The different varieties of the Suyama-Yamaguchi consistency relation and
  its violation as a signal of statistical inhomogeneity
  We present the different consistency relations that can be seen as variations
of the well known Suyama-Yamaguchi (SY) consistency relation \tau_{NL}
\geqslant ((6/5) f_{NL})^2. It has been claimed that the following variation:
\tau_{NL} ({\bf k}_1, {\bf k_3}) \geqslant (6/5)^2 f_{NL} ({\bf k}_1) f_{NL}
({\bf k}_3), which we call "the fourth variety", in the collapsed (for
\tau_{NL}) and squeezed (for f_{NL}) limits is always satisfied independently
of any physics; however, the proof depends sensitively on the assumption of
scale-invariance which only applies for cosmological models involving
Lorentz-invariant scalar fields (at least at tree level), leaving room for a
strong violation of this variety of the consistency relation when non-trivial
degrees of freedom, for instance vector fields, are in charge of the generation
of \zeta. With this in mind as a motivation, we explicitly state under which
conditions the SY consistency relation has been claimed to hold in its
different varieties (implicitly) presented in the literature; as a result, we
show for the first time that the variety \tau_{NL} ({\bf k}_1, {\bf k}_1)
\geqslant ((6/5) f_{NL} ({\bf k}_1))^2, which we call "the fifth variety", is
always satisfied even when there is strong scale-dependence as long as
statistical homogeneity holds: thus, an observed violation of this specific
variety would prevent the comparison between theory and observation, shaking
this way the foundations of cosmology as a science. Later, we concern about the
existence of non-trivial degrees of freedom, concretely vector fields for which
the levels of non-gaussianity have been calculated for very few models, finding
that the fourth variety of the SY consistency relation is indeed strongly
violated for some specific wavevector configurations while the fifth variety
continues to be well satisfied. (Abridged)
