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The (not so) squeezed limit of the primordial 3-point function

Paolo Creminelli, Guido D'Amico, Marcello Musso, Jorge Noreña

TL;DR

This work establishes that, for generic single-field inflationary models, the squeezed-limit bispectrum obeys a tight consistency relation with corrections only at quadratic order in the long-to-short mode ratio $k_L/k_S$. It provides two complementary derivations of the leading result, analyzes gradient and constraint contributions, and shows that even models with features or varying sound speed respect the no-linear-correction property. The authors extend the discussion to multi-field scenarios, noting that linear corrections are absent only when all fields are light, while quasi-single-field setups can alter the squeezed-limit scaling. They connect these theoretical findings to observations of scale-dependent bias in large-scale structure, illustrating how the squeezed limit shapes bias predictions and outlining observational prospects. Overall, the paper clarifies when a detection of certain squeezed-limit behavior would challenge broad classes of inflationary models and informs the interpretation of upcoming surveys.

Abstract

We prove that, in a generic single-field model, the consistency relation for the 3-point function in the squeezed limit receives corrections that vanish quadratically in the ratio of the momenta, i.e. as (k_L/k_S)^2. This implies that a detection of a bispectrum signal going as 1/k_L^2 in the squeezed limit, that is suppressed only by one power of k_L compared with the local shape, would rule out all single-field models. The absence of this kind of terms in the bispectrum holds also for multifield models, but only if all the fields have a mass much smaller than H. The detection of any scale dependence of the bias, for scales much larger than the size of the haloes, would disprove all single-field models. We comment on the regime of squeezing that can be probed by realistic surveys.

The (not so) squeezed limit of the primordial 3-point function

TL;DR

This work establishes that, for generic single-field inflationary models, the squeezed-limit bispectrum obeys a tight consistency relation with corrections only at quadratic order in the long-to-short mode ratio . It provides two complementary derivations of the leading result, analyzes gradient and constraint contributions, and shows that even models with features or varying sound speed respect the no-linear-correction property. The authors extend the discussion to multi-field scenarios, noting that linear corrections are absent only when all fields are light, while quasi-single-field setups can alter the squeezed-limit scaling. They connect these theoretical findings to observations of scale-dependent bias in large-scale structure, illustrating how the squeezed limit shapes bias predictions and outlining observational prospects. Overall, the paper clarifies when a detection of certain squeezed-limit behavior would challenge broad classes of inflationary models and informs the interpretation of upcoming surveys.

Abstract

We prove that, in a generic single-field model, the consistency relation for the 3-point function in the squeezed limit receives corrections that vanish quadratically in the ratio of the momenta, i.e. as (k_L/k_S)^2. This implies that a detection of a bispectrum signal going as 1/k_L^2 in the squeezed limit, that is suppressed only by one power of k_L compared with the local shape, would rule out all single-field models. The absence of this kind of terms in the bispectrum holds also for multifield models, but only if all the fields have a mass much smaller than H. The detection of any scale dependence of the bias, for scales much larger than the size of the haloes, would disprove all single-field models. We comment on the regime of squeezing that can be probed by realistic surveys.

Paper Structure

This paper contains 13 sections, 36 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Single-field consistency relation with subleading corrections
  3. Review of the standard leading order computation
  4. A non-standard proof with standard methods
  5. Including gradients: proof of the absence of linear corrections
  6. The decay of the decaying mode
  7. Constraints on the constraints
  8. Explicit check in various models
  9. Modulations and features in the inflaton potential
  10. Non-standard single-field models with scale invariant spectrum
  11. Multi-field case
  12. The squeezing of the scale-dependent bias.
  13. Conclusions

Figures (2)

  • Figure 1: Expression \ref{['eq:bias']} for the modification to the halo bias in the presence of local non-Gaussianity multiplied by $k^2$ before integration over $k_1$, for a smoothing length of $R = 2\,h^{-1}\,\mathrm{Mpc}$ at redshift $z = 1$.
  • Figure 2: Comparison of the non-Gaussian correction from the local template to the halo bias from a "modified" local template with the extreme squeezed limit removed (see text). The blue solid line is a "conservative" estimate for a smoothing scale corresponding to $2.5\times 10^{12} M_\odot\, h^{-1}$, the red dashed line is an "optimistic" estimate for larger scales and smaller objects with a mass of $3.1\times 10^{11} M_\odot\,h^{-1}$. The choice of scale for the "optimistic" case corresponds roughly to the largest scale accessible to the planned EUCLID survey.