Barotropic matter-vacuum acceleration transition ($(1+z_{\mathrm{tr}}{)}^{3(1+w)}=2{\Omega }_{\Lambda 0}/[(1+3w){\Omega }_{m0}]$)

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.