Gravitational waves are a new messenger in astronomy: there is a lot of unexplored information that we can obtain from them. Gravitational lensing, already observed with light, is the deflection of a wave due to the gravity of a massive object. It is one of the few ways gravitational waves interact with matter. Although lensed gravitational waves have not been detected yet, they are expected to be observed at any moment. Wave effects (interference, diffraction), which are very difficult to observe in lensing of light, are expected to be more commonly observed in gravitational waves due to their long wavelengths and coherence. The imprint (signature) of gravitational lensing on gravitational-wave signals can give us additional information about otherwise invisible lenses.
Can gravitational lensing help distinguish between different environments where gravitational waves from compact binary mergers come from? We study self-lensing by lenses in the same astrophysical environment as the gravitational wave source.
First, we quantify the probability of self-lensing in different environments (through the optical depth τ). [+]
In star clusters (globular and nuclear clusters), the probability of lensing by a stellar-mass black hole is low (τ~10-7), even considering the existence of resonant interactions (τ~10-5) which could decrease the distance between the lens and the source.
In star clusters with a central massive black hole (intermediate to supermassive), self-lensing by the massive black hole can reach τ~10-4, for mergers induced by von Zeipel-Lidov-Kozai oscillations or in single-single gravitational wave captures close to the massive black hole.
Active Galactic Nuclei (AGN) disks are the environment with a higher self-lensing probability. This is due to the assumption that binary black holes can migrate to close distances to the central supermassive black hole. The probability of self-lensing is high (τ~10-2) both for lensing by the central supermassive black hole and for lensing by any "lateral" intermediate-mass black hole that could be formed in the AGN disk.
Then, we analyze the imprint of lensing in each case and its detectability.
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Lensing by stellar-mass objects has a marginal imprint, requiring the mass of the lens to be larger than the mass of the source to be detectable in most cases.
Lensing by a massive black hole gives a detectable imprint, either as an interference pattern or multiple lensed images.
Lensing in AGN disks has an additional characteristic signature: having preferential h+ polarization.
By combining the self-lensing probabilities and imprints with other signatures such as polarization and eccentricity, it is possible to contrain the environment where the source comes from:
We don't expect self-lensing by stellar-mass black holes in star clusters.
If eccentricity and multiple images are present, it is likely to come from a single-single gravitational wave capture in a star cluster with a massive lens (unless both images come with h+ polarization, in which case the environment is an AGN disk).
If h+ polarisation is present in the signal (with a lensing imprint either as an interference pattern or as multiple images), it is likely to come from an AGN disk. Additionally, eccentricity is expected to also be high in this environment.
We aim to understand how a gravitationally-lensed gravitational wave would "look"* like at current (2023) detectors. For that, we use a more complete gravitational-wave template as a source.
Then, we explore the detectability of wave effects as a function of the system parameters (mass of the lens, position of the source) for a point mass lens. We find that signals with an interference pattern would be potentially detectable, while signals with the frequency-dependent amplification due to diffraction would be very faint to detect.
Finally, we explore how much the observable horizon could increase with the wave-optics lens effect, and obtain an upper limit on the probability of lensing.
We study the frontier where the Geometrical Optics approximation breaks down and wave effects need to be taken into account (for a point mass lens). This frontier does not only depend on the relation between the wavelength of gravitational waves and the Schwarzschild radius of the lens, but also on the source position y.
We explore the parameters which would give rise to the interference pattern. For LIGO-Virgo-KAGRA sources, the lenses that would imprint an interference pattern would be about 1000-100000 solar masses (assuming a moderate value of the source position, e.g. y=0.25). Some hypothetical astrophysical objects of these masses, such as intermediate-mass black holes and compact dark matter clumps, could be probed/constrained in this interference regime.
We provide analytical predictions for the separation and amplitude of the oscillations, which might enable us to distinguish wave-optics lensing from similar effects such as eccentricity, precession or echoes. The positions of these oscillations provide information about the mass of the lens and the source position.
Are you curious to know how a lensed gravitational wave would "sound"* like?
You can hear the sonifications we made in collaboration with Jordi Espuny, for interfering gravitational waves.
*In reality, human senses do not allow us to directly see or hear gravitational waves. Even if we could "feel" the passage of gravitational waves, they are too small to be noticeable. Nevertheless, that does not prevent us from interpreting them with mathematical and computational tools. We can "see" or "hear" the representation of the received data in graphics or sonifications, respectively.
Bridging Art&Science: Exploring the Cosmos through Sonification at the University of Barcelona. Luri et al., Highlights of Spanish Astrophysics XII (2025). [Proceedings, online version]
Contribution to review articles
Multi-messenger Gravitational Lensing. Smith et al. Philosophical Transactions of the Royal Society A, 383, 2295 (2025). [arXiv] [journal]
A selection of recent communication of our research, where you can explore some of the material.
2025
Self-lensing signatures to constrain the environment of binary mergers [Poster, 6.44 MB pdf]Prospects in Theoretical Physics 2025. Institute for Advanced Study (IAS), Princeton (NJ, USA).