[Paper Review] Note on the bispectrum and one-loop corrections in single-field inflation with primordial black hole formation

Jason Kristiano, Jun’ichi Yokoyama|arXiv (Cornell University)|Mar 1, 2023

Cosmology and Gravitation Theories49 references29 citations

TL;DR

The paper analyzes the bispectrum and one-loop power-spectrum corrections in single-field inflation with a temporary ultraslow-roll period that aims to form primordial black holes, confirming Maldacena’s theorem in this context and comparing two methods for the one-loop correction.

ABSTRACT

Primordial black holes can be formed from the collapse of large-amplitude perturbation on small scales in the early Universe. Such an enhanced spectrum can be realized by introducing a flat region in the potential of single-field inflation, which makes the inflaton go into a temporary ultraslow-roll period. In this paper, we calculate the bispectrum of curvature perturbation in such a scenario. We explicitly confirm that bispectrum satisfies Maldacena's theorem. At the end of the ultraslow-roll period, the bispectrum is generated by bulk interaction and field redefinition. At the end of inflation, bispectrum is generated only by bulk interaction. We also calculate the one-loop correction to the power spectrum from the bispectrum, called the source method. We find it consistent with the calculation of the one-loop correction from the second-order expansion of in-in perturbation theory.

Motivation & Objective

• Motivate the PBH formation scenario via a flat potential region inducing ultraslow-roll during inflation.
• Compute the curvature perturbation bispectrum arising from cubic self-interactions in this setting.
• Evaluate the one-loop correction to the large-scale power spectrum using two independent methods.
• Demonstrate consistency with Maldacena’s theorem in a non-attractor USR context.
• Address criticisms and clarify the correct application of Maldacena’s theorem.

Proposed method

• Derive the curvature perturbation dynamics through SR and USR phases, and express the Mukhanov-Sasaki variable with matching conditions at SR→USR and USR→SR transitions.
• Compute the bispectrum from the cubic self-interaction and from field redefinitions, focusing on the dominant H_int term in the USR transition regime.
• Apply in-in perturbation theory to evaluate the three-point function and isolate bulk and boundary contributions to the bispectrum.
• Show that the squeezed-limit bispectrum of the curvature perturbation obeys Maldacena’s theorem when expressed for ζ, not for the auxiliary variable ζ̃.
• Compute the one-loop correction to the large-scale power spectrum via two approaches (source method and second-order in-in expansion) and compare results.
• Respond to criticisms by highlighting the correct use of Maldacena’s theorem and the equivalence of the two loop-calculation methods.

Figure 1: Schematic picture of the inflaton potential realizing PBH formation. 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 period. It enters the S

Experimental results

Research questions

• RQ1 Does a temporary ultraslow-roll phase in single-field inflation lead to a large small-scale power spectrum capable of PBH formation without violating perturbativity?
• RQ2 What is the form and magnitude of the curvature perturbation bispectrum during and after the USR phase, and does it respect Maldacena’s theorem?
• RQ3 How does the one-loop correction to the large-scale power spectrum arise from the small-scale enhancement, and do different calculation methods agree?
• RQ4 Can the claimed no-go result for PBH formation in single-field inflation be reconciled with careful treatment of non-attractor phases and boundary contributions?

Key findings

• The bispectrum during and after USR is generated by bulk interaction and field redefinition, with Maldacena’s theorem confirmed for ζ (not ζ̃).
• The squeezed-limit bispectrum coefficient for ζ matches the spectral tilt, once field-redefinition effects are included, consistent with Maldacena’s theorem.
• Two independent calculations of the one-loop large-scale power spectrum (source method and second-order in-in) yield the same result, supporting the originally claimed perturbativity issue.
• The analysis argues that a large small-scale power (order 0.01) would induce a one-loop large-scale correction comparable to the tree-level term, challenging PBH formation in single-field inflation under these dynamics.
• The response to criticism clarifies that Maldacena’s theorem must be applied to ζ, and, after correcting a misapplication, the one-loop correction is reproduced by the source method as well.

Figure 2: Power spectrum of the curvature perturbation. At CMB scale, $k\ll k_{s}$ , the power spectrum is almost scale invariant. $p_{*}=0.05~{}\mathrm{Mpc}^{-1}$ is the pivot scale with amplitude $\Delta_{s(\mathrm{SR})}^{2}(p_{*})=2.1\times 10^{-9}$ , based on observational result [ 11 ] . At sma