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

TL;DR

The paper investigates whether the scalar consistency relation generalizes to three-point functions involving tensors in single-field inflation. It defines the dimensionless non-Gaussianity parameters $C_{NL}^{R}$, $C_{NL}^{\gamma}$, and $h_{NL}$ and derives squeezed-limit relations linking them to the scalar and tensor spectral indices $n_S$ and $n_T$ and to the power spectra ${\mathcal P}_S$ and ${\mathcal P}_T$. Analytic demonstrations are provided for power-law inflation and the Starobinsky model, while numerical checks are performed in three slow-roll-violating scenarios with features in the scalar power spectrum. The results support the validity of tensor-related consistency relations in single-field inflation and suggest these relations as a diagnostic of the single-clock paradigm for upcoming tensor non-Gaussianity data.

Abstract

It is well known that the non-Gaussianity parameter $f_{_{\rm NL}}$ 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 $C_{_{\rm NL}}^{\mathcal R}$ and $C_{_{\rm NL}}^{\mathcal γ}$ 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 $h_{_{\rm NL}}$ 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)

Figures (2)

• Figure 1: The non-Gaussianity parameter $C_{_{\rm NL}}^{\mathcal{\gamma}}$ in the Starobinsky model, evaluated in the squeezed limit, has been plotted as a function of $k/k_0$. The solid blue curve represents the parameter arrived at from the analytical results for the scalar-tensor-tensor cross-correlation (obtained using the Maldacena formalism) and the scalar and the tensor power spectra. The dashed red curve corresponds to the non-Gaussianity parameter obtained from consistency condition (\ref{['eq:cnlt-cr']}), with the tensor spectral index being determined numerically. Evidently, there is good agreement between the two results, indicating that the consistency relation holds even when departures from slow roll occur. Note that we have worked with the following values of the potential parameters in arriving at these results: $\phi_0/M_{_{\rm Pl}}=0.707$, $V_0/M_{_{\rm Pl}}^4=2.37\times 10^{-12}$, $A_+/M_{_{\rm Pl}}^3=3.35\times 10^{-14}$ and $A_-/M_{_{\rm Pl}}^3=7.26\times10^{-15}$. These values have been chosen so that the assumptions of the Starobinsky model, under which the analytical results have been arrived at, are valid (in this context, see, for instance, Ref. martin-2012).
• Figure 2: The non-Gaussianity parameters, $C_{_{\rm NL}}^{\mathcal{R}}$ (the top row), $C_{_{\rm NL}}^{\mathcal{\gamma}}$ (the middle row) and $h_{_{\rm NL}}$ (the bottom row), arrived at using the Maldacena formalism and from the scalar and tensor spectral indices through the consistency conditions, have been plotted as a function of the wavenumber for the three inflationary models of our interest, viz. punctuated inflation (the left column), the quadratic potential with a step (the middle column) and the axion monodromy model (the right column). While the solid blue lines correspond to the numerical results for the parameters obtained using the Maldacena formalism, the dashed red lines represent the values arrived at from the spectral indices and the consistency relations. The two results match at the level of $3\%$ or better in the cases of punctuated inflation and the quadratic potential with the step, with the largest differences arising for the smallest wavenumbers due to the inherent limitation in implementing the squeezed limit numerically. Though the match is slightly poorer in the axion monodromy model (with differences of the order of $7\%$ for certain wavenumbers), we find that the match can be improved by integrating for a longer duration. At a first look, the difference in the cases of the model with the step and the axion monodromy model in the last row may seem striking. But, we should clarify here that it is simply due to the fact that the non-Gaussianity parameter $h_{_{\rm NL}}$ has been plotted in these cases over a rather small range in amplitude to highlight the mild variations. Needless to add, these figures confirm the validity of the consistency relations in single field inflationary models even in situations that allow strong departures from slow roll.