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On Squeezed Limits in Single-Field Inflation - Part I

Raphael Flauger, Daniel Green, Rafael A. Porto

TL;DR

The paper investigates whether single-field inflation can exhibit observable violations of the squeezed-limit consistency relation when the initial state is excited. By analyzing non-adiabatic evolution, Bogoliubov initial states, and EFT implementations, it shows that any non-trivial squeezed-limit signal is tightly bounded: typically $k_S/k_L \lesssim 10^2$, and in many viable scenarios the signal is suppressed or dominated by power-spectrum features, making detection unlikely with Planck or Euclid. The results unify constraints across slow-roll dynamics, resonant/non-adiabatic effects, and EFT descriptions, and indicate that observable deviations from the standard single-field prediction are not expected within these controlled frameworks. The work highlights folded and resonant shapes as more promising avenues for distinction, while delineating the regime where excited-state effects remain theoretically interesting but observationally elusive.

Abstract

The n-point correlation functions in single-field inflation obey a set of consistency conditions in the exact squeezed limit which are not present in multi-field models, and thus are powerful tools to distinguish between the two. However, these consistency conditions may be violated for a finite range of scales in single-field models, for example by departures from the Bunch-Davies state. These it excited states may be the consequence of interactions during inflation, or may be a remnant of the era that preceded inflation. In this paper we analyze the bispectrum, and show that in the regime of theoretical control the resulting signal in the squeezed limit remains undetectably small in all known models which continuously excite the state. We also show that the signal remains undetectably small if the initial state is related to the Bunch-Davies state by a Bogoliubov transformation and the energy density in the state is small enough so that the usual slow-roll conditions are obeyed. Bogoliubov states that lead to violations of the slow-roll conditions, as well as more general excited states, require more careful treatment and will be discussed in a separate publication.

On Squeezed Limits in Single-Field Inflation - Part I

TL;DR

The paper investigates whether single-field inflation can exhibit observable violations of the squeezed-limit consistency relation when the initial state is excited. By analyzing non-adiabatic evolution, Bogoliubov initial states, and EFT implementations, it shows that any non-trivial squeezed-limit signal is tightly bounded: typically , and in many viable scenarios the signal is suppressed or dominated by power-spectrum features, making detection unlikely with Planck or Euclid. The results unify constraints across slow-roll dynamics, resonant/non-adiabatic effects, and EFT descriptions, and indicate that observable deviations from the standard single-field prediction are not expected within these controlled frameworks. The work highlights folded and resonant shapes as more promising avenues for distinction, while delineating the regime where excited-state effects remain theoretically interesting but observationally elusive.

Abstract

The n-point correlation functions in single-field inflation obey a set of consistency conditions in the exact squeezed limit which are not present in multi-field models, and thus are powerful tools to distinguish between the two. However, these consistency conditions may be violated for a finite range of scales in single-field models, for example by departures from the Bunch-Davies state. These it excited states may be the consequence of interactions during inflation, or may be a remnant of the era that preceded inflation. In this paper we analyze the bispectrum, and show that in the regime of theoretical control the resulting signal in the squeezed limit remains undetectably small in all known models which continuously excite the state. We also show that the signal remains undetectably small if the initial state is related to the Bunch-Davies state by a Bogoliubov transformation and the energy density in the state is small enough so that the usual slow-roll conditions are obeyed. Bogoliubov states that lead to violations of the slow-roll conditions, as well as more general excited states, require more careful treatment and will be discussed in a separate publication.

Paper Structure

This paper contains 24 sections, 76 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Origin of a non-trivial squeezed limit
  3. Bounds on squeezed limits in slow-roll inflation
  4. Non-adiabatic evolution
  5. Resonances
  6. Dissipation
  7. Bogliubov initial states
  8. Bound on $k_S/k_L$ for Bogoliubov initial states
  9. Generalization to small speed of sound
  10. Bounds on more general models: The EFT of inflation
  11. Non-adiabatic evolution
  12. Generalized bound on $k_S/k_L$
  13. Bogoliubov initial states
  14. Generalized bound on $k_S/k_L$
  15. Prospects for observations
  16. ...and 9 more sections

Figures (3)

  • Figure 1: Sample processes responsible for correlations in the three- and four-point function given in equations \ref{['phiflat']} and \ref{['phiflat2']}
  • Figure 2: The green and red curves show the scale dependent halo bias as a function of $k$ with and without the bound \ref{['ratiok']} enforced, respectively. We have chosen $M=10^{12} h^{-1}\, M_\text{sol}$, $\beta_{{\vec{k}}_\star}=0.3$, $k_\star=1\,Mpc^{-1}$, and slow-roll parameters $\epsilon=\eta=10^{-2}$.
  • Figure 3: This plot shows $\Delta\chi^2$ as defined in \ref{['eq:dchi2']} as a function of $\beta_{{\vec{k}}_\star}$ with the phase $\theta_{\vec{k}}$ and $k_\star$ chosen to maximize the signal. The red and green curves show the results without and with the bound \ref{['ratiok']}, respectively. The shaded region indicates the region that is ruled out by the power spectrum when \ref{['ratiok']} is enforced.