[Paper Review] One-loop Corrections in Power Spectrum in Single Field Inflation

Hassan Firouzjahi|arXiv (Cornell University)|Mar 21, 2023

Cosmology and Gravitation Theories4 citations

TL;DR

This paper calculates one-loop quantum corrections to the curvature perturbation power spectrum in single-field inflation with an ultra-slow-roll (USR) phase, including contributions from cubic and quartic interactions. It finds that for an arbitrarily sharp transition from USR to slow-roll, one-loop corrections become unphysically large, invalidating perturbation theory, but suggests these may be washed out in milder transitions, with implications for primordial black hole formation.

ABSTRACT

We revisit the one-loop correction in curvature perturbation power spectrum in models of single field inflation which undergo a phase of ultra slow-roll (USR) inflation. We include the contributions from both the cubic and quartic interaction Hamiltonians and calculate the one-loop corrections on the spectrum of the CMB scale modes from the small scale modes which leave the horizon during the USR phase. It is shown that the amplitude of one-loop corrections depends on the sharpness of the transition from the USR phase to the final slow-roll phase. For an arbitrarily sharp transition, the one-loop correction becomes arbitrarily large, invalidating the perturbative treatment of the analysis. We speculate that for a mild transition, the large one-loop corrections are washed out during the subsequent evolution after the USR phase. The implications for primordial black holes formation are briefly reviewed.

Motivation & Objective

• To investigate one-loop quantum corrections to the curvature perturbation power spectrum in single-field inflation with an ultra-slow-roll (USR) phase.
• To assess the role of both cubic and quartic interaction Hamiltonians in generating these corrections from small-scale modes during USR.
• To examine how the sharpness of the transition from USR to final slow-roll phase affects the magnitude of one-loop corrections.
• To evaluate the viability of USR models for primordial black hole (PBH) formation in light of potential large quantum corrections.
• To determine whether large one-loop effects are physically meaningful or washed out during the post-USR evolution.

Proposed method

• Uses the effective field theory (EFT) of inflation to construct cubic and quartic interaction Hamiltonians for curvature perturbations.
• Applies the in-in formalism to compute one-loop corrections to the power spectrum, integrating over loop momenta and time during the USR phase.
• Considers two integration strategies: restricting to modes that become superhorizon during USR, or integrating over the full time range.
• Performs analytical calculations of nested in-in integrals using asymptotic mode function behavior and computational tools (Maple).
• Evaluates contributions from both $/pi\pi^{\prime 2}$ and $ \pi^{2}\partial^{2}\pi$ interaction terms, identifying subleading contributions.
• Compares results with prior work, particularly Kristiano et al., focusing on the divergence of corrections under sharp transitions.

Figure 1 : The Feynman diagrams for the one-loop correction in curvature perturbation power spectrum $\langle{\cal{R}}({\bf{p}}_{1}){\cal{R}}({\bf{p}}_{2})\rangle$ . The left and right panel represent the contribution of the cubic Hamiltonian and quartic Hamiltonian respectively.

Experimental results

Research questions

• RQ1 How do one-loop corrections from small-scale modes during the USR phase affect the power spectrum of CMB-scale curvature perturbations?
• RQ2 What is the dependence of one-loop corrections on the sharpness of the transition from ultra-slow-roll to slow-roll inflation?
• RQ3 Why do large one-loop corrections arise in USR models despite the usual intuition that superhorizon modes are unaffected by subhorizon modes?
• RQ4 Can large one-loop corrections be physically meaningful, or are they artifacts of an unphysical sharp transition?
• RQ5 What are the implications of these corrections for primordial black hole formation in USR models?

Key findings

• One-loop corrections to the curvature perturbation power spectrum become arbitrarily large for an arbitrarily sharp transition from the USR phase to the final slow-roll phase.
• The divergence of corrections in the sharp transition limit invalidates the perturbative treatment of the theory, suggesting a breakdown of the one-loop analysis.
• For a mild transition, the large one-loop corrections may be washed out during the subsequent evolution after the USR phase, preserving perturbative control.
• The amplitude of one-loop corrections depends critically on the transition dynamics, with the correction scaling as $\Delta N e^{6\Delta N}$, where $\Delta N$ is the duration of the USR phase.
• The calculation confirms previous results on the bispectrum using the EFT approach, validating the method for higher-order corrections.
• Contributions from the $\pi^{2}\partial^{2}\pi$ interaction term are subleading compared to $\pi\pi^{\prime 2}$, but contribute to the overall correction when included.