Parity-violating and anisotropic correlations in pseudoscalar inflation

A pseudo-scalar inflaton field can have interesting phenomenological signatures associated with parity violation. The existing analyses of these signatures typically assume statistical isotropy. In the present work we instead investigate the possibility that a pseudo-scalar inflaton is coupled to a vector field carrying a small but non-negligible vacuum expectation value (vev) coherent over our Hubble patch. We show that, in such case, correlators involving the primordial curvature perturbations and gravitational waves violate both statistical isotropy and parity symmetry. We compute the Cosmic Microwave Background (CMB) temperature anisotropies (T) and polarization (E/B) generated by these primordial modes. The CMB two-point correlation functions present distinct signals of broken rotational and parity invariance. Specifically, we find non-vanishing TT, TE, EE and BB correlators between $\ell_1$ and $\ell_2 = \ell_1 \pm 1$ multipoles, and non-vanishing TB and EB correlators between $\ell_1$ and $\ell_2 = \ell_1 \pm 2$ multipoles. Such signatures are specific of the models under consideration and they cannot be generated if one of parity and isotropy is preserved. As a specific example we consider the simple case in which the vector field has just an "electric" background component decaying in the standard way as $a^{-2}$. In this case a strong scale-dependent quadrupolar modulation of the primordial power spectra is generated and we find that almost noiseless data of the large-scale temperature and E-mode polarization anisotropies (like, e.g., the ones provided by WMAP or $Planck$) should be able to constrain the quadrupolar amplitude coefficients $g_{2M}$ of the primordial scalar power spectrum (normalized at the pivot scale comparable to the present horizon size $k_0^{-1} = 14~{\rm Gpc}$) down to $g_{2M} = 30$ (68%CL).