Approximate $w_\phi\sim\Omega_\phi$ Relations in Quintessence Models

TL;DR

This paper derives an analytic approximation for the relation between dark energy equation of state and density in quintessence models, enabling potential constraints directly from observations without numerical solutions.

Contribution

It introduces a full tracker equation approach to approximate the $w_$-$\Omega_$ relation across all regimes, improving analysis of quintessence evolution.

Findings

Derived stable fixed points for $w_$ and $\Omega_$ from the full tracker equation.

Established an analytic $w_ ext{-}\Omega_$ relation valid for most quintessence potentials.

Provided inequalities linking $w_$, $\\Omega_$ with their fixed points, aiding observational constraints.

Abstract

Quintessence field is a widely-studied candidate of dark energy. There is "tracker solution" in quintessence models, in which evolution of the field $\phi$ at present times is not sensitive to its initial conditions. When the energy density of dark energy is neglectable ($\Omega_\phi\ll1$), evolution of the tracker solution can be well analysed from "tracker equation". In this paper, we try to study evolution of the quintessence field from "full tracker equation", which is valid for all spans of $\Omega_\phi$. We get stable fixed points of $w_\phi$ and $\Omega_\phi$ (noted as $\hat w_\phi$ and $\hat\Omega_\phi$) from the "full tracker equation", i.e., $w_\phi$ and $\Omega_\phi$ will always approach $\hat w_\phi$ and $\hat\Omega_\phi$ respectively. Since $\hat w_\phi$ and $\hat\Omega_\phi$ are analytic functions of $\phi$, analytic relation of $\hat w_\phi\sim\hat\Omega_\phi$ can be obtained, which is a good approximation for the $w_\phi\sim\Omega_\phi$ relation and can be obtained for the most type of quintessence potentials. By using this approximation, we find that inequalities $\hat w_\phi<w_\phi$ and $\hat\Omega_\phi<\Omega_\phi$ are statisfied if the $w_\phi$ (or $\hat w_\phi$) is decreasing with time. In this way, the potential $U(\phi)$ can be constrained directly from observations, by no need of solving the equations of motion numerically.