From this article : https://arxiv.org/pdf/1611.09787.pdf , I try to deduce the equation that my teacher told me which links 3 quantities :

1) the global number density of galaxies

2) the local number density of galaxies

3) the contrast of Dark matter density

The relation that I would like to find (the relation given by my teacher) is very simple :

$$N_{1} = n_{1} b_{1}\,\delta_{\text{DM}}\quad\quad(1)$$

where $N_{1}$ is the local number density of galaxies in Universe, $n_{1}$ is the global number density, $b_{1}$ is the bias (cosmological bias of galaxies) and $\delta_{\text{DM}}$ the contrast in dark matter density. When I say "local", I mean in the volume of scale that I consider (in a cluster of galaxies for example, doesn't it ?)

for the moment, I can't find this equation.

Into the article above, they define the bias by doing the relation $(1.1)$ :

$$\delta_{g}(\vec{x}) = \dfrac{n_{g(\vec{x})}}{\overline{n_{g}}}-1 = b_{1}\,\delta_{\text{DM}}(\vec{x}) = b_{1}\big(\dfrac{\rho_{m}(\vec{x})}{\overline{\rho_{m}}}-1\big)\quad\quad(2)$$

with $b_{1}$ the bias.

As you can see, in this article, authors are reasoning with the contrast of density number of galaxies ($\delta_{g}(\vec{x}))$ and the contrast of matter density of Dark matter ($\delta_{\text{DM}}(\vec{x})$).

I tried to modify this equation $(2)$ to get $(1)$ but I am stuck by the following difference : on one side, one takes number densities and on the other one, they take contrasts of density (with contrast density number and Dark matter contrast).

Multiplying the both by the volume $V$ is not enough since there is the value "-1" in the definition of contrast : $\text{[Global Number of galaxies into volume $V$]} = \overline{n_{g}}\,V$. I think that I have to use the following relations : $N_{\text{local}}\equiv N_{1}$ and $\overline{n_{g}}=n_{1}$ in the relation of my teacher but I am not sure.

Anyone could help me to find the equation (1) from the equation (2) of article cited ?



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