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 $d{w}_Q/da$ and the second derivative $d^2{w}_Q/d{a}^2$ 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 $\Delta \to 0$, 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 $d{w}_Q/da$.

15 pages

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