A Generalized Doppler and Aberration Kernel for Frequency-Dependent Cosmological Observables

We introduce a $\textit{frequency-dependent}$ Doppler and aberration transformation kernel for the harmonic multipoles of a general cosmological observable with spin weight $s$, Doppler weight $d$ and arbitrary frequency spectrum. In the context of Cosmic Microwave Background (CMB) studies, the frequency-dependent formalism allows to correct for the motion-induced aberration and Doppler effects on individual frequency maps with different masks. It also permits to deboost background radiations with non-blackbody frequency spectra, like extragalactic foregrounds and CMB spectra with primordial spectral distortions. The formalism can also be used to correct individual E and B polarization modes and account for motion-induced E/B mixing of polarized observables with $d\neq1$ at different frequencies. We apply the generalized aberration kernel on polarized and unpolarized CMB specific intensity at 100 and 217 GHz and show that the motion-induced effects typically increase with the frequency of observation. In all-sky CMB experiments, the frequency-dependence of the motion-induced effects for a blackbody spectrum are overall negligible. However in a cut-sky analysis, ignoring the frequency dependence can lead to percent level error in the polarized and unpolarized power spectra over all angular scales. In the specific cut-sky used in our analysis ($b > 45^\circ, f_\text{sky}\simeq14\%$), and for the dipole-inferred velocity $\beta=0.00123$ typically attributed to our peculiar motion, the Doppler and aberration effects can change polarized and unpolarized power spectra of specific intensity in the CMB rest frame by $1-2\%$, but we find the polarization cross-leakage between E and B modes to be negligible.