Scale and shape dependent non-Gaussianity in the presence of
  inflationary vector fields
  We consider cosmological inflationary models in which vector fields play some
role in the generation of the primordial curvature perturbation $\zeta$. Such
models are interesting because the involved vector fields naturally seed
statistical anisotropy in the primordial fluctuations which could eventually
leave a measurable imprint on the cosmic microwave background fluctuations. In
this article, we estimate the scale and shape dependent effects on the
non-Gaussianity (NG) parameters due to the scale dependent statistical
anisotropy in the distribution of the fluctuations. For concreteness, we use a
power spectrum (PS) of the fluctuations of the quadrupolar form:
$P_\zeta(\vec{k})\equiv P_\zeta(k)[1+g_\zeta(k)(\hat{n} \cdot \hat{k})^2 ]$,
where $g_{\zeta}(k)$ is the only quantity which parametrizes the level of
statistical anisotropy and $\hat{n}$ is a unitary vector which points towards
the preferred direction. Then, we evaluate the contribution of the running of
$g_{\zeta}(k)$ on the NG parameters by means of the $\delta N$ formalism. We
focus specifically on the details for the $f_{\rm NL}$ NG parameter, associated
with the bispectrum $B_\zeta$, but the structure of higher order NG parameters
is straightforward to generalize. Although the level of statistical anisotropy
in the PS is severely constrained by recent observations, the importance of
statistical anisotropy signals in higher order correlators remains to be
determined, this being the main task that we address here. The precise
measurement of the shape and scale dependence (or running) of statistical
parameters such as the NG parameters and the statistical anisotropy level could
provide relevant elements for model building and for the determination of the
presence (or nonpresence) of inflationary vector fields and their role in the
inflationary mechanism.
