Quest for potentials in the quintessence scenario

TL;DR

This paper derives the third derivative of the dark energy equation of state in quintessence models to help distinguish potential types and predict future observations for understanding dark energy dynamics.

Contribution

It provides a general formula for the third derivative of w for various scalar potentials, aiding in identifying the potential form from observational data.

Findings

Third derivative of w derived for multiple potentials.

At least n+2 observations needed to identify potential with n parameters.

Numerical analysis shows potential to distinguish freezing and thawing modes.

Abstract

The time variation of the equation of state $w$ for quintessence scenario with a scalar field as dark energy is studied up to the third derivative ($d^3w/da^3$) with respect to the scale factor $a$, in order to predict the future observations and specify the scalar potential parameters with the observables. The third derivative of $w$ for general potential $V$ is derived and applied to several types of potentials. They are the inverse power-law ($V=M^{4+\alpha}/Q^{\alpha}$), the exponential ($V=M^4\exp{(\beta M/Q)}$), the cosine ($V=M^4(\cos (Q/f)+1)$) and the Gaussian types ($V=M^4\exp(-Q^2/\sigma^2)$), which are prototypical potentials for the freezing and thawing models. If the parameter number for a potential form is $ n$, it is necessary to find at least for $n+2$ independent observations to identify the potential form and the evolution of the scalar field ($Q$ and $ \dot{Q} $). Such observations would be the values of $ \Omega_Q, w, dw/da. \cdots $, and $ dw^n/da^n$. Since four of the above mentioned potentials have two parameters, it is necessary to calculate the third derivative of $w$ for them to estimate the predict values. If they are tested observationally, it will be understood whether the dark energy could be described by the scalar field with this potential. Numerical analysis for $d^3w/da^3$ are made under some specified parameters in the investigated potentials. It becomes possible to distinguish the freezing and thawing modes by the accurate observing $dw/da$ and $d^2w/da^2$ in some parameters.