The mildly nonlinear imprint of structure on the CMB: the $O(\ell )$ couplings are lensing and aberration

Abstract

The exact 1+3 covariant multipole hierarchy for the CMB brightness contains kinematic couplings between the shear, acceleration and vorticity of the observers and the temperature multipoles, whose coefficients grow linearly in $\ell$. They are products of first-order quantities, so linearization about FLRW removes them. Here they are kept as sources in a weakly nonlinear extension of the almost-FLRW temperature anisotropies, and evaluated in the aligned-wavevector, weak-coupling limit. In the Newtonian frame the acceleration terms cancel at leading order in $1/\ell$. In the total (CDM) frame the shear terms cancel in the same way, to a residual of about $3/\ell$ of either term. Earlier versions of this paper reported that the total-frame terms do not cancel, and inferred a mildly nonlinear Rees-Sciama imprint near $\ell \sim 100-300$. That result came from a sign misprint in the $\ell -2$ shear coupling and is withdrawn; see the Note added. The $O(\ell )$ couplings are the multipole form of gravitational lensing and observer aberration, both already in the standard treatment. The explicit kinematic couplings written here are only part of the full second-order propagation source; the perturbed free-streaming contribution is linearized away in this paper.

23 pages, no figures. Replacement: central claim withdrawn (a sign error in the l-2 shear coupling); see the Note added. Ancillary file verify.py regenerates key algebraic statements (sympy)

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