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Simple Bounds from the Perturbative Regime of Inflation

Louis Leblond, Sarah Shandera

TL;DR

The paper derives perturbative validity conditions for inflationary fluctuations in single-field and mildly multi-field setups with general sound speed, showing that the ratio of cubic to quadratic actions imposes a lower bound on $c_s$ tied to the observed power spectrum: $c_s^4 > \mathcal{P}_\zeta$. For scale-dependent $c_s$, this bound constrains the permissible duration of the perturbative regime and can limit viable inflationary histories, particularly in DBI-like models. The analysis also demonstrates that eternal inflation cannot be perturbatively realized in small-$c_s$ scenarios and provides a simple large-$N$ bound on the number of light fields, $N < M_p^2/H^2$, aligning with black-hole and Planck-mass renormalization limits. Together, these results offer practical consistency checks for reconstructing inflationary actions and have implications for non-Gaussianity, e-fold counts, and the viability of scale-dependent sound-speed models.

Abstract

We examine the conditions under which a perturbative expansion around an inflating background is valid. When inflation is driven by a single field with a general sound speed, we find a lower limit on the sound speed related to the amplitude of the inflationary power spectrum. Generalizing the sound speed constraints to include scale dependence can limit the number of e-folds obtained in the perturbative regime and restrict otherwise apparently viable models. We also show that for models with a low sound speed, eternal inflation cannot occur in the perturbative regime.

Simple Bounds from the Perturbative Regime of Inflation

TL;DR

The paper derives perturbative validity conditions for inflationary fluctuations in single-field and mildly multi-field setups with general sound speed, showing that the ratio of cubic to quadratic actions imposes a lower bound on tied to the observed power spectrum: . For scale-dependent , this bound constrains the permissible duration of the perturbative regime and can limit viable inflationary histories, particularly in DBI-like models. The analysis also demonstrates that eternal inflation cannot be perturbatively realized in small- scenarios and provides a simple large- bound on the number of light fields, , aligning with black-hole and Planck-mass renormalization limits. Together, these results offer practical consistency checks for reconstructing inflationary actions and have implications for non-Gaussianity, e-fold counts, and the viability of scale-dependent sound-speed models.

Abstract

We examine the conditions under which a perturbative expansion around an inflating background is valid. When inflation is driven by a single field with a general sound speed, we find a lower limit on the sound speed related to the amplitude of the inflationary power spectrum. Generalizing the sound speed constraints to include scale dependence can limit the number of e-folds obtained in the perturbative regime and restrict otherwise apparently viable models. We also show that for models with a low sound speed, eternal inflation cannot occur in the perturbative regime.

Paper Structure

This paper contains 10 sections, 62 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Set-up, Gauges and Conventions
  3. The Variance of $\zeta$
  4. Perturbative Limits During Inflation
  5. The Expansion Parameter to Third Order
  6. Implications for Eternal inflation
  7. Higher Order Terms
  8. Scale Dependence
  9. Multiple Fields
  10. Conclusions

Figures (2)

  • Figure 1: (a) Contours of number of e-folds obtained before the bound is violated, as a function of the sound speed at some initial scale $k_0$ and the scale-dependence $s$. (b) Contours of number of e-folds obtained in the DBI model before the bound is violated, as a function of $f^{eff}_{NL}\approx-0.32/c_s^2$ at scale $k_0$ and with scale-dependence $n_{NG}$.
  • Figure 2: (a) Scaling of $\hat{\mathcal{L}}_3$ (solid blue), $\hat{\mathcal{L}}_4$ (long-dashed red), $\hat{\mathcal{L}}_5$ (short-dashed green) with coefficients that obey the bound in Eq.(\ref{['Lnbound']}), with $s=-0.1$. (b) Same as before, but with coefficients that do not obey the bound. Note that when the equality is saturated, all terms would meet at the point where $\hat{\mathcal{L}}_3$ crosses 1.