Conformal Riccati equation for matter and a coasting fluid ($2{\mathcal{H}}^{\prime }+{\mathcal{H}}^2=H_0^2(1-{\Omega }_m)$)

The conformal Friedmann equation is ${\mathcal{H}}^2=H_0^2[{\Omega }_m/a+1-{\Omega }_m]$. Differentiating and using $a^{\prime }=a\mathcal{H}$ eliminates the matter term and yields a first-order Riccati equation. The expanding Big-Bang branch for $0<{\Omega }_m<1$ has $\mathcal{H}=\alpha \coth (\alpha \tau /2)$.