Effect of aberration on partial-sky measurements of the cosmic microwave background temperature power spectrum

Our motion relative to the cosmic microwave background (CMB) rest frame deflects light rays giving rise to shifts as large as ℓ→ℓ(1± β), where β=0.00123 is our velocity (in units of the speed of light) on measurements of CMB fluctuations. Here we present a novel harmonic-space approach to this CMB aberration that improves upon prior work by allowing us to (i) go to higher orders in β, thus extending the validity of the analysis to measurements at ℓâ‰β-1â‰800; and (ii) treat the effects of window functions and pixelization in a more accurate and computationally efficient manner. We calculate precisely the magnitude of the systematic bias in the power spectrum inferred from the partial sky and show that aberration shifts the multipole moment by Δℓ/ℓ cos, with cos averaged over the survey footprint. Such a shift, if ignored, would bias the measurement of the sound-horizon size θ* at the 0.01% level, which is comparable to the measurement uncertainties of Planck. The bias can then propagate into cosmological parameters such as the angular-diameter distance, Hubble parameter and dark-energy equation of state. We study the effect of aberration for current Planck, South Pole Telescope (SPT) and Atacama Cosmology Telescope (ACT) data and show that the bias cannot be neglected. On the other hand, the aberration effect yields the opposite sign of the discrepancy and cannot account for the small tension between ACT and SPT. An Appendix shows how the near constancy of the full-sky power spectrum under aberration follows from unitarity of the aberration kernel.