Exact scaling solutions and fixed points for general scalar field

TL;DR

This paper derives exact scaling solutions and fixed points for general scalar field models, showing that k-essence encompasses quintessence and tachyon models, and identifies potentials leading to specific dark energy behaviors.

Contribution

It provides a comprehensive analysis of scaling solutions and fixed points in scalar field dark energy models, including the derivation of exact solutions and potential forms.

Findings

k-essence is the most general dark energy model with scaling solutions

Inverse squared potential yields tracking solutions

Scalar field dominant fixed point identified

Abstract

We show that the most general dark energy model that possesses a scaling solution $\rho_\phi\propto a^n$ is the k-essence model, which includes both of the quintessence and tachyon models. The exact scaling solutions are then derived. The potential that gives the tracking solution in which dark energy exactly tracks the background matter field is the inverse squared potential. The quintessence field with exponential potential can be obtained from the k-essence field with the inverse squared potential. We also find the fixed points and study their main properties, whereby the scalar field dominant fixed point is identified.