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

This paper derives new constraints on the primordial curvature power spectrum within the previously underexplored small-scale regime of $3\times 10^{18} \lesssim k \lesssim 4.5\times 10^{21}~\rm Mpc^{-1}$ by utilizing recent theoretical advancements in black hole physics and existing limits on the initial mass fraction of light primordial black holes.

Original authors: Yupeng Yang

Published 2026-02-10

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CC BY 4.0

Original authors: Yupeng Yang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Cosmic "Static" and the Tiny Black Holes: A Simple Guide

Imagine you are listening to an old radio. Most of the time, you hear clear music (the large-scale structure of our universe). But if you turn the dial to a very specific, high-frequency setting, you might hear a faint, fuzzy static.

In cosmology, that "static" is called the Primordial Curvature Power Spectrum ( $P_R$ ). It represents the tiny ripples in the fabric of space that existed at the very beginning of time. Scientists have studied the "big songs" (large scales) very well, but they are still trying to figure out what the "tiny static" (small scales) looks like.

This paper is like a new, high-tech way to listen to that tiny static by looking for the "footprints" left behind by microscopic black holes.

1. The Tiny Detectives: Primordial Black Holes (PBHs)

Think of the early universe as a giant, bubbling pot of soup. Occasionally, a clump of that soup gets so thick and heavy that it collapses under its own weight, forming a tiny, microscopic black hole. These are Primordial Black Holes (PBHs).

If these tiny black holes existed, they would act like little "energy grenades." As they sit in space, they slowly evaporate (leak energy) until they eventually pop. By studying how much energy was released by these "grenades," scientists can work backward to figure out how much "static" (ripples) was in the soup to begin with.

2. The Plot Twist: The "Memory Burden" Effect

For a long time, scientists thought these tiny black holes were "ghosts"—they were so small that they would have evaporated and vanished long before the universe even got started. They thought they were too insignificant to leave any evidence.

But this paper introduces a game-changer: The "Memory Burden."

Imagine you are running a marathon. Usually, you lose energy steadily as you run. But imagine that as you get tired, your body suddenly develops a "memory" of how much energy you’ve already spent, and it forces you to slow down your energy loss to conserve what's left.

The paper suggests that as these tiny black holes evaporate, a quantum effect acts like a "memory burden," slowing down their evaporation. Instead of vanishing instantly, they linger. They stay around much longer than we expected, acting like tiny, long-lasting lanterns in the dark.

3. How We Catch Them: Neutrinos and Gamma Rays

Because these black holes are "lingering" instead of vanishing, they are constantly leaking high-energy light (Gamma Rays) and ghostly particles called Neutrinos.

The author uses data from massive, Earth-based detectors (like IceCube in Antarctica, which is basically a giant ear listening to the universe) to see if we can detect these leaks.

If we see a lot of these particles, it means there were many tiny black holes.
If we don't see them, it means there couldn't have been many black holes.

4. The Conclusion: Setting the Speed Limit

By using these "detectors" (the observations of light and neutrinos), the author has set new "speed limits" on the primordial static.

He has essentially said: "Based on the fact that we don't see a massive flood of high-energy particles hitting our detectors, the 'static' at the beginning of the universe couldn't have been any louder than X."

In short: By studying the "lingering" tiny black holes that refuse to disappear, we have gained a brand-new way to peek into the very first moments of the universe, providing much stricter rules for how the cosmos began.

This paper, authored by Yupeng Yang, presents new constraints on the primordial curvature power spectrum ( $P_{\mathcal{R}}$ ) at extremely small scales, specifically in the wavenumber range $3 \times 10^{18} < k < 4.5 \times 10^{21} \text{ Mpc}^{-1}$ .

1. The Problem

The primordial curvature power spectrum is well-constrained on large scales via the Cosmic Microwave Background (CMB) and Large Scale Structure (LSS). However, on very small scales ( $k > 10^{17} \text{ Mpc}^{-1}$ ), observational data is scarce. Historically, constraints in this regime have relied on the limits imposed by the abundance of stable Planck-mass relics or the production of Lightest Supersymmetric Particles (LSPs) from Primordial Black Hole (PBH) evaporation. These previous limits are relatively weak and leave a large "underexplored" window in the power spectrum amplitude.

2. Methodology

The author utilizes recent theoretical advancements in black hole physics to bridge the gap between PBH phenomenology and the primordial power spectrum. The methodology follows these steps:

Incorporating the "Memory Burden" Effect: Unlike standard Hawking radiation, which predicts that light PBHs ( $M_{PBH} < 10^{15} \text{ g}$ ) evaporate completely in the early universe, the author incorporates the "memory burden" effect. This back-reaction effect suggests that as a black hole loses mass, its evaporation rate slows down significantly, potentially allowing light PBHs to survive until the present day.
Utilizing Multi-Messenger Constraints: The paper leverages two recent types of observational constraints on the abundance of these surviving light PBHs:
1. Big Bang Nucleosynthesis (BBN): For PBHs with $10^8 < M_{PBH} < 10^9 \text{ g}$ , evaporation modifies the expansion rate and the baryon-to-photon ratio, affecting the primordial abundances of Deuterium and Helium.
2. High-Energy Particle Fluxes: For surviving PBHs, the author uses observations of high-energy gamma-rays and neutrinos (specifically from the IceCube-EHE and IceCube merger scenarios) to limit the present-day fraction of PBHs ( $f_{PBH}$ ).
Mathematical Mapping: The author uses the Press-Schechter formalism to relate the initial mass fraction ( $\beta$ ) of PBHs to the mass variance ( $\sigma^2$ ), which is in turn a function of the primordial density perturbation power spectrum ( $P_\delta$ ), and finally to the curvature power spectrum ( $P_{\mathcal{R}}$ ).

3. Key Contributions

New Scale Exploration: The paper identifies and probes a specific regime ( $3 \times 10^{18} < k < 4.5 \times 10^{21} \text{ Mpc}^{-1}$ ) that was previously under-constrained.
Integration of Modern BH Physics: By moving beyond the standard Hawking evaporation model to include the "memory burden" effect, the paper provides a more physically complete way to use modern astrophysical observations to probe the very early universe.
Comparative Analysis: The work provides a direct comparison between new constraints derived from particle physics/astrophysics (neutrinos/gamma-rays) and traditional cosmological constraints (relic density/LSP).

4. Results

The results are summarized in the derived upper limits for $P_{\mathcal{R}}$ :

Tightest Constraints: The most stringent limits on $P_{\mathcal{R}}$ come from the merger of memory-burdened PBHs detected via high-energy neutrinos (IceCube), yielding $P_{\mathcal{R}} \lesssim 10^{-1.8}$ in the range $3 \times 10^{18} < k < 5 \times 10^{20} \text{ Mpc}^{-1}$ .
Intermediate Constraints: In the range $5 \times 10^{20} < k < 1.8 \times 10^{21} \text{ Mpc}^{-1}$ , the limits are driven by non-merged memory-burdened PBHs (using both neutrino and gamma-ray data).
BBN Contribution: While BBN provides a robust constraint on the range $4.5 \times 10^{18} < k < 7 \times 10^{18} \text{ Mpc}^{-1}$ , it is generally weaker than the constraints derived from the memory-burdened PBH scenarios.
Improvement over Previous Work: The new constraints are significantly tighter than the previous limits set by DM relic density or LSP production.

5. Significance

This paper is significant because it demonstrates that high-energy neutrino and gamma-ray astronomy can serve as a powerful "microscope" to probe the primordial universe at scales far smaller than those accessible by the CMB. It establishes a new benchmark for the maximum allowable amplitude of primordial fluctuations at ultra-small scales, which is crucial for testing various inflationary models and theories of the early universe. The author concludes that future detectors like IceCube-Gen2 and GRAND200k will likely push these constraints even further.