A consistency relation for the CMB B-mode polarization in the squeezed limit

A large-scale temperature perturbation has a non-zero correlation with the power spectrum of B-modes of cosmological origin on short scales while the corresponding correlation is expected to be zero for B-modes sourced by galactic foregrounds. We thus compute the squeezed limit of a three-point function in which one correlates the temperature fluctuations at large scales with two polarization modes at small scales. In the particular case of the B-mode polarization we obtain a relation that connects the squeezed limit of the TBB three-point function with the cosmological B-mode power spectrum, which can be used as a consistency relation. This could in principle help to distinguish a primordial signal from that induced by inter-stellar dust.