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

TL;DR

This paper derives exact general solutions for a scalar field evolving in a vacuum-energy dominated expansion ($w_B=-1$), extending prior exact-solution work that focused on $w_B>-1$. It identifies solvable cases with linear ODEs for constant, linear, and quadratic potentials, and derives exact forms (including a first integral) for three nonlinear potentials: $V=V_0 \ln(\phi)$, $V=V_0 \phi^{1/2}$, and $V=V_0/\phi$; the logarithmic case, in particular, yields a usable closed-form first integral and a parametric solution. The authors also generalize the slow-roll condition to vacuum domination, showing it applies to a wide class of sufficiently flat potentials, in contrast to the $w_B>-1$ case where slow-roll never applies. Overall, the work completes the exact-solution program for vacuum-dominated cosmologies, clarifying the role of exact first integrals and the practical utility of slow-roll in this regime.

Abstract

We derive exact general solutions (as opposed to attractor particular solutions) for the evolution of a scalar field $φ$ 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 $V = V_0\ln φ$, $V = V_0 φ^{1/2}$ and $V = V_0/φ$, 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 $w_B > -1$.