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

Abstract

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

10 pages, 1 figure, discussion, references, and phase-plane figure added

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