Primordial black holes: constraints, potential evidence and prospects

This review summarizes current observational constraints on primordial black holes (PBHs) across the full mass range, discusses their formation scenarios and mass functions, highlights potential evidence for their existence as dark matter, and outlines future search prospects, particularly through gravitational-wave observatories.

The Gist

Imagine the universe as a giant, cosmic ocean. We know there are massive islands in this ocean called stars and galaxies, and we know there are invisible currents called dark matter that hold everything together. But what if there were also tiny, invisible pebbles floating in the water that we can't see, but whose gravity affects everything around them?

This paper is a massive "detective report" about a specific type of invisible pebble called a Primordial Black Hole (PBH). Unlike the black holes formed when massive stars die (like a giant tree falling over), PBHs are like "cosmic dust" that formed instantly in the very first split-second of the Big Bang.

Here is the breakdown of the paper's main points, translated into everyday language:

1. The Big Mystery: Are These Pebbles the Dark Matter?

For decades, scientists have wondered if these ancient black holes could be the missing ingredient that makes up Dark Matter (the invisible glue holding galaxies together).

• The Idea: If the universe was born with a "shower" of these tiny black holes, they could be everywhere, making up all the dark matter we can't see.
• The Problem: It's hard to prove they exist because they are invisible and don't emit light.

2. The "Goldilocks" Zones (Mass Ranges)

The paper maps out the entire "family tree" of these black holes, from the size of a grain of sand to the size of a galaxy. They divide them into three main groups:

• The "Too Hot" Zone (Tiny Pebbles): These are so small that they would have evaporated (disappeared) long ago, exploding like tiny fireworks. We can't find them now because they are gone.
• The "Too Big" Zone (Giant Monsters): These are so massive that if they were everywhere, they would have messed up the orbits of stars or made the universe look very different. We've looked for them and haven't found enough to be the dark matter.
• The "Just Right" Zone (The Asteroid Window): This is the most exciting part. There is a specific size range (about the size of an asteroid) where these black holes should be invisible to our current telescopes. They are too heavy to evaporate, but too small to be seen by standard star-mapping. This is the "Holy Grail" window where PBHs could be hiding as all the dark matter.

3. The Detective Work: How We Hunt Them

Since we can't see them, the paper explains how we look for their "footprints" using different methods:

• The "Flashlight" Test (Lensing): Imagine holding a flashlight behind a glass marble. The light bends. If a PBH passes in front of a distant star, it acts like a magnifying glass, briefly making the star look brighter. We've checked this, but for the "Asteroid Window," the effect is too subtle for our current eyes.
• The "Eating" Test (Accretion): If a black hole eats gas, it gets hot and glows. We've checked the sky for this glow, but the "Asteroid" ones are too small to eat enough to glow brightly.
• The "Rumble" Test (Gravitational Waves): This is the new, high-tech method. When two black holes crash, they send ripples through space-time (like dropping two stones in a pond).
  • The Good News: We are now hearing these ripples (thanks to LIGO and other detectors).
  • The Clue: Some of the black holes we've detected are in "mass gaps"—sizes that normal stars shouldn't be able to make. This suggests they might be the ancient PBHs we've been looking for!

4. The "Ghost" Signals (Scalar-Induced Waves)

The paper introduces a clever new trick. If the universe was "bumpy" enough to create these black holes, those bumps should have also created a background hum of gravitational waves, even if the black holes never crashed.

• The Analogy: Think of a drum. If you hit it hard enough to make a drumstick (a black hole), the whole drum skin vibrates (gravitational waves).
• The Future: New detectors (like LISA, a space-based telescope) are coming online soon. They might hear this "hum," which would be the smoking gun proving these black holes exist.

5. The "Non-Gaussian" Twist

The paper spends a lot of time on a complex math concept called "Non-Gaussianity."

• Simple Explanation: Imagine throwing darts at a board. If you throw them randomly, they form a perfect bell curve (Gaussian). But if the universe was "clumpy" or "biased" (Non-Gaussian), the darts would cluster in weird ways.
• Why it matters: If the universe was "clumpy," it changes the rules. It might mean we need fewer black holes to explain what we see, or it might mean we need more. It's like realizing the wind was blowing harder than we thought, changing how far the darts flew.

6. The Verdict: What's Next?

The paper concludes with a hopeful outlook:

• The "Asteroid Window" is still open: We haven't ruled out the idea that tiny, asteroid-sized black holes are the dark matter.
• New Tools are coming: We are about to get much better "ears" (Gravitational Wave detectors) and "eyes" (X-ray telescopes).
• The Goal: In the next few years, we might finally catch a glimpse of these ancient ghosts. If we do, it would solve one of the biggest mysteries in physics: What is the invisible stuff holding our universe together?

In a nutshell: This paper is a roadmap for finding invisible, ancient black holes. It tells us where to look, how to look, and why the "Asteroid-sized" ones are the most likely candidates to be the dark matter that makes up our universe. We are just about to turn on the lights and see if they are really there.

Technical Summary

1. Problem Statement

Primordial Black Holes (PBHs) are hypothetical black holes formed in the early Universe (prior to recombination) from the gravitational collapse of density perturbations. They are compelling candidates for Dark Matter (DM), potential progenitors of Supermassive Black Holes (SMBHs), and sources of observed Gravitational Waves (GWs).

However, the field faces two major challenges:

1. Fragmented Constraints: Observational constraints on PBH abundance ($f_{PBH}$) are typically derived assuming a monochromatic mass function (all PBHs have the same mass). This is unrealistic, as most formation mechanisms predict extended mass functions (a distribution of masses). Translating monochromatic limits to extended distributions is non-trivial and often leads to incorrect conclusions.
2. The "Asteroid-Mass" Gap: A significant mass window ($10^{-17} M_\odot$ to $10^{-10} M_\odot$) has historically lacked robust constraints, yet it remains a prime target for PBHs constituting 100% of DM.
3. Emerging Evidence vs. Constraints: Recent observations (e.g., LIGO-Virgo-KAGRA events, NANOGrav pulsar timing array signals, microlensing anomalies) suggest potential PBH signatures, but these must be reconciled with stringent existing limits.

2. Methodology

The authors employ a comprehensive review and analytical framework to address these issues:

• Mass Function Formalism: They define the PBH mass function $\psi(M_{PBH})$ and establish a rigorous method to translate monochromatic constraints into constraints on extended mass functions. They utilize a linear functional approach where an observable $A$ is expressed as an integral over the mass function:
  $$A[\psi] = A_0 + \int d \ln M_{PBH} A_1(M_{PBH}) \psi(M_{PBH})$$
  This allows them to combine constraints from different observables (evaporation, lensing, dynamics, etc.) using a quadrature sum (Eq. 29) to determine the maximum allowed DM fraction for a given mass distribution.
• Formation Scenarios: They analyze various formation mechanisms, including:
  • Critical Collapse: Relating the density contrast $\delta$ to PBH mass via a scaling law $m \propto (\delta - \delta_{th})^\gamma$.
  • Non-Gaussianity (NG): Explicitly modeling the impact of primordial non-Gaussianity (both induced by non-linear relations and intrinsic to the inflationary model, e.g., Curvaton or Ultra-Slow-Roll models) on the PBH abundance and mass function.
  • QCD Phase Transition: Incorporating the softening of the Equation of State (EoS) at $T \sim 200$ MeV, which creates a distinct peak in the mass function around $1 M_\odot$.
  • Non-Inflationary Mechanisms: Phase transitions, topological defects (cosmic strings, domain walls), and Q-ball collapse.
• Data Compilation: The authors digitized existing constraints and provided a Mathematica notebook (hosted on GitHub) to evaluate limits for arbitrary extended mass functions, moving beyond the limitations of previous "monochromatic" reviews.

3. Key Contributions

A. Theoretical Framework for Extended Mass Functions

The paper provides the definitive prescription for applying observational constraints to broad mass distributions. It demonstrates that:

• Spreading a mass distribution over many decades can evade specific constraints at a single mass point but may violate limits at other masses.
• For broad distributions, constraints generally become stronger than for monochromatic ones because the "tail" of the distribution often hits sensitive limits (e.g., evaporation for low masses or lensing for high masses).
• The "Asteroid-mass window" ($10^{-17} - 10^{-10} M_\odot$) closes for log-normal distributions with width $\sigma \gtrsim 2.5$.

B. Impact of Non-Gaussianity (NG)

A major contribution is the quantification of how NG alters PBH predictions:

• Abundance: NG can change the predicted PBH abundance by many orders of magnitude compared to Gaussian assumptions. For instance, in Curvaton models, the quadratic approximation often fails to capture the true abundance, especially for broad power spectra.
• Scalar-Induced Gravitational Waves (SIGWs): While NG significantly alters the PBH abundance (tail of the distribution), its effect on the SIGW spectrum is subtler, primarily introducing spectral features rather than massive amplitude shifts. However, the interplay is crucial: GW observations constrain the scalar power spectrum amplitude, which, when combined with NG models, tightly constrains the resulting PBH abundance.

C. The Asteroid-Mass Window and New Probes

The authors identify that the asteroid-mass window is currently unconstrained by standard microlensing (due to wave optics and finite source size effects) but is accessible via:

• X-ray Microlensing: Using compact X-ray pulsars (e.g., NICER, STROBE-X) to bypass source-size limitations.
• Solar System Ephemerides: Detecting transient perturbations in planetary orbits caused by PBH flybys.
• Stellar Destruction: Re-evaluating constraints from white dwarf and neutron star survival, noting that previous constraints based on dynamical friction ignition in white dwarfs may be invalid under realistic hydrodynamic conditions.

D. Gravitational Wave Prospects

The paper highlights GWs as the most promising future probe:

• Binary Mergers: LVK data constrains stellar-mass PBHs, but future detectors (Einstein Telescope, Cosmic Explorer) will probe high-redshift mergers and subsolar masses, distinguishing PBHs from astrophysical black holes.
• SIGWs: The detection of a stochastic GW background by NANOGrav (nano-Hz) and LISA (milli-Hz) could directly probe the scalar power spectrum required to generate PBHs. The paper shows that current PTA data already excludes $f_{PBH}=1$ for solar-mass PBHs in critical collapse scenarios, especially when NG is considered.
• GW Lensing: Future GW detectors can detect lensing effects (diffraction) from PBHs, probing masses down to $10^{-2} M_\odot$.

4. Results

• Constraints: The authors present updated, digitized constraints covering the mass range $10^{-24} M_\odot$ to $10^{20} M_\odot$. They show that for extended mass functions, the allowed fraction of DM ($f_{PBH}$) drops significantly compared to monochromatic assumptions, particularly for broad distributions.
• QCD Peak: The QCD phase transition creates a robust feature in the mass function around $1 M_\odot$. While this allows for a "bump" in PBHs, it is heavily constrained by LIGO merger rates and microlensing.
• NG Effects: The inclusion of non-Gaussianity (specifically $f_{NL} \approx -2$) can suppress PBH formation in the solar-mass range by orders of magnitude, making it harder to explain LIGO events as PBHs unless the power spectrum amplitude is finely tuned. Conversely, strong NG can enhance formation in specific regimes.
• Evidence:
  • Microlensing: Excess events in OGLE and MACHO data suggest a population of compact objects, potentially PBHs in the planetary to stellar mass range.
  • GW Events: Events in the "mass gaps" (2–5 $M_\odot$ and 60–120 $M_\odot$) and subsolar-mass candidates are consistent with a PBH origin, though astrophysical explanations remain possible.
  • NANOGrav: The stochastic GW background could be SIGWs from PBH formation, but this requires specific power spectrum shapes and NG levels.

5. Significance

This paper serves as a critical bridge between theoretical PBH models and observational cosmology. Its significance lies in:

1. Standardization: It provides the community with a standardized, computational tool (the Mathematica notebook) to test any extended mass function against the full suite of observational constraints, moving the field beyond the oversimplified monochromatic assumption.
2. Clarifying the "Asteroid" Window: It rigorously defines the current status of the asteroid-mass window, highlighting that while it is not yet ruled out, it requires specific, next-generation observational strategies (X-ray microlensing, planetary ephemerides) to probe.
3. Interplay of NG and GWs: It elucidates the complex relationship between primordial non-Gaussianity, PBH abundance, and Scalar-Induced Gravitational Waves, showing that GW observations are not just a probe of the background but a direct test of the inflationary physics that creates PBHs.
4. Future Roadmap: It outlines a clear path for the next decade of research, emphasizing that the combination of GW detectors (LISA, ET, PTA) and high-precision electromagnetic surveys will either confirm the existence of PBHs as a major DM component or definitively close the remaining mass windows.

In summary, the paper argues that while PBHs remain a viable and intriguing DM candidate, the "easy" windows are closing. Future progress depends on accurately modeling extended mass functions, accounting for non-Gaussianity, and leveraging the new era of multi-messenger astronomy (GWs + EM).