Examining the consistency relations describing the three-point functions involving tensors

Abstract

It is well known that the non-Gaussianity parameter $f_{{}_{\mathrm{NL}}}$ characterizing the scalar bi-spectrum can be expressed in terms of the scalar spectral index in the squeezed limit, a property that is referred to as the consistency relation. In this work, we consider the consistency relations associated with the three-point cross-correlations involving scalars and tensors as well as the tensor bi-spectrum in inflationary models driven by a single, canonical, scalar field. Characterizing the cross-correlations in terms of the dimensionless non-Gaussianity parameters $C_{{}_{\mathrm{NL}}}^{\mathcal{R}}$ and $C_{{}_{\mathrm{NL}}}^{\gamma }$ that we had introduced earlier, we express the consistency relations governing the cross-correlations as relations between these non-Gaussianity parameters and the scalar or tensor spectral indices, in a fashion similar to that of the purely scalar case. We also discuss the corresponding relation for the non-Gaussianity parameter $h_{{}_{\mathrm{NL}}}$ used to describe the tensor bi-spectrum. We analytically establish these consistency relations explicitly in the following two situations: a simple example involving a specific case of power law inflation and a non-trivial scenario in the so-called Starobinsky model that is governed by a linear potential with a sharp change in its slope. We also numerically verify the consistency relations in three types of inflationary models that permit deviations from slow roll and lead to scalar power spectra with features which typically result in an improved fit to the data than the more conventional, nearly scale invariant, spectra. We close with a summary of the results we have obtained. (Abridged)

v1: 25 pages, 2 figures; v2: 25 pages, 2 figures, minor changes, references added, accepted for publication in JCAP

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