# Hubble expansion rate and reaction rates

I've come across sentences like

...the reaction rates (the number of reactions per particle per unit time) must be larger than the cosmic expansion rate $$H(t)$$ in order for the particles to maintain equilibrium.

The quote is from the book "Extragalactic Astronomy and Cosmology" by Peter Schneider, page 195. Why does the reaction rate have to be greater than $$H(t)$$ and not $$\dot a$$? $$\dot a$$ feels like it should be the cosmic expansion rate.

A more intuitive way to understand the comparison between the reaction rate $$\Gamma$$ and the Hubble rate $$H$$ is to take the reciprocal. $$\Gamma^{-1}$$ is the time between reactions. $$H^{-1}$$ is approximately the age of the Universe. If $$\Gamma\gg H$$, then $$\Gamma^{-1}\ll H^{-1}$$, i.e. the time between reactions is much shorter than the age of the Universe. If on the other hand $$\Gamma\ll H$$, then $$\Gamma^{-1}\gg H^{-1}$$, i.e. the time between reactions is much longer than the age of the universe. In the latter case, clearly equilibrium cannot be maintained.
• The scale factor $a$ is indeed dimensionless and normalized to unity at the present epoch.