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Quantum corrections to slow roll inflation and new scaling of superhorizon fluctuations

D. Boyanovsky, H. J. de Vega, N. G. Sanchez

TL;DR

This work develops a renormalized EFT treatment of single-field slow-roll inflation to compute quantum loop corrections to the inflaton's equation of motion, the Friedmann equation, and superhorizon fluctuations. The leading corrections are driven by the near scale-invariance of scalar fluctuations, producing infrared enhancements regulated by the slow-roll parameter $\Delta = \eta_V - \epsilon_V$ and manifesting as a novel $1/\Delta$ structure in the effective potential and dynamics. The authors derive explicit one-loop results for the corrected EOM, the effective potential in terms of CMB observables, and the modified scaling and decay of superhorizon modes, including extensions to couplings with a light scalar field. They show that these quantum effects, while small in EFT, can compete with higher-order slow-roll corrections and are essential for precision constraints on inflationary parameters, with future work to include gravitational fluctuations.

Abstract

Precise cosmological data from WMAP and forthcoming CMB experiments motivate the study of the quantum corrections to the slowroll inflationary parameters.We find the quantum (loop) corrections to the equations of motion of the classical inflaton, its quantum fluctuations and the Friedmann equation in general single field slow roll inflation.We implement a renormalized effective field theory EFT approach based on an expansion in (H/M_{Pl})^2 and slow roll parameters epsilon_V,eta_V,sigma_V, xi_V.We find that the leading order quantum corrections to the inflaton effective potential and its equation of motion are determined by the power spectrum of scalar fluctuations. Its near scale invariance introduces a strong infrared behavior naturally regularized by the slow roll parameter Delta = eta_V-epsilon_V=(n_s-1)/2+r/8.To leading order in the EFT and slow roll expansions we find V_{eff}(Phi_0)=V_R(Phi_0)[1+(Delta^2_T/32)(n_s-1+3r/8) /(n_s-1+r/4)+higher orders]where n_s and r=Delta^2_T/Delta^2_R are the CMB observables that depend implicitly on Phi_0, and V_R(Phi_0) is the renormalized classical inflaton potential.This effective potential during slow roll inflation is strikingly different from the Minkowski space-time result.Superhorizon scalar field fluctuations grow for late times eta -> 0^- as |η|^{-1+Delta-d_} where d_ is a novel quantum correction to the scaling exponent related to the self decay of superhorizon inflaton fluctuations eta is the conformal time. We generalize this to the case of the inflaton interacting with a light scalar field. These quantum corrections arising from interactions will compete with higher order slow-roll corrections and must be taken into account for the precision determination of inflationary parameters.

Quantum corrections to slow roll inflation and new scaling of superhorizon fluctuations

TL;DR

This work develops a renormalized EFT treatment of single-field slow-roll inflation to compute quantum loop corrections to the inflaton's equation of motion, the Friedmann equation, and superhorizon fluctuations. The leading corrections are driven by the near scale-invariance of scalar fluctuations, producing infrared enhancements regulated by the slow-roll parameter and manifesting as a novel structure in the effective potential and dynamics. The authors derive explicit one-loop results for the corrected EOM, the effective potential in terms of CMB observables, and the modified scaling and decay of superhorizon modes, including extensions to couplings with a light scalar field. They show that these quantum effects, while small in EFT, can compete with higher-order slow-roll corrections and are essential for precision constraints on inflationary parameters, with future work to include gravitational fluctuations.

Abstract

Precise cosmological data from WMAP and forthcoming CMB experiments motivate the study of the quantum corrections to the slowroll inflationary parameters.We find the quantum (loop) corrections to the equations of motion of the classical inflaton, its quantum fluctuations and the Friedmann equation in general single field slow roll inflation.We implement a renormalized effective field theory EFT approach based on an expansion in (H/M_{Pl})^2 and slow roll parameters epsilon_V,eta_V,sigma_V, xi_V.We find that the leading order quantum corrections to the inflaton effective potential and its equation of motion are determined by the power spectrum of scalar fluctuations. Its near scale invariance introduces a strong infrared behavior naturally regularized by the slow roll parameter Delta = eta_V-epsilon_V=(n_s-1)/2+r/8.To leading order in the EFT and slow roll expansions we find V_{eff}(Phi_0)=V_R(Phi_0)[1+(Delta^2_T/32)(n_s-1+3r/8) /(n_s-1+r/4)+higher orders]where n_s and r=Delta^2_T/Delta^2_R are the CMB observables that depend implicitly on Phi_0, and V_R(Phi_0) is the renormalized classical inflaton potential.This effective potential during slow roll inflation is strikingly different from the Minkowski space-time result.Superhorizon scalar field fluctuations grow for late times eta -> 0^- as |η|^{-1+Delta-d_} where d_ is a novel quantum correction to the scaling exponent related to the self decay of superhorizon inflaton fluctuations eta is the conformal time. We generalize this to the case of the inflaton interacting with a light scalar field. These quantum corrections arising from interactions will compete with higher order slow-roll corrections and must be taken into account for the precision determination of inflationary parameters.

Paper Structure

This paper contains 10 sections, 113 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Effective field theory (EFT) and slow roll expansion.
  3. Quantum corrections to the equation of motion for the inflaton.
  4. Renormalized effective field theory: renormalization counterterms
  5. Quantum corrections to the Friedmann equation: the effective potential
  6. Quantum corrections to slow roll parameters
  7. Quantum corrections to superhorizon modes: a new scaling dimension
  8. Inflaton coupling to other light scalars.
  9. Conclusions and further questions
  10. One loop effective potential and equations of motion in Minkowski space-time: a comparison

Figures (1)

  • Figure 1: One-loop self energy contributions. $\lambda=\frac{1}{6} \; V^{(IV)}(\Phi_0)~,~g=\frac{1}{2} \; V^{"'}(\Phi_0)$. The square box in diagram (c) represents $- C^3(\eta)\chi\left[ \ddot{\Phi}_0+3 \, H \; \dot{\Phi}_0+V^{'}_R(\Phi_0)+\mathcal{C}_2[\Lambda,H] \; V^{"'}_R(\Phi_0)\right]$. The sum of diagrams (c) and (d) is proportional to the equation of motion (\ref{['1lupeqn']}) and vanishes. Only diagrams (a) and (b) contribute to the self-energy.