New operator approach to the CMB aberration kernels in harmonic space

Abstract

Aberration kernels describe how harmonic-space multipole coefficients of cosmic microwave background (CMB) observables transform under Lorentz boosts of the reference frame. For spin-weighted CMB observables, transforming like the CMB temperature (i.e. Doppler weight $d = 1$), we show that the aberration kernels are the matrix elements of a unitary boost operator in harmonic space. Algebraic properties of the rotation and boost generators then give simple, exact recursion relations that allow us to raise or lower the multipole quantum numbers $\ell$ and $m$, and the spin weight $s$. Further recursion relations express kernels of other Doppler weights $d \neq 1$ in terms of the $d = 1$ kernels. From those we show that on the full sky, to all orders in $β=v/c$, $E$- and $B$-mode polarization observables do not mix under aberration if and only if $d = 1$. The new relations, fully non-linear in the boost velocity $β$, form the basis of a practical recursive algorithm to accurately compute any aberration kernel. In addition, we develop a second, fast algorithm in which aberration kernels are obtained using a set of ordinary differential equations. This system can also be approximately solved at small scales, providing simple asymptotic formulae for the aberration kernels. The results of this work will be useful for further studying the effect of aberration on future CMB temperature and polarization analysis, and might provide a basis for relativistic radiative transfer schemes.