Flux-limited galaxy selection ($V_{\max }\propto L^{3/2}$)

For a nearby Euclidean survey with limiting flux $f_{\min }$, a galaxy of luminosity $L$ is detectable out to $d_{\max }=\sqrt{L/(4\pi f_{\min })}$. Its accessible volume is therefore proportional to $L^{3/2}$. With a Schechter function $\varphi (L)\propto L^{\alpha }{e}^{-L/L_{\ast }}$, counts per logarithmic interval are proportional to $L^{\alpha +5/2}{e}^{-L/L_{\ast }}$ and peak at $(\alpha +5/2)L_{\ast }$ when $\alpha >-5/2$. This selection suppresses faint objects relative to a volume-limited census; finite survey boundaries and cosmological evolution modify the simple law.