[Paper Review] Constraining Primordial Black Hole Formation from Single-Field Inflation

Jason Kristiano, Jun’ichi Yokoyama|arXiv (Cornell University)|Nov 7, 2022

Cosmology and Gravitation Theories142 references54 citations

TL;DR

The paper computes the one-loop correction to the large-scale power spectrum in single-field inflation models that produce primordial black holes (PBHs) and shows that such models induce nonperturbative couplings on CMB scales, effectively ruling out PBH formation from this mechanism.

ABSTRACT

The most widely studied formation mechanism of a primordial black hole is collapse of large-amplitude perturbation on small scales generated in single-field inflation. In this Letter, we calculate one-loop correction to the large-scale power spectrum in a model with sharp transition of the second slow-roll parameter. We find that models producing an appreciable amount of primordial black holes induce nonperturbative coupling on a large scale probed by cosmic microwave background radiation. Our result implies that a small-scale power spectrum can be constrained by large-scale cosmological observations.

Motivation & Objective

• Motivate PBH formation from enhanced small-scale perturbations during inflation as a viable dark matter or GW source
• Assess the perturbative consistency of single-field inflation models that generate large small-scale power
• Quantify the backreaction of amplified small-scale modes on the large-scale power spectrum via loop corrections
• Derive bounds on the small-scale power spectrum to maintain perturbativity and consistency with observed CMB data
• Generalize the constraints to broader PBH formation scenarios within single-field inflation

Proposed method

• Quantize curvature perturbations using the Mukhanov-Sasaki variable and solve mode functions in SR and USR regimes
• Compute the leading one-loop correction to the large-scale power spectrum from cubic self-interactions of ζ in in-in perturbation theory
• Evaluate time integrals assuming η is sharply transitioning at USR boundaries, isolating dominant contributions
• Regularize and renormalize the loop correction to obtain a finite, physically meaningful power spectrum
• Extract phenomenological bounds on the PBH-amplified power, k-e/k-s ratio, and related parameters from perturbativity constraints

Experimental results

Research questions

• RQ1 Can single-field inflation with an ultraslow-roll phase generating PBHs remain perturbative at one-loop?
• RQ2 What upper bound on the small-scale power spectrum is required to keep loop corrections subdominant?
• RQ3 How do loop corrections constrain the viability of PBH formation mechanisms within standard single-field inflation?

Key findings

• A one-loop correction to the large-scale power spectrum is generated by the USR-induced enhancement of small-scale perturbations
• The leading one-loop contribution scales with (Δη)^2 Δ_s(PBH)^2 and a logarithmic/k-scale factor, leading to strong backreaction on CMB scales
• Imposing Δ_s(1)^2 ≪ Δ_s(0)^2 yields k_e/k_s ≪ 15, which translates to Δ_s(PBH)^2 ≪ 0.03
• Renormalization confirms that the power spectrum at one-loop is finite only when the PBH-amplified part obeys perturbativity, i.e., (Δη)^2 Δ_s(PBH)^2 ≪ 1
• The results suggest PBH formation from cosmological perturbation theory in single-field inflation with USR dynamics is ruled out
• The constraint applies broadly to PBH formation models in single-field inflation that rely on sharp η transitions or related mechanisms