# Cosmology : Formula for the bias of galaxies

first I have posted originally on the physics stackexchange but I have not had answers about my question, so I try my luck on this specific forum (if forums leaders want to delete the original post on physics exchange, they can do it without problems, I don't want to make duplicates).

From this article : https://arxiv.org/pdf/1611.09787.pdf , I try to deduce the equation that my teacher told me which links 2 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 $$(1.1)$$ (equation reference on the article) :

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

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 : I don't know if I have to write :

$$\text{Global Number of galaxies} = \overline{n_{g}}\quad V$$

or

$$\text{Local Number of galaxies} = \overline{n_{g}}\quad V$$

???

I think that I have to use the following relations : $$N_{g}\equiv N_{1}$$ 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 ?

EDIT 1:

If I take the relation eq$$(2)$$, I can write :

$$n_{g(\vec{x})} = \overline{n_{g}}\,b_{1}\,\delta_{\text{DM}}+\overline{n_{g}}\quad\quad(3)$$

As you can see, $$(3)$$ is not equal to the equation $$(1)$$ that I would like to get (since a second term $$\overline{n_{g}}$$)

With the notations of the equation$$(1)$$, in order to be coherent, I think that I have to assimilate $$N_{1}$$ to $$n_{g}(\vec{x})$$ (local density) and $$n_{1}$$ to $$\overline{n_{g}}$$ (global or mean density).

How can I circumvent this issue about the presence of this second term into eq$$(3)$$ compared to eq$$(1)$$ ?

I am near from the equality between both, this is frustrating . Maybe it is a problem of convention about the factor $$b_{1}$$ called "bias" ?

Edit: This is also cross-posted at https://physics.stackexchange.com/questions/489437/demonstration-about-large-scale-bias-of-galaxies

• Cross posted in Physics SE Jul 8 '19 at 10:50