2026/01/21 by Patrick Hu, Robert J. Scherrer
#gr-qc #astro-ph.CO #hep-th #math-ph #math.MP
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 wB = -1, extending earlier work on exact solutions with wB > -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 = V0ln ϕ, V = V0 ϕ1/2 and V = V0/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 wB > -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 wB > -1.