The Observed Squeezed Limit of Cosmological Three-Point Functions

TL;DR

This paper develops a comprehensive framework to relate the primordial bispectrum in the squeezed limit to late-time cosmological observables, clarifying the role of projection effects and confirming the absence of primordial signals in standard inflation models.

Contribution

It introduces a coordinate transformation-based formalism using Fermi Normal Coordinates to connect primordial bispectra with observable quantities in cosmology.

Findings

Primordial contributions to the squeezed CMB bispectrum vanish in standard slow-roll inflation.

No primordial fossil correlations are found between long-wavelength tensors and small-scale perturbations.

Observed correlations are solely due to projection effects like lensing and redshift perturbations.

Abstract

The squeezed limit of the three-point function of cosmological perturbations is a powerful discriminant of different models of the early Universe. We present a conceptually simple and complete framework to relate any primordial bispectrum in this limit to late time observables, such as the CMB temperature bispectrum and the scale-dependent halo bias. We employ a series of convenient coordinate transformations to capture the leading non-linear effects of cosmological perturbation theory on these observables. This makes crucial use of Fermi Normal Coordinates and their conformal generalization, which we introduce here and discuss in detail. As an example, we apply our formalism to standard slow-roll single-field inflation. We show explicitly that Maldacena's results for the squeezed limits of the scalar bispectrum [proportional to (ns-1) in comoving gauge] and the tensor-scalar-scalar bispectrum lead to no deviations from a Gaussian universe, except for projection effects. In particular, the primordial contributions to the squeezed CMB bispectrum and scale dependent halo bias vanish, and there are no primordial "fossil" correlations between long-wavelength tensor perturbations and small-scale perturbations. The contributions to observed correlations are then only due to projection effects such as gravitational lensing and redshift perturbations.