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All Dark Matter as Primordial Black Holes: a Gravitational-Wave Test

This paper proposes a novel test for the hypothesis that dark matter consists entirely of asteroidal-mass primordial black holes by predicting a detectable stochastic gravitational-wave background from their coalescence onto rapidly spinning supermassive black holes, which future space-based observatories like LISA and DECIGO could observe.

Original authors: A. Ianniccari, A. J. Iovino, G. Perna, D. Perrone, A. Riotto

Published 2026-10-28

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CC BY 4.0

Original authors: A. Ianniccari, A. J. Iovino, G. Perna, D. Perrone, A. Riotto

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

For decades, astronomers have known that the visible stars and gas in our universe make up only a tiny fraction of the cosmos. The rest is an invisible substance called dark matter, which holds galaxies together through its gravity but refuses to reveal itself through light or any other direct signal. While many theories propose that this missing mass consists of exotic, undiscovered particles, a simpler and older idea suggests it might be made of black holes. These would not be the remnants of dead stars, but rather primordial black holes, born in the chaotic first moments after the Big Bang. If these ancient black holes exist in vast numbers, they could account for all the dark matter, specifically those with masses comparable to large asteroids, ranging from one ten-quadrillionth to one hundred-trillionth the mass of our Sun.

The question of whether these asteroid-sized black holes are the true face of dark matter has been difficult to answer because they are too small to be seen directly and too light to cause the gravitational lensing effects that reveal larger black holes. However, a new study proposes a way to test this hypothesis by listening to the universe rather than looking at it. The researchers suggest that if these primordial black holes exist, they would not just drift aimlessly; they would be drawn into the centers of galaxies, where they would accumulate around the supermassive black holes that anchor them. As these small black holes spiral inward, they would generate a faint, continuous hum of gravitational waves—ripples in the fabric of space-time—that could be detected by future observatories.

The team, led by A. Ianniccari and colleagues, set out to calculate the strength of this cosmic hum. They focused on a specific scenario where the dark matter forms a dense, spherical cloud, or "spike," around a supermassive black hole. As the supermassive black hole grows over billions of years, it drags this cloud of primordial black holes inward, creating a steep density gradient. Within this spike, the small black holes orbit the giant one, slowly losing energy by emitting gravitational waves. Eventually, they spiral all the way in and merge with the central black hole. While a single merger is too weak to be heard, the combined effect of billions of these events happening in galaxies across the entire universe creates a stochastic background—a constant, overlapping noise of gravitational waves.

The researchers modeled this process using the laws of general relativity, accounting for the spin of the central black holes and the various angles at which the small black holes might orbit. They found that the strength of the resulting signal depends heavily on how steep the density spike is. If the spike is very steep, meaning the dark matter is packed tightly near the center, the signal becomes much louder. Their calculations show that for the most optimistic scenarios, where the dark matter forms a very dense spike, the signal would be strong enough to be detected by the Laser Interferometer Space Antenna, or LISA, a space-based gravitational wave observatory planned for the 2030s. LISA is designed to listen for ripples in the millihertz frequency range, which is exactly where these inspiraling black holes would sing.

The study also looked at a more sensitive future mission called DECIGO, which would be capable of detecting the signal even if the dark matter spikes are less dense. The results indicate that the detectability of this signal is not a simple yes or no; it hinges on the specific properties of the dark matter distribution. If the primordial black holes are indeed the dark matter, the signal should be there, but its loudness depends on how the dark matter is arranged around the supermassive black holes. The researchers found that for the signal to be clearly heard by LISA, the density of the dark matter spike must be quite high, a condition that is theoretically possible but not guaranteed.

Crucially, the paper does not claim to have found this signal yet, nor does it prove that dark matter is made of black holes. Instead, it provides a concrete test. If future observatories like LISA or DECIGO detect this specific background hum, it would be strong evidence that the universe is filled with asteroid-mass primordial black holes. If they listen and hear nothing, it would place strict limits on how many of these black holes can exist, effectively ruling out the idea that they make up all the dark matter. The work transforms a theoretical possibility into a measurable prediction, offering a new way to probe the invisible architecture of the cosmos. By listening for the collective whisper of countless tiny black holes falling into the giants at the hearts of galaxies, scientists may finally be able to determine the true nature of the dark matter that shapes our universe.

Technical Summary: All Dark Matter as Primordial Black Holes: a Gravitational-Wave Test

Problem Statement
The hypothesis that the Universe's dark matter (DM) consists entirely of asteroidal-mass primordial black holes (PBHs) is currently constrained to a narrow mass window ( $10^{-16} M_\odot \lesssim m \lesssim 10^{-11} M_\odot$ ) by Hawking evaporation limits and microlensing observations. While various formation channels exist, confirming this hypothesis requires complementary probes. This paper addresses the potential for detecting such a population through the stochastic gravitational-wave (GW) background generated by PBHs accumulating in density spikes around supermassive black holes (SMBHs) and subsequently inspiraling into them. Although individual extreme mass-ratio inspirals (EMRIs) with mass ratios $q \sim 10^{-22}–10^{-17}$ are too weak to be resolved, their incoherent superposition across the cosmic population of galaxies may produce a detectable background in the millihertz frequency band.

Methodology
The authors develop a framework to compute the GW background from PBHs in DM spikes surrounding spinning (Kerr) SMBHs. The methodology proceeds through several stages:

Spike Modeling and Evolution:
- The DM spike is modeled as a spherically symmetric power-law distribution, $\rho_{PBH}(r) \propto r^{-\gamma}$ , with slopes $\gamma$ ranging from 1 to 3. The inner boundary is defined by the innermost stable spherical orbit (ISSO), which depends on the SMBH spin $\chi$ and orbital inclination.
- The temporal evolution of the spike is calculated under radiation reaction. PBHs lose energy via GWs and spiral inward. The authors derive the radius within which PBHs have merged by a given cosmic time, accounting for the "frozen" nature of the spike for most SMBH masses in the LISA band, while noting that evolution is significant for lower-mass SMBHs ( $M \sim 10^4 M_\odot$ ).
- The total mass of the spike is constrained by the influence radius of the SMBH, relating the spike fraction $\Delta$ to the slope $\gamma$ .
GW Emission Calculation:
- Numerical Approach: The authors compute the GW flux for generic inclined spherical orbits in Kerr geometry using the Black Hole Perturbation Toolkit. They solve the Teukolsky equation for perturbations, summing over multipoles up to $\ell=8$ (specifically modes $\tilde{m} = \pm \ell$ ).
- Analytical Approximation: An analytical limit is derived assuming equatorial prograde orbits ( $x=1$ ) and using the Newtonian quadrupole formula. This serves as a benchmark to understand parameter dependencies, though it is noted to overestimate the peak amplitude by an order of magnitude due to strong-field effects.
- The spectral luminosity is integrated over the PBH distribution, orbital inclinations, and the SMBH mass function (adopting an optimistic, occupation-corrected local mass function).
Observational Prospects:
- The resulting stochastic GW background, $\Omega_{GW}(f)$ , is compared against the projected sensitivities of the space-based interferometers LISA and DECIGO.
- Signal-to-noise ratio (SNR) calculations are performed for LISA, incorporating astrophysical foregrounds (e.g., double white dwarfs, black hole binaries) as additional noise sources.

Key Contributions and Results

Spectral Characteristics: The GW background is concentrated in the millihertz band, determined primarily by the SMBH mass. The spectrum exhibits a peak frequency corresponding to the ISSO and a high-frequency cutoff.
Dependence on Spike Slope ( $\gamma$ ): The signal amplitude is highly sensitive to the spike slope. Steeper profiles (higher $\gamma$ ) place more PBHs in the high-frequency, high-power inner regions. Increasing $\gamma$ from 2.5 to 2.75 raises the peak amplitude by an order of magnitude.
Dependence on Mass and Spin:
- The signal scales linearly with PBH mass $m$ at low frequencies, but the peak amplitude grows more slowly as heavier PBHs deplete the innermost orbits faster.
- Higher SMBH spins ( $\chi$ ) bring the ISSO closer to the horizon, increasing orbital frequencies and emitted power, thereby boosting the signal peak and extending the spectrum to higher frequencies.
Detectability with LISA:
- For a fiducial spike slope of $\gamma = 2.5$ (the steepest profile from adiabatic growth), the signal remains slightly below LISA's sensitivity curve for the upper mass limit ( $m = 10^{-11} M_\odot$ ).
- Detectability (SNR $\geq 10$ ) requires steeper slopes ( $\gamma \gtrsim 2.55$ for $m = 10^{-11} M_\odot$ ) or higher PBH masses.
- The presence of astrophysical foregrounds reduces the SNR by approximately 25%, but the contours with and without foregrounds remain similar.
Detectability with DECIGO: Due to its superior sensitivity at higher frequencies, DECIGO is predicted to detect the signal for all the parameter configurations studied, provided that a proper understanding of the spin distribution of low-mass SMBHs is achieved, as this distribution affects the shape of the final spectrum.

Significance and Claims
The paper proposes a novel test for the asteroidal-mass PBH dark matter hypothesis. The authors claim that the detection of this stochastic background would provide evidence for PBHs as the totality of dark matter, independent of their specific formation mechanism, although the signal amplitude itself depends on the spike slope which is shaped by the formation history.

The significance of the work lies in identifying the spike slope as the critical parameter governing detectability. The authors modestly note that while their fiducial predictions ( $\gamma = 2.5$ ) may fall just below LISA's threshold, theoretical uncertainties (such as the inclusion of all emission modes, the full SMBH mass range, and Kerr geometry corrections to the spike density) could bring the signal within reach. Conversely, the detection of such a background would strongly constrain the properties of DM spikes around SMBHs. The study concludes that while LISA offers a potential test contingent on steep spikes, DECIGO provides a robust probe for this scenario across the considered parameter space, contingent on knowledge of the low-mass SMBH spin distribution.

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