🔭 astrophysics

Probability of gravitational-wave lensing by intermediate-mass black holes and globular clusters

This paper concludes that gravitational-wave lensing by intermediate-mass black holes within globular clusters is highly unlikely, with relative rates as low as 1/10,000, thereby disfavoring such an explanation for the GW231123 event and suggesting lensed waves are instead a more promising probe for dark matter substructures and primordial black holes.

Original authors: Luka Vujeva, Jose María Ezquiaga, Rico K. L. Lo, Lorenz Zwick

Published 2026-08-11

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Luka Vujeva, Jose María Ezquiaga, Rico K. L. Lo, Lorenz Zwick

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

Imagine the universe as a giant, cosmic funhouse mirror. When light or gravitational waves travel across the vastness of space, they sometimes pass near massive objects like galaxies. These heavyweights warp the fabric of space-time, bending the path of the waves like a lens bending light. This is called "gravitational lensing." Usually, this just makes the signal brighter or splits it into a few distinct copies, like seeing multiple reflections of yourself in a funhouse mirror. But what if there were smaller, hidden bumps on that mirror? What if, right in the middle of a galaxy's lensing power, there was a tiny, invisible "bump" made of a heavy, compact object?

Scientists are hunting for these bumps because they could reveal secrets about the universe's invisible stuff, known as dark matter. If a gravitational wave hits a small, dense object on its way to Earth, the signal might get distorted in a very specific way: it could split into two copies that arrive almost instantly one after the other, or the shape of the wave could get wobbly. It's like if a smooth, rolling ball hit a tiny pebble on a ramp; the ball might bounce slightly or split its path. By studying these distortions, astronomers hope to find things we can't see with telescopes, like "intermediate-mass black holes" (black holes that are heavier than stars but lighter than the giants at galaxy centers) or "globular clusters" (dense balls of stars).

This paper asks a very specific question: How likely is it that a gravitational wave, already magnified by a huge galaxy, will also get messed up by one of these smaller, hidden bumps? The authors, a team of researchers from the Niels Bohr Institute, ran detailed simulations to see if this double-lensing effect is a common occurrence or a cosmic rarity. They focused on two main suspects for these "bumps": intermediate-mass black holes (IMBHs) and globular clusters (GCs). They wanted to know if these objects are frequent enough to explain a mysterious signal detected recently, or if they are too rare to be the culprit.

The team built a model of the universe where galaxies are surrounded by halos of dark matter and filled with thousands of globular clusters. They imagined a gravitational wave traveling from a distant pair of colliding black holes, getting magnified by a galaxy, and then passing through a globular cluster that might contain a heavy black hole. They calculated the odds of this happening, looking for specific signs: would the two copies of the wave arrive within one second of each other, and would the second copy be bright enough to be noticed?

The results were surprisingly clear. The authors found that while it is theoretically possible for a globular cluster with a black hole to distort a gravitational wave, it is incredibly unlikely to happen. In their most optimistic scenarios, the chance of this occurring is only about 1 in 10,000. Even in the best-case situations where the signal is magnified a huge amount, the rate only climbs to about 1 in 100. The authors emphasize that these events are so rare that they are essentially invisible to our current detectors.

This finding has a major implication for a specific event called GW231123, a gravitational wave signal that some scientists thought might have been lensed by a black hole of about 1,000 times the mass of our Sun. The authors' calculations suggest that if this signal were caused by a black hole inside a globular cluster, it would be a statistical miracle. The odds are so low that the paper argues we should probably rule out this specific explanation. Instead, the authors suggest that if we do see these distortions in the future, they are more likely to be caused by something else entirely, such as clumps of dark matter or primordial black holes formed at the very beginning of the universe.

In short, the paper acts as a reality check. It tells us that while the idea of black holes in star clusters messing up gravitational waves is exciting, the universe is likely too empty of these specific setups for us to see them often. This is actually good news for scientists hunting for dark matter, because it means that if we do see a distorted wave, we can be more confident it's not just a random star cluster, but perhaps a clue to the mysterious dark matter that makes up most of the universe. The authors conclude that while we might not find these IMBHs this way, the search for lensed gravitational waves remains a powerful tool for exploring the hidden corners of the cosmos.

Technical Summary: Probability of Gravitational-Wave Lensing by Intermediate-Mass Black Holes and Globular Clusters

Problem Statement
Strongly lensed gravitational waves (GWs) are anticipated to become frequent detections with current and next-generation ground-based detectors, offering unique probes for cosmology, modified gravity, and binary black hole (BBH) formation. A specific candidate event, GW231123, has been interpreted as a strongly lensed signal distorted by an intermediate-mass black hole (IMBH) of $\sim 1000 M_\odot$ . However, the astrophysical likelihood of such a configuration—specifically an IMBH embedded within a globular cluster (GC) acting as a secondary lens on a galaxy-scale lensed GW—remains unconstrained. This paper investigates the probability that a GW, already strongly lensed by a galaxy-scale halo, undergoes additional wave-optics lensing by an IMBH residing in a GC. The study aims to determine if such events are frequent enough to explain candidates like GW231123 or if they are too rare, thereby favoring alternative interpretations (e.g., dark matter substructures or primordial black holes).

Methodology
The authors construct a compound lens model consisting of a primary galaxy-scale halo and a secondary substructure comprising a GC and a central IMBH.

Population Modeling: The study focuses on IMBHs associated with GCs, as other IMBH populations (nuclear dwarf galaxies or wandering IMBHs) are either observationally unconstrained or have negligible lensing cross-sections. The GC population is modeled using empirical scaling relations for Milky Way-like galaxies, assuming a total GC mass of $\sim 3 \times 10^{-5} M_h$ and a characteristic mass of $\sim 1.5 \times 10^5 M_\odot$ . GCs are modeled as truncated singular isothermal spheres (SIS) with a half-light radius of 3 pc.
Lens Configuration: The IMBH is treated as a point mass ( $M_\bullet$ ) at the center of the GC ( $M_{GC}$ ), with the mass ratio parameterized as $f_{mass} = M_\bullet / M_{GC}$ . The system is embedded in an external convergence ( $\kappa_{ext}$ ) and shear ( $\gamma_{ext}$ ) generated by the main halo.
Lensing Criteria: The authors define a "lensing event" as a scenario where the compound lens produces repeated chirps with a time delay $\Delta t < 1$ second and a relative magnification $\mu_{rel} > 0.1$ . These criteria ensure the signals overlap in time (characteristic of wave-optics effects) and are distinguishable as a lensed pair.
Calculation: The optical depth ( $\tau$ ) is computed by integrating the lensing cross-sections over the redshift range $z_S \in [0.5, 3.0]$ and the halo mass function. The conditional probability $P^{Lens}_{GC}$ is derived as the ratio of the optical depth for GC+IMBH lensing to that of the main halo alone. Numerical validation is performed using the lenstronomy code.

Key Contributions and Results

Suppressed Rates: The study finds that the relative rate of GWs lensed by IMBHs in GCs is extremely low. For a fiducial optimistic scenario, the relative rate is at most $1/1000$ and decays to $1/10,000$ . Even at high magnifications ( $\mu > 100$ ), where the relative rate approaches 1%, the absolute number of such events is intrinsically rare.
Dependence on Parameters: The fractional rate is strongly dependent on the minimum external magnification threshold, varying from $10^{-4}$ to $10^{-2}$ as the threshold increases from 2 to 100. The rate is less sensitive to the source redshift and the IMBH-to-GC mass ratio.
Time Delay Constraints: The rate increases with the time delay threshold but plateaus at $\sim 10^{-3}$ for delays of tens of minutes. The fiducial 1-second threshold results in a rate an order of magnitude lower.
Future Projections: Translating optical depths into detection forecasts (Fig. 3), the authors project that even with next-generation (XG) detectors, the number of detectable GWs lensed by IMBH+GC configurations will remain well below 1 detection over the coming decades.
Implications for GW231123: By calculating the Bayes factor and odds ratio, the authors conclude that the astrophysical prior for an IMBH+GC lens is insufficient to support the lensing hypothesis for GW231123. The inferred lensing parameters ( $\mu_{rel}, \Delta t$ ) for GW231123 lie in a region where the IMBH+GC model predicts rare occurrences.

Significance and Claims
The paper claims that GW lensing by IMBHs and GCs is highly unlikely, effectively disfavoring this specific astrophysical interpretation for the GW231123 candidate. This conclusion has two primary implications:

Exclusion of Known Substructure: The low rate implies that searches for lensing signatures in GWs are statistically unlikely to be confused with lensing by known astrophysical potentials like GCs and IMBHs.
Clean Probe for New Physics: Because the "background" of lensing by known substructures is so low, strongly lensed GWs serve as a clean probe for other sub-galactic structures. The authors highlight that lensed GWs could effectively constrain dark matter substructures (specifically subhalos in cold dark matter models) and primordial black holes (PBHs), as the rates for these scenarios can be orders of magnitude higher than those for IMBHs in GCs.

The authors maintain a modest tone regarding future implications, noting that while stellar fields near critical curves can cause stochastic deviations, these are qualitatively different from the massive perturbers studied here and only become significant at densities comparable to GCs. The study does not propose new experimental setups but rather provides a rigorous astrophysical prior to interpret existing and future GW lensing candidates.

More like this