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 $ϕ$ in a universe dominated by a background fluid with equation of state parameter $w_B$. In addition to the previously-examined linear [$V(ϕ) = V_0 ϕ$] and quadratic [$V(ϕ) = V_0 ϕ^2$] potentials, we show that exact solutions exist for the power law potential $V(ϕ) = V_0 ϕ^n$ with $n = 4(1+w_B)/(1-w_B) + 2$ and $n = 2(1+w_B)/(1-w_B)$. These correspond to the potentials $V(ϕ) = V_0 ϕ^6$ and $V(ϕ) = V_0 ϕ^2$ for matter domination and $V(ϕ) = V_0 ϕ^{10}$ and $V(ϕ) = V_0 ϕ^4$ for radiation domination. The $ϕ^6$ and $ϕ^{10}$ 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, $w_ϕ$, for this case.

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