Quadratic-inflaton Hubble oscillations ($H=H_m+\frac{3H_m^2}{4m}\sin (2mt)+\cdots$)

For a homogeneous canonical inflaton oscillating near $V=m^2{\varphi }^2/2$, use reduced-Planck units and require $m\gg H$. The slowly varying ansatz $\varphi =A(t)\cos mt$ gives $H_m^2\simeq m^2{A}^2/6$. The Nonincreasing Hubble rate for a canonical inflaton becomesCycle averaging gives ${\dot{H}}_m=-3H_m^2/2$, so $H_m=2/[3(t-t_{\ast })]$. 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 $A\propto a^{-3/2}$.