Fast and accurate computation of the aberration kernel for the cosmic microwave background sky

It is well known that our motion with respect to the cosmic microwave background (CMB) est frame introduces a large dipolar CMB anisotropy, with an amplitude alpha beta = v/c similar to 10(-3). In addition it should lead to a small breaking of statistical isotropy which becomes most notable at higher multipoles. In principle this could be used to determine our velocity with respect to the CMB rest frame using high angular resolution data from Planck, without directly relying on the amplitude and direction of the CMB dipole, allowing us to constrain cosmological models in which the cosmic dipole arises partly from large-scale isocurvature perturbations instead of being fully motion-induced. Here, we derive simple recursion relations that allow precise computation of the motion-induced coupling between different spherical harmonic coefficients. Although the lowest order approximations for the coupling kernel can be deficient by factors of 2-5 at multipoles l similar to 1000-3000, using our results for the aberration kernel we explicitly confirm that for a statistical detection of the aberration effect only first-order terms in beta matter. However, the expressions given here are not restricted to beta similar to 10(-3), but can be used at much higher velocities. We demonstrate the robustness of these formulae, illustrating the dependence of the kernel on beta, as well as the spherical harmonic indices l and m.