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Closing the Mass Window for Stupendously Large Black Holes

This paper demonstrates that primordial black holes with masses exceeding $10^{11}\,M_\odot$ are excluded as a significant dark matter component because their associated isocurvature perturbations violate current Planck Cosmic Microwave Background constraints.

Original authors: Christopher Gerlach, Yann Gouttenoire, Antonio J. Iovino, Nicholas Leister

Published 2026-10-30

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CC BY 4.0

Original authors: Christopher Gerlach, Yann Gouttenoire, Antonio J. Iovino, Nicholas Leister

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 universe is filled with invisible matter that holds galaxies together, a substance we call dark matter. While we cannot see it directly, we know it is there because of how its gravity pulls on stars and light. For decades, scientists have wondered what this mysterious stuff actually is. One compelling idea suggests that dark matter might be made of primordial black holes—tiny, ancient black holes that formed in the first fraction of a second after the Big Bang, long before the first stars ever ignited. These objects would be scattered throughout the cosmos, invisible and silent, yet massive enough to account for the missing weight of the universe.

If these ancient black holes exist in vast numbers, they would leave a specific fingerprint on the early universe. Because they formed from clumps of matter rather than a smooth, even distribution, they would create local pockets of extra density surrounded by empty space. As the universe expanded, these pockets would tug on the surrounding gas and radiation, creating ripples in the fabric of space-time. These ripples, known as isocurvature perturbations, would eventually leave a distinct mark on the cosmic microwave background, the faint afterglow of the Big Bang that we can still detect today. By studying this ancient light, astronomers can test whether the universe was born with these specific kinds of clumps, effectively using the early cosmos as a laboratory to weigh the possible ingredients of dark matter.

A recent study by researchers at the Mainz Institute for Theoretical Physics and New York University has used this method to close the door on a particularly massive class of these hypothetical objects. The team focused on "stupendously large" black holes, a theoretical category defined as having masses greater than one hundred billion times that of our Sun. While these giants are too heavy to live inside individual galaxies, some theories had suggested they could still make up a significant portion of the universe's total dark matter, perhaps explaining the presence of supermassive black holes in the early universe or the gravitational waves detected by observatories. The researchers set out to see if such massive objects could exist without contradicting the precise measurements we have of the cosmic microwave background.

To test this, the team modeled how a universe filled with these enormous black holes would behave. They calculated that if these stupendously large black holes existed in the numbers required to be a major part of dark matter, their gravitational influence would have created far too much disorder in the early universe. Specifically, the local overdensities caused by these black holes would have generated isocurvature perturbations that are far stronger than what we observe in the cosmic microwave background today. The data from the Planck satellite, which maps the temperature of the early universe with extreme precision, shows a smoothness that simply cannot accommodate the chaotic fluctuations these massive black holes would have produced.

The researchers found that even if these black holes made up only a small fraction of the total dark matter, their sheer size would still generate signals that exceed current observational limits. By comparing their theoretical models against the strict constraints provided by the Planck mission and other measurements of the large-scale structure of the universe, they demonstrated that the mass window for these stupendously large black holes is effectively closed. In other words, if these objects exist at all, they are far too rare to play any significant role in the composition of dark matter. This conclusion holds true even when accounting for the most conservative estimates of how these black holes might have clustered or evolved over time.

This finding has important implications for several other active areas of astrophysics. For instance, it suggests that the gravitational wave background recently detected by pulsar timing arrays cannot be caused by the mergers of these specific types of black holes. It also challenges theories that try to explain the existence of surprisingly massive galaxies observed by the James Webb Space Telescope as being the result of fluctuations seeded by such black holes. The study does not rule out the existence of smaller primordial black holes, which remain a viable candidate for dark matter in a very narrow mass range, but it firmly excludes the possibility that the universe is populated by a sea of these stupendously large giants.

The work relies on a straightforward physical principle: the universe we see today is the result of how matter behaved in its earliest moments. If the ingredients were different, the final picture would look different. By showing that the inclusion of stupendously large black holes would distort the cosmic microwave background beyond recognition, the researchers have provided a clear and robust reason to reject them as a primary component of dark matter. Their analysis confirms that the universe's early history was remarkably smooth, leaving little room for the kind of massive, chaotic seeds that these hypothetical black holes would have provided.

Technical Summary: Closing the Mass Window for Stupendously Large Black Holes

Problem Statement
Primordial Black Holes (PBHs) remain a candidate for dark matter (DM) and a potential explanation for gravitational wave events. While constraints tightly limit PBHs as a dominant DM component in most mass ranges, the "Stupendously Large Black Hole" (SLAB) regime, defined as $M_{\text{PBH}} \gtrsim 10^{11} M_\odot$ , has been proposed as a viable window where PBHs could constitute a significant fraction of DM ( $f_{\text{PBH}} \in [10^{-3}, 1]$ ) without violating existing astrophysical limits. This paper addresses whether such massive PBHs can exist as a significant DM component by analyzing the isocurvature perturbations they generate.

Methodology
The authors model the total cold dark matter as a mixture of PBHs and particle dark matter (PDM), characterized by the fraction $f_{\text{PBH}}$ . Unlike adiabatic perturbations, the discrete nature of PBHs induces primordial isocurvature perturbations ( $\delta_{\text{iso}}$ ), where local overdensities in PBHs are initially compensated by radiation deficits.

The analysis proceeds through the following steps:

Perturbation Theory: The authors derive the evolution of the DM density contrast, $\delta_{\text{DM}}$ , separating adiabatic and isocurvature modes. They define the DM isocurvature perturbation $I_{\text{DM}}$ as the fluctuation in the DM-to-photon number density ratio.
Power Spectrum Calculation: Assuming a Poisson distribution for PBHs, the authors calculate the primordial isocurvature power spectrum. They treat PBHs as point-like objects initially, introducing a white-noise spectrum characterized by the interspacing scale $k_{\text{PBH}}$ .
Non-Linear Cutoffs: To ensure validity within the linear regime (required for CMB constraints), the authors introduce two cutoff prescriptions for the power spectrum:
- Case A: The cutoff occurs when PBH fluctuations themselves become non-linear ( $k_{\text{PBH}}^{\text{NL}}$ ).
- Case B: The cutoff occurs when the total DM fluid fluctuations become non-linear ( $k_{\text{DM}}^{\text{NL}}$ ).
- A second cutoff is applied based on the inverse Schwarzschild radius to account for PBH exclusion effects.
Evolution and Suppression: The authors account for the fact that PBHs form at a finite time ( $y_f$ ) during the radiation era, rather than being initialized deep in the radiation era. They calculate a suppression factor, $S(y_f)$ , by numerically solving the coupled evolution of the Newtonian potential and the isocurvature mode from PBH formation to recombination. This factor reduces the amplitude of the isocurvature power spectrum at the time of recombination compared to standard primordial assumptions.
Constraint Application: The derived isocurvature power spectra are compared against 95% Confidence Level (CL) constraints from the Planck satellite (CMB temperature, polarization, and lensing) and Baryonic Acoustic Oscillation (BAO) data.

Key Contributions and Results

Exclusion of SLABs: The primary result is that PBHs in the SLAB mass range ( $M_{\text{PBH}} \gtrsim 10^{11} M_\odot$ ) generate isocurvature perturbations that exceed current CMB and BAO limits, even when assuming a conservative non-linear cutoff (Case A). Consequently, SLABs are ruled out as a significant component of the dark matter.
Mass Range Constraints: The authors establish stringent upper limits on the PBH fraction $f_{\text{PBH}}$ for masses in the range $M_{\text{PBH}} \in [10^8, 10^{19}] M_\odot$ . The exclusion holds for $f_{\text{PBH}}$ values as low as $10^{-3}$ specifically for masses around $M_{\text{PBH}} \sim 10^{10} M_\odot$ , while higher fractions are excluded at higher masses.
Refined Cutoffs: The paper provides explicit derivations for the non-linear cutoff scales ( $k_{\text{PBH}}^{\text{NL}}$ and $k_{\text{DM}}^{\text{NL}}$ ) specific to the PBH-PDM mixture, distinguishing between the onset of non-linearity in the PBH sub-component versus the total DM fluid.
Finite-Time Generation Effect: The study quantifies the suppression of isocurvature constraints due to the finite formation time of PBHs. While this effect weakens the bounds compared to assuming primordial initial conditions, the resulting constraints remain sufficient to exclude SLABs as a dominant DM component.

Significance and Claims
The paper claims to "close the mass window" for Stupendously Large Black Holes as a significant dark matter candidate. By demonstrating that the isocurvature perturbations generated by such massive objects are incompatible with CMB observations, the work excludes a previously open parameter space.

The authors state that their results:

Strengthen the conclusion that the nano-Hertz gravitational-wave background detected by Pulsar Timing Arrays cannot be sourced by PBH mergers in this mass range.
Challenge scenarios attributing early massive galaxy candidates (observed by the James Webb Space Telescope) to PBH-induced fluctuations.
Provide a robust exclusion that holds even under conservative assumptions regarding non-linear structure formation and the timing of PBH generation.

The paper notes that while it assumes a monochromatic mass function, the results are expected to hold for extended mass distributions. It also suggests that future CMB experiments (LiteBIRD, Simons Observatory) will further tighten these constraints.

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