[Paper Review] Constant-rate inflation: primordial black holes from conformal weight transitions

Kin‐Wang Ng, Yi-Peng Wu|arXiv (Cornell University)|Feb 10, 2021

Cosmology and Gravitation TheoriesPhysics and Astronomy94 references44 citations

TL;DR

This paper explains the steepest growth and continuous scaling of primordial black hole (PBH) power spectra in constant-rate inflation via conformal weight continuity across phase transitions. By analyzing curvature perturbations in de Sitter spacetime using dilatation symmetry, it shows that entropy production during adiabatic-to-non-adiabatic transitions triggers k⁴ growth via subleading modes with next-to-lowest conformal weights, while continuous scaling arises from preserved conformal weights, offering a unified analytic framework for PBH formation models with enhanced small-scale power spectra.

ABSTRACT

Constant-rate inflation, including ultra-slow-roll as a special case, has been widely applied to the formation of primordial black holes with significant deviation from the standard slow-roll conditions at both the growing and decaying phases of the power spectrum. We derive analytic solutions for the curvature perturbations with respect to the late-time scaling dimensions (conformal weights) constrained by the dilatation symmetry of the de Sitter background and show that the continuity of conformal weights across different rolling phases is protected by the adiabatic condition of the inflaton perturbation. The temporal excitation of subleading states (with the next-to-lowest conformal weights), recorded as the "steepest growth" of the power spectrum, is triggered by the entropy production in the transition from slow-roll to constant-rate phases.

Motivation & Objective

• To resolve the 'steepest growth' problem (PR ∼k⁴) in constant-rate inflation models for PBH formation.
• To explain the 'continuous scaling' of the power spectrum (PR ∼k⁶⁺²δ) across phase transitions despite changing rolling rates.
• To establish that conformal weight continuity across inflationary phases constrains PBH formation scenarios.
• To clarify the role of entropy production in triggering non-adiabatic transitions and spectral enhancements.
• To unify analytic solutions for curvature perturbations across multiple inflationary phases using de Sitter isometries.

Proposed method

• Derives analytic solutions for curvature perturbations using the de Sitter background's dilatation symmetry and conformal invariance.
• Defines conformal weights ∆i = 3/2 − νi with νi = |3/2 + δi|, linking them to the rolling rate δi of the inflaton.
• Applies boundary matching conditions at phase transitions to enforce continuity of the conjugate momentum πR, ensuring conformal weight continuity.
• Identifies subleading modes with next-to-lowest conformal weights as the source of k⁴ growth via entropy production during adiabatic-to-non-adiabatic transitions.
• Introduces a generalized adiabatic condition based on ∆ and δ, distinguishing adiabatic (AN = 0) and non-adiabatic (AN ≠ 0) phases.
• Uses bulk solutions in de Sitter spacetime to validate the analytic behavior of the power spectrum across phases.

Experimental results

Research questions

• RQ1 How does the k⁴ growth of the power spectrum arise in constant-rate inflation, particularly in the ultra-slow-roll limit?
• RQ2 Why does the power spectrum exhibit continuous scaling (PR ∼k⁶⁺²δ) even when the rolling rate δ changes from negative to positive values?
• RQ3 What physical mechanism ensures the continuity of conformal weights across different inflationary phases?
• RQ4 How is entropy production related to the excitation of subleading modes and the onset of steepest growth?
• RQ5 What constraints do conformal weight continuity and adiabaticity impose on PBH formation models?

Key findings

• The steepest growth of the power spectrum (PR ∼k⁴) is triggered by entropy production during the transition from slow-roll to constant-rate inflation, corresponding to the excitation of subleading modes with next-to-lowest conformal weights.
• The continuous scaling of the power spectrum (PR ∼k⁶⁺²δ) is a consequence of conformal weight continuity across phases, even when the rolling rate δ changes sign.
• Conformal weight continuity across phase transitions is enforced by the continuity of the conjugate momentum πR, which acts as a boundary condition in the analytic solutions.
• The transition from an adiabatic to a non-adiabatic phase (e.g., slow-roll to ultra-slow-roll) is the only case where conformal weight violation occurs, driven by entropy production.
• The generalized adiabatic condition is defined by AN = ∆N + δN, where AN = 0 for adiabatic phases and AN = 2δN + 3 for non-adiabatic phases (δN < −3/2), with AN ≠ 0 signaling non-adiabatic evolution.
• The final power spectrum's momentum dependence is fully determined by the conformal weight ∆, with PR ∼k⁶⁺²δ = k²∆, linking the spectral shape to the late-time scaling dimension.