Dark energy from a quintessence (phantom) field rolling near potential minimum (maximum)

TL;DR

This paper derives a general analytical expression for the dark energy equation of state in quintessence and phantom models near potential extrema, revealing how curvature influences its evolution and oscillatory behavior.

Contribution

It provides a unified analytical framework for understanding dark energy dynamics near potential minima or maxima, including oscillatory solutions and observational constraints.

Findings

Analytical expression for w(a) near potential extrema

Oscillatory behavior of w(a) for certain potential curvatures

Agreement within 1% between analytical and numerical results

Abstract

We examine dark energy models in which a quintessence or a phantom field, $\phi$, rolls near the vicinity of a local minimum or maximum, respectively, of its potential $V(\phi)$. Under the approximation that $(1/V)(dV/d\phi) \ll 1$, [although $(1/V)(d^2 V/d\phi^2)$ can be large], we derive a general expression for the equation of state parameter $w$ as a function of the scale factor for these models. The dynamics of the field depends on the value of $(1/V)(d^2 V/d\phi^2)$ near the extremum, which describes the potential curvature. For quintessence models, when $(1/V)(d^2 V/d\phi^2)<3/4$ at the potential minimum, the equation of state parameter $w(a)$ evolves monotonically, while for $(1/V)(d^2 V/d\phi^2)>3/4$, $w(a)$ has oscillatory behavior. For phantom fields, the dividing line between these two types of behavior is at $(1/V)(d^2 V/d\phi^2) = -3/4$. Our analytical expressions agree within 1% with the exact (numerically-derived) behavior, for all of the particular cases examined, for both quintessence and phantom fields. We present observational constraints on these models.