Formula for the large-scale bias of galaxies

[fab13]

TL;DR

I would like to infer the relation between the local density of galaxies and the global density in Universe.

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 $(1.1)$ :

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

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 : $\text{Global Number of galaxies} = \overline{n_{g}}\,V$. 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 ?

Regards

 

[kimbyd]

The conversion is in the $b_1$ term which relates the number density contrast to the matter density contrast. I seem to remember this term being called the density bias.

 

[fab13]

@kimbyd thanks for your quick answer.

Which $b_{1}$ do you talk about ? this of eq$(1)$ or eq$(2)$. ?

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 $\overline{n_{g}}$)

How can I circumvent this issue ?

Any help is welcome, Regards

 

[fab13]

Given I have exceeded the edit deadline, I just wanted to add at the end of my post above :

@kimbyd thanks for your quick answer.

Which $b_{1}$ do you talk about ? this of eq$(1)$ or eq$(2)$. ?

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$(1)$ ?

Any help is welcome, Regards

 

[fab13]

I don't want to be insistent but I really need help about the issue between the eq$(3)$ and eq$(1)$, especially how to suppress the presence of the second term of eq$(3)$ in order to find equation$(1)$.

Thanks