Angular dependence of primordial trispectra and CMB spectral distortions

Under the presence of anisotropic sources in the inflationary era, the trispectrum of the primordial curvature perturbation has a very specific angular dependence between each wavevector that is distinguishable from the one encountered when only scalar fields are present, characterized by an angular dependence described by Legendre polynomials. We examine the imprints left by curvature trispectra on the $TT\mu$ bispectrum, generated by the correlation between temperature anisotropies (T) and chemical potential spectral distortions ($\mu$) of the Cosmic Microwave Background (CMB). Due to the angular dependence of the primordial signal, the corresponding $TT\mu$ bispectrum strongly differs in shape from $TT\mu$ sourced by the usual $g_{\rm NL}$ or $\tau_{\rm NL}$ local trispectra, enabling us to obtain an unbiased estimation. From a Fisher matrix analysis, we find that, in a cosmic-variance-limited (CVL) survey of $TT\mu$, a minimum detectable value of the quadrupolar Legendre coefficient is $d_2 \sim 0.01$, which is 4 orders of magnitude better than the best value attainable from the $TTTT$ CMB trispectrum. In the case of an anisotropic inflationary model with a $f(\phi)F^2$ interaction (coupling the inflaton field $\phi$ with a vector kinetic term $F^2$), the size of the curvature trispectrum is related to that of quadrupolar power spectrum asymmetry, $g_*$. In this case, a CVL measurement of $TT\mu$ makes it possible to measure $g_*$ down to $10^{-3}$.