Clusters 209and the matter overdensity are related. Mass profile estimation is thus possible once
the infall pattern of galaxies is known.
Dark matter is usually dissected from baryons in lensing analyses by first fitting the
lensing features to obtain a map of the total matter distribution and then subtracting
the gas mass fraction as inferred from X-ray observations. The total mass map can
then be obtained with parametric models in which the contribution from cluster-sized
DM halos is considered together with the main galactic DM halos. Mass in stars and in
stellar remnants is estimated converting galaxy luminosity to mass assuming suitable
stellar mass to light ratios.
One may go one step further by exploiting a parametric model which has three
kinds of components: cluster-sized DM halos, galaxy-sized (dark plus stellar) matter
halos, and a cluster-sized gas distribution. In systems of merging clusters DM may
become spatially segregated from baryonic matter and thus observable. We shall meet
several such cases later in this Chapter.
The Local Supercluster (LSC). The autocorrelation function휉(푟)in Equation (9.8)
was defined for distances푟in real space. In practice, distances to galaxies are mea-
sured in redshifts, and then two important distortions enter. To describe the sepa-
ration of galaxy pairs on the surface of the sky, let us introduce the coordinate휎,
transversal to the line of sight, and휋radial. In redshift space the correlation function
is then described by휉(휎,휋)or its spherical average휉(푠),where푠=
√
휋^2 +휎^2.
The transversal distance휎is always accurate, but the radial redshift distance휋
is affected by velocities other than the isotropic Hubble flow. For relatively nearby
galaxies,푟⩽2Mpc, the random peculiar velocities make an unknown contribution to
휋so that휉(푠)is radially distorted. The undistorted correlation function휉(푟)is seen
isotropic in(휎,휋)-space in the top left panel of Figure 9.2. The lower left panel of
Figure 9.2 shows the distortion to휉(푠)as an elongation in the휋direction.
Over large distances where the peculiar velocities are unimportant relative to the
Hubble flow (tens of Mpc), the galaxies in the LSC feel its attraction, as is manifested
by their infall toward its center with velocities in the range 150–450kms−^1. From this
one can derive the local gravitational field and the mass excess훿푀concentrated in the
LSC. The infall velocities cause another distortion to휉(푠): a flattening as is shown in
the top right panel of Figure 9.2. When both distortions are included, the correlation
function in(휎,휋)-space looks like the bottom right panel of Figure 9.2. The narrow
peaks in the휋direction have been seen for a long time, and are calledFingers of God.
If galaxy formation is a local process, then on large scales galaxies must trace mass
(on small scales galaxies are less clustered than mass), so that휉gal(푟)and휉mass(푟)are
proportional:
휉gal(푟)=푏^2 휉mass(푟).Here푏is the linearbias: bias is when galaxies are more clustered than mass, and
anti-biasis the opposite case;푏=1 corresponds to the unbiased case. The presence
of bias is an inevitable consequence of the nonlinear nature of galaxy formation. The
distortions in휉(푠)clearly depend on the mass density훺mwithin the observed volume.