Mock Catalogs of Strongly Lensed Gravitational Waves via A Halo Model Approach with Ground-based Detectors

This paper presents the Gravitational Waves-Lensing Mock Catalog (GW-LMC), a comprehensive suite of simulated strongly lensed gravitational wave events derived from a composite halo model, which forecasts hundreds of annual detections (including doublets, quadruplets, subhalo-lensed systems, and central images) for future third-generation ground-based detector networks and provides essential statistical priors for their identification.

The Gist

Imagine the universe is a giant, dark ocean, and gravitational waves are like ripples created when two massive objects (like black holes) crash into each other. For a long time, we've been listening to these ripples with our "ears" (detectors like LIGO). But now, we are building super-sensitive "super-ears" (the Einstein Telescope and Cosmic Explorer) that will hear ripples from the very beginning of time.

This paper is a forecast for what we expect to hear with these new super-ears. Specifically, it predicts how often we will hear a special kind of echo caused by gravitational lensing.

Here is the breakdown using simple analogies:

1. The "Cosmic Funhouse Mirror" (Gravitational Lensing)

Imagine you are looking at a streetlamp through a thick, wavy glass bottle. The light doesn't just go straight; it bends, creating multiple, distorted images of the same lamp.

• In space: Massive objects like galaxies act as that glass bottle. When a gravitational wave passes near a galaxy, the galaxy's gravity bends the wave's path.
• The Result: Instead of hearing one "crash," you might hear the same crash multiple times, arriving at different times and with different volumes. This is a lensed gravitational wave.

2. The "Recipe Book" (The Mock Catalog)

Since we haven't built the super-ears yet, the authors couldn't just go out and count the events. Instead, they wrote a simulation recipe (a "Mock Catalog").

• They created a virtual universe with millions of fake black hole collisions.
• They programmed the rules of gravity, including how dark matter (the invisible stuff holding galaxies together) is distributed.
• They ran the simulation to see how many of these fake collisions would get "lensed" and how loud they would sound when they reached Earth.

3. The "New Ingredients" (What Makes This Different)

Previous studies were like cooking with only flour and sugar (simple galaxy models). This paper adds complex ingredients:

• Dark Matter Sub-halos: Imagine a galaxy isn't just a smooth ball of light, but a ball filled with smaller, invisible clumps of dark matter. These clumps act like tiny lenses within the big lens. The authors found that about 100 events a year will be lensed specifically by these tiny clumps. This is huge news because it helps us map the invisible dark matter.
• The "Ghost" Image: In optical astronomy (light), if a galaxy lenses a star, it usually creates 2 or 4 images. Physics says there should be an odd number (3 or 5), but the middle image is usually so dim and hidden behind the galaxy's glare that we can't see it.
  • The Twist: Gravitational waves don't get blinded by galaxy glare. The authors predict that with our new super-ears, we will finally hear this "ghost" central image! This allows us to "see" the very center of the lensing galaxy.

4. The "Volume Knob" (Magnification)

Sometimes, the lens doesn't just split the sound; it turns the volume up.

• High Magnification: The lens acts like a megaphone, making a faint, distant signal loud enough to hear.
• The Discovery: The authors predict we will find about 360 events a year that are super-bright because of this effect. This is like finding a whisper from a billion miles away because someone shouted it through a megaphone. It lets us study the very first stars and black holes in the universe.

5. The "Echo Chamber" (Time Delays and Patterns)

When you hear a sound echo, the second sound is usually quieter and arrives later.

• The Pattern: The authors analyzed the "echoes" (the multiple images). They found that for double echoes, the second one is usually quieter.
• The Surprise: For complex echoes (4 images), the second or third sound is often the loudest, not the first one!
• Why it matters: If we only listen for the first sound, we might miss the rest of the story. This paper gives scientists a "cheat sheet" (statistical priors) so they know what to expect: "If you hear a loud crash, check your archives for a quieter one that happened 60 days ago, or look for a louder one that might have happened 10 days later."

6. The "Big Numbers" (What to Expect)

With the combined power of the Einstein Telescope (ET) and Cosmic Explorer (CE):

• ~400 pairs of double echoes per year.
• ~36 sets of four echoes per year.
• ~100 events lensed by tiny dark matter clumps.
• ~20 events where we hear the "ghost" central image.
• ~600 total events where at least one part of the echo is loud enough to hear.

The Bottom Line

This paper is a blueprint for the future. It tells astronomers: "Don't just look for one sound. Look for echoes. Look for hidden central sounds. Look for sounds that are louder than they should be."

By providing this detailed "forecast," the authors are giving the scientific community the tools to distinguish a real cosmic echo from a random coincidence, helping us unlock secrets about dark matter, the expansion of the universe, and the nature of gravity itself. All their data and simulations are now public, like a shared library for anyone to use.

Technical Summary

Here is a detailed technical summary of the paper "Mock Catalogs of Strongly Lensed Gravitational Waves via A Halo Model Approach with Ground-based Detectors."

1. Problem Statement

As the field of gravitational wave (GW) astronomy transitions toward third-generation (3G) detectors (Einstein Telescope and Cosmic Explorer), the detection rate of strongly lensed GW events is expected to increase from rare occurrences to hundreds annually. However, current identification strategies face significant challenges:

• Model Limitations: Previous simulations relied heavily on simplified lens models (e.g., Singular Isothermal Sphere/Ellipsoid) restricted to galaxy scales, neglecting the complex mass distributions of galaxy groups, clusters, and dark matter subhalos.
• Source Population Uncertainty: Many studies used simplified power-law mass distributions for binary black holes (BBHs), failing to capture observed substructures like mass peaks.
• Detection Blind Spots: Optical lensing studies often discard "central images" (the odd-numbered image in 3 or 5-image systems) due to demagnification and host galaxy glare. In GW astronomy, these images may be detectable due to high signal-to-noise ratios (SNR), but their statistical properties were not well-characterized.
• Identification Complexity: Distinguishing lensed pairs from chance coincidences requires robust statistical priors regarding time delays, SNR ratios, and magnification biases, which are currently lacking for 3G detector networks.

2. Methodology

The authors constructed a comprehensive mock catalog (GW-LMC) using a multi-scale, physically motivated framework:

A. Source Population Modeling

• Redshift Distribution: Adopted the merger rate density from Dominik et al. (2013), cross-validated against GWTC-4.0 (LVK 2025) data, covering redshifts up to $z \sim 21.5$.
• Mass and Spin Models:
  • BBH: Utilized the "Broken Power Law + 2 Peaks" model (LVK 2025 fiducial) to capture primary mass peaks at $\sim 10 M_\odot$ and $\sim 35 M_\odot$, replacing older simple power laws.
  • BNS & NSBH: Employed specific peak models and hierarchical frameworks (Biscoveanu et al. 2022) respectively.
  • Spins: Assumed aligned spins with orbital angular momentum (negligible precession) to simplify analysis, using truncated normal or uniform distributions.
• Sampling: Used Monte Carlo sampling with Posterior Predictive Distributions (PPD) to incorporate hyperparameter uncertainties from LVK data.

B. Deflector (Lens) Modeling

• Composite Halo Model: Replaced simple SIS/SIE models with a multi-component framework (based on Abe et al. 2025) comprising:
  1. Host Dark Matter Halos (NFW profile).
  2. Central Galaxies (Hernquist profile).
  3. Dark Matter Subhalos (analytic approach with tidal stripping).
  4. Satellite Galaxies.
  5. External Shear.
• Subhalo Importance: Explicitly modeled subhalos to capture small-scale lensing effects, which are critical for GWs due to their point-source nature and high time-domain resolution.

C. Simulation Pipeline

• SL-Hammocks Code: Adapted the optical lensing code SL-Hammocks for GWs by creating an "Equivalent Luminosity Function." This maps GW SNR and redshift to an equivalent absolute magnitude, allowing the code to filter events based on detection thresholds.
• Detector Networks: Simulated four configurations:
  • PlusNetwork (A+): Current generation upgrades.
  • ET (Einstein Telescope): Triangular, 10km arms, 1 Hz low-frequency cutoff.
  • CE (Cosmic Explorer): L-shaped, 40km arms, 5 Hz cutoff.
  • ET+CE: Combined network.
• Earth Rotation: Crucially accounted for Earth's rotation during the time delay between images, which alters the detector antenna pattern and thus the observed SNR of each image.

3. Key Contributions

1. First Comprehensive GW-LMC: A public catalog containing $\sim 4 \times 10^5$ source events and $\sim 6,000$ lensed systems, covering BBH, BNS, and NSBH populations across multiple detector configurations.
2. Subhalo Lensing Quantification: Provided the first robust statistical prediction for GW events lensed primarily by dark matter subhalos, a feature previously ignored in GW lensing studies.
3. Central Image Detection: Identified a unique class of "Central Image" events where the typically demagnified central image (in 3 or 5-image systems) becomes detectable in the GW band, offering a new probe for galactic core density profiles.
4. High-Magnification Bias Analysis: Quantified how magnification bias extends the observable horizon to higher redshifts ($z > 15$), particularly for high-magnification events.
5. Statistical Priors for Identification: Generated detailed distributions for SNR ratios ($\rho_{late}/\rho_{early}$) and time delays ($\Delta t$) to serve as Bayesian priors for future lensed signal authentication algorithms.

4. Key Results

Detection Rates (ET+CE Network, Annual)

• Total Lensed Events: $\sim 617$ events (where at least one image exceeds SNR > 8).
• Multi-Image Systems:
  • Doublets: $\sim 395$ events.
  • Quadruplets: $\sim 36$ events.
  • Subhalo-Lensed: $\sim 107$ events (approx. 25% of multi-image systems).
• Central Images: $\sim 20$ events where the central image is detectable (totaling 3 or 5 detectable images).
• High-Magnification ($\mu > 3$): $\sim 360$ events.
• Source Dominance: BBHs constitute $\sim 95\%$ of lensed events; BNS and NSBH are rare ($\sim 4.8$ and $\sim 13.4$ yr$^{-1}$ respectively).

Statistical Distributions

• Redshift: Mean source redshift $\langle z_s \rangle \approx 6.0$, mean lens redshift $\langle z_l \rangle \approx 1.1$. High-magnification events shift the source peak to $z \approx 4.5$.
• Time Delays:
  • Doublets: Median $\Delta t \approx 65.5$ days; Mean $\approx 157$ days (skewed by rare long delays).
  • Quadruplets: The second and third images often arrive with higher SNR than the first (due to caustic crossings), with median delays of $\sim 12$ and $\sim 20$ days respectively. The fourth image is typically demagnified.
• SNR Ratios:
  • For doublets, the second image is usually fainter ($\rho_2/\rho_1 < 1$ in 88% of cases).
  • For quadruplets, the "flux inversion" ($\rho_{late} > \rho_{early}$) is common for the 2nd and 3rd images (78% and 63% respectively) but rare for the 4th.
• Model Sensitivity:
  • Event rates are highly sensitive to the binary evolution model (e.g., Optimistic CE model yields 3x more events than Standard; High BH Kicks yields 1/20th).
  • The "Broken Power Law + 2 Peaks" mass model is robust against mass range variations, unlike simple Power Law models which overestimate rates by artificially extending the high-mass tail.

5. Significance

• Dark Matter Probes: The ability to detect subhalo-lensed events provides a unique, independent method to constrain the abundance of low-mass dark matter halos and test the Cold Dark Matter (CDM) paradigm, free from the spatial resolution limits of optical astronomy.
• Cosmology: The detection of central images and high-redshift lensed events offers new avenues to measure the Hubble constant ($H_0$) and probe the inner mass distribution of galaxies (cuspy vs. cored profiles).
• Algorithm Development: The provided distributions of time delays and SNR ratios establish a rigorous statistical foundation for developing machine learning and Bayesian inference tools to identify lensed GWs in future 3G data, reducing false alarm rates from chance coincidences.
• Observational Strategy: The results suggest that for quadruple systems, the brightest signal may not be the first to arrive. This necessitates "pre-trigger" searches in archival data to recover the initial, potentially sub-threshold signal, and continuous monitoring for at least six months to capture full event sets.

The GW-LMC catalog is publicly available on GitHub, serving as a standardized benchmark for the GW community to prepare for the era of strongly lensed gravitational waves.