= Closed constant-equation-of-state Friedmann scale factor
{title2=$a=\alpha[\sin(q(\tau-\tau_0))]^{1/q}$}
For a closed <Friedmann equation> normalized to $kc^2=1$, constant equation-of-state parameter $w$, and <conformal time> $\tau$, let $q=(1+3w)/2$. The <cosmological continuity equation> and <Friedmann acceleration equation> give $\mathcal H'=-q(1+\mathcal H^2)$, where $\mathcal H=a'/a$. For $q\ne0$ 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 $a=Ae^{H_0\tau}$ with $H_0^2+1=8\pi G\rho_0/3$; the sine-power formula is inapplicable.
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