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

Posted: April 13 2013
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.

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

Posted: April 14 2013
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.

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

Posted: April 14 2013
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.

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

Posted: April 17 2013
However, what if b=3 (positive bias), but delta_m=-0.5? In this case, delta_g=b*delta_m=-1.5, still $$<-1$$.

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

Posted: July 16 2013
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).