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?


1 Answer 1


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.

  • $\begingroup$ Ah ok, this makes sense to me intuitively. Is there any way to represent this in mathematical terms, like some volume integrals or something? I ask because I'd like to later generalise to other possible distributions, where intuition would not be enough. $\endgroup$
    – John Doe
    May 1, 2018 at 10:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.