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 can not 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 _ { 2 M } of the primordial scalar power spectrum ( normalized at the pivot scale comparable to the present horizon size k _ { 0 } ^ { -1 } = 14 ~ { } { Gpc } ) down to g _ { 2 M } = 30 ( 68 \% CL ) .