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 universe dominated by background fluid with equation state parameter $w_B$. In addition previously-examined linear [$V(\phi) = V_0 \phi$] quadratic \phi^2$] potentials, we show that exist power law potential $V(\phi) \phi^n$ $n 4(1+w_B)/(1-w_B) + 2$ 2(1+w_B)/(1-w_B)$. These correspond potentials \phi^6$ \phi^2$ matter domination \phi^{10}$ \phi^4$ radiation domination. The $\phi^6$ $\phi^{10}$ can yield either oscillatory or non-oscillatory evolution, use determine how initial conditions map onto each form evolution. exponential yields an solution stiff/kination ($w_B 1$) background. this analytic expression parameter, $w_\phi$, case.