Time variation of the Equation of State for Dark Energy

Abstract: 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 $dw_Q/da$.