 Closed constant-equation-of-state Friedmann scale factor

For a closed Friedmann equation normalized to $k{c}^2=1$, constant equation-of-state parameter $w$, and conformal time $\tau$, let $q=(1+3w)/2$. The cosmological continuity equation and Friedmann acceleration equation give ${\mathcal{H}}^{\prime }=-q(1+{\mathcal{H}}^2)$, where $\mathcal{H}=a^{\prime }/a$. For $q\ne 0$ this integrates to the stated scale factor on any interval where the sine is positive, with ${\alpha }^{2q}=8\pi G{\rho }_0/3$. For $w=-1/3$ the separate solution is $a=A{e}^{H_0\tau }$ with $H_0^2+1=8\pi G{\rho }_0/3$; the sine-power formula is inapplicable.