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Quantum radiative corrections to slow-roll inflation

Ante Bilandzic, Tomislav Prokopec

TL;DR

The paper develops a renormalization-group improved one-loop framework to calculate quantum radiative corrections to slow-roll inflation in a nonminimally coupled ${\lambda}\varphi^4$ model in de Sitter space. By combining the Chernikov-Tagirov propagator with RG-improved effective potentials, it derives quantum corrections to the slow-roll parameters $\epsilon$ and $\eta$, the scalar spectral index $n_s$, the tensor spectral index $n_g$, and the tensor-to-scalar ratio $r$, showing an infrared enhancement that scales with the number of e-foldings $N$. The leading quantum contributions are suppressed by the small coupling $\lambda$ (as $\lambda N^2$ or $\lambda N$ depending on the observable), leading to unobservable corrections for realistic parameter values ($\lambda \sim 10^{-12}$, $N\sim 50$–60), though the framework provides explicit, RG-consistent expressions for these corrections. The work highlights the role of curvature couplings and infrared effects in curved-space inflation and outlines avenues to extend the analysis to quasi-de Sitter backgrounds and other matter sectors.

Abstract

We consider the nonminimally coupled lambda phi^4 scalar field theory in de Sitter space and construct the renormalization group improved renormalized effective theory at the one-loop level. Based on the corresponding quantum Friedmann equation and the scalar field equation of motion, we calculate the quantum radiative corrections to the scalar spectral index n_s, gravitational wave spectral index n_g and the ratio r of tensor to scalar perturbations. When compared with the standard (tree-level) values, we find that the quantum contributions are suppressed by lambda N^2 where N denotes the number of e-foldings. Hence there is an N^2 enhancement with respect to the naive expectation, which is due to the infrared enhancement of scalar vacuum fluctuations characterising de Sitter space. Since observations constrain lambda to be very small lambda ~ 10^(-12) and N ~ 50-60, the quantum corrections in this inflationary model are unobservably small.

Quantum radiative corrections to slow-roll inflation

TL;DR

The paper develops a renormalization-group improved one-loop framework to calculate quantum radiative corrections to slow-roll inflation in a nonminimally coupled model in de Sitter space. By combining the Chernikov-Tagirov propagator with RG-improved effective potentials, it derives quantum corrections to the slow-roll parameters and , the scalar spectral index , the tensor spectral index , and the tensor-to-scalar ratio , showing an infrared enhancement that scales with the number of e-foldings . The leading quantum contributions are suppressed by the small coupling (as or depending on the observable), leading to unobservable corrections for realistic parameter values (, –60), though the framework provides explicit, RG-consistent expressions for these corrections. The work highlights the role of curvature couplings and infrared effects in curved-space inflation and outlines avenues to extend the analysis to quasi-de Sitter backgrounds and other matter sectors.

Abstract

We consider the nonminimally coupled lambda phi^4 scalar field theory in de Sitter space and construct the renormalization group improved renormalized effective theory at the one-loop level. Based on the corresponding quantum Friedmann equation and the scalar field equation of motion, we calculate the quantum radiative corrections to the scalar spectral index n_s, gravitational wave spectral index n_g and the ratio r of tensor to scalar perturbations. When compared with the standard (tree-level) values, we find that the quantum contributions are suppressed by lambda N^2 where N denotes the number of e-foldings. Hence there is an N^2 enhancement with respect to the naive expectation, which is due to the infrared enhancement of scalar vacuum fluctuations characterising de Sitter space. Since observations constrain lambda to be very small lambda ~ 10^(-12) and N ~ 50-60, the quantum corrections in this inflationary model are unobservably small.

Paper Structure

This paper contains 12 sections, 133 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Propagator in de Sitter space
  3. de Sitter space
  4. Scalar propagator in de Sitter space
  5. Effective potential
  6. Renormalization group analysis
  7. Slow-roll parameters
  8. Classical contributions $\epsilon_C$ and $\eta_C$
  9. Quantum contributions $\epsilon_Q$ and $\eta_Q$
  10. Tensor and scalar spectral indices $n_g$ and $n_s$
  11. The spectrum of curvature perturbation ${\cal P}_{\cal R}$ and the tensor-to-scalar ratio $r$
  12. Discussion

Figures (2)

  • Figure 1: The embedding of de Sitter space into a five dimensional flat space-time. The vertical line corresponds to the time coordinate, $X_0=T$, and the radial coordinate $R=\sqrt{X_1^2+X_2^2+X_3^2+X_4^2}$. At each point $(T,R)$ there is a unit 3-sphere $S^3$, which is for the sake of clarity represented by a circle $S^1$ erected at each point $(T,R)$. The Hubble radius $R_H=1/H$ is the coordinate distance $R$ of the hyperboloid from the origin at $T=0$.
  • Figure 2: The one-loop diagram (vacuum bubble) contributing to the scalar effective theory (\ref{['effective action:1 loop']}) in a curved background.