Statistical anisotropy in CMB spectral distortions

Measurements of the cosmic microwave background (CMB) spectral $y$-distortion anisotropy offer a test for the statistical isotropy of the primordial density perturbations on $0.01\lesssim k\mathrm{M}\mathrm{p}\mathrm{c}\lesssim 1$. We compute the 1-point ensemble averages of the $y$-distortion anisotropies which vanish for the statistically isotropic perturbations. For the quadrupole statistical anisotropy, we find $4\pi\langle y_{2m}\rangle=-6.8A_2\times 10^{-9}Y_{2m}(\mathbf d )$ with the quadruple Legendre coefficient of the anisotropic powerspectrum $A_2$ and the $\mathrm{\ell }=2$ spherical harmonics $Y_{2m}(\mathbf{d})$ for the preferred direction $\mathbf{d}$. Also, we discuss the cosmic variance of the $y$-distortion anisotropy in the statistically anisotropic Universe.