The different varieties of the Suyama-Yamaguchi consistency relation and its violation as a signal of statistical inhomogeneity

TL;DR

This paper explores various forms of the Suyama-Yamaguchi consistency relation in cosmology, identifying conditions under which they hold or are violated, especially in models involving non-trivial degrees of freedom like vector fields.

Contribution

It clarifies the conditions for the validity of different Suyama-Yamaguchi relation variants and demonstrates potential violations in models with vector fields, impacting cosmological tests.

Findings

The fifth variety of the SY relation is always satisfied under statistical homogeneity.

The fourth variety can be strongly violated in models with vector fields.

Violations of these relations can challenge the foundations of cosmological inference.

Abstract

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)