= Flat barotropic matter-vacuum solution
{title2=$H=H_\Lambda\coth[3(1+w)H_\Lambda t/2]$}
For positive constant <cosmological constant>, zero <cosmological spatial curvature> and a separate positive-density fluid with constant <equation-of-state parameter> $w>-1$, set $H_\Lambda=\sqrt{\Lambda/3}$ in the time-unit convention. Eliminating the fluid <density> gives $\dot H=-3(1+w)(H^2-H_\Lambda^2)/2$. On the expanding <Big Bang> branch,
$$
H=H_\Lambda\coth\left[\frac32(1+w)H_\Lambda t\right],\qquad
a\propto\sinh^{2/[3(1+w)]}\left[\frac32(1+w)H_\Lambda t\right].
$$
The origin is the <Big Bang>. The fluid's early-time power-law expansion approaches exponential vacuum domination at late times.
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