Era

expression

Pre-transition

Spatial expansion

$H_s^2=\frac{8\pi G{\varrho }_m}{3}=\frac{1}{t^2}$

Pre-transition

Volumetric expansion

$H_v^2={\left(\frac{2}{3t}\right)}^2={\left(\frac{2}{3}\right)}^2\left(\frac{8\pi G{\varrho }_m}{3}\right)$

Post transition

Spatial expansion

$H_s^2=\frac{8\pi G{\varrho }_m}{3}=\frac{1}{t^2}$

(the same as pre-transition)

Post transition

volumetric expansion

$H_v=\frac{c}{r_{ue}}$ ( $r_{ue}$ : equivalent spherical radius)