Parity-violating CMB correlators with non-decaying statistical anisotropy

We examine the effect induced on cosmological correlators by the simultaneous breaking of parity and of statistical isotropy. As an example of this, we compute the scalar-scalar, scalar-tensor, tensor-tensor and scalar-scalar-scalar cosmological correlators in presence of the coupling = f(\u3c6) ( - 1/4 F2 + \u3b3/4 F F ) between the inflaton \u3c6 and a vector field with vacuum expectation value A. For a suitably chosen function f, the energy in the vector field \u3c1A does not decay during inflation. This results in nearly scale-invariant signatures of broken statistical isotropy and parity. Specifically, we find that the scalar-scalar correlator of primordial curvature perturbations includes a quadrupolar anisotropy, P\u3b6(k) = P(k)[1+g 17(c\uc2)2], and a (angle-averaged) scalar bispectrum that is a linear combination of the first 3 Legendre polynomials, B\u3b6(k1, k2, k3) = 11L cL PL (1 c 2) P(k1) P(k2) + 2 perms , with c0:c1:c2=2-3:1 (c10 is a consequence of parity violation, corresponding to the constant 0\u3b3 ). The latter is one of the main results of this paper, which provides for the first time a clear example of an inflationary model where a non-negligible c1 contribution to the bispectrum is generated. The scalar-tensor and tensor-tensor correlators induce characteristic signatures in the Cosmic Microwave Background temperature anisotropies (T) and polarization (E/B modes); namely, non-diagonal contributions to a\u21131m1a 17\u21132m2, with |\u21131 - \u21132| = 1 in TT, TE, EE and BB, and |\u21131 - \u21132| = 2 in TB and EB. The latest CMB bounds on the scalar observables (g 17, c0, c1 and c2), translate into the upper limit \u3c1A / \u3c1\u3c610-9 at 0\u3b3=. We find that the upper limit on the vector energy density becomes much more stringent as \u3b3 grows