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

TL;DR

This paper derives exact solutions for scalar field evolution in a universe with background fluid, covering various potentials and background equations of state, providing new analytical tools for cosmological models.

Contribution

It presents the first integrals and exact solutions for scalar fields with power law and exponential potentials in background-dominated universes, extending previous work.

Findings

Exact solutions for $V() = V_0 ^n$ with specific n values.

Solutions for exponential potential in stiff background.

Mapping initial conditions to evolution types.

Abstract

We derive exact general solutions (as opposed to attractor particular solutions) and corresponding first integrals for the evolution of a scalar field $\phi$ in a universe dominated by a background fluid with equation of state parameter $w_B$. In addition to the previously-examined linear [$V(\phi) = V_0 \phi$] and quadratic [$V(\phi) = V_0 \phi^2$] potentials, we show that exact solutions exist for the power law potential $V(\phi) = V_0 \phi^n$ with $n = 4(1+w_B)/(1-w_B) + 2$ and $n = 2(1+w_B)/(1-w_B)$. These correspond to the potentials $V(\phi) = V_0 \phi^6$ and $V(\phi) = V_0 \phi^2$ for matter domination and $V(\phi) = V_0 \phi^{10}$ and $V(\phi) = V_0 \phi^4$ for radiation domination. The $\phi^6$ and $\phi^{10}$ potentials can yield either oscillatory or non-oscillatory evolution, and we use the first integrals to determine how the initial conditions map onto each form of evolution. The exponential potential yields an exact solution for a stiff/kination ($w_B = 1$) background. We use this exact solution to derive an analytic expression for the evolution of the equation of state parameter, $w_\phi$, for this case.