Primordial tensor bispectra in p,-CMB cross-correlations

Cross-correlations between Cosmic Microwave Background (CMB) temperature and polarization anisotropies and mu-spectral distortions have been considered to measure (squeezed) primordial scalar bispectra in a range of scales inaccessible to primary CMB bispectra. In this work we address whether it is possible to constrain tensor non-Gaussianities with these cross-correlations. We find that only primordial tensor bispectra with statistical anisotropies leave distinct signatures, while isotropic tensor bispectra leave either vanishing or highly suppressed signatures. We discuss how the angular dependence of squeezed bispectra in terms of the short and long momenta determine the non-zero cross-correlations. We also discuss how these non-vanishing configurations are affected by the way in which primordial bispectra transform under parity. By employing the so-called BipoSH formalism to capture the observational effects of statistical anisotropies, we make Fisher-forecasts to assess the detection prospects from mu T, mu E and mu B cross-correlations. Observing statistical anisotropies in squeezed (gamma gamma gamma) and (gamma gamma zeta) bispectra is going to be challenging as the imprint of tensor perturbations on mu-distortions is subdominant to scalar perturbations, therefore requiring a large, independent amplification of the effect of tensor perturbations in the mu-epoch. In absence of such a mechanism, statistical anisotropies in squeezed (zeta zeta gamma) bispectrum are the most relevant sources of mu T, mu E and mu B cross-correlations. In particular, we point out that in anisotropic inflationary models where (zeta zeta zeta) leaves potentially observable signatures in mu T and mu E, the detection prospects of (zeta zeta gamma) from mu B are enhanced. JCAP02(2022)004