= Quadratic-inflaton Hubble oscillations
{title2=$H=H_m+\frac{3H_m^2}{4m}\sin(2mt)+\cdots$}
For a homogeneous canonical <inflaton> oscillating near $V=m^2\phi^2/2$, use reduced-Planck units and require $m\gg H$. The slowly varying ansatz $\phi=A(t)\cos mt$ gives $H_m^2\simeq m^2A^2/6$. The <Nonincreasing Hubble rate for a canonical inflaton> becomes
$$
\dot H\simeq-\frac32H_m^2(1-\cos2mt).
$$
Cycle averaging gives $\dot H_m=-3H_m^2/2$, so $H_m=2/[3(t-t_*)]$. Integrating the rapidly varying term while holding its coefficient approximately constant gives the displayed correction. It is smaller than the mean by $O(H_m/m)$. Although the correction oscillates, the total <Hubble parameter> remains nonincreasing. Equal average kinetic and <potential energies> give pressureless-matter-like expansion, with amplitude $A\propto a^{-3/2}$.
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