Anisotropic CMB distortions from non-Gaussian isocurvature perturbations

We calculate the CMB $\mu$-distortion and the angular power spectrum of its cross-correlation with the temperature anisotropy in the presence of the non-Gaussian neutrino isocurvature density (NID) mode. While the pure Gaussian NID perturbations give merely subdominant contribution to $<\mu >$ and vanishing $<\mu T>$, the latter quantity can be large enough to be detected in the future when the NID perturbations $\mathcal{S}(\mathbf{x})$ are proportional to the square of a Gaussian field $g(\mathbf{x})$, i.e. $\mathcal S({\mathbf x})\propto g^2({\mathbf x})$\mathrm{.}Inparticular,large$< \mu T>$ can be realized since Gaussian-squared perturbations can yield a relatively large bispectrum, satisfying the constraints from the power spectrum of CMB anisotropies, i.e. ${\mathcal{P}}_{\mathcal{S}\mathcal{S}}(k_0)\sim {\mathcal{P}}_g^2(k_0)\lesssim {10}^{-10}$ at $k_0=0.05$ Mpc${}^{-1}$. We also forecast constraints from the CMB temperature and E-mode polarisation bispectra, and show that ${\mathcal{P}}_g(k_0)\lesssim {10}^{-5}$ would be allowed from Planck data. We find that $<\mu >$ and $\mathrm{\mid }l(l+1)C_l^{\mu T}\mathrm{\mid }$ can respectively be as large as ${10}^{-9}$ and ${10}^{-14}$ with uncorrelated scale-invariant NID perturbations for ${\mathcal{P}}_g(k_0)={10}^{-5}$. When the spectrum of the Gaussian field is blue-tilted (with spectral index $n_g\simeq 1.5$), $<\mu T>$ can be enhanced by an order of magnitude.