[Paper Review] Generating large primordial fluctuations in single-field inflation for PBH formation

Jason Kristiano, Jun’ichi Yokoyama|arXiv (Cornell University)|May 20, 2024

Cosmology and Gravitation Theories4 citations

TL;DR

This paper proposes a single-field inflation model with sequential slow-roll (SR), ultraslow-roll (USR), and SR phases to generate large primordial curvature perturbations on small scales—necessary for primordial black hole (PBH) formation—while preserving small-scale fluctuations consistent with CMB observations. It demonstrates that quantum one-loop corrections to the power spectrum and bispectrum from nonlinear interactions during the USR transition impose strong constraints on such models, limiting the feasibility of large-scale PBH formation despite observational permissibility on small scales.

ABSTRACT

In order to produce appreciable amount of primordial black holes (PBHs), the square amplitude of curvature perturbation must take a large value of $\mathcal{O}(0.01)$, namely, seven digits larger than the value observed by cosmic microwave background radiation (CMB) on large scales. Such a large fluctuation can be achieved by violating the slow-roll (SR) condition within a short duration. The best known of such possibilities is the ultraslow-roll (USR) inflation. We calculate the power spectrum of curvature perturbation in a simple single-field inflation model which evolves through the SR-USR-SR regimes so that both large-amplitude small-scale fluctuation for PBH formation and small-amplitude large-scale fluctuation as observed by CMB are realized. We further calculate the bispectrum and one-loop correction to the power spectrum induced by the third-order action of curvature perturbation as the beginning of precision cosmology on small scales. As a result, we show that single-field inflation model realizing PBH formation can be constrained by the quantum correction.

Motivation & Objective

• To construct a viable single-field inflation model that simultaneously produces large primordial curvature perturbations on small scales (for PBH formation) and small perturbations on large scales (consistent with CMB observations).
• To investigate the role of nonlinear quantum corrections—specifically one-loop corrections to the power spectrum and bispectrum—in constraining models that generate large primordial fluctuations via violation of the slow-roll condition.
• To analyze the impact of a sharp transition from ultraslow-roll (USR) to slow-roll (SR) on the higher-order correlation functions of curvature perturbations.
• To resolve discrepancies in the literature regarding the treatment of total time derivative and equation-of-motion terms in one-loop calculations of the power spectrum.
• To establish that even observationally viable PBH-forming models are constrained by quantum corrections, particularly in the USR-to-SR transition regime.

Proposed method

• Formulates a single-field inflation model with three distinct phases: slow-roll (SR), ultraslow-roll (USR), and back to SR, using a potential with a flat region enabling USR dynamics.
• Calculates the curvature perturbation power spectrum using the $δ N$ formalism and linearized cosmological perturbation theory in the comoving gauge.
• Derives the bispectrum from the third-order action of curvature perturbations, distinguishing contributions from bulk interactions and field redefinitions.
• Evaluates one-loop corrections to the power spectrum using the effective action approach, including contributions from cubic self-interactions and quartic terms in the Hamiltonian.
• Applies the $δ N$ formalism and stochastic inflation techniques to analyze the effects of sharp transitions in the second slow-roll parameter $\epsilon_2$.
• Compares results with prior works, particularly resolving inconsistencies in the treatment of total time derivative and equation-of-motion terms in one-loop computations.

Figure 1 : Schematic picture of the inflaton potential realizing PBH formation from USR condition. When the inflaton is around $\phi_{\mathrm{CMB}}$ , scales probed by CMB observations leave the horizon and it is in the SR regime. It enters an extremely flat region at $t=t_{s}$ undergoing an USR per

Experimental results

Research questions

• RQ1 Can a single-field inflation model with a transient USR phase generate large primordial curvature perturbations on small scales while remaining consistent with CMB observations on large scales?
• RQ2 What is the role of the bispectrum, particularly from field redefinition and bulk interactions, in constraining models with a sharp USR-to-SR transition?
• RQ3 How do one-loop quantum corrections to the power spectrum affect the viability of PBH-forming inflation models, especially in the presence of a rapid change in the second slow-roll parameter?
• RQ4 Why do some previous works on one-loop corrections yield conflicting results, and what is the correct treatment of total time derivative and equation-of-motion terms in the effective action?
• RQ5 To what extent do nonlinear effects in the curvature perturbation, such as those from cubic self-interactions, lead to backreaction that constrains the amplitude of small-scale power spectra?

Key findings

• A SR-USR-SR transition in a single-field inflation model successfully generates a large-amplitude power spectrum peak on small scales (amplitude ~0.01), suitable for PBH formation, while maintaining small-scale fluctuations consistent with CMB observations.
• The bispectrum in the USR-to-SR transition regime reaches an $\mathcal{O}(1)$ nonlinear parameter, indicating strong non-Gaussianity that cannot be neglected in precision cosmology.
• One-loop corrections to the power spectrum—induced by cubic self-interactions and field redefinition—introduce a significant backreaction that constrains the allowed amplitude of small-scale power spectra.
• The model shows that even if a PBH-forming scenario is observationally viable on small scales, quantum corrections from the USR transition phase can render it inconsistent with quantum field theory constraints.
• The paper resolves a discrepancy in the literature by showing that prior works [145, 146] incorrectly treat total time derivative and equation-of-motion terms in the one-loop calculation, leading to unphysical results.
• The one-loop correction is found to be sensitive to the sharpness of the transition in the second slow-roll parameter $\epsilon_2$, with divergent behavior in the limit $|\epsilon_2| \to \infty$, indicating a critical role for transition dynamics.

Figure 2 : Sketch of $\epsilon_{2}(\tau)$ for sharp and smooth transition from USR to SR period are shown by the gray and blue curve, respectively. Brown curve shows sharp transition from SR-USR-SR period, which will be discussed in Sec. 4 .