= Flat matter-coasting-fluid Friedmann solution
{title2=$a=\frac{\Omega_m}{2(1-\Omega_m)}[\cosh(\alpha\tau)-1]$}
A flat universe with separately conserved <pressureless matter> and a <coasting fluid> satisfies $a'^2=H_0^2[\Omega_m a+(1-\Omega_m)a^2]$. For both densities positive, choosing the <Big Bang> at zero <conformal time> gives $a=\Omega_m[\cosh(\alpha\tau)-1]/[2(1-\Omega_m)]$, with $\alpha=H_0\sqrt{1-\Omega_m}$. Physical time follows by integrating $dt=a\,d\tau$. The hyperbolic parametrization is formally like the open dust solution, but the spatial geometry here is flat.
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