# Total number density of galaxies and problematic expression in practise

This post comes from physics exchange forum, I have transfered it here, maybe I will be luckier since I have not received answers on the other forum.

Here is the issue :

I am asked to give the formal expression of the total number density of galaxies and explain why is this expression problematic in practice.

From what I saw from my research and into my lectures, I have found the follwing relation which gives the number of galaxies $$N$$ with magnitude $$(m < M)$$ (number counts) :

$$\text{log}\,N(m

$$N(m

I don't know what the teacher wants to highlight, i.e by saying that the formal expression of the total number density of galaxies is problematic in practice, given that I even haven't the kind of expression or equation which gives the estimation of galaxy density.

1) Concerning this distribution to use, do you advise me to use a distribution as a function of magnitude, mass or a distribution as function of luminosity (like Schechter) ?

2) I have seen on web the HMF (Halo Mass function) but it seems to be about the dark matter halos : can we count one dark matter halo per galaxy ?

3) But the problem is here is that a dark matter halo is more massive than the galaxy hosted in this halo : how can we deal with this ?

If someone could help me, this would be kind.

PS: I have also to explain why the estimation of galaxies density is difficult and problematic.

4) What are the issues when we want to estimate density of galaxies ?

Concerning the distribution as a function of luminosity, I have found the Schechter luminosity function :

$$N(L)\ \mathrm {d} L=\phi^{*}\left({\frac{L}{L^{*}}}\right)^{\alpha}\mathrm {e}^{-L/L^{*}}{\frac{\mathrm {d} L}{L^{*}}}$$

UPDATE 1: my apologies, $$m$$ and $$M$$ represent apparent magnitude and not masses.

But the questions remain about the counting of galaxies and the computation of total density at large scale.

In practise, what are the major problems when we want to estimate this density ?