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Are Primordial Black Holes Truly Fine-Tuned?

A. J. Iovino, A. Riotto

TL;DR

The paper addresses whether PBH production from single-field inflation with an USR phase is technically natural. It adopts a Wilsonian naturalness criterion, defining $\gamma = c/\bar{c}$, to quantify fine-tuning and applies it to three benchmark USR models: a toy Starobinsky-like dip, a minimally coupled polynomial potential, and a non-minimally coupled polynomial potential. The curvature power spectrum and PBH abundance are computed via the Mukhanov-Sasaki equation and threshold statistics of the compaction function, yielding $f_{\rm PBH}$ values from $\sim 10^{-7}$ to order unity while maintaining $n_s \simeq 0.96$ and $r \lesssim 0.06$. Across all models, $\gamma$ remains ${\cal O}(1)$, indicating the PBH scenarios are not technically unnatural and suggesting that PBH production in single-field USR frameworks can be natural; extensions to multi-field or spectator sectors are proposed for future work.

Abstract

Single-field inflationary models which generate primordial black holes through the enhancement of the curvature primordial power at small scales are commonly criticized and frequently dismissed because they require a large amount of fine-tuning in the parameters setting the ultra slow-roll phase. However, the standarly adopted definition of fine-tuning has a clear drawback: the more the primordial black hole abundance is small and cosmologically harmless, the larger the parameter space is fine-tuned. A reliable measure of fine-tuning should deliver a large value when the primordial black hole abundance is fine-tuned and at the same time reduce to something close to unity when it encounters typical sensitivity. Motivated by such arguments, we use the (modified version of) Wilson's naturalness criterion for quantifying the fine-tuning and naturalness and we show that the primordial black hole models are not technically unnatural.

Are Primordial Black Holes Truly Fine-Tuned?

TL;DR

The paper addresses whether PBH production from single-field inflation with an USR phase is technically natural. It adopts a Wilsonian naturalness criterion, defining , to quantify fine-tuning and applies it to three benchmark USR models: a toy Starobinsky-like dip, a minimally coupled polynomial potential, and a non-minimally coupled polynomial potential. The curvature power spectrum and PBH abundance are computed via the Mukhanov-Sasaki equation and threshold statistics of the compaction function, yielding values from to order unity while maintaining and . Across all models, remains , indicating the PBH scenarios are not technically unnatural and suggesting that PBH production in single-field USR frameworks can be natural; extensions to multi-field or spectator sectors are proposed for future work.

Abstract

Single-field inflationary models which generate primordial black holes through the enhancement of the curvature primordial power at small scales are commonly criticized and frequently dismissed because they require a large amount of fine-tuning in the parameters setting the ultra slow-roll phase. However, the standarly adopted definition of fine-tuning has a clear drawback: the more the primordial black hole abundance is small and cosmologically harmless, the larger the parameter space is fine-tuned. A reliable measure of fine-tuning should deliver a large value when the primordial black hole abundance is fine-tuned and at the same time reduce to something close to unity when it encounters typical sensitivity. Motivated by such arguments, we use the (modified version of) Wilson's naturalness criterion for quantifying the fine-tuning and naturalness and we show that the primordial black hole models are not technically unnatural.
Paper Structure (6 sections, 28 equations, 2 figures)

This paper contains 6 sections, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Benchmark models
  3. Computing the Power Spectra
  4. Computing the PBH abundance
  5. Results
  6. Conclusions

Figures (2)

  • Figure 1: On the main panel, we show power spectra of curvature perturbations over the entire range of scales covered by the inflationary dynamics generate in the Toy model with an artificial dip, Eq.\ref{['eq:StaroDip']} tuning the parameter $A_s$ while we fix the others parameters. We also plot the region excluded by CMB anisotropy measurements, Ref. Planck:2018jri, the FIRAS bound on CMB spectral distortions, Ref. Chluba:2012we (see also Ref. Jeong:2014gnaIovino:2024tyg) and the bound obtained from Lyman-$\alpha$ forest data Bird:2010mp. The magenta inset shows the same Power spectra of curvature perturbations as a function of comoving wavenumbers, zoomed in on the region of scales relevant for PBH productions.
  • Figure 2: Evolution of the naturalness parameter $\gamma$ for the toy model with an artificial dip, Eq.\ref{['eq:StaroDip']}, (top panel), minimally coupled polynomial model, Eq.\ref{['eq:MC']}, (middle panel) and non-minimally coupled polynomial model, Eq.\ref{['eq:NMC']}, (bottom panel) as function of the amplitude of the main peak $A$ (left panels) and of the corresponding PBH abundance $f_{\rm PBH}$ (right panels).