## Fingers of God

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Fingers of God (or Finger of God) effect is one of the redshift-space distortions and lies in a stretching of the two-point correlation function, $\xi (\sigma ,\pi )$, of galaxies (or flatteninging of the power spectrum correspondingly) along the line of sight on the small scales due to random peculiar (i.e. non-Hubble) velocities of galaxies. These small-scale velocities can be the virial velocities of galaxies in clusters as far as any other random velocities obtained by galaxies in course of their evolution.
Firstly this effect was observed not in two-point scorrelation functions, but in galaxy clusters which appear to be stretched along the line of sight due to contamination' of the measured galaxies redshifts by their peculiar velocities — similar to fingers of one hand. Note, that concept Fingers of God' is related to galaxy clusters, and `Finger of God' to (2D) two-point correlation function of galaxies in redshift space.
It is worth to note, that the similar effect of stretching of the correlation function (or galaxy clusters) can be caused also by the errors in redshift measurement.
To model this effect the two-point correlation function, $\xi (\sigma ,\pi )$, is convolved with the pairwise velocity distribution $f(w_{z})$, assuming that the line-of-sight distance in redshift space is $\pi '=\pi -w_{z}(1+z)/H(z)$:
where $H(z)$ is the Hubble parameter, $\sigma$ is the projection of the distance onto the plane perpendicular to the line of sight, $f(w_{z})$ is the pairwise velocity distribution and $\langle w_{z}^{2}\rangle ^{1/2}$ is the line-of-sight pairwise velocity dispersion. If due to peculiar velocities $f(w_{z})$ can be best described by an exponential distribution (see e.g. [1]):
and if the redshift measurement errors dominate then the distribution can by better described by a Gaussian (see e.g. [2]):
Here $\pi$ means the number $\pi$, not the radial distance as before.
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