## What is the meaning of bias and galaxy overdensity?

YinZhe Ma
Posts: 11
Joined: October 09 2008
Affiliation: University of KwaZulu-Natal

### What is the meaning of bias and galaxy overdensity?

Hi Guys,

I am confused about the bias and galaxy overdensity.

First, galaxy overdensity is related to matter density contrast through:
$\delta_{\rm{g}}=b*\delta_{\rm{m}}$, where $b$ is the bias. $\delta_{\rm{m}}$ cannot be less than -1, because it is defined as $(\rho-\overline{\rho})/\overline{\rho}$, since $\rho \geq 0$, $\delta_{\rm{m}} \geq -1$.

However, $b$ can take any value. It can be $-3, -2$ (void), or very positive number, such as $2$ in Table 5 and 6 of 1303.4486. Therefore, $\delta_{\rm{g}}$ can be any value. For example, If $\delta_{\rm{m}}=-0.8$, $b=2$, then $\delta_{\rm{g}}=-1.6$. Then what is the definition of $\delta_{\rm{g}}$ if it can take any value?

It certainly cannot be defined as the matter density contrast because it can be less than -1. Then how to understand its physical meaning?

Thanks.

Maciej Bilicki
Posts: 21
Joined: May 12 2010
Affiliation: Center for Theoretical Physics PAS, Warsaw
Contact:

### What is the meaning of bias and galaxy overdensity?

The bias as defined here is linear. Voids with \delta=-0.8 are pretty non-linear.

Galaxy underdensity, as any other underdensity, cannot -- by definition -- be smaller than -1.

You can find more basics on the bias in for instance the classic review by Strauss & Willick (1995).

I am quite sure that the bias can't be negative.

Wojciech Hellwing
Posts: 1
Joined: May 17 2006
Affiliation: Institute for Computational Cosmology, Durham

### What is the meaning of bias and galaxy overdensity?

A negative bias would be totally unphysical when one concerns density. However an anti-bias (0<b<1) is possible and in the fact is present for small galaxies/haloes.

As Maciej had noticed You have used a definition of the linear bias. In general we have:

$\delta_g= f(\delta)$

in particular we can express the non-linear function f in Tylor series:
$\delta_g = \sum_{k=0}^\infty {b_k\over k!} \delta^k$
(see e.g. Fry&Gaztanaga 1994)
hence in regions where $b_1>1$ and $\delta<0.8$ rest of the bias parameters will have values making the $\delta_g\geq -1$ in the end.

YinZhe Ma
Posts: 11
Joined: October 09 2008
Affiliation: University of KwaZulu-Natal

### What is the meaning of bias and galaxy overdensity?

However, what if b=3 (positive bias), but delta_m=-0.5? In this case, delta_g=b*delta_m=-1.5, still $<-1$.

Boud Roukema
Posts: 87
Joined: February 24 2005
Affiliation: Institute of Astronomy, Nicolaus Copernicus University
Contact:

### What is the meaning of bias and galaxy overdensity?

As Maciej and Wojciech said, the discussion concerns linear perturbation theory, meaning $|\delta| \ll 1$. Your value of $\delta_m=-0.5$ is highly non-linear, so the linear theory is no longer valid. If you apply it anyway, then you get unphysical results.

Similarly, when the overall virialisation fraction at low redshifts $f_{vir}(z)$ fails to satisfy $f_{vir}(z) \ll 1$, the underlying homogeneity assumption fails and an artefact - would-be "dark energy" - arises if the homogeneous (FLRW) metric is used to interpret the observations despite the invalidity of the homogeneity assumption (1303.4444).