Solution (source code)

= Solution

A <flux-limited galaxy selection> samples a much larger volume for bright <galaxies>: in the nearby Euclidean limit $d_{\max}\propto\sqrt L$, hence $V_{\max}\propto L^{3/2}$, so very numerous faint <dwarf galaxies> are usually detectable only nearby. With a <Schechter function> $\phi(L)\propto L^{\alpha_S}e^{-L/L_*}$, detected counts per logarithmic <luminosity> interval scale as $L^{\alpha_S+5/2}e^{-L/L_*}$ and, for $\alpha_S\sim-1$ to $-1.3$, peak near $L\sim(1.2\text{--}1.5)L_*$ rather than at the numerous faint end. \b[The competition between increasing survey volume and the bright-end cutoff concentrates detected luminosities near $L_*$], with the precise width depending on the selection and survey depth.