The anisotropic power spectrum and bispectrum in the f(phi) F^2
  mechanism
  A suitable coupling of the inflaton phi to a vector kinetic term F^2 gives
frozen and scale invariant vector perturbations. We compute the cosmological
perturbations zeta that result from such coupling by taking into account the
classical vector field that unavoidably gets generated at large scales during
inflation. This generically results in a too anisotropic power spectrum of
zeta. Specifically, the anisotropy exceeds the 1% level (10% level) if
inflation lasted ~5 e-folds (~50 e-folds) more than the minimal amount required
to produce the CMB modes. This conclusion applies, among others, to the
application of this mechanism for magnetogenesis, for anisotropic inflation,
and for the generation of anisotropic perturbations at the end of inflation
through a waterfall field coupled to the vector (in this case, the unavoidable
contribution that we obtain is effective all throughout inflation, and it is
independent of the waterfall field). For a tuned duration of inflation, a 1%
(10%) anisotropy in the power spectrum corresponds to an anisotropic bispectrum
which is enhanced like the local one in the squeezed limit, and with an
effective local f_{NL} ~3 (~30). More in general, a significant anisotropy of
the perturbations may be a natural outcome of all models that sustain higher
than 0 spin fields during inflation.
