general relativity - Cosmology - how to do the $V/V_{max}$ test?

In Cosmology, we have the co-moving distance (assuming $\Omega_k=0$), $$D_C=\frac c{H_0}\int_0^z\frac{dz'}{\sqrt{\Omega_m(1+z)^3+\Omega_\Lambda}}$$ and we also have the total co-moving volume formula $$V=\frac{4\pi}3 D_C^3$$Then we can use what is called the $\langle V/V_{max}\rangle$ test to test if a sample of objects has uniform co-moving density & luminosity that is constant in time. The number of objects per unit co-moving volume with luminosity in the range $(L,L+dL)$ is given by $\Phi(L)$. Then the total number of objects in the sample is $$\int_0^\infty\Phi(L)\int_0^{V_{max}(L)}\mathrm dV\,\mathrm dL$$ If we have a uniform distribution of objects, then the value of this expectation should be $1/2$, as described by this similar question asked on Student Room, which did not get any responses.

This test is supposedly well known, but I can't find any questions about it here, nor can I find a simple article about the actual test, or this result. There are many articles online instead showing generalisations to this test which seem very abstract to me.

Question: From these definitions, I don't know how to get a value for $\langle V/V_{max}\rangle$, nor do I know what the explicit formula is. What is the explicit formula for $\langle V/V_{max}\rangle$? Is it that integral? Clearly, $V$ depends only on the value of $z$, but I don't know what a uniform distribution of objects implies about the distribution of $z$. Can someone help me understand this a bit better?

Answer

I am not so sure about the cosmological application, but the principle is straightforward.

If you have an estimated distance $D$ to an object, then that defines a volume of $$ V =\frac{4\pi}{3} D^3$$

If your survey is capable of detecting such objects to a distance $D_{\rm max}$, then this defines a volume $V_{\rm max}$.

So for each object you can calculate $V/V_{\rm max}$. If the source population is uniform in space (and hence in time for cosmological sources), then the average $\langle V/V_{\rm max}\rangle = 0.5$. In fact you can go further and say that $V/V_{\rm max}$ ought to be uniformly distributed between 0 and 1.

This can be done as a function of source type, or luminosity or whatever.