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) >= (6/5 f(NL))(2), the latter involving the levels of non-gaussianity f(NL) and tau(NL) in the primordial curvature perturbation zeta. It has been (implicitly) claimed that the following variation: tau(NL) (k(1), k(3)) >= (6/5)(2) f(NL) (k(1)) f(NL) (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 (expressing this way the fourth variety of the SY consistency relation as tau(NL) >= (6/5 f(NL))(2)) 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 the primordial curvature perturbation. With this in mind as a motivation, we explicitly state, in the first part of this work, under which conditions the SY consistency relation has been claimed to hold in its different varieties (implicitly) presented in the literature since its inception back in 2008; as a result, we show for the first time that the variety tau(NL) (k(1), k(1)) >= (6/5 f(NL) (k(1)))(2), which we call the fifth variety, is always satisfied even when there is strong scale-dependence and high levels of statistical anisotropy 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. In the second part, 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; among them, and by making use of the delta N formalism at tree level, we study a class of models that includes the vector curvaton scenario, vector inflation, and the hybrid inflation with coupled vector and scalar waterfall field where zeta is generated at the end of inflation, 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. Finally, as a byproduct of our investigation, we draw attention to a quite recently demonstrated variety of the SY consistency relation: tau(iso)(NL) >= (6/5 f(NL)(iso))(2), in scenarios where scalar and vector fields contribute to the generation of the primordial curvature perturbation; this variety of the SY consistency relation is satisfied although the isotropic pieces of the non-gaussianity parameters receive contributions from the vector fields. We discuss further implications for observational cosmology.