[Paper Review] One loop to rule them all: Perturbativity in the presence of ultra slow-roll dynamics

Gabriele Franciolini, Antonio J. Iovino+2 more|arXiv (Cornell University)|May 5, 2023

Cosmology and Gravitation Theories88 references26 citations

TL;DR

The paper analyzes perturbativity of single-field inflation with a USR phase intended to boost primordial black hole production, computing one-loop corrections to the curvature power spectrum using the in-in formalism and SR/USR/SR background dynamics. It finds loop effects remain subdominant but non-negligible, and identifies a tree-level dip as an artifact.

ABSTRACT

We discuss the issue of perturbativity in single-field inflationary models with a phase of ultra slow-roll (USR) tailor suited to generate an order-one abundance of primordial black holes (PBHs). More in detail, we impose the condition that loop corrections made up of short-wavelength modes enhanced by the USR dynamics do not alter the tree-level power spectrum of curvature perturbations. In our analysis, the USR phase is preceded and followed by two stages of ordinary slow-roll (SR), and we model the resulting SR/USR/SR dynamics using both instantaneous and smooth transitions. Focusing on scales relevant for CMB observations, we find that it is not possible, with these arguments, to rule out the scenario of PBH formation via USR, not even in the limit of instantaneous transition. However, we also find that loop corrections of short modes on the power spectrum of long modes, even though not large enough to violate perturbativity requirements, remain appreciable and, most importantly, are not tamed in realistic realisations of smooth SR/USR/SR transitions. This makes perturbativity a powerful theoretical tool to constrain USR dynamics. We extend the analysis at any scale beyond those relevant for CMB observations. We find that loop corrections of short modes remain within the few percent if compared to the tree-level power spectrum. However, we also find one notable exception of phenomenological relevance: we show that the so-called dip in the power spectrum of curvature perturbation is an artifact of the tree-level computation.

Motivation & Objective

• Motivate the study of perturbativity in single-field inflation with a USR phase tailored to produce primordial black holes (PBHs).
• Quantify loop corrections from short-wavelength modes enhanced by USR on the tree-level curvature power spectrum.
• Determine how SR/USR/SR transitions (instantaneous and smooth) affect perturbativity and PBH viability.
• Extend the analysis to a broad range of scales beyond CMB, including short and long modes, and assess phenomenological implications.

Proposed method

• Use the in-in formalism to compute quantum corrections to correlators in an interacting theory of curvature perturbations.
• Construct the SR/USR/SR background with controlled transitions via a semi-analytical model for epsilon, eta, and dη/dN.
• Derive the cubic and quartic interaction Hamiltonians for zeta fluctuations.
• Compute the one-loop correction to the curvature power spectrum arising from H_int at cubic and quartic orders.
• Solve the Mukhanov-Sasaki equation for mode functions with BD initial conditions across the SR/USR/SR sequence.
• Analyze perturbativity by requiring P_tree(k) [1+ΔP_1-loop(k)] with ΔP_1-loop(k) < 1 across various k.

Figure 1: Classical (top panel) and quantum (central panel) dynamics in the context of an explicit single-field model of inflation that exhibits the presence of a phase of USR in between the time interval $N_{\textrm{\rm in}}<N<N_{\textrm{\rm end}}$ (cf. ref. Ballesteros et al. ( 2020 ) ). In this s

Experimental results

Research questions

• RQ1 Does the USR phase induce large loop corrections that threaten perturbativity of the curvature power spectrum?
• RQ2 How do SR/USR/SR transitions (instantaneous vs smooth) influence the size and scale dependence of one-loop corrections?
• RQ3 Are short-mode loop corrections capable of significantly modifying long-wavelength (CMB) curvature perturbations?
• RQ4 Is the PBH-forming scenario via USR compatible with perturbativity when considering a complete SR/USR/SR evolution?
• RQ5 How do loop corrections behave across scales beyond those probed by the CMB, including short and intermediate modes?

Key findings

• Loop corrections from short modes enhanced by USR remain appreciable but do not violate perturbativity (i.e., do not exceed tree-level power).
• Even with instantaneous transitions, short-mode loops on long-mode power spectra are not large enough to break perturbativity in the studied setups.
• Realistic smooth SR/USR/SR transitions yield loop corrections that are not tamed, underscoring perturbativity as a powerful constraint on USR dynamics.
• The so-called dip in the tree-level power spectrum is shown to be an artifact of the tree-level computation rather than a physical feature when loops are included.
• Across scales beyond CMB, loop corrections stay within a few percent of the tree-level spectrum for the cases studied.

Figure 3: Left panel: Tree-level power spectrum in the minimal dynamics of section II.2 . The numerical values of the other parameters are $\eta_{\textrm{\rm II}}=3.5$ , $\eta_{\textrm{\rm III}}=0$ and $N_{\textrm{\rm end}}-N_{\textrm{\rm in}}=2.5$ . In our parametrization, we go beyond the instanta