On the Quintessence Scalar Field Potential

TL;DR

This paper introduces an analytical method to derive scalar field potentials in cosmological models with perfect fluid and quintessence, providing explicit formulas for different universe geometries and transition redshifts.

Contribution

It presents a new analytical approach to determine scalar field potentials in FRW cosmologies with constant equation of state parameters, extending to open and closed universes.

Findings

Derived explicit scalar field potentials for flat, open, and closed universes.

Provided a formula for the transition redshift from deceleration to acceleration.

Compared transition redshift with the standard $\\Lambda$CDM model.

Abstract

In this work we propose a new analytical method for determining the scalar field potential $V(\phi)$ in FRW type cosmologies containing a mixture of perfect fluid plus a quintessence scalar field. By assuming that the equation of state parameters of the perfect fluid, $\gamma -1 \equiv p_{\gamma}/\rho_{\gamma}$ and the quintessence, $\omega \equiv p/\rho$ are constants, it is shown that the potential for the flat case is $V(\phi) = A\rho_{\phi_{0}}\sinh^{B}(\lambda \phi)$, where $A$, $B$ and $\lambda$ are functions of $\gamma$ and $\omega$. This general result is a pure consequence of the Einstein field equations and the constancy of the parameters. Applying the same method for closed and open universes, the corresponding scalar field potentials are also explicitly obtained for a large set of values of the free parameters $\gamma$ and $\omega$. A formula yielding the transition redshift from a decelerating to an accelerating regime is also determined and compared to the $\Lambda$CDM case.