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One-loop corrections to the power spectrum in general single-field inflation

Nicola Bartolo, Emanuela Dimastrogiovanni, Alberto Vallinotto

TL;DR

This work computes the one-loop corrections to the curvature power spectrum in general single-field inflation with non-canonical kinetics $P(X,\phi)$, focusing on models with a small sound speed $c_s$ that amplify interactions and non-Gaussianity. Using the $\delta N$ formalism and a careful in-in perturbative expansion in the ADM framework, the authors derive the full third- and fourth-order actions and the corresponding interaction Hamiltonians, including tensor modes. They show that tensor-loop contributions are subdominant to scalar loops and provide explicit scaling for the leading diagrams, culminating in a one-loop correction to $P_\zeta$ with a logarithmic UV term $\ln(\Lambda/H_*)$. By enforcing perturbativity, they obtain a theoretical lower bound $c_s^4 \geq 2\mathcal{P}_{\zeta}$, which translates to $c_s \gtrsim 9\times 10^{-3}$, a result that is closely compatible with current observational constraints and highlights the interplay between loop corrections and primordial non-Gaussianity in constraining inflationary models.

Abstract

We perform a thorough computation of the one-loop corrections from both scalar and tensor degrees of freedom to the power spectrum of curvature fluctuations for non-canonical Lagrangians in single-field inflation. We consider models characterized by a small sound speed c_{s}, which produce large non-Gaussianities. As expected, the corrections turn out to be inversely proportional to powers of c_{s}; evaluating their amplitudes it is then possible to derive some theoretical bounds on the sound speed by requesting the conditions necessary for perturbation theory to hold.

One-loop corrections to the power spectrum in general single-field inflation

TL;DR

This work computes the one-loop corrections to the curvature power spectrum in general single-field inflation with non-canonical kinetics , focusing on models with a small sound speed that amplify interactions and non-Gaussianity. Using the formalism and a careful in-in perturbative expansion in the ADM framework, the authors derive the full third- and fourth-order actions and the corresponding interaction Hamiltonians, including tensor modes. They show that tensor-loop contributions are subdominant to scalar loops and provide explicit scaling for the leading diagrams, culminating in a one-loop correction to with a logarithmic UV term . By enforcing perturbativity, they obtain a theoretical lower bound , which translates to , a result that is closely compatible with current observational constraints and highlights the interplay between loop corrections and primordial non-Gaussianity in constraining inflationary models.

Abstract

We perform a thorough computation of the one-loop corrections from both scalar and tensor degrees of freedom to the power spectrum of curvature fluctuations for non-canonical Lagrangians in single-field inflation. We consider models characterized by a small sound speed c_{s}, which produce large non-Gaussianities. As expected, the corrections turn out to be inversely proportional to powers of c_{s}; evaluating their amplitudes it is then possible to derive some theoretical bounds on the sound speed by requesting the conditions necessary for perturbation theory to hold.

Paper Structure

This paper contains 12 sections, 121 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Background equations of motion
  3. Useful relations in terms of slow-variation parameters
  4. Some worked examples
  5. $\delta$N formalism applied to $P(X,\phi)$ models
  6. Perturbative expansion of the action
  7. Leading terms in the third and fourth order actions
  8. Calculation of the interaction hamiltonian
  9. Calculation of the loop corrections
  10. Order of magnitude of all leading order diagrams
  11. Final results
  12. Conclusions

Figures (1)

  • Figure 5: Diagrammatic representation of the one-vertex interaction diagrams. The dots on each end of a line indicate temporal derivatives of the corresponding wavefunctions.