= Source counts on an FLRW past light cone
{title2=$dN=4\pi a^2r^2n\,dt$}
For proper <number density> $n(t)$ in an <FLRW metric> $ds^2=dt^2-a^2[dr^2/(1-kr^2)+r^2d\Omega^2]$, a radial shell has proper volume $4\pi a^3r^2|dr|/\sqrt{1-kr^2}$. A radial <null geodesic> obeys $|dr|=\sqrt{1-kr^2}\,dt/a$. Multiplication gives the displayed shell count on a local radial branch of the <past light cone>. Spatially uniform <number density> need not be time independent or have conserved <comoving number density>.
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