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 <w< 3 . While the most model independent ing. But one can show [16,17] that considering the combination analysis suggest that the acceleration of the universe to be be- of quintessence and phantom in a joint model, the transition low the de Sitter value [6], it is certainly true that the body of can be fulfilled. This model, dubbed quintom, can produce a observational data allows for a wide parameter space compati- better fit to the data than more familiar models with w −1. ble with an acceleration larger than the de Sitter’s [8,9]. If even- In the other term the quintom model of dark energy represents tually this proves to be the case, the fluid driving the expansion a transition of dark energy equation of state from w>−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 dom- inated 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 dom- inated 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.