Wide-Angle Redshift Distortions Revisited

We explore linear redshift distortions in wide-angle surveys from the point of view of symmetries. We show that the redshift-space two-point correlation function can be expanded into tripolar spherical harmonics of zero total angular momentum S_l1l2l3(x₁,x₂,x). The coefficients of the expansion B_l1l2l3 are analogous to the C_l of the angular power spectrum and express the anisotropy of the redshift-space correlation function. Moreover, only a handful of B_l1l2l3 are nonzero: the resulting formulae reveal a hidden simplicity comparable to the distant observer limit. The B_l1l2l3 depend on spherical Bessel moments of the power spectrum and f=Ω^0.6/b. In the plane-parallel limit, the results of Kaiser and Hamilton are recovered. The general formalism is used to derive useful new expressions. We present a particularly simple trigonometric polynomial expansion, which is arguably the most compact expression of wide-angle redshift distortions. These formulae are suitable for inversion because of the orthogonality of the basis functions. An alternative Legendre polynomial expansion was obtained as well. This can be shown to be equivalent to the results of Szalay and coworkers. The simplicity of the underlying theory will admit similar calculations for higher order statistics as well.