A spherical harmonic analysis of redshift space

Abstract

We re-examine the effects of redshift space distortion in all-sky galaxy redshift surveys in the formalism of spherical harmonics. This natural decomposition of the density field into radial and angular eigenfunctions of the Laplacian operator complements both the spherical symmetry of the survey geometry and the dynamical basis for the redshift distortion. Within this framework we show how one can treat the distortion exactly to first order. We also analyse the effect of small-scale non-linear clustering on the distortion. In general, the distortion introduces statistical anisotropy in the clustering pattern, which is absent in real space, according to the Cosmological Principle. This can be exploited to constrain the cosmological density parameter, via $$\beta\equiv\Omega_0^{0.6}/b$$, where b is the galaxy bias parameter. The method also allows in principle a determination of the power spectrum of perturbations, requiring no assumptions beyond that of linear theory. The method offers significant advantages over Fourier techniques. The advantages are transparent when dealing with all-sky surveys, but the exact linear treatment of the distortion and the natural angular and radial definition of redshift surveys mean that the method is still very powerful, even for partial sky coverage. In this paper, we use a maximum likelihood analysis, and apply it to both simulated data and real data, for which we use the IRAS 1.2-Jy galaxy catalogue, finding a maximum likelihood $$\beta\simeq1.1\pm0.3$$, and a real-space amplitude corresponding to fractional fluctuations in an 8 h^–1 Mpc sphere of $$\sigma_{8,IRAS}=0.68\pm0.05$$. The 1σ errors should be treated cautiously and are discussed in the paper. We also relax the gravitational instability assumption, to find a more general determination of the velocity power spectrum required to reconcile the anisotropic redshift-space map with the assumed isotropic real-space map.