⚛️ general relativity

Primordial black holes within Higgs hybrid metric-Palatini approach

This paper investigates the formation of primordial black holes as dark matter candidates within the Higgs hybrid metric-Palatini framework, demonstrating that enhanced primordial curvature perturbations can lead to a PBH abundance capable of accounting for all or part of the universe's dark matter depending on the coupling constant and e-folds number.

Original authors: Brahim Asfour, Farida Bargach, Yahya Ladghami, Ahmed Errahmani, Taoufik Ouali

Published 2026-01-22

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CC0 1.0

Original authors: Brahim Asfour, Farida Bargach, Yahya Ladghami, Ahmed Errahmani, Taoufik Ouali

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 Big Picture: Finding the "Missing" Weight of the Universe

Imagine the universe is a giant, invisible backpack. We know this backpack is heavy because stars and galaxies move as if they are being pulled by something heavy inside. But when we look inside, we only see regular stuff (stars, gas, planets) which accounts for only a tiny fraction of the weight. The rest is "Dark Matter," a mysterious substance we can't see.

This paper asks a specific question: Could the "missing weight" be made of tiny, invisible black holes created right at the very beginning of time?

The authors say "Yes, it's possible," but only if the universe behaved in a very specific way during its first split-second of existence. They use a special mathematical recipe (the "Higgs hybrid metric-Palatini model") to see if this scenario works.

1. The Recipe: Mixing Two Kinds of Gravity

To understand how these black holes form, the authors had to tweak the standard rules of how gravity works.

The Standard View: Usually, scientists think of gravity as a smooth fabric (like a trampoline) that bends when you put a heavy ball on it.
The Paper's View: The authors mixed two different ways of describing that fabric. Think of it like baking a cake where you combine two different types of flour. One type is the standard "Metric" flour, and the other is the "Palatini" flour.
The Secret Ingredient: They added a special spice called the Higgs field (the same field that gives particles mass). In their recipe, this spice is tightly linked to the "Palatini" flour. This combination creates a unique environment where the universe expands rapidly (a phase called "Inflation").

2. The Explosion of Bubbles: Creating the Seeds

During the universe's rapid expansion (Inflation), tiny quantum fluctuations (like tiny bubbles in boiling water) happened everywhere.

The Problem: In most models, these bubbles are too small and weak to turn into black holes.
The Paper's Solution: Because of their special "flour and spice" recipe, the authors found that at very small scales, these fluctuations get super-charged.
The Analogy: Imagine a calm pond. Usually, the ripples are tiny. But in this model, the authors found a way to make the ripples in certain spots grow into massive waves. When these "super-waves" (density perturbations) re-entered the normal universe after inflation, they were so heavy and dense that they collapsed instantly under their own gravity, forming Primordial Black Holes (PBHs).

3. The Goldilocks Zone: Not Too Big, Not Too Small

The authors had to make sure their recipe didn't break the universe.

Too much chaos: If the waves were too big, the universe would have collapsed into black holes everywhere, and we wouldn't be here.
Too little chaos: If the waves were too small, no black holes would form, and they couldn't explain the Dark Matter.
The Result: They found a "Goldilocks" setting. By adjusting two "dials" in their recipe—the coupling constant (how strong the spice is mixed) and the number of e-folds (how long the inflation lasted)—they could create just the right amount of black holes.

4. The Final Test: Are They the Dark Matter?

The authors ran the numbers to see if these black holes could be the "missing weight" in our cosmic backpack.

Scenario A (The "All-In" Bet): If they set the dials to a specific setting (a lower coupling value), the model predicts that these tiny black holes could make up 100% of the Dark Matter. It's as if the entire missing weight of the universe is just a sea of these invisible, ancient black holes.
Scenario B (The "Partial" Bet): If they tweaked the dials slightly differently (a higher coupling value), these black holes would only make up a small fraction (about 1.8% to 4.5%) of the Dark Matter. In this case, they are just a side dish, not the main course.

Summary of Findings

The paper concludes that:

It works: Their specific gravity recipe allows for the creation of primordial black holes without breaking the laws of physics we already know.
It fits the data: The predictions match what we see in the Cosmic Microwave Background (the "afterglow" of the Big Bang) from observatories like Planck and ACT.
It's a candidate: Depending on how you tune the math, these black holes could either be the entire explanation for Dark Matter or just a part of it.

In short: The authors built a theoretical machine that turns the early universe's tiny ripples into a swarm of ancient black holes, showing that this swarm could be the invisible glue holding our universe together.

Technical Summary: Primordial Black Holes within the Higgs Hybrid Metric-Palatini Approach

Problem Statement
The fundamental nature of dark matter (DM) remains an open question in modern cosmology. While Primordial Black Holes (PBHs) are viable candidates—being cold, non-baryonic, and stable—their formation requires sufficiently large primordial curvature perturbations generated during inflation. Standard single-field inflationary models typically produce a nearly scale-invariant power spectrum, which is insufficient to trigger the gravitational collapse necessary for PBH formation. Furthermore, while various modified gravity theories (e.g., Gauss-Bonnet, Brans-Dicke) have been explored to enhance these perturbations, there is a need to investigate PBH production within the specific context of the Higgs hybrid metric-Palatini model. This model combines metric and Palatini formalisms to overcome the shortcomings of pure theories, yet its implications for PBH abundance and their potential role as dark matter have not been fully characterized.

Methodology
The authors adopt the Higgs hybrid metric-Palatini inflation model, where the inflaton field (identified as the Higgs field) is non-minimally coupled to the Palatini curvature. The study proceeds through the following analytical steps:

Model Dynamics: The authors define the action involving the Einstein-Hilbert curvature ( $R$ ) and the Palatini curvature ( $\hat{R}$ ), treating the metric and connection as independent variables. They derive the Friedmann and Klein-Gordon equations for a flat Friedmann-Robertson-Walker (FRW) background, incorporating the non-minimal coupling constant $\xi$ .
Inflationary Parameters: Using slow-roll approximations, the authors calculate the scalar spectral index ( $n_s$ ) and the tensor-to-scalar ratio ( $r$ ). They test the Higgs potential $V(\phi) = \frac{1}{4}\lambda\phi^4$ against observational constraints from Planck, ACT, and DESI collaborations.
Curvature Power Spectrum: The comoving curvature perturbation ( $\mathcal{R}$ ) is derived using the Mukhanov-Sasaki variable. The authors analyze the primordial curvature power spectrum, $\mathcal{P}_\mathcal{R}(k)$ , specifically looking for enhancements at small scales (large wavenumbers $k$ ) which are necessary for PBH formation.
PBH Formation Probability: The mass of a PBH formed at time $t$ is related to the horizon mass. The authors employ the Press-Schechter formalism to calculate the mass variance, $\sigma(M_{PBH})$ , and the mass fraction, $\beta(M_{PBH})$ , representing the portion of energy density collapsing into PBHs. They assume a Gaussian distribution for density fluctuations and utilize a critical density contrast $\delta_c \approx 0.414$ for the radiation-dominated era.
Dark Matter Abundance: Finally, the authors compute the fractional abundance of PBHs ( $f_{PBH}$ ) relative to the total dark matter density, comparing their results against observational constraints from evaporation, microlensing, gravitational waves, and accretion.

Key Contributions and Results

Consistency with Observational Data: The study demonstrates that the Higgs hybrid metric-Palatini model is consistent with current cosmological observations. For specific values of the coupling constant ( $\xi = 10^{-3.9}$ and $\xi = 10^{-4.1}$ ) and e-folds numbers ( $35 \leq N \leq 75$ ), the predicted spectral index ( $n_s$ ) and tensor-to-scalar ratio ( $r$ ) fall within the 1 $\sigma$ and 2 $\sigma$ confidence regions of Planck, ACT, and DESI data.
Enhancement of Curvature Power Spectrum: The analysis reveals a significant enhancement in the primordial curvature power spectrum at small scales (large $k$ ). This enhancement is more pronounced for smaller values of the coupling constant $\xi$ and the e-folds number $N$ . This behavior satisfies the condition required for the gravitational collapse of overdense regions into PBHs.
Mass Variance and Formation Probability: The mass variance $\sigma(M_{PBH})$ remains below the critical threshold ( $\sigma_{thresh} = 0.08$ ) required to avoid overproduction of PBHs, while still allowing for formation. The mass fraction $\beta(M_{PBH})$ is found to be consistent with observational limits ( $< 10^{-20}$ ) within the allowed mass ranges.
PBHs as Dark Matter: The study investigates whether PBHs can constitute the entirety or a fraction of the current dark matter:
- For a coupling constant $\xi = 10^{-4}$ and e-folds $N=52$ or $N=55$ , the model predicts PBH masses in the range of $1 \times 10^{17}$ to $3.75 \times 10^{19}$ g. In this "golden window," PBHs can account for 100% of the current dark matter content.
- For a slightly higher coupling constant $\xi = 10^{-4.1}$ , the model predicts much more massive PBHs ( $1.36 \times 10^{25}$ to $1 \times 10^{29}$ g). In this scenario, PBHs contribute only a fraction of the dark matter (approximately 4.5% for $N=52$ and 1.8% for $N=55$ ).

Significance and Claims
The paper claims that the Higgs hybrid metric-Palatini approach provides a viable framework for generating primordial black holes without violating current cosmological constraints. The primary significance lies in demonstrating that the interplay between the non-minimal coupling constant ( $\xi$ ) and the number of e-folds ( $N$ ) can be tuned to produce PBHs that either fully explain the dark matter density or constitute a significant sub-component of it. The authors conclude that the mass of the resulting PBHs is highly sensitive to these model parameters, offering a mechanism to test modified gravity theories through the lens of PBH abundance and dark matter composition. The work reinforces the utility of PBHs as a probe to constrain modified gravity theories, specifically hybrid metric-Palatini models.