Parity-violating CMB correlators with non-decaying statistical anisotropy

We examine the cosmological correlators induced by the simultaneous breaking of parity and of statistical isotropy, e.g., in presence of the coupling ${\cal L} = f(\phi) ( - \frac{1}{4} F^2 + \frac{\gamma}{4} F \tilde{F} )$ between the inflaton $\phi$ and a vector field with vacuum expectation value ${\bf A}$. For a suitably chosen function $f$, the energy in the vector field $\rho_{\rm A}$ 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, $P_\zeta ( {\bf k}) = P(k)[ 1 + g_* ( \hat{\bf k} \cdot \hat{\bf A})^2]$, and a (angle-averaged) scalar bispectrum that is a linear combination of the first $3$ Legendre polynomials, $B_\zeta(k_1, k_2, k_3) = \sum_L c_L P_L (\hat{\bf k}_1 \cdot \hat{\bf k}_2) P(k_1) P(k_2) + 2~{\rm perms}$, with $c_0 : c_1 : c_2 = 2 : -3 : 1$ ($c_1 \neq 0$ is a consequence of parity violation, corresponding to the constant $\gamma \neq 0$). 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 $c_1$ 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 $\langle a_{\ell_1 m_1} a_{\ell_2 m_2}^* \rangle$, with $|\ell_1 - \ell_2| = 1$ in TT, TE, EE and BB, and $|\ell_1 - \ell_2| = 2$ in TB and EB. The latest CMB bounds on the scalar observables ($g_*$, $c_0$, $c_1$ and $c_2$), translate into the upper limit $\rho_{\rm A} / \rho_\phi \lesssim 10^{-9}$ at $\gamma=0$. We find that the upper limit on the vector energy density becomes much more stringent as $\gamma$ grows.