Paper detail

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 $Ω_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 ($V=M^{4+α}/Q^α$) and the other is an exponential one ($V=M^4\exp{(βM/Q)}$). 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 $Δ=w_Q+1$ and $Ω_Q$, with the assuming for $Q_0$. In the limit $Δ\rightarrow 0$, the first derivative is null and, under the tracker approximation, the second derivative also becomes null. For the inverse power potential $V=M^{4+α}/Q^α$, the observed first and second derivatives are used to determine the potential parameter $M$ and $α$. For the potential of $V=M^4\exp{(βM/Q)}$, the second derivative is estimated by the observed parameters $Δ$, $Ω_Q$ and $dw_Q/da$.