Spherical Redshift Distortions

Peculiar velocities induce apparent line of sight displacements of galaxies in redshift space, distorting the pattern of clustering in the radial versus transverse directions. On large scales, the amplitude of the distortion yields a measure of the dimensionless linear growth rate $\beta \approx \Omega^{0.6}/b$, where $\Omega$ is the cosmological density and $b$ the linear bias factor. To make the maximum statistical use of the data in a wide angle redshift survey, and for the greatest accuracy, the spherical character of the distortion needs to be treated properly, rather than in the simpler plane parallel approximation. In the linear regime, the redshift space correlation function is described by a spherical distortion operator acting on the true correlation function. It is pointed out here that there exists an operator, which is essentially the logarithmic derivative with respect to pair separation, which both commutes with the spherical distortion operator, and at the same time defines a characteristic scale of separation. The correlation function can be expanded in eigenfunctions of this operator, and these eigenfunctions are eigenfunctions of the distortion operator. Ratios of the observed amplitudes of the eigenfunctions yield measures of the linear growth rate $\beta$ in a manner independent of the shape of the correlation function. More generally, the logarithmic derivative $\partial/\partial\ln r$ with respect to depth $r$, along with the square $L^2$ and component $L_z$ of the angular momentum operator, form a complete set of commuting operators for the spherical distortion operator acting on the density. The eigenfunctions of this complete set of operators are spherical waves about the observer, with radial part lying in logarithmic real or Fourier space.