The impact of strong lensing on Hubble constant measurements with gravitational-wave dark sirens

This paper proposes a novel method for measuring the Hubble constant by combining strong lensing of gravitational-wave dark sirens with galaxy lensing data, demonstrating that just eight such lensed events can improve measurement precision by approximately 50% compared to 250 unlensed events.

The Gist

Imagine the universe is a giant, expanding balloon. For decades, scientists have been trying to measure exactly how fast this balloon is inflating. This speed is called the Hubble Constant ($H_0$).

Here's the problem: When we look at the "baby pictures" of the universe (the Cosmic Microwave Background), we get one speed. When we look at the "adult pictures" (nearby exploding stars and pulsating stars), we get a different, faster speed. This disagreement is known as the "Hubble Tension," and it's like two mechanics arguing over how fast a car is going, even though they are both looking at the same car.

This paper proposes a new, high-tech way to settle the argument using Gravitational Waves (ripples in space-time caused by colliding black holes) and a cosmic trick called Strong Lensing.

The Cast of Characters

1. Dark Sirens: These are colliding black holes. They scream in gravitational waves but are completely silent in light. We can hear them, but we can't see them. Because they are "dark," we don't know exactly where they are or how far away they are with perfect precision.
2. The Lens: Imagine a massive galaxy sitting between us and a black hole. Its gravity acts like a giant, cosmic magnifying glass.
3. The Trick (Strong Lensing): Just like a magnifying glass can make a small object look bigger and brighter, a galaxy can bend the gravitational waves from a black hole. This creates multiple images of the same event. Instead of hearing one "chirp," we might hear the same chirp four times, arriving at slightly different times and with different volumes.

The Big Idea: Using the "Cosmic Zoom"

The authors of this paper realized that these "lensed" dark sirens are actually super-stars for measuring the universe's expansion. Here is why, using a simple analogy:

The "Blurry Photo" vs. The "Sharp Zoom"

• Normal (Unlensed) Events: Imagine trying to guess how far away a car is by looking at a blurry photo taken from a mile away. You have a rough idea, but there's a lot of guesswork. In astronomy, this means we have to look at thousands of these blurry events to get a decent answer.
• Lensed Events: Strong lensing is like having a super-powerful zoom lens. It sharpens the image and tells us exactly where the car is. Because the gravity of the lens galaxy bends the light (and waves) in a predictable way, we can use the lens itself as a ruler.

How They Did the Math (The "Cosmic Detective" Work)

The team created a massive computer simulation. They didn't just guess; they built a virtual universe (using a catalog called MICECAT) filled with millions of galaxies and simulated black hole collisions.

They tested two scenarios:

1. The Single Detective: They took one single, perfectly lensed event (specifically one that split into four images). By using the lens galaxy to pinpoint the location and the "magnification" to fix the distance, they could calculate the expansion speed of the universe with incredible precision from just one event.
2. The Team of Detectives: They combined a small group of these lensed events (only 8 of them!) and compared them to a huge group of normal, un-lensed events (250 of them).

The Result:
The small group of 8 lensed events gave a much sharper, more accurate answer than the group of 250 normal events. In fact, using these lensed events improved the precision by about 50%. It's like getting a better answer from 8 people using high-tech tools than from 250 people using binoculars.

The Catch: The "Missing Pages" Problem

The paper also warns about a potential pitfall: Incomplete Catalogs.

Imagine you are trying to solve a puzzle, but the box is missing some pieces. In astronomy, our "catalog" of galaxies isn't perfect; we miss the faint, dim ones.

• If we miss the lens galaxy or the host galaxy in our data, our calculation gets wobbly.
• The authors found that even with missing data, the lensed method still outperformed the normal method, but the "missing pieces" made the answer slightly less precise.

The Danger of Mistakes

The paper also highlights a funny but dangerous scenario: Mixing up the signals.

• If you accidentally think a normal event is lensed, your math breaks, and you get a wildly wrong answer.
• However, if you accidentally think a lensed event is normal, the answer is just a little less precise, but not totally wrong.
• Lesson: It is much more important to be careful about identifying which events are lensed than to worry about missing a few lensed ones.

The Takeaway

This paper is a roadmap for the future. As our gravitational wave detectors get better (like the upcoming "Einstein Telescope"), we will start seeing these "cosmic magnifying glass" events more often.

By treating these lensed black hole collisions as precision rulers, we can finally solve the Hubble Tension. We won't need thousands of blurry events; we just need a handful of these super-sharp, lensed events to tell us exactly how fast the universe is growing.

In short: Strong lensing turns a blurry cosmic whisper into a clear, loud shout, allowing us to measure the universe's expansion with a precision we've never had before.

Technical Summary

1. Problem Statement

The paper addresses the persistent Hubble tension, the significant discrepancy between early-Universe measurements of the Hubble constant ($H_0$) from the Cosmic Microwave Background (CMB) and late-Universe measurements from the cosmic distance ladder (Cepheids and Type Ia supernovae).

While gravitational waves (GWs) from compact binary coalescences (CBCs) offer a "standard siren" method to measure $H_0$ independently, current "dark siren" methods (where no electromagnetic counterpart is identified) suffer from large statistical uncertainties. These uncertainties arise primarily from:

1. Poor sky localization: Leading to a large number of potential host galaxies in the error box.
2. Luminosity distance degeneracy: The distance is inferred from waveform amplitude but is subject to measurement noise.
3. Catalog incompleteness: Real galaxy catalogs do not contain all potential host galaxies, introducing selection biases.

The authors investigate whether Strongly Lensed Gravitational Waves (SLGWs) can overcome these limitations. Strong lensing creates multiple images of a single GW event, providing redundant measurements that can improve sky localization and luminosity distance estimates, while the lensing geometry offers additional constraints on cosmology.

2. Methodology

Data Simulation

• Galaxy Catalogs: The authors utilize the MICECATv2 (MICE-Grand Challenge) light-cone halo and galaxy catalog.
  • UGC0: A complete, volume-limited subset of $\sim 6.2 \times 10^7$ galaxies ($M_r < -20$).
  • LGC0: A simulated catalog of strongly lensed galaxies derived from UGC0, assuming Singular Isothermal Ellipsoid (SIE) mass profiles for lenses. It contains $\sim 4.6 \times 10^4$ double-image and $\sim 1.0 \times 10^4$ quadruple-image systems.
  • Incompleteness: To mimic real surveys, magnitude cuts ($m_{th} = 21.87$) are applied to create incomplete catalogs (UGC1 and LGC1), reducing the sample sizes by roughly 50% and 66%, respectively.
• GW Signals:
  • Dark Sirens: Binary Black Hole (BBH) populations are simulated using the POWERLAW+PEAK mass distribution model (based on GWTC-3) and a star-formation-rate-based merger rate.
  • Lensing: Lensing is simulated by pairing BBH sources with lens galaxies from LGC0. The amplification factor $F(f)$ is applied to generate lensed waveforms.
  • Detection: Signals are injected into a network of LIGO-Livingston, LIGO-Hanford, and Virgo detectors (O4 sensitivity). Only events with a network signal-to-noise ratio ($\rho_{net}$) $> 8$ are considered detected.

Inference Framework

The authors employ the gwcosmo pipeline for Bayesian hierarchical inference.

1. Single Event Analysis (Section 4.1):
   • For a single lensed event, the lensing observables (relative magnification $\mu_{rel}$ and time delay $\Delta t$) are used to reconstruct the lens model.
   • This reconstruction allows for delensing: correcting the apparent luminosity distance ($d_L^{eff}$) to retrieve the true distance ($d_L$) using $d_L^{eff} = d_L / \sqrt{\mu}$.
   • The source redshift ($z_s$) is obtained via the Hubble-Lemaître law once the host galaxy is uniquely identified through the tight sky localization provided by lensing.
2. Multiple Event Analysis (Section 4.2):
   • A joint likelihood is constructed for $N$ lensed events.
   • The framework incorporates Line-of-Sight (LoS) redshift priors derived from cross-matching GW sky localization maps with the galaxy catalogs (UGC/LGC).
   • The likelihood accounts for selection effects, including the probability of detecting lensed systems (which depends on magnification and $H_0$) and catalog incompleteness.

3. Key Contributions

• Novel Methodology: Proposes a combined approach using strongly lensed dark sirens and galaxy-galaxy lensing catalogs to infer $H_0$, moving beyond the standard unlensed dark siren approach.
• Quantification of Bias: Systematically quantifies the systematic biases introduced by:
  • Incorrect lens reconstruction (e.g., assuming a Singular Isothermal Sphere (SIS) model when the true profile is SIE).
  • Misclassification of events (treating unlensed events as lensed, or vice versa).
• Impact of Catalog Incompleteness: Explicitly models how the incompleteness of galaxy catalogs affects the precision of $H_0$ inference in both lensed and unlensed scenarios.
• Delensing Accuracy: Demonstrates that while double-image systems provide some improvement, quadruple-image systems are critical for uniquely identifying host galaxies and achieving high-precision delensing.

4. Key Results

Precision Improvements

• Single Events: A single quadruply lensed dark siren can constrain $H_0$ to a precision of $\sim 20\%$ (relative uncertainty $\delta H_0 \approx 0.19$), comparable to bright sirens. Double-image systems yield broader constraints ($\delta H_0 \approx 0.34 - 1.22$) due to degeneracies in lens reconstruction.
• Multiple Events:
  • Using 8 strongly lensed events (6 double + 2 quadruple) analyzed with a dedicated lensing catalog (LGC0) improves the precision of $H_0$ by roughly 50% compared to analyzing 250 unlensed events (UGC0).
  • The relative uncertainty drops from $\delta H_0 \approx 0.34$ (250 unlensed) to $\delta H_0 \approx 0.23$ (8 lensed).
  • To reach the few-percent precision ($\delta H_0 < 0.1$) typical of supernova measurements, the framework suggests $\sim 40$ lensed events are required.

Impact of Incompleteness

• Incomplete catalogs (UGC1/LGC1) broaden the $H_0$ posteriors due to increased uncertainty in host identification.
• However, even with incomplete catalogs, the lensed sample (8 events) outperforms the unlensed sample (250 events) in precision, though the gain is slightly reduced compared to the complete catalog scenario.

Systematic Biases

• Lens Reconstruction Bias: If the lens model is incorrect (e.g., using SIS instead of SIE), the bias in $H_0$ is subtle for a single event but accumulates significantly in hierarchical analyses, leading to large systematic offsets.
• Misidentification Bias:
  • Unlensed $\to$ Lensed: If unlensed events are misclassified as lensed and delensed, the inferred distances are artificially increased, biasing $H_0$ lower.
  • Lensed $\to$ Unlensed: If lensed events are treated as unlensed (ignoring magnification), the distances are underestimated, biasing $H_0$ higher.
  • The paper finds that misclassifying a small fraction of lensed events as unlensed causes a shift of only $\Delta H_0 \sim 1$ km s$^{-1}$ Mpc$^{-1}$, whereas misclassifying unlensed events as lensed can cause larger shifts depending on the sample size.

5. Significance and Future Outlook

• Complementary Probe: Strong lensing transforms dark sirens into a highly competitive probe for $H_0$, potentially resolving the Hubble tension with fewer events than the unlensed approach requires.
• Critical Requirements: The study highlights that accurate lens modeling (breaking the mass-sheet degeneracy via stellar kinematics or wave-optics effects) and robust lens identification are paramount. Failure to correctly identify lensing or model the lens mass distribution introduces systematic errors that can invalidate the measurement.
• Next-Generation Detectors: With upcoming detectors (Einstein Telescope, Cosmic Explorer) and space-based observatories (LISA), the detection rate of SLGWs is expected to increase significantly. The authors note that while these detectors will provide better sky localization, the challenge of galaxy catalog incompleteness at high redshifts will become increasingly critical, requiring improved photometric surveys and population modeling.

In conclusion, the paper establishes that strong lensing is a powerful amplifier for GW cosmology, capable of delivering high-precision $H_0$ measurements with a small number of events, provided that lens reconstruction is accurate and selection biases are rigorously controlled.