The Observed Squeezed Limit of Cosmological Three-Point Functions

TL;DR

This work develops a simple, complete framework to translate the squeezed-limit of any primordial three-point function into late-time observables, such as the CMB bispectrum and halo bias, by employing conformal Fermi Normal Coordinates to separate local non-linear evolution from projection effects. The method shows that in single-field slow-roll inflation, Maldacena's squeezed-limit consistency relations cause the primordial squeezed contribution to vanish in the conformal FNC frame, leaving observable signals dominated by projection effects like lensing and redshift perturbations. The paper provides explicit transformations for two- and three-point functions, derives the squeezed-limit behavior in both global and $\overline{\mathrm{FNC}}$ frames, and connects these results to late-time observations via transfer functions and standard ruler formalism. It also discusses extensions to resonant non-Gaussianity and multifield models, highlighting how the framework can identify genuine observational signatures beyond projection effects.

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.

Figures (2)

• Figure 1: Sequence of coordinates employed in the computation of observables predicted by a primordial bispectrum in the squeezed limit. The arrows represent the change from one set of coordinates to the next, indicating when this transformation is most conveniently performed and with a reference to the relevant equations. $\tau_{\rm em}$ and $\tau_{\rm obs}$ denote the conformal times at which the observed photons were emitted and observed, respectively.
• Figure 2: Illustration of the $\overline{\mathrm{FNC}}$ patch throughout cosmic history, in comoving units. The wavy lines indicate perturbations. The solid circles denote the comoving horizon $1/aH$, which coincides with the size of the usual Fermi coordinate patch during inflation. The dashed circles denote the size of the $\overline{\mathrm{FNC}}$ patch, i.e. the region within which the metric is of the form $\bar{g}_{\mu\nu}^F = a^2 \eta_{\mu\nu}$ with small corrections. a) During inflation, when the long wavelength perturbation $h_{ij}(\mathbf{k}_L)$ is generated (i.e. leaves the horizon), thin red line. b) Later on during inflation; $h_{ij}(\mathbf{k}_L)$ is far outside the horizon when the short wavelength modes $\vartheta(\mathbf{k})$ are generated (thick blue line). The perturbations are generated well within the $\overline{\mathrm{FNC}}$ patch corresponding to the long-wavelength mode. c) Near the end of inflation. All perturbations are far outside the horizon. Nevertheless, the small-scale modes are still well within the $\overline{\mathrm{FNC}}$ patch. d) At observation time, after the long-wavelength mode has reentered the horizon (solid circle indicating present horizon). The $\overline{\mathrm{FNC}}$ patch now coincides with the usual Fermi coordinate patch, which is much larger than the small-scale modes which have been processed by nonlinear evolution since horizon entry (distorted thick wavy line).