What is the meaning of bias and galaxy overdensity?
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:
[tex]\delta_{\rm{g}}=b*\delta_{\rm{m}}[/tex], where [tex]b[/tex] is the bias. [tex]\delta_{\rm{m}}[/tex] cannot be less than −1, because it is defined as [tex](\rho\overline{\rho})/\overline{\rho}[/tex], since [tex]\rho \geq 0[/tex], [tex]\delta_{\rm{m}} \geq 1[/tex].
However, [tex]b[/tex] can take any value. It can be [tex]3, 2[/tex] (void), or very positive number, such as [tex]2[/tex] in Table 5 and 6 of 1303.4486. Therefore, [tex]\delta_{\rm{g}}[/tex] can be any value. For example, If [tex]\delta_{\rm{m}}=0.8[/tex], [tex]b=2[/tex], then [tex]\delta_{\rm{g}}=1.6[/tex]. Then what is the definition of [tex]\delta_{\rm{g}}[/tex] 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.
I am confused about the bias and galaxy overdensity.
First, galaxy overdensity is related to matter density contrast through:
[tex]\delta_{\rm{g}}=b*\delta_{\rm{m}}[/tex], where [tex]b[/tex] is the bias. [tex]\delta_{\rm{m}}[/tex] cannot be less than −1, because it is defined as [tex](\rho\overline{\rho})/\overline{\rho}[/tex], since [tex]\rho \geq 0[/tex], [tex]\delta_{\rm{m}} \geq 1[/tex].
However, [tex]b[/tex] can take any value. It can be [tex]3, 2[/tex] (void), or very positive number, such as [tex]2[/tex] in Table 5 and 6 of 1303.4486. Therefore, [tex]\delta_{\rm{g}}[/tex] can be any value. For example, If [tex]\delta_{\rm{m}}=0.8[/tex], [tex]b=2[/tex], then [tex]\delta_{\rm{g}}=1.6[/tex]. Then what is the definition of [tex]\delta_{\rm{g}}[/tex] 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.

 Posts: 19
 Joined: May 12 2010
 Affiliation: University of Cape Town
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What is the meaning of bias and galaxy overdensity?
The bias as defined here is linear. Voids with δ=−0.8 are pretty nonlinear.
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.
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.

 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 antibias (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:
[tex]\delta_g= f(\delta)[/tex]
in particular we can express the nonlinear function f in Tylor series:
[tex]\delta_g = \sum_{k=0}^\infty {b_k\over k!} \delta^k[/tex]
(see e.g. Fry&Gaztanaga 1994)
hence in regions where [tex]b_1>1[/tex] and [tex]\delta<0.8[/tex] rest of the bias parameters will have values making the [tex]\delta_g\geq 1[/tex] in the end.
As Maciej had noticed You have used a definition of the linear bias. In general we have:
[tex]\delta_g= f(\delta)[/tex]
in particular we can express the nonlinear function f in Tylor series:
[tex]\delta_g = \sum_{k=0}^\infty {b_k\over k!} \delta^k[/tex]
(see e.g. Fry&Gaztanaga 1994)
hence in regions where [tex]b_1>1[/tex] and [tex]\delta<0.8[/tex] rest of the bias parameters will have values making the [tex]\delta_g\geq 1[/tex] in the end.
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 [tex]<1[/tex].

 Posts: 84
 Joined: February 24 2005
 Affiliation: Torun Centre for Astronomy, University of Nicolaus Copernicus
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What is the meaning of bias and galaxy overdensity?
As Maciej and Wojciech said, the discussion concerns linear perturbation theory, meaning [tex]\delta \ll 1[/tex]. Your value of [tex]\delta_m=0.5[/tex] is highly nonlinear, 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 [tex]f_{vir}(z)[/tex] fails to satisfy [tex]f_{vir}(z) \ll 1[/tex], the underlying homogeneity assumption fails and an artefact  wouldbe "dark energy"  arises if the homogeneous (FLRW) metric is used to interpret the observations despite the invalidity of the homogeneity assumption (1303.4444).
Similarly, when the overall virialisation fraction at low redshifts [tex]f_{vir}(z)[/tex] fails to satisfy [tex]f_{vir}(z) \ll 1[/tex], the underlying homogeneity assumption fails and an artefact  wouldbe "dark energy"  arises if the homogeneous (FLRW) metric is used to interpret the observations despite the invalidity of the homogeneity assumption (1303.4444).