⚛️ general relativity

Strong lensing cosmography using binary-black-hole mergers: Prospects for the near future

This paper investigates the potential of using upcoming upgraded LIGO-Virgo-KAGRA (LVK) observations to perform strong lensing cosmography by incorporating detector selection effects, demonstrating that even a modest number of lensed binary-black-hole detections can yield cosmological constraints while confirming that previous forecasts for next-generation detectors remain valid.

Original authors: Koustav N. Maity, Souvik Jana, Tejaswi Venumadhav, Ankur Barsode, Parameswaran Ajith

Published 2026-02-09

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CC BY 4.0

Original authors: Koustav N. Maity, Souvik Jana, Tejaswi Venumadhav, Ankur Barsode, Parameswaran Ajith

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 is a giant, echoing canyon. Usually, when a "sound" (a gravitational wave) is made by two black holes crashing together, we hear it once. But sometimes, a massive galaxy or cluster of galaxies sits between us and the crash, acting like a giant, cosmic magnifying glass. This is called gravitational lensing.

When this happens, the sound doesn't just get louder; it gets split. You might hear the same crash twice, or even more times, with a pause in between. It's like shouting in a canyon and hearing your voice bounce back as two distinct echoes arriving at different times.

This paper is about using those "echoes" to measure the size and shape of the universe itself.

The Big Problem: The "Hubble Tension"

Right now, scientists are arguing about how fast the universe is expanding. One group of measurements (using old supernovae) says one speed, and another group (using the leftover heat from the Big Bang) says a different speed. They disagree so much that it's causing a crisis in physics. We need a new, independent way to measure this speed to see who is right.

The New Tool: Listening for Echoes

The authors of this paper propose a new way to solve this puzzle using the "echoes" of black hole collisions. Here is how they plan to do it:

Counting the Echoes: If the universe is expanding at a certain speed, there will be a specific number of these "echoing" black hole collisions that our detectors can hear. If the universe is expanding faster or slower, that number changes.
Timing the Gaps: The time delay between the first echo and the second echo depends on the geometry of space. By measuring exactly how long we have to wait between the two sounds, we can calculate the distance to the black holes and the lenses, which tells us about the universe's expansion.

The Catch: The "Ear" Isn't Perfect

In previous studies, scientists assumed our detectors were perfect ears that could hear every echo, no matter how faint or when it arrived. The authors of this paper realized that's not true.

Think of our current detectors (like LIGO and Virgo) as a team of people with hearing aids.

The Volume Problem: They can only hear the loudest crashes from relatively nearby. Faint, distant echoes might be too quiet to hear.
The Schedule Problem: The detectors don't run 24/7. They have maintenance breaks. If the first echo arrives while the detector is "asleep," and the second echo arrives while it's "awake," we miss the pair entirely. We can't count it as an echo if we only hear one half of the conversation.

The authors spent a lot of time building a computer model that accounts for these "sleeping" periods and "quiet" limits. They wanted to know: Even with these imperfect ears, can we still learn something about the universe?

What They Found

They simulated the future of gravitational wave astronomy, looking at:

Now and Soon (O4, O5, O6): The current and slightly upgraded detectors.
The Future (Voyager, XG): Super-sensitive detectors coming in the next few decades.

The Results:

Modest but Real: Even with the current and near-future detectors (which will only catch a "dozen" or so of these echoing pairs), we can start to put limits on the expansion rate of the universe. It won't be a perfect answer yet, but it's a start.
The Future is Bright: By the time we get to the "Next-Generation" (XG) detectors, which will hear millions of collisions, the number of echoes will jump to tens of thousands. At that point, this method will be just as powerful as our best current tools for measuring the universe.
A New Perspective: This method is special because it looks at the universe at a "middle age" (intermediate redshift), a time period that other methods don't look at very well. It's like taking a photo of a person's life at age 40, whereas other methods only look at their birth or their retirement.

The Bottom Line

This paper is a "reality check" and a "roadmap." It tells us that even though our current listening equipment has limitations (like gaps in time and sensitivity limits), we can still use the rare "echoes" of black hole collisions to measure the universe's expansion.

As we upgrade our "ears" over the next decade, this method will become a powerful, independent way to solve the mystery of how fast the universe is growing, potentially helping us finally settle the argument between the different measurements we have today.

Technical Summary: Strong Lensing Cosmography Using Binary-Black-Hole Mergers

Problem Statement
The paper addresses the potential of using strongly lensed gravitational wave (GW) signals from binary black hole (BBH) mergers as a probe for cosmological parameters. While next-generation (XG) detectors are expected to detect millions of GW events, the path to these facilities involves intermediate upgrades (e.g., O5, O6, Voyager) with varying sensitivities and network configurations. Previous work by Jana et al. [1] proposed using the number of lensed events and their time-delay distributions to constrain cosmology but assumed an idealized detector network where all BBH mergers are detectable. This study identifies a critical gap: the need to incorporate realistic detector network selection effects (including sensitivity limits, duty cycles, and observation gaps) when forecasting constraints from current and near-future detectors. The authors investigate whether modest constraints on cosmological parameters can be derived from upcoming observing runs of the LIGO-Virgo-KAGRA (LVK) network, where the horizon distance is limited compared to XG facilities.

Methodology
The authors employ a Bayesian inference framework to forecast constraints on the parameters of a flat $\Lambda$ CDM model: the Hubble constant ( $H_0$ ), matter density ( $\Omega_m$ ), and matter clustering amplitude ( $\sigma_8$ ).

Detector Modeling: The study models a sequence of observing runs (O4 through XG), accounting for specific detector locations, sensitivity curves (noise amplitude spectral densities), and duty cycles. Crucially, it incorporates "gaps" between observing runs and varying sensitivities across different network configurations (e.g., the addition of LIGO-India or KAGRA).
Source and Lens Population:
- Sources: Intrinsic BBH merger rates are modeled using two astrophysical scenarios: the "Dominik" model (incorporating stellar evolution effects) and "MD without delay" (based on the Madau-Dickinson star formation rate). These are calibrated to low-redshift constraints from GWTC-3 data.
- Lenses: Lenses are modeled as Singular Isothermal Spheres (SIS). The lens population is derived from a Halo Mass Function (HMF) (Behroozi model) converted to velocity dispersion.
Selection Effects and Detection Rates:
- The authors calculate the expected number of detectable BBH mergers ( $\Lambda$ ) and strongly lensed pairs ( $\Lambda_\ell$ ) by convolving intrinsic rates with network selection functions ( $S(z_s)$ ).
- A key innovation is the calculation of the selection function for lensed pairs ( $S_L$ ). Unlike unlensed events, a lensed event is only identified if both images are detected. The authors account for the possibility that one image arrives during a detector gap or falls below the signal-to-noise ratio (SNR) threshold due to demagnification or changing antenna patterns (Earth's rotation).
- They compute the joint probability distribution of the signal-to-noise ratios for the two images, marginalizing over source parameters, arrival times, and time delays ( $\Delta t$ ).
Bayesian Inference:
- The likelihood function is constructed using two observables: the lensing fraction ( $u_{det} = N_\ell/N$ ) and the distribution of measured time delays ( $\{\Delta t_i\}$ ).
- The lensing fraction likelihood is derived from the convolution of Poisson distributions for total and lensed event counts.
- The time-delay likelihood is the product of the probability density functions for individual time delays, which depend on cosmological parameters via distance measures and the lensing optical depth.
- Posterior distributions for cosmological parameters are obtained via Monte Carlo sampling, assuming flat priors.

Key Contributions

Incorporation of Selection Effects: This work is the first to rigorously include detector network selection effects (sensitivity evolution, duty cycles, and observation gaps) in forecasts for lensing cosmography. It demonstrates that neglecting these effects leads to inaccurate forecasts for current-generation and intermediate-upgrade detectors.
Lensed Pair Detection Criteria: The authors explicitly model the requirement that both lensed images must be detected within the operational windows of the network, revealing that observation gaps significantly reduce the detectable fraction of lensed events, particularly in the near term.
Revised Forecasts for Near-Future Runs: The paper provides specific forecasts for the O5, O6, and Voyager observing runs, showing that even with "modest" numbers of lensed detections (dozens), cosmological constraints are achievable.
Validation of XG Forecasts: The study confirms that for XG detectors, where selection effects are less dominant due to high sensitivity, the revised forecasts align with earlier idealized predictions by Jana et al. [1].

Results

Event Rates: The authors forecast tens of thousands of lensed GW events in the XG era. For intermediate runs (O5, O6), they expect dozens of lensed pairs. The fraction of lensed events is non-monotonic with time; it drops when new observing runs begin after gaps because the first image of a pair might be detected in a new run while the second arrives during a gap or in a previous run with different sensitivity.
Cosmological Constraints:
- Time-Delay Distribution: The distribution of time delays is sensitive to cosmological parameters. Increasing $H_0$ shifts the peak to smaller time delays, while increasing $\Omega_m$ or $\sigma_8$ shifts it to larger delays.
- Parameter Precision: By the end of the Voyager and XG eras, the constraints on $H_0$ , $\Omega_m$ , and $\sigma_8$ tighten significantly. The precision approaches that of current Cosmic Microwave Background (CMB) observations and is comparable to expected constraints from GW standard sirens.
- Impact of Selection Effects: For near-future detectors (O5, O6), ignoring selection effects would overestimate the constraining power. Including them shows that constraints are modest but non-negligible.
- Cross-Run Pairs: The analysis highlights the importance of including lensed pairs where the two images are detected in different observing runs (cross-run pairs). Excluding these pairs (considering only concurrent detections) significantly degrades the constraints.

Significance and Claims
The paper claims to demonstrate the potential of lensing cosmography even with relatively modest numbers of lensed detections from upgraded LVK detectors. It establishes that the number of lensed events and their time-delay distribution encode imprints of cosmological parameters, particularly at intermediate redshifts ( $z \sim 1-10$ ), a regime not commonly probed by other observations.

The authors position this work as a necessary step toward making the concept of lensing cosmography a practical observational program. They emphasize that while their current lens model (SIS) is simplified, the core finding—that selection effects must be accounted for and that cosmological information is extractable from lensing observables—remains robust. They conclude that in the XG era, the precision of lensing cosmography will be competitive with other leading cosmological probes, offering a complementary method to resolve tensions such as the Hubble tension. The paper does not claim to solve these tensions immediately but provides a rigorous framework for how future GW data will contribute to this goal.