- #1

fab13

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- TL;DR Summary
- 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

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

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