= Halo virial-velocity conversion between epochs
{title2=$v_2/v_1=[\Delta_{c,2}H_2^2/(\Delta_{c,1}H_1^2)]^{1/6}$}
For a fixed mass under a spherical-overdensity definition, $R_h\propto(\Delta_cH^2)^{-1/3}$ and $v_h^2=GM_h/R_h$. Thus equivalent virial mass scales at two epochs have the displayed velocity ratio. With fixed overdensity during matter domination, $v_h\propto(1+z)^{1/2}$. This converts an epoch-specific mass–velocity calibration; it does not require a previously collapsed bound halo to physically expand with the later background density.
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