dV + Vdρ. (9) when we consider two scalar fields with no coupling potential From Eqs. (3), (6) we have term. In our calculations we use c = 8πGN = 1. P + ρ =−φ˙2 +˙σ 2 (10) 2. The quintom model of dark energy and the Friedman constraint equation will be The quintom model of dark energy [16] is of new models 1 −φ˙2 σ˙ 2 H 2 = + V + + V . (11) proposed to explain the new astrophysical data, due to transition 3 2 φ 2 σ from w>−1tow<−1, i.e. transition from quintessence dominated universe to phantom dominated universe. Here we con- So, using relations (7) and (11), it is seen that sider the spatially flat Friedman–Robertson–Walker universe, 1 −1 H˙ = φ˙2 −˙σ 2 = (P + ρ). (12) where has following space–time metric 2 2 ˙2 ˙ 2 ˙ ds2 =−dt2 + a(t)2 dr2 + r2 dΩ2 . (1) Thus, if φ < σ then H<0, i.e. for the quintessence dominated universe and if φ˙2 > σ˙ 2 then H>˙ 0, for the phantom Containing the normal scalar field σ and negative kinetic scalar dominated universe. Rewriting the first law of thermodynamics field φ, the action which describes the quintom model is ex- = 4 3 with respect to relations above and using V 3 πRh , in which pressed as the following form the R is the event of horizon, one can obtain h √ 4 R 1 μν TdS=−2HdV˙ + Vdρ=−8πR2HdR˙ + 8πR3HdH, S = d x −g − g ∂μφ∂νφ h h h 2 2 (13) 1 μν where T is the temperature of the quintom fluid. Therefore, the + g ∂μσ∂νσ + V(φ,σ) , (2) 2 time derivative of normal entropy will have the following form where we have not considered the Lagrangian density of matter 8πHR˙ 2 S˙ = h (H R − R˙ ). (14) field. In the spatially flat Friedman–Robertson–Walker (FRW) T h h universe, the effective energy density, ρ, and the effective pres- As we know, the quintom is the combination of normal scalar sure, P , of the scalar fields can be described by filed, i.e. quintessence and phantom scalar field. From the defi- 1 1 nition of event of horizon ρ =− φ˙2 + σ˙ 2 + V(φ,σ), (3) 2 2 ts ts dt dt 1 ˙2 1 2 = ∞ P =− φ + σ˙ − V(φ,σ). (4) Rh a(t) , < , (15) 2 2 a(t ) a(t ) t t 132 M.R. Setare / Physics Letters B 641 (2006) 130–133 where for different spacetimes ts has different values, e.g. for de (b) Quintessence dominated: ˙ ˙ Sitter space–time ts =∞, Rh satisfies the following equation In this case Rh 0, and H<0 so the sum of normal entropy which is true for both scalar fields individually and horizon entropy could be positive, if T>0 then the condition for validity of GSL is ˙ = − Rh HRh 1, (16) ˙ ˙ T Rh ˙ ˙ |H | (23) where Rh 0 for phantom dominated universe [22] and Rh 0 2Rh for quintessence dominated universe [23]. As the final form, we using Eq. (21) this condition has following form write the time derivative of normal entropy of the quintom fluid | ˙ | using relation (13) 4π H Rh ; b ˙ (24) H Rh 8πHR˙ 2 S˙ = h . (17) (c) Phase transition from quintessence to phantom: ˙ ˙ T As Rh 0 in quintessence model and Rh 0 in phantom As it is seen from relation (14), it is shown that, the sign of S˙ model, and assuming that Rh variates continually one can ex- ˙ = depends on the sign of H˙ , hence for quintessence dominated pect that in transition from quintessence to phantom Rh 0. universe S<˙ 0 and for phantom dominated universe S>˙ 0. So the horizon entropy in transition time will be zero, also in ˙ = ˙ = The entropy of a black hole is proportional to the area of its transition time H 0, using Eq. (17), we obtain S 0. There- event horizon is well understood, it has deep physical mean- fore in the transition time the total entropy is differentiable and ing. The status of an entropy associated to a cosmological event continuous. horizon is not well established. In some cases like the case a de Sitter horizon this seems plausible, with some caveats, but in 4. Conclusion general this is a topic of current research; see [24]. If, the hori- = 2 In order to solve cosmological problems and because the zon entropy is taken to be Sh πRh, the generalized second law stated that lack of our knowledge, for instance to determine what could be the best candidate for DE to explain the accelerated ex- ˙ ˙ pansion of universe, the cosmologists try to approach to best S + Sh 0. (18) results as precise as they can by considering all the possibili- Thus, we will have ties they have. Investigating the principles of thermodynamics and specially the second law—as global accepted principle in 8πHR˙ 2 S˙ + S˙ = h + 2πR R˙ 0. (19) the universe—in different models of DE, as one of these pos- h T h h sibilities, has been widely studied in the literature, since this To investigate the validity of Eq. (19), we will consider three investigation can constrain some of parameters in studied mod- different cases, the first case we dominate the phantom fluid, els, say, P.C. Davies [23] studied the change in event horizon the second, the quintessence will be dominated, and the third, area in cosmological models that depart slightly from de Sitter the transition from quintessence to phantom. space and showed that for this models the GSL is respected for the normal scalar field, provided the fluid to be viscous. (a) Phantom dominated: In the present Letter we have considered total entropy as the ˙ ˙ ˙ entropy of a cosmological event horizon plus the entropy of a In this case Rh 0 and H>0, then Sh < 0. If the phantom fluid temperature T>0 the condition for validity of GSL is as normal scalar field σ and ghost scalar field φ. In the quintom model of dark energy H˙ isgivenbyEq.(12), for the phantom ˙ ˙ T R˙ dominated case H>0, in this case Rh 0, then the horizon ˙ h ˙ H . (20) entropy is constant or decreases with time, i.e. Sh 0, there- 2Rh fore the phantom entropy must increases with expansion so long If the temperature is assumed to be proportional to the de Sitter as T>0. In fact the phantom fluids possess negative entropy temperature [23] and equals to minus the entropy of black hole of radius Rh. In contrast with the previous case in the quintessence domi- bH ˙ ˙ ˙ T = (21) nated case, H<0 and Rh 0, then Sh 0. By considering 2π the influence of the transition from the quintessence to phan- where b is a parameter, the GSL hold when tom dominated universe on the GSL, one can obtain that the time derivative of the future event horizon and the entropy must ˙ be zero at the transition time. In the summary, we have exam- 4πHRh b ˙ (22) ined the quintessence and phantom dominated universe, and we H|Rh| have shown that by satisfying the conditions (20), (23) the total = 1 in de Sitter space–time case Rh H , then b 1. 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