Odd-parity bipolar spherical harmonics

Abstract

Bipolar spherical harmonics (BiPoSHs) provide a general formalism for quantifying departures in the cosmic microwave background (CMB) from statistical isotropy (SI) and from Gaussianity. However, prior work has focused only on BiPoSHs with even parity. Here we show that there is another set of BiPoSHs with odd parity, and we explore their cosmological applications. We describe systematic artifacts in a CMB map that could be sought by measurement of these odd-parity BiPoSH modes. These BiPoSH modes may also be produced cosmologically through lensing by gravitational waves (GWs), among other sources. We derive expressions for the BiPoSH modes induced by the weak lensing of both scalar and tensor perturbations. Our estimate of the expected signal-to-noise with which we could measure the weak lensing of GWs from the correlation of two BiPoSH modes is respectable, indicating that such a measurement may be merited. We also investigate the possibility to detect parity-breaking physics, such as chiral GWs, by cross-correlating opposite parity BiPoSH modes with multipole moments of the CMB polarization. We nd that the expected signal-to-noise of such a detection is modest, with errors dominated by cosmic variance.