Time variation of the Equation of State for Dark Energy

The time variation of the equation of state ($w_Q$) for the dark energy is analyzed by the current values of parameters $\Omega_Q $, $w_Q $ and their time derivatives. In the future, detailed feature of the dark energy could be observed, so we have considered the second derivatives of $w_Q$ for two types of potential: One is an inverse power-law type ($V=M^{4+\alpha}/Q^{\alpha}$) and the other is an exponential one ($V=M^4\exp{(\beta M/Q)}$). The first derivative $dw_Q/da$ and the second derivative $d^2 w_Q/da^2$ for both potentials are derived. The first derivative is estimated by the observed two parameters $\Delta=w_Q+1$ and $\Omega_Q$, with the assuming for $Q_0$. In the limit $\Delta \rightarrow 0$, the first derivative is null and, under the tracker approximation, the second derivative also becomes null. For the inverse power potential $V=M^{4+\alpha}/Q^{\alpha}$, the observed first and second derivatives are used to determine the potential parameter $M$ and $\alpha$. For the potential of $V=M^4\exp{(\beta M/Q)}$, the second derivative is estimated by the observed parameters $\Delta$, $\Omega_Q$ and $dw_Q/da$.


I. INTRODUCTION
Even though it has passed almost one and half decade after the detection of the acceleration of the universe, it is not well understood the dark energy [1]. We do not yet know whether it is the cosmological constant or not [2,3]. Then it is searched to observe the variation of the equation of state (w = w Q ) for the dark energy. Many works have been done on the study of dark energy in the form of a slowly rolling scalar field and time variation of the equation of state for the dark energy. Usually it is taken the parameter to denote the variation of w as [1, [4][5][6][7][8][9] w(a) = w 0 + w a (1 − a), where a, w 0 and w a are the scale factor (a = 1 at present), the present value of w(a) and the first derivative of w(a) by w a = −dw/da, respectively.
Even now it is not so much well known about the value of w 0 and w a , we extend the parameter space as w(a) = w 0 + w a (1 − a) + 1 2 including the second derivative of w(a) as w a2 = d 2 w/da 2 . Although it may be hard to observe the parameter w a2 , it must be a good clue to understand the feature of the dark energy in the future.
We follow the single scalar field formalism of Steinhardt et al. [10] and take the potential of two type as V = M 4+α /Q α and V = M 4 exp(βM/Q). These potentials are supported by Wang et al. [11].
In §2, the equation of state for scalar field and parameters to describe potentials are presented. The First derivatives of w Q for two type potentials are calculated in §3. In §4, the second derivatives are presented, where the detailed calculations are shown in the Appendix A The results and discussion are presented in §5.

A. Scalar field
For the dark energy, we consider the scalar field Q(x, t), where the action for this field in the gravitational field is described by [10] where S M is the action of matter field and G is the gravitational constant, occasionally putting G = 1. Neglecting the coordinate dependence, the equation for Q(t) becomes as where H is the Hubble parameter and V ′ is the derivative of V by Q. Being κ = 8π/3, H satisfies the following equation where ρ B , ρ Q and ρ c are the energy density of the background, scalar field and the critical density of the universe. The energy density and pressure for the scalar field are written as and respectively. Then the parameter w Q for the equation of state is described by

B. Time variation of w Q
It is assumed that the present value of w Q is slightly different from a negative unity by ∆(> 0) as By using Eq.(8),Q 2 is written byQ which is also written by using the density parameter Ω Q = ρ Q /ρ c aṡ Combining Eq.(10) and Eq.(11), V is given by From Eq.(11) and Eq.(12),Q is given aṡ If we determine the potential, the parameters to describe the evolution of the scalar field is the value of Q andQ at some fixed time, because Eq.(4) is the second derivative equation.
Then the evolution or backword variation could be estimated from this fixed point. In the following we take this fixed time is at present and estimate backward the accelerating behavior in the near past.
As ρ c is given from the observation through Hubble parameter H,Q is determined by Ω Q and ∆ which also determine the value of V . If we adopt the form and parameter of the potential, the value of V could be used to estimate the value of Q. In reality, the evolution of H in Eq. (4) depends on the background densities which include radiation density. The effect of radiation density could be ignored in the near past (z ≤ 10 3 ) and so is not considered in this work.
The parameters of the potential are M and α. If we take M = M * , V becomes as From Eq. (12) as M 4 If we take Q = Q 0 M pl at present, M pl being the planck mass, M * becomes Then Q 0 , Ω Q , ∆ and α determine the parameter M * , which means that parameters to determine the accelerating behavior are Q 0 , Ω Q , ∆ and α.
The difference of the observed value ρ c and the value M 4 pl is described by the observed value N as being N ≃ 122, [3] M * becomes as The problem is how to estimate Q 0 and α.
In essence, as β is combined with M, the parameter of this potential is β. From Eq. (12), Then Q is estimated as If we take Q = Q 0 M pl at present, Q 0 determines the parameter β as, In this potential, the parameters to determine the accelerating behavior are Q 0 , Ω Q and ∆.
The problem how to estimate Q 0 is also left.

III. FIRST DERIVATIVE OF w Q
To investigate the variation of w Q , we calculate dw Q /da, using Eqs. (4), (6) and (7), In the limit ∆ → 0 whereQ = ∆ρ c Ω Q → 0, dw Q /da becomes null. To investigate further, we must consider the potential form.
To investigate the signature of dw Q /da, we must estimate the following term, To estimate Q 0 , we consider the tracker approximation that w Q is almost constant as which is given by Eq. (9) of Steinhardt et al. [10] If we adopt this approximation, Q/M pl becomes as We approximate the present value Q 0 as Q 0 = (1+ε)×Q α . Then from Eq. (25), 24π∆ then Q 0 is given by If dw Q /da is observed, Q 0 /α is determined by the observed values Ω Q , ∆, and dw Q /da.
From Eq. (30), Q is determined as Then Q 0 is estimated by observable parameters Ω Q , ∆, and dw q /da as Q 0 does not depend on the potential parameter β, which is determined by Eq. (22).

IV. THE SECOND DERIVATIVE OF w Q
From Eq. (23), the second derivative of w Q is given by The time derivatives of p Q and ρ Q are written asṗ whereä/a = −4πG(ρ c + p Q ). By using these equations, the detailed calculation of Eq.(34) is described in the Appendix A.
From Eq.(A21), d 2 w/da 2 becomes From this equation, we estimate d 2 w Q /da 2 for each potential in the following.
A. V = M 4+α /Q α As the following relations are derived we put them in Eq.(37) and get If dw Q /da is observed, Q/α is determined by Eq. (29). If d 2 w Q /da 2 is observed, one could estimate the value of α from the above equation.
If we take the tracker approximation as 24π∆ In the limit ∆ → 0 where Q α → ∞, dw 2 Q /da 2 becomes null in this approximation.
As the following relations are derived we put them in Eq.(37) and get If we take the tracker approximation as in Eq. (31), the part within [ ] of the above equation becomes 24π∆ In the limit ∆ → 0 where Q β → ∞, dw 2 Q /da 2 becomes null.

V. CONCLUSIONS AND DISCUSSION
It is important to know the variation of the equation of state w of the back ground field for the investigation of the expansion of the universe. It is known thatä is described by the following equationä = − 4πG 3 (1 + 3w)aρ.
At present, it has been proceed the look back observation of large scale structure of the universe to estimate w Q at the age (1 + z). [5] For the moment, it has been persued the value of w(a = a 0 ) and dw/da It could be expected the observation of the second derivative of w in the future w(a) = w(a 0 = 1) + dw da da + 1 2 so we estimate the second derivative of w with a for typical two potentials in this work.
The first derivative dw Q /da and the second derivative d 2 w Q /da 2 for the power inverse and exponential potentials are calculated. The first derivative is estimated by the observed two parameters ∆ = w Q + 1 and Ω Q , with the assuming parameters Q 0 . In the limit ∆ → 0, the first derivative is null and, under the tracker approximation, the second derivative also If we assume matter dominant approximation, we could estimate α and M * from the attractor solution [12] which is outlined in the Appendix B.