What do we learn by mapping dark energy to a single value of $w$?

Abstract

We examine several dark energy models with a time-varying equation of state parameter, $w(z)$, to determine what information can be derived by fitting the distance modulus in such models to a constant equation of state parameter, $w_*$. We derive $w_*$ as a function of the model parameters for the Chevallier-Polarski-Linder (CPL) parametrization, and for the Dutta-Scherrer approximation to hilltop quintessence models. We find that all of the models examined here can be well-described by a pivot-like redshift, $z_{pivot}$ at which the value of $w(z)$ in the model is equal to $w_*$. However, the exact value of $z_{pivot}$ is a model-dependent quantity; it varies from $z_{pivot} = 0.22-0.25$ for the CPL models to $z_{pivot} = 0.17-0.20$ for the hilltop quintessence models. Hence, for all of the models considered here, a constant-$w$ fit gives the value of $w$ for $z$ near 0.2. However, given the fairly wide variation in $z_{pivot}$ over even this restricted set of models, the information gained by fitting to a constant value of $w$ seems rather limited.