## Kaiser effect

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Kaiser effect (or $\beta$-distortion, or sometimes `The Bull's eye' effect) is one of the redshift-space distortions and lies in a flattening of the two-point correlation function (or stretching of the matter power spectrum correspondingly) of galaxies along the line of sight due to the gravitational infall of galaxies to density inhomogeneities.
In the linear theory of structure formation the perturbation of the Hubble parameter is
Here $H$ is the Hubble parameter, $\Delta \rho /\rho$ is the density contrast, and $f(\Omega _{M})=H^{-1}\frac {d\ln {D(t)}}{dt}$, where $D(t)$ is the growing mode of density perturbations (see e.g. ). Hence for the observer outside the region of overdensity this region will appear smaller in line-of-sight direction in redshift space. Kaiser  showed that the matter power spectrum$P(\vec {k})$ in real ($r$) and redshift space ($s$) are related to each other with the following formula:
where $\mu _{k}$ is a cosine of angle between $\vec {k}$ and line of sight. Later $f(\Omega _{M})$ was replaces by $\beta =f(\Omega _{M})/b$, where $b$ is the bias parameter. The similar relation for two-point correlation functions was obtained by Hamilton  and generalised (for a case $z\neq 0$) by Matsubara & Suto  in the following form:
where distance in redshift space $s=\sqrt {\pi ^{2}+\sigma ^{2}}$, $\pi$ and $\sigma$ are its projections onto the line of sight and the plane, perpendicular to it, correspondingly, $\mu ={\rm acos}(\pi /s)$, $P_{m}$ are Legendre polynomials of order $m$ and $\xi _{2l}(r)$ are $2l$-order moments of the real-spacecorrelation function$\xi (r)$, the form of which depends on the $\xi (r)$ model. In general case they are given in  as:

References:
• ^ Peebles P. J. E. The Large-Scale Structure of the Universe, Princeton University Press, Princeton, New Jersey, 1980
• ^ Kaiser N., 1987, MNRAS, 227, 1, 1987MNRAS.227....1K
• ^ Hamiltom A. J. S., 1992, ApJ, 358, L5, 1992ApJ...385L...5H
• ^ab Matsubara T. & Suto Ya., 1996, ApJ, 470, L1, 1996ApJ...470L...1M
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