= Constant-equation-of-state conformal Riccati equation
{title2=$\mathcal H'+\alpha(\mathcal H^2+k)=0,\quad\alpha=(1+3w)/2$}
For a <perfect fluid in general relativity> with constant <equation-of-state parameter> $w$, use <conformal time> $dt=a\,d\tau$ and the <conformal Hubble parameter> $\mathcal H=a'/a=aH$. The <Friedmann acceleration equation> gives $\mathcal H'=a\ddot a=-4\pi Ga^2(1+3w)\rho/3$. The <Friedmann equation> eliminates $\rho$ and gives this <Riccati equation>. For positive <energy density>, $\alpha\ne0$ and appropriately shifted <conformal time>, its usual branches are $\cot(\alpha\tau)$ for $k=1$, $1/(\alpha\tau)$ for $k=0$, and $\coth(\alpha\tau)$ for $k=-1$. Their <scale factors> are proportional to $\sin^{1/\alpha}(\alpha\tau)$, $(\alpha\tau)^{1/\alpha}$ and $\sinh^{1/\alpha}(\alpha\tau)$ respectively on intervals where the bases are positive. At $\alpha=0$ the <conformal Hubble parameter> is constant and the separate solution is $a=Ae^{h\tau}$.
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