Flat barotropic matter-vacuum solution ($H=H_{\Lambda }\coth [3(1+w)H_{\Lambda }t/2]$)

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,The origin is the Big Bang. The fluid's early-time power-law expansion approaches exponential vacuum domination at late times.

For a constant-$w$ material fluid and positive vacuum energy, the Friedmann acceleration equation gives $\ddot{a}/(a{H}_0^2)={\Omega }_{\Lambda 0}-(1+3w){\Omega }_{m0}(1+z{)}^{3(1+w)}/2$. If $w>-1/3$, its zero determines the displayed transition. A past transition additionally requires the right-hand side of the displayed equation to exceed one. The equation does not imply the same transition for a fluid that already has sufficiently negative pressure.

The flat barotropic matter-vacuum solution has $H_{\Lambda }/H_0=\sqrt{{\Omega }_{\Lambda 0}}$ and therefore the displayed age, measured from its Big Bang. At fixed $H_0$ and density fractions it is $1/(1+w)$ times the dust-plus-vacuum age. Its $w=0$ case is the age of a flat matter-Lambda universe.