Redshift-space Distortions

• Two important effects occur in redshift space. Although redshift (velocity) corresponds to true distance according to the Hubble Law, small peculiar velocities not associated with the Hubble flow can cause distortions in redshift space. The most evident of these is the Fingers-of-God effect, where long thin filaments in redshift space point directly back at observer. We should know by now that we are not privileged observers, thus this effect must be unphysical. It can be easily seen in this slice from the older CfA survey:

SAO, The CfA Redshift Survey Team (1998)

• The Fingers-of-God effect is attributed to random velocity dispersions in galaxy clusters that deviate a galaxy's velocity from pure Hubble flow, stretching out a cluster in redshift space. Since this affects only redshift and not position on the sky, the stretching occurs only radially (hence why fingers point back to observer)
• The other important redshift distortion is the Kaiser Effect, which is more subtle and difficult to quantify. The Kaiser Effect describes the peculiar velocities of galaxies bound to a central mass as they undergo infall. This differs from the Fingers-of-God in that the peculiar velocities are coherent, not random, towards the central mass, though the effect is very subtle. Once again, these peculiar velocities deviate measured redshifts from a pure Hubble Law and cause distortions in redshift space. This effect can only be detected on large scales
• When computing the correlation function ξ(r) in redshift space, the r coordinate is usually separated into the transverse component σ and the radial component π. The transverse component σ is a true measure of distance, while π is distorted by peculiar velocities as explained above. On small transverse scales, random dispersion velocities cause the Finger-of-God effect which is easily seen in ξ(r). On large scales, the Kaiser Effect causes a "flattening" of the correlation function, due to the subtle infall motion of galaxies. This is illustrated below:

The 2dF Galaxy Redshift Survey Team (2001)

• The correlation function computed in redshift-space from 2dF data is shown below, as a function of π and σ (Peacock et al. 2001). To emphasize deviations from circular symmetry, the first quadrant has been mirrored along both axes. The contours represent model predictions, with ξ = 10,5,2,1,0.5,0.2 and 0.1. The Fingers-of-God effect is clearly seen, and the Kaiser flattening can just be seen at large radii. (Note: The redshift-space ξ(r) can be translated into real-space ξ(r), though I will not discuss it here)

The 2dF Galaxy Redshift Survey Team (2001)

• Kaiser flattening allows β and σ_p to be determined, where:
  β ≡ (Ω_m)^0.6/b
  Recall δ_galaxies = b (δ_mass)
  σ_p is the random dispersion in relative velocities of galaxies
  
• In order to do this, one measures the quadrupole-to-monopole ratio (i.e. the stretching or squashing) of the redshift-space correlation function. For small separations, the Fingers-of-God effect dominates which causes a stretching (positive rato). On large scales, Kaiser flattening is dominant (negative ratio). ξ₂/ξ₀ can be expressed as a function of β and σ_p. The parameters can be determining by finding the best fit model. This is shown below (solid lines have varying β and σ_p=400 km/s, dashed lines have varying σ_p and β=0.4):

Peacock et al. and The 2dF Galaxy Redshift Survey Team (2001)

• Once model fitting yields values for β, we can determine b from studies of clustering in nonlinear regime (bispectrum - see next page) or by comparing data with models allowed by CMB constraints (need to convert power spectrum from redshift-space to real-space; this relation depends on β and σ_p). The 2dF team find (Peacock et al., 2001) using this CMB model fitting, that: Ω_m ≅ 0.3
This may seem like an obvious result given the measurement obtained from P(k), but this provides a totally independent method of determining Ω_m and can be used to establish consistency with the results.
• Recall that ξ(r) is dependent on galaxy type, and that "red"/early-type galaxies tend to cluster more heavily. This is reflected in the 2dF measurements of ξ(r), as shown below. The correlation function is indeed greater for early-type galaxies:

Lahav et al. and The 2dF Galaxy Redshift Survey Team (2002)

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