Kaiser effect (or -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 is the Hubble parameter, is the density contrast, and , where is the growing mode of density perturbations (see e.g. [1]). Hence for the observer outside the region of overdensity this region will appear smaller in line-of-sight direction in redshift space. Kaiser [2] showed that the matter power spectrum in real () and redshift space () are related to each other with the following formula: where is a cosine of angle between and line of sight. Later was replaces by , where is the bias parameter. The similar relation for two-point correlation functions was obtained by Hamilton [3] and generalised (for a case ) by Matsubara & Suto [4] in the following form: where distance in redshift space , and are its projections onto the line of sight and the plane, perpendicular to it, correspondingly, , are Legendre polynomials of order and are -order moments of the real-spacecorrelation function, the form of which depends on the model. In general case they are given in [4] as:

In the linear theory of structure formation the perturbation of the Hubble parameter is Here is the Hubble parameter, is the density contrast, and , where is the growing mode of density perturbations (see e.g. [1]). Hence for the observer outside the region of overdensity this region will appear smaller in line-of-sight direction in redshift space. Kaiser [2] showed that the matter power spectrum in real () and redshift space () are related to each other with the following formula: where is a cosine of angle between and line of sight. Later was replaces by , where is the bias parameter. The similar relation for two-point correlation functions was obtained by Hamilton [3] and generalised (for a case ) by Matsubara & Suto [4] in the following form: where distance in redshift space , and are its projections onto the line of sight and the plane, perpendicular to it, correspondingly, , are Legendre polynomials of order and are -order moments of the real-spacecorrelation function, the form of which depends on the model. In general case they are given in [4] as:

References:

- [1]^ Peebles P. J. E. The Large-Scale Structure of the Universe, Princeton University Press, Princeton, New Jersey, 1980
- [2]^ Kaiser N., 1987, MNRAS, 227, 1, 1987MNRAS.227....1K
- [3]^ Hamiltom A. J. S., 1992, ApJ, 358, L5, 1992ApJ...385L...5H
- [4]^
^{a}^{b}Matsubara T. & Suto Ya., 1996, ApJ, 470, L1, 1996ApJ...470L...1M