We develop a methodology for estimating parity-odd bispectra in the cosmic microwave background (CMB). This is achieved through the extension of the original separable modal methodology to parity-odd bispectrum domains (ℓ 1 + ℓ 2 + ℓ 3 = odd). Through numerical tests of the parity-odd modal decomposition with some theoretical bispectrum templates, we verify that the parity-odd modal methodology can successfully reproduce the CMB bispectrum, without numerical instabilities. We also present simulated non-Gaussian maps produced by modal-decomposed parity-odd bispectra, and show the consistency with the exact results. Our new methodology is applicable to all types of parity-odd temperature and polarization bispectra.Bispectrum estimation of the cosmic microwave background (CMB) is one of the most powerful ways to explore the non-Gaussianity of primordial fluctuations. While standard single-field slow-roll inflation predicts a tiny amount non-Gaussianity (NG) of the primordial curvature perturbations [1,2], this is no longer true for a large number of extensions of the simplest inflationary paradigm (see e.g., refs. [3, 4] and references therein). Measurements of primordial NG thus provide a stringent test of the standard single-field slow roll scenario, and allow to put stringent constraints on alternative models. The most stringent constraints on primordial NG to date have been obtained through bispectrum measurements of Planck temperature data [5]. Future analyses, including correlations with E-mode polarization (and thus additional CMB bispectra of the type T T E , T EE and EEE ), will bring in further improvement on the current observational bounds [6,7].All CMB NG searches so far have been focused on parity-even bispectra, in which the condition ℓ 1 + ℓ 2 + ℓ 3 = even is enforced. This is because, as long as we consider the bispectrum of primordial curvature perturbations, parity cannot be broken, due to the spin-0 nature of the scalar mode. On the other hand, several interesting models predict bispectra generated by vector or tensor perturbations. In these cases the parity-even condition might have to be removed, since the vector or tensor modes can create parity-odd NG due to their spin dependence. For example, Early Universe models with some parity-violating or parityodd sources, such as the gravitational and electromagnetic Chern-Simons actions [8][9][10][11][12], or large-scale helical magnetic fields [13,14], generate NG with sizable CMB bispectrum signals in parity-odd configurations (ℓ 1 + ℓ 2 + ℓ 3 = odd) [15][16][17][18]. Vector or tensor modes also induce B-mode polarization. B-mode bispectra can thus be useful to prove tensor NG [18,19]. At the same time, the parity-odd property of the B-mode field can generate ℓ 1 + ℓ 2 + ℓ 3 = odd configurations in T T B , T EB , EEB and BBB bispectra, even when primordial NG has even parity. B-mode bispectra are also generated via secondary CMB lensing effects [20]. These theoretical predictions motivate us to investigate the CMB signals in ℓ 1 + ℓ 2 + ℓ 3 =
