Evolution of CMB spectral distortion anisotropies and tests of
  primordial non-Gaussianity
  Anisotropies in distortions to the frequency spectrum of the cosmic microwave
background (CMB) can be created through spatially varying heating processes in
the early Universe. For instance, the dissipation of small-scale acoustic modes
does create distortion anisotropies, in particular for non-Gaussian primordial
perturbations. In this work, we derive approximations that allow describing the
associated distortion field. We provide a systematic formulation of the problem
using Fourier-space window functions, clarifying and generalizing previous
approximations. Our expressions highlight the fact that the amplitudes of the
spectral-distortion fluctuations induced by non-Gaussianity depend also on the
homogeneous value of those distortions. Absolute measurements are thus required
to obtain model-independent distortion constraints on primordial
non-Gaussianity. We also include a simple description for the evolution of
distortions through photon diffusion, showing that these corrections can
usually be neglected. Our formulation provides a systematic framework for
computing higher order correlation functions of distortions with CMB
temperature anisotropies and can be extended to describe correlations with
polarization anisotropies.
