CosmoCoffee

YinZhe Ma · 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:

δ g = b * δ m

, where

b

is the bias.

δ 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,

δ g

can be any value. For example, If

δ m = - 0.8

,

b = 2

, then

δ g = - 1.6

. Then what is the definition of

δ 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 · Posted: April 14 2013

The bias as defined here is linear. Voids with

δ

=−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 · 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:

δ g = f (δ)

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

δ < 0.8

rest of the bias parameters will have values making the $\delta_g\geq -1$ in the end.

YinZhe Ma · 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

.

Boud Roukema · Posted: July 16 2013

As Maciej and Wojciech said, the discussion concerns linear perturbation theory, meaning $|\delta| \ll 1$ . Your value of

δ 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is highly non-linear, so the linear theory is no longer valid.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 v i r (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).