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

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.