Euclidean accessible-volume luminosity weighting ($V_{\max }\propto L^{3/2}$)

For a flux threshold $f_{\min }$ and isotropic point-source luminosity $L$, the detection radius is $r_{\max }=\sqrt{L/(4\pi f_{\min })}$. In a homogeneous Euclidean population the accessible volume is proportional to $L^{3/2}$, so the selected luminosity distribution is proportional to $\phi (L)L^{3/2}$. This assumes flux selection and inverse-square propagation, not an arbitrary surface-brightness-selected diffuse field.