= Cosmic-time derivative of the comoving Hubble radius
{title2=$d[(aH)^{-1}]/dt$}
With $\mathcal H=aH=\dot a$, the <Friedmann acceleration equation> gives
$$
\frac d{dt}\frac1{\mathcal H}=-\frac{\ddot a}{\dot a^2}=\frac\Omega{2a}(1+3w).
$$
The coefficient is positive when $a>0$, $\rho>0$ and $H\ne0$. Thus growth of the <comoving Hubble radius> and linear growth of departures from flatness have the same sign criterion in an expanding universe. This local criterion is distinct from a global <particle horizon>, which depends on the whole past history.
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