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Constraints on the primordial curvature power spectrum at small scales between $3\times 10^{18}$ and $4.5\times 10^{21}~\rm Mpc^{-1}$

Yupeng Yang

TL;DR

This work addresses the challenge of constraining the primordial curvature power spectrum on ultra-small scales by leveraging PBH physics, including memory-burden backreaction and PBH mergers. It establishes the theoretical link between the initial PBH mass fraction $\beta(M_{\rm PBH})$ and the curvature power spectrum $\mathcal{P}_{\mathcal{R}}$ for Gaussian perturbations, and updates constraints using light PBHs in the mass range $10^{3} \lesssim M_{\rm PBH} \lesssim 2\times 10^{9}~\mathrm{g}$ from BBN, gamma-ray and neutrino observations, and merger scenarios. The resulting bounds on $\mathcal{P}_{\mathcal{R}}$ span $3\times 10^{18} \lesssim k \lesssim 4.5\times 10^{21}~\mathrm{Mpc}^{-1}$, with the strongest limits ($\mathcal{P}_{\mathcal{R}} \lesssim 10^{-1.8}$) driven by memory-burdened PBHs; neutrino and gamma-ray channels further shape the bounds in specific subranges. The results extend small-scale probes of the primordial power spectrum and point to future prospects with IceCube-Gen2 and GRAND200k for even tighter constraints.

Abstract

The primordial curvature power spectrum $\mathcal{P}_\mathcal{R}$ has been measured with high precision on large scales $10^{-4}\lesssim k\lesssim 3~\rm Mpc^{-1}$ based on observations of the cosmic microwave background, Lyman-$α$ forest and large scale structure. On small scales $3\lesssim k \lesssim 10^{23}~\rm Mpc^{-1}$, constraints are primarily derived from studies on primordial black holes (PBHs). In particular, for very small scales $10^{17}\lesssim k\lesssim 10^{23}~{\rm Mpc^{-1}}$, current limits come exclusively from investigations of the lightest supersymmetric particles produced by PBH radiation and the stable Planck-mass relics after their evaporation. Recent findings also indicate that the evaporation of light PBHs ($M_{\rm PBH}\lesssim 10^{9}~\rm g$) can modify the expansion rate of the Universe and the baryon-to-photon ratio, thereby affecting the primordial abundance of light nuclei. Moreover, it has been proposed that the ``memory burden'' effect can slow down the mass loss rate of black holes, allowing light PBHs to survive until the present day. Based on recent theoretical advancements in black hole physics and existing constraints on the initial mass fraction of light PBHs with masses $10^{3}\lesssim M_{\rm PBH}\lesssim 2\times 10^{9}~\rm g$, we derive new constraints on $\mathcal{P}_\mathcal{R}$ on small scales $3\times 10^{18}\lesssim k\lesssim 4.5\times 10^{21}~\rm Mpc^{-1}$, a regime that has been underexplored in previous literature.

Constraints on the primordial curvature power spectrum at small scales between $3\times 10^{18}$ and $4.5\times 10^{21}~\rm Mpc^{-1}$

TL;DR

This work addresses the challenge of constraining the primordial curvature power spectrum on ultra-small scales by leveraging PBH physics, including memory-burden backreaction and PBH mergers. It establishes the theoretical link between the initial PBH mass fraction and the curvature power spectrum for Gaussian perturbations, and updates constraints using light PBHs in the mass range from BBN, gamma-ray and neutrino observations, and merger scenarios. The resulting bounds on span , with the strongest limits () driven by memory-burdened PBHs; neutrino and gamma-ray channels further shape the bounds in specific subranges. The results extend small-scale probes of the primordial power spectrum and point to future prospects with IceCube-Gen2 and GRAND200k for even tighter constraints.

Abstract

The primordial curvature power spectrum has been measured with high precision on large scales based on observations of the cosmic microwave background, Lyman- forest and large scale structure. On small scales , constraints are primarily derived from studies on primordial black holes (PBHs). In particular, for very small scales , current limits come exclusively from investigations of the lightest supersymmetric particles produced by PBH radiation and the stable Planck-mass relics after their evaporation. Recent findings also indicate that the evaporation of light PBHs () can modify the expansion rate of the Universe and the baryon-to-photon ratio, thereby affecting the primordial abundance of light nuclei. Moreover, it has been proposed that the ``memory burden'' effect can slow down the mass loss rate of black holes, allowing light PBHs to survive until the present day. Based on recent theoretical advancements in black hole physics and existing constraints on the initial mass fraction of light PBHs with masses , we derive new constraints on on small scales , a regime that has been underexplored in previous literature.

Paper Structure

This paper contains 7 sections, 5 equations, 2 figures.

Table of Contents

  1. Introduction
  2. constraints on light PBHs and primordial curvature perturbation power spectrum on small scales
  3. The initial mass fraction of PBHs
  4. Constraints on the (initial) mass fraction of light PBHs
  5. Constraints on the primordial curvature perturbation power spectrum on small scales
  6. Summary
  7. Acknowledgements

Figures (2)

  • Figure 1: New constraints on the (initial) mass fraction of light primordial black holes (PBHs) for the mass range $1\times 10^{3}\lesssim M_{\rm PBH}\lesssim 2\times10^{9}~\rm g$ are presented: (i) the initial mass fraction ($\beta_{\rm PBH}$) from big bang nucleosynthesis (BBN) using the deuterium abundance ("BBN(D)", green shaded area, right $y$-axis) Boccia:2024nly; (ii) the present mass fraction ($f_{\rm PBH}$) of memory-burdened PBHs from gamma-ray observations ("Gamma-rays", blue shaded area) Thoss:2024hsr; (iii) the present mass fraction ($f_{\rm PBH}$) of memory-burdened PBHs from high-energy neutrino observations using 7-year IceCube-EHE data ("IceCube-EHE", red shaded area) Chianese:2024rsn; (iv) the present mass fraction ($f_{\rm PBH}$) of merged memory-burdened PBHs from high-energy neutrino observations using 7-year IceCube data for energies above 60 TeV ("Merger(IceCube)", black shaded area) Zantedeschi:2024ram. For comparison, previous constraints on the initial PBH mass fraction ($\beta_{\rm PBH}$, right $y$-axis) are also plotted: a stable Planck-mass relic from PBH evaporation ("DM relic density", dotted line) and the lightest supersymmetric particles ("LSP", hashed line) produced by PBH evaporation Josan:2009. Both scenarios are consistent with the upper limits for the cold dark matter density. All shaded areas are excluded by the corresponding observations.
  • Figure 2: Constraints on the power spectrum of the primordial curvature perturbation ($\mathcal{P}_\mathcal{R}$) for small scales $3\times 10^{18}~{\rm Mpc^{-1}}\lesssim k \lesssim 4.5\times 10^{21}~\rm Mpc^{-1}$. Line styles follow those in Fig. \ref{['fig:cons_frac']}. For comparison, previous constraints are also shown: (i) a stable Planck-mass relic from primordial black hole evaporation, consistent with the upper limits for the cold dark matter density ("DM relic density", dotted line); (ii) the lightest supersymmetric particles generated by PBH evaporation("LSP", dashed line), consistent with the upper limits for the cold dark matter density Josan:2009.