Exact general solutions for cosmological scalar field evolution in a vacuum-energy dominated expansion

TL;DR

This paper derives exact solutions for scalar field evolution in a vacuum-energy dominated universe, extending previous work and analyzing the applicability of the slow-roll approximation in such scenarios.

Contribution

It provides new exact solutions for scalar fields with specific potentials in a vacuum-dominated universe and generalizes the slow-roll approximation to these conditions.

Findings

Exact solutions for linear, quadratic, and certain nonlinear potentials.

Logarithmic potential yields an exact first integral.

Slow-roll approximation applies to flat potentials in vacuum domination, not in $w_B > -1$ cases.

Abstract

We derive exact general solutions (as opposed to attractor particular solutions) for the evolution of a scalar field $\phi$ in a universe dominated by a background fluid with equation of state parameter $w_B = -1$, extending earlier work on exact solutions with $w_B > -1$. Straightfoward exact solutions exist when the evolution is described by a linear differential equation, corresponding to constant, linear, and quadratic potentials. In the nonlinear case, exact solutions are derived for $V = V_0\ln \phi$, $V = V_0 \phi^{1/2}$ and $V = V_0/\phi$, and the logarithmic potential also yields an exact first integral. These complicated parametric solutions are considerably less useful than those derived previously for a universe dominated by a barotropic fluid such as matter or radiation with $w_B > -1$. However, we generalize the slow-roll approximation and show that it applies to all sufficiently flat potentials in the case of a vacuum-dominated expansion, while it never applies when the universe is dominated by a background fluid with $w_B > -1$.