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.