The Quadratic Approximation for Quintessence with Arbitrary Initial Conditions

TL;DR

This paper develops an analytic approximation for quintessence dark energy models near potential extrema, allowing for nonzero initial velocities, and compares it with numerical results to constrain model parameters using observational data.

Contribution

It generalizes previous quadratic potential approximations by including nonzero initial velocities and derives bounds on the dark energy equation of state from observations.

Findings

Analytic approximation for w(a) matches numerical simulations.

Upper bounds on present-day w derived from model parameters.

Constraints on quintessence models from observational data.

Abstract

We examine quintessence models for dark energy in which the scalar field, $\phi$, evolves near the vicinity of a local maximum or minimum in the potential $V(\phi)$, so that $V(\phi)$ be approximated by a quadratic function of $\phi$ with no linear term. We generalize previous studies of this type by allowing the initial value of $d \phi/dt$ to be nonzero. We derive an analytic approximation for $w(a)$ and show that it is in excellent agreement with numerical simulations for a variety of scalar field potentials having local minima or maxima. We derive an upper bound on the present-day value of $w$ as a function of the other model parameters and present representative limits on these models from observational data. This work represents a final generalization of previous studies using linear or quadratic approximations for $V(\phi)$.