Cosmological continuity equation

Energy conservation in FLRW gives $\dot{\rho }=-3H(\rho +P)$ when $\rho$ and $P$ are energy density and pressure.

A cosmological fluid with constant equation-of-state parameter $w$ has $P=w\rho$ and $\rho \propto a^{-3(1+w)}$.

Quintessence is dynamical dark energy with negative pressure, conventionally $-1<w<-1/3$.

For $w=-2/3$, zero cosmological constant, and negative curvature, $a(t)=\sqrt{\beta }t+\gamma t^2/4$ with density and curvature parameters $\gamma ,\beta >0$.

For $w=-2/3$, a quintessence-dominated scale factor grows quadratically and $tH$ approaches two.