The w = -1 crossing of the quintom model with slowly-varying potentials

Considering the quintom model with arbitrary potential, it is shown that there always exists a solution which evolves from w>-1 region to w<-1 region. The problem is restricted to the slowly varying potentials, i.e. the slow-roll approximation. It is seen that the rate of this phase transition only depends on the energy density of matter at transition time, which itself is equal to the kinetic part of quintom energy density at that time. The perturbative solutions of the fields are also obtained.


Introduction
Recent astrophysical data indicate that the expansion of the universe is accelerating [1]. This acceleration can be related to the presence of a perfect fluid with negative pressure, known as dark energy, which constitutes two third of our present universe.
One candidate for dark energy is the cosmological constant: a constant quantum vacuum energy density which fills the space homogeneously, corresponding to a fluid with a constant equation of state, EOS, parameter ω d = −1 (in this paper ω d and ω denote the EOS parameters of dark energy component and the universe, respectively). This model suffers from fine tuning and coincidence problem [2]. Alternatively, dynamical homogeneous fields, e.g. scalar fields with canonical and non-canonical kinetic terms and with various potentials, have been proposed as the origin of dark energy [3,4]. In these models, in contrast to the cosmological constant model, the EOS parameter may vary with time. The EOS parameter satisfying −1 < ω < −1/3, which determines the quintessence era, can be achieved by introducing a normal scalar field φ, known as quintessence scalar field [3].
Due to the fact that the observations have shown some mild preference for an equation of state parameter ω < −1, the phantom scalar field σ with a wrong sign kinetic term was introduced in the literature [4]. Depending on phantom scalar field potential, different solutions such as asymptotic de Sitter, big rip, etc. may be obtained [5].
Some astrophysical data seem to slightly favor an evolving dark energy EOS parameter ω d and show a recent ω d = −1 crossing [8]. As the dark energy model with a quintessence field has an equation of state ω d > −1 and for the phantom ghost field we always have ω d < −1, the phase transition from quintessence to phantom era does not occur in models with only one scalar field, either phantom or quintessence [6]. To study such a transition, we need models which are at least composed of two scalar fields [9], known as hybrid models. One of these models is the quintom model which assumes that the cosmological fluid, besides the matter and radiation, is composed of a quintessence and a phantom scalar fields [7]. In [10], a phase-space analysis of a spatially flat Friedman-Robertson-Walker universe containing a barotropic fluid and phantom-scalar fields with exponential potentials has been presented. This model has late time phantom attractor solution. In [11] the same calculation has been done by introducing an interaction term between the phantom and the quintessence fields. The quantum stability of quintom models is also an important problem whose some aspects have been discussed in [12].
Recently a new view of quintom model, known as hessence model, has been introduced. In this model, the dark energy is described by a single field with an internal degree of freedom rather than two independent real scalar fields. Hessence model can avoid the difficulty of the Q-ball formation which gives trouble to the spintessence model [13]. The evolution of ω in this model has been studied in [14] and [15], via phase-space analysis. Although this model allows ω to cross −1, but it avoids the late time singularity or the "big rip".
In the recent paper [16], we have investigated the necessary conditions for occurrence of the ω = −1 crossing in the quintom model ( which also implies the ω d = −1 crossing ), in special the transition from quintessence to phantom phase. This transition has been checked for two specific quintom potentials, the power-law and exponential potentials, and has been shown that the ω = −1 crossing really exists in these examples. It has been assumed that the fields have slowly varying behavior, i.e. the problem has been solved in slow-roll (SR) approximation.
In this paper, we are going to prove that this phase transition occurs for an arbitrary quintom potential, as long as we consider the SR approximation. We think that this is an important result which is in agreement with the present data.
The scheme of the paper is as follows. In section 2, we briefly review the main results of [16] and write down the necessary conditions needed for occurrence of the quintessence to phantom phase transition in the quintom model. In section 3, by solving the Friedman equations for arbitrary potential around the transition point and in the SR approximation, it is shown that the desired transition occurs for the general quintom model. The general form of SR conditions, needed for consistency of the equations, is stated in terms of the time derivatives of quintom fields. In section 4, we obtain the explicit perturbative solution of quintom fields for arbitrary potential, from which the SR approximation can be expressed in terms of the derivatives of the potential, which is more physical. This is discussed in appendix A. We check our perturbative results with the exact expressions known for special cases.
We use the units = c = G = 1 throughout the paper.

Transition conditions in quintom model
Consider a spatially flat Friedman-Robertson-Walker universe with scale factor a(t), filled with (dark) matter and quintom dark energy. The evolution equation of matter density ρ m isρ in which γ m = 1 + ω m . ω m is the equation of state parameter of matter field defined by ω m = P m /ρ m , where P m is the pressure of matter field. H(t) = a(t)/a(t) is the Hubble parameter and "dot" represents the time derivative. The quintom dark energy consists of a normal scalar field φ, i.e. the quintessence field, and a negative kinetic energy scalar field σ, called the phantom field. The energy density ρ D and pressure P D of the homogenous quintom dark energy are [7,10] and their evolution equations arë The Friedman equations, obtained from Einstein equations, are Note that eqs. (3), (4) and, (5) are not independent. The equation of state parameter ω = (P D + P m )/(ρ D + ρ m ) can be expressed in terms of Hubble parameter as ForḢ < 0, the system is in the quintessence phase ω > −1, and whenḢ > 0, it is in the phantom phase with ω < −1. So crossing the ω = −1 line is, in principle, possible in quintom model. If we are interested in situation in which the quintessence to phantom phase occurs in some instant of time t = t 0 , then H(t) must have a local minimum at that time, i.e.Ḣ(t 0 ) = 0. So at t < t 0 ,Ḣ < 0 and ω > −1 and at t > t 0 ,Ḣ > 0 with ω < −1. If we restrict ourselves to t − t 0 << h −1 0 , where h 0 = H(t 0 ) and h −1 0 is of order of the age of the universe, H(t) can be taken as α ≥ 2 is the order of the first non-vanishing derivative of H(t) at t = t 0 and The desired phase transition occurs when α is an even positive integer and h 1 > 0.
For arbitrary quintom potential V (φ, σ), we want to study if this situation ( even α and positive h 1 ) exists or not. As usual, we restrict ourselves to SR approximation in which the first terms of equations (3) are negligible: φ << Hφ, σ << Hσ.

Seeking for transition solution
Following [16], we are going to find any solution to eqs. (3)- (5), when H(t) given by eq. (7) and the fields vary slowly, i.e. eq. (8), with the desired property, that is h 1 > 0 and α = even. Let us begin with eq.(5) and expand it near t 0 ≡ 0. One finds in which In terms of the dimensionless variablesΦ =φ/h 0 ,Σ =σ/h 0 , R m = ρ m /h 2 0 , τ = h 0 t, and H 1 = h 1 /h α+1 0 , eq.(9) can be written as We will continue our study with eq.(9), reminding that shifting to dimensionless variables is always possible. The matter density ρ m (t) can be found by solving eq.(1) with H(t) given by eq. (7). The result is For α ≥ 2, the first equation obtained from eq. (9) is For second relation, we first note thaṫ But in SR approximation (8), we have where in the last equality we use eq. (14). Therefore, excluding special γ m << 1 cases, one finds −4πβ(0) = 12πh 0 γ 2 m ρ m (0), which is a non-zero positive quantity. This shows that for all quintom potentials, one has α = 2 which proves the ω = −1 crossing for all quintom models in SR approximation, as long as the remaining relations which are obtained from eq.(4), are satisfied, up to the lowest order, consistently.
The important observation is that in all quintom models in SR approximation, the rate of the phase transition (h 1 ) only depends on the (dark) matter energy density ρ m (0), which itself is determined by the kinetic part of quintom energy density (eq. (14)).
It is worth noting that if there is no dark matter field, the eq.(18), and therefore eq.(19), can not be used directly. In this case, we can not neglect the first two-terms of eq.(15) and therefore h 1 becomes The above equation shows that the ω = −1 crossing is in principle possible in dark-matter free quintom models.
The first two relations obtained from eq.(21) are which determines h 0 in terms of V (0) and ρ m (0), andδ(0) = 0, which can be written as in which the equation of motion (3) in SR approximation has been used. So eq.(24) can be written as which is reduced to eq. (14) in SR approximation.
The last equation that must be checked in the lowest order approximation, is one obtained from t 2 -term in the right-hand-side of eq.(21). Using eqs. (13) and (25) andḢ (0) where the SR approximation has been used in the second equality. But the natural extension of SR approximation (8) to higher order terms is So the eq.(27) reduces to where eqs. (15) and (9), with α = 2, have been used. Now this is exactly equal to the second term in the left-hand-side of eq.(21) for α = 2. Let us briefly express what we have done. We have shown that the eqs.(3)-(5) have a transition solution when H(t) behaves as (7) with α = 2. It has been shown that eqs. (4) and (5) result three independent relations: Eq. (14), which relates ρ m (0) to the kinetic energy of quintom fields, and eqs. (19) and (23) which determine the Hubble parameters h 0 and h 1 in terms of matter field and potential. For t << h −1 0 and in SR approximation, it has been shown that all remaining relations are consistent with the mentioned three ones. This completes the proof of ω = −1 crossing of all quintom models in SR approximation.

Solution of equations of motion
In special cases, like ones considered in [16], one can, in principle, solve the equations of motion (3) in SR approximation and studies their physical behaviors. For arbitrary potential, as expected, we can not solve these equations without knowing the functional form of the potential, but instead we try to find a perturbative solution, from which one can deduce some physics behind the problem. Here we focus on the equation of motion of the quintessence field. The solution of phantom field can be simply obtained from final relations by replacing the derivatives from φ to σ and transforming V → −V .
Consider the equation of motion of quintessence field in SR approximation φ << Hφ with H = h 0 + h 1 t 2 : Using the Lerchphi function Φ(z, a, b) defined as it can be easily shown Eq.(31) then leads to where and λ is the constant of integration. Note that in this stage, we assume that ∂V /∂φ does not depend on σ. Expanding both sides of eq.(34) around t = 0 up to order t 3 , results in which leads to 3G(0) + λ = 0, ...
As an example, we consider the exponential potential Eq.(40) then results in: (42) But it can be easily shown that the above expression is same as one obtained from expanding the exact expression: [16] in which At the end, let us discuss a possible ambiguity. It may be argued that as we take H(t) up to order t 2 in eq.(31), keeping the t 3 -terms for φ(t) is not correct. To answer this question, we keep the t 3 -term in H(t): In SR approximation, we have dt Using (45), the left-hand-side of eq.(46), up to order t 3 , is So the coefficient h 2 only contributes to t 4 -term and therefore the expansion (40) is correct. Acknowledgement: We would like to thank the "center of excellence in structure of matter" of the Department of Physics for partial financial support.