The Different Varieties of the Suyama-Yamaguchi Consistency Relation

TL;DR

This paper explores various forms of the Suyama-Yamaguchi consistency relation in primordial non-Gaussianity, identifying conditions under which they hold or are violated, with implications for cosmological observations and theory validation.

Contribution

It systematically analyzes different variants of the SY relation, introducing the fifth variety and demonstrating its universal validity under certain conditions.

Findings

The fifth variety of the SY relation always holds with statistical homogeneity.

Violations of the fifth variety would challenge the comparison between theory and observations.

The paper clarifies conditions for the validity of SY relation variants in cosmology.

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 (\frac{6}{5} f_{NL})^2, the latter involving the levels of non-gaussianity f_{NL} and \tau_{NL} in the primordial curvature perturbation \zeta, they being scale-invariant. We explicitly state 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} ({\bf k}_1, {\bf k}_1) \geqslant (\frac{6}{5} f_{NL} ({\bf 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.