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

Abstract

We derive exact general solutions (as opposed to attractor particular solutions) and corresponding first integrals for the evolution of a scalar field $\varphi$ in a universe dominated by a background fluid with equation of state parameter $w_B$. In addition to the previously-examined linear [$V(\varphi )=V_0\varphi$] and quadratic [$V(\varphi )=V_0{\varphi }^2$] potentials, we show that exact solutions exist for the power law potential $V(\varphi )=V_0{\varphi }^n$ with $n=4(1+w_B)/(1-w_B)+2$ and $n=2(1+w_B)/(1-w_B)$. These correspond to the potentials $V(\varphi )=V_0{\varphi }^6$ and $V(\varphi )=V_0{\varphi }^2$ for matter domination and $V(\varphi )=V_0{\varphi }^{10}$ and $V(\varphi )=V_0{\varphi }^4$ for radiation domination. The ${\varphi }^6$ and ${\varphi }^{10}$ 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, $w_{\varphi }$, for this case.

13 pages, 3 figures, references added

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