Galaxy bias: The ratio between the galaxy overdensity field and the dark matter overdensity field.
The clustering of galaxies can be quantified in terms of the projected correlation function of galaxies
as done in Zehavi et al. (2010, Z10 hereafter) or the three dimensional power spectrum of galaxies
as measured by Tegmark et al. (2004, T04 hereafter). To obtain the galaxy bias, the clustering of matter in the Universe has to be specified. If the cosmological parameters are specified, then one can use analytical expressions from Eisenstein & Hu (1998) to calculate the linear dark matter power spectrum, or the Smith et al. (2005) prescription to get the non-linear power spectrum.
This webpage is meant to compare the galaxy biases obtained by these two methods. I have been in touch with Zheng Zheng and Idit Zehavi about this.
In Z10, there are three measurements of the galaxy bias: (i) based on the HOD modelling of galaxies, (ii) based on the ratio of galaxy clustering to dark matter clustering on scales of 2.7
and (iii) based on a fit to this ratio on scales between 2.7 to 30
In an email communication, Zheng mentions that the galaxy bias was obtained by taking the ratio of the galaxy clustering to the projected nonlinear matter clustering obtained from the nonlinear matter clustering of Smith et al. at z=0.1 . He presented his own version of the following figure:
Figure 1: Galaxy bias
The agreement between T04 and Z10 does not look that terrible in this figure. Let us first understand how this figure was obtained.
The T04 results were obtained from a table in Seljak et al. (2005) and there they are given in the form of
which is actually
if one carefully thinks about how the galaxy bias was measured by T04. They also quote the normalization for the bias of
galaxies (with magnitude equal to -20.83),
Therefore to compare with the result from Z10, Zheng obtained
The bias was obtained by multiplying
by
In principle this procedure gives
which is what Z10 would measure if they found the galaxy bias by dividing by the z=0.1 matter clustering.
I have to check the following things with this figure:
The redshift at which the matter clustering was used to infer the galaxy bias.
The shape parameter used in T04 and Z10 are different. So it is not fully fair to compare in this manner.
The SGW was not excluded in T04 but it was in Z10.
I am not sure that the redshift 0.1 matter clustering was used to find the galaxy bias for the
measurement. The reason for me doubting that the galaxy bias was found by dividing with the z=0.1 matter clustering is the following figure:
Figure 2: Non-linear matter clustering at what redshift was used to calculate the galaxy bias?
Here I present the different ways of calculating the galaxy bias at radius = 2.67
The squares are from the figure in Z10. The triangles is what one get when dividing by redshift 0.1, while the hexagons use for redshift 0. The redshift 0 part agrees pretty well with what the measurements of Z10 are. However, it seems the measurements of
were done with the z=0.1 matter power spectrum. Here is the figure which shows this:
Reproducing Zehavi et al. results. The b_{fit} measurements were obtained by dividing by z=0.1 clustering.
The above figures show that we can reproduce the Z10 results for
and
measurements.
So the final thing to be done is to use the T04 cosmological parameters to compare and use the Sloan great wall included figure, because that is what you want to compare.
Figure 4: Galaxy bias-luminosity relation with the same cosmological parameters adopted for both analyses.
The small scale clustering measurements give systematically larger values for galaxy bias than the large scale power spectrum measurements of galaxy bias. The reason why the black points here moved up with respect to Figure 1 can be understood by looking at the following figure which shows the comparison between the projected non-linear matter power in the cosmology assumed by T04 and by Z10, for both models we assume
.
Figure 5: The nonlinear matter clustering projected along the line of sight up to 60 hinv Mpc.
This clearly shows that the bias that will be inferred by using the T04 cosmology will be larger!
The correlation function was integrated along the line-of-sight up to a distance of
(and upto 40 in some cases) to calculate
Norberg et al. (2009) have shown in their paper that the integral when carried out in redshift space upto a finite value of line-of-sight separation gives values that are systematically larger than when the integral is carried out in real space. This effect can cause a systematic bias in the measured values of the galaxy bias as calculated by Z10.
It seems that the reason for this is the Kaiser effect which is caused by the infall of matter (or galaxies) towards high density regions. This causes the iso-correlation function contours in the
space to flatten on large scales.
Figure 6: The projected non-linear correlation function with changing value of the limit of line-of-sight integration. The Kaiser effect is taken in to account while calculating the integration in redshift space.
Figure 6 shows the difference between integrating in real space (black line) and integrating the correlation function in redshift space along the line of sight up to different values of
shown in the caption. These curves was calculated in the following manner.
The power spectrum of galaxies calculated in redshift space gets distorted due to the peculiar motions of galaxies. In real space, the correlation can be written as a function of
and
. On small scales, iso-correlation contours get elongated along the line-of-sight (the finger-of-god effect). On large scales, these contours get flattened due to the infall of galaxies towards over dense regions (the Kaiser effect). When the correlation function is integrated to a sufficiently large line-of-sight distance,
, the finger of god effect gets integrated out. Kaiser (1987) showed that in the presence of the large scale redshift distortions, the power spectrum is given by
where,
denotes the cosine of the angle between the vector k and the line-of-sight, and
. This equation holds in the linear regime. The corresponding expression for the correlation function was given by Hamilton (1992). This is the equation we use to calculate the
. Then we can integrate this correlation function along
direction. Note that for the right hand panel of the above figure, we have assumed the equation given by Kaiser (1987) for P(k) also holds for the non-linear power spectrum.
For the figure below we are going to report the galaxy bias in units of
The figures will therefore be slightly different from what were presented above to compare with Zheng's results. In those figures the galaxy bias was
Figure 7: Including the Kaiser effect moves the galaxy bias points by roughly one-sigma lower but the discrepancy between T04 and Z10 still persists..
This figure clearly shows that by including the Kaiser effect correction moves the
points by roughly
Thorsten Naab
Atlas 3d.
Slow rotators: round, have perhaps two disks counter-rotating. 36/260
However, most of them are fast rotators. 224/260, Most of them are S0's. Are major mergers playing any role? Most of them are they like fading disks?
lambda, rotation parameter: Low mass early types are fast rotators, high mass early types are fast rotators.
lambda vs ellipticity: lambda=0.31 sqrt(e) gives a rough division between fast and slow rotators.
Simulation do not produce very fast rotators and have trouble producing slow rotators.
Only high mass galaxies with a number of minor mergers forms slow rotators.
The environments of Sauron galaxies, may be important to know.
Oser
Size growth of elliptical galaxies by dry minor merging.
121 Academic Resources Center (ARCenter) University of California, Santa Cruz 1156 High Street Santa Cruz, CA 95064
Feldmann:
Morphological transformations happen before satellites fall into groups, before they turn red. Contradiction with Bundy observations where the satellites turn red first, and then undergo morphological transformations.
Why were the Mstar as a function of redshift have spikes during mergers?
Wetzel:
Using the group catalog look at the quenched fractions as a function of stellar mass, halo mass. Halo mass is the dominant effect.
The clustering of galaxies can be quantified in terms of the projected correlation function of galaxies
as done in Zehavi et al. (2010, Z10 hereafter) or the three dimensional power spectrum of galaxies
as measured by Tegmark et al. (2004, T04 hereafter). To obtain the galaxy bias, the clustering of matter in the Universe has to be specified. If the cosmological parameters are specified, then one can use analytical expressions from Eisenstein & Hu (1998) to calculate the linear dark matter power spectrum, or the Smith et al. (2005) prescription to get the non-linear power spectrum.
This webpage is meant to compare the galaxy biases obtained by these two methods. I have been in touch with Zheng Zheng and Idit Zehavi about this.
In Z10, there are three measurements of the galaxy bias: (i) based on the HOD modelling of galaxies, (ii) based on the ratio of galaxy clustering to dark matter clustering on scales of 2.7
and (iii) based on a fit to this ratio on scales between 2.7 to 30
In an email communication, Zheng mentions that the galaxy bias was obtained by taking the ratio of the galaxy clustering to the projected nonlinear matter clustering obtained from the nonlinear matter clustering of Smith et al. at z=0.1 . He presented his own version of the following figure:
The agreement between T04 and Z10 does not look that terrible in this figure. Let us first understand how this figure was obtained.
The T04 results were obtained from a table in Seljak et al. (2005) and there they are given in the form of
which is actually
if one carefully thinks about how the galaxy bias was measured by T04. They also quote the normalization for the bias of
galaxies (with magnitude equal to -20.83),
Therefore to compare with the result from Z10, Zheng obtained
The bias was obtained by multiplying
by
In principle this procedure gives
which is what Z10 would measure if they found the galaxy bias by dividing by the z=0.1 matter clustering.
I have to check the following things with this figure:
I am not sure that the redshift 0.1 matter clustering was used to find the galaxy bias for the
measurement. The reason for me doubting that the galaxy bias was found by dividing with the z=0.1 matter clustering is the following figure:
Here I present the different ways of calculating the galaxy bias at radius = 2.67
The squares are from the figure in Z10. The triangles is what one get when dividing by redshift 0.1, while the hexagons use for redshift 0. The redshift 0 part agrees pretty well with what the measurements of Z10 are. However, it seems the measurements of
were done with the z=0.1 matter power spectrum. Here is the figure which shows this:
The above figures show that we can reproduce the Z10 results for
and
measurements.
So the final thing to be done is to use the T04 cosmological parameters to compare and use the Sloan great wall included figure, because that is what you want to compare.
The small scale clustering measurements give systematically larger values for galaxy bias than the large scale power spectrum measurements of galaxy bias. The reason why the black points here moved up with respect to Figure 1 can be understood by looking at the following figure which shows the comparison between the projected non-linear matter power in the cosmology assumed by T04 and by Z10, for both models we assume
.
The correlation function was integrated along the line-of-sight up to a distance of
(and upto 40 in some cases) to calculate
Norberg et al. (2009) have shown in their paper that the integral when carried out in redshift space upto a finite value of line-of-sight separation gives values that are systematically larger than when the integral is carried out in real space. This effect can cause a systematic bias in the measured values of the galaxy bias as calculated by Z10.
It seems that the reason for this is the Kaiser effect which is caused by the infall of matter (or galaxies) towards high density regions. This causes the iso-correlation function contours in the
space to flatten on large scales.
Figure 6 shows the difference between integrating in real space (black line) and integrating the correlation function in redshift space along the line of sight up to different values of
shown in the caption. These curves was calculated in the following manner.
The power spectrum of galaxies calculated in redshift space gets distorted due to the peculiar motions of galaxies. In real space, the correlation can be written as a function of
and
. On small scales, iso-correlation contours get elongated along the line-of-sight (the finger-of-god effect). On large scales, these contours get flattened due to the infall of galaxies towards over dense regions (the Kaiser effect). When the correlation function is integrated to a sufficiently large line-of-sight distance,
, the finger of god effect gets integrated out. Kaiser (1987) showed that in the presence of the large scale redshift distortions, the power spectrum is given by
where,
denotes the cosine of the angle between the vector k and the line-of-sight, and
. This equation holds in the linear regime. The corresponding expression for the correlation function was given by Hamilton (1992). This is the equation we use to calculate the
. Then we can integrate this correlation function along
direction. Note that for the right hand panel of the above figure, we have assumed the equation given by Kaiser (1987) for P(k) also holds for the non-linear power spectrum.
For the figure below we are going to report the galaxy bias in units of
The figures will therefore be slightly different from what were presented above to compare with Zheng's results. In those figures the galaxy bias was
This figure clearly shows that by including the Kaiser effect correction moves the
points by roughly
Thorsten Naab
Atlas 3d.
Slow rotators: round, have perhaps two disks counter-rotating. 36/260
However, most of them are fast rotators. 224/260, Most of them are S0's. Are major mergers playing any role? Most of them are they like fading disks?
lambda, rotation parameter: Low mass early types are fast rotators, high mass early types are fast rotators.
lambda vs ellipticity: lambda=0.31 sqrt(e) gives a rough division between fast and slow rotators.
Simulation do not produce very fast rotators and have trouble producing slow rotators.
Only high mass galaxies with a number of minor mergers forms slow rotators.
The environments of Sauron galaxies, may be important to know.
Oser
Size growth of elliptical galaxies by dry minor merging.
121 Academic Resources Center (ARCenter)
University of California, Santa Cruz
1156 High Street Santa Cruz, CA 95064
Feldmann:
Morphological transformations happen before satellites fall into groups, before they turn red. Contradiction with Bundy observations where the satellites turn red first, and then undergo morphological transformations.
Why were the Mstar as a function of redshift have spikes during mergers?
Wetzel:
Using the group catalog look at the quenched fractions as a function of stellar mass, halo mass. Halo mass is the dominant effect.