Probing the scale dependence of non-Gaussianity with spectral
  distortions of the cosmic microwave background
  Many inflation models predict that primordial density perturbations have a
nonzero three-point correlation function, or bispectrum in Fourier space. Of
the several possibilities for this bispectrum, the most commmon is the
local-model bispectrum, which can be described as a spatial modulation of the
small-scale (large-wavenumber) power spectrum by long-wavelength density
fluctuations. While the local model predicts this spatial modulation to be
scale-independent, many variants have some scale-dependence. Here we note that
this scale dependence can be probed with measurements of frequency-spectrum
distortions in the cosmic microwave background (CMB), in particular
highlighting Compton-$y$ distortions. Dissipation of primordial perturbations
with wavenumbers $50\,{\rm Mpc}^{-1} \lesssim k \lesssim 10^4\,{\rm Mpc}^{-1}$
give rise to chemical-potential ($\mu$) distortions, while those with
wavenumbers $1\,{\rm Mpc}^{-1} \lesssim k \lesssim 50\,{\rm Mpc}^{-1}$ give
rise to Compton-$y$ distortions. With local-model non-Gaussianity, the
distortions induced by this dissipation can be distinguished from those due to
other sources via their cross-correlation with the CMB temperature $T$. We show
that the relative strengths of the $\mu T$ and $yT$ correlations thus probe the
scale-dependence of non-Gaussianity and estimate the magnitude of possible
signals relative to sensitivities of future experiments. We discuss the
complementarity of these measurements with other probes of squeezed-limit
non-Gaussianity.
