Constraining the Hubble Constant using Cross-Correlation of Gravitational Wave Events with Flux-Limited Galaxy Catalog

This paper presents a novel Bayesian method that utilizes the 3D cross-correlation of gravitational wave events with flux-limited galaxy catalogs to constrain the Hubble constant to approximately 9% precision using 300 simulated binary black hole mergers detectable by Advanced LIGO and Virgo at O4 sensitivity.

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. Getting this number right is crucial because it tells us the age of the universe and how it will end.

Usually, to measure this speed, you need two things for a specific event:

1. How far away it is.
2. How fast it is moving away from us.

The Problem: The "Dark" Sirens

In 2017, scientists heard a "siren" (a gravitational wave) from two neutron stars crashing together. Because they saw the light (electromagnetic signal) from the crash, they knew exactly which galaxy it came from. They could measure the distance and the speed, and calculate the Hubble Constant.

But most of these crashes happen between Black Holes. Black holes don't emit light. They are "Dark Sirens."

• The Good News: Gravitational waves tell us exactly how far away the crash happened.
• The Bad News: Because we can't see the light, we don't know which galaxy the black holes were in. Without knowing the galaxy, we don't know the speed (redshift). It's like hearing a siren in a foggy city but not knowing which street it's on.

The Old Way: Guessing the Neighborhood

Previously, scientists tried to guess the location by looking at the "mass" of the black holes and assuming they live in certain types of galaxies. It's a bit like trying to guess which house a person lives in just by knowing their shoe size. It works okay, but it's not very precise.

The New Idea: The "Crowd" Connection

This paper proposes a smarter way to find the location of these Dark Sirens. Instead of guessing based on shoe size, they use a crowd-sourcing method.

Here is the analogy:
Imagine you are in a massive, dark stadium (the universe). You hear a shout (the gravitational wave) from somewhere in the stands, but you can't see who shouted.

• The Old Way: You guess the shout came from the VIP section because the shout sounded "heavy."
• The New Way: You look at the crowd. You know that people tend to shout in groups. If you see a dense cluster of people (galaxies) in a specific section of the stadium, and your shout seems to come from that general direction, it's highly likely the shout came from that group.

The scientists are using a 3D map of galaxies (the crowd) and checking where the gravitational waves (the shouts) overlap with the densest parts of the galaxy map.

The Twist: The "Floodlight" Problem

The paper introduces a realistic challenge. In the real world, our galaxy maps aren't perfect. We have a "flux-limited" catalog.

• The Analogy: Imagine you are trying to map the stadium crowd using a flashlight. You can see everyone clearly up close. But as you look further away, the flashlight gets dimmer. You can only see the people wearing bright neon shirts (bright galaxies); the people in dark clothes (faint galaxies) disappear into the darkness.
• The Result: Your map of the distant crowd is incomplete. You might miss a whole section of the stadium because your flashlight wasn't strong enough.

The authors asked: "Does our 'dim flashlight' (flux-limited catalog) ruin our ability to measure the universe's expansion speed?"

What They Did

They ran a massive computer simulation:

1. They created a fake universe with 300 "Dark Sirens" (black hole crashes).
2. They tried to find the Hubble Constant using two maps:
   • Map A: A perfect map where they could see every single galaxy, no matter how far or faint.
   • Map B: A realistic "flux-limited" map where they could only see the bright galaxies (the neon-shirted crowd).

The Results

• The Perfect Map: As expected, it gave a very precise answer.
• The "Dim Flashlight" Map: The answer was a bit "fuzzier" (less precise) at first, especially for distant events, because the map was missing faint galaxies.
• The Good News: Even with the imperfect map, the method still worked! By adding up more and more "shouts" (more gravitational wave events), the fuzziness disappeared. With about 300 events, they could measure the Hubble Constant with about 9% precision.

Why This Matters

This is a "proof of concept." It shows that even if we can't see the faint galaxies in the distance, we can still use the bright ones to find the location of black hole crashes and measure the expansion of the universe.

In simple terms:
We used to be blind to where black holes lived. Now, we have a new trick: we look at where the "bright neighbors" are clustered to guess where the black holes are hiding. Even if our flashlight isn't perfect and we miss some neighbors, if we listen to enough black hole crashes, we can still figure out exactly how fast the universe is growing.

This method is a powerful new tool that doesn't require us to wait for a rare, bright light to appear. We can use the "dark" events that happen all the time to solve one of the biggest mysteries in physics.

Technical Summary

1. Problem Statement

The Hubble constant ($H_0$) is a fundamental cosmological parameter, but its measurement remains subject to tension between early- and late-universe probes. Gravitational Waves (GWs) from compact binary coalescence offer a "standard siren" method to measure $H_0$ directly via luminosity distance.

• The Challenge: While Binary Neutron Star (BNS) mergers often have electromagnetic (EM) counterparts allowing direct redshift measurement, Binary Black Hole (BBH) mergers (the majority of GW events) are "dark sirens" lacking EM counterparts.
• The Limitation: Without a host galaxy identification, the redshift of BBH events is unknown. Current statistical methods attempt to infer $H_0$ by fitting source mass models to the GW population but often fail to explicitly utilize the spatial clustering of galaxies.
• The Gap: Real-world galaxy catalogs are flux-limited (only galaxies brighter than a certain magnitude are detected), whereas previous theoretical proofs-of-concept often assumed idealized, volume-limited catalogs. The impact of flux limits on $H_0$ inference via cross-correlation had not been rigorously tested.

2. Methodology

The authors employ a novel Bayesian formalism to infer $H_0$ by cross-correlating GW events with a galaxy catalog in configuration space.

• Core Approach: The method utilizes the 3D cross-correlation function between GW events and galaxies with known redshifts. Since GW sources and galaxies trace the same underlying Large Scale Structure (LSS), a non-zero cross-correlation is expected at the true redshift of the GW source.
• Simulation Setup:
  • GW Events: 300 simulated GW events distributed within 1 Gpc, modeled with colored Gaussian noise corresponding to Advanced LIGO and Advanced Virgo detectors operating at O4 sensitivity.
  • Galaxy Catalogs: Two catalogs were constructed from the BigMDPL simulation:
    1. Complete Catalog: A volume-limited catalog (idealized).
    2. Flux-Limited Catalog: A realistic catalog constructed by calculating apparent magnitudes ($m$) and selecting only galaxies with $m \le 17.77$ (mimicking SDSS spectroscopic limits). Random Gaussian errors ($\sigma = 0.5$) were added to absolute magnitudes to simulate scatter in the $M-v_{max}$ relation.
• Key Parameters:
  • Angular Scale: A maximum angular separation of $\theta_{max} = 0.02$ radians was used, identified as optimal in previous work.
  • Galaxy Bias ($b_g$): A critical adjustment was made for the flux-limited catalog. Unlike the complete catalog where bias is constant, the flux-limited catalog exhibits redshift-dependent bias. As redshift increases, only brighter (more massive) galaxies are visible, increasing the bias. This bias was calculated using dark matter halo masses derived from the simulation.
• Statistical Framework: The Bayesian framework models galaxy overdensity for each GW event, accounting for the volume uncertainty region of the GW localization.

3. Key Contributions

• Extension to Realistic Catalogs: The study extends the cross-correlation method proposed in Ref. [7] from idealized volume-limited catalogs to flux-limited catalogs, which are representative of real observational data (e.g., SDSS).
• Bias Modeling: The authors explicitly model the evolution of galaxy bias ($b_g$) as a function of redshift for flux-limited catalogs. They demonstrate that this bias increases with redshift due to the decreasing number of observable galaxies and the selection of intrinsically brighter objects.
• Quantification of Precision Loss: The paper quantifies the statistical penalty incurred by using flux-limited data compared to complete data, showing how the variance of overdensity fluctuations behaves differently at high redshifts.

4. Results

• Overdensity Fluctuations ($\sigma_\delta$):
  • In the complete catalog, $\sigma_\delta$ decreases with redshift as the cosmic volume increases, accommodating more galaxies.
  • In the flux-limited catalog, $\sigma_\delta$ initially follows the complete catalog at low redshifts but increases at higher redshifts. This is because the flux limit excludes fainter galaxies, reducing the sample size and increasing variance.
• Hubble Constant ($H_0$) Constraints:
  • 250 GW Events: The posterior distribution for $H_0$ using the complete catalog is precise. In contrast, the posterior using the flux-limited catalog is significantly broader, indicating a loss of statistical power due to the reduced galaxy sample at high redshifts.
  • 300 GW Events: Increasing the number of GW events improves the precision for the flux-limited catalog. The study demonstrates that with 300 events, the method can constrain $H_0$ with a precision of $\sim 9\%$ (90% highest density interval).
• Figure 2 Analysis: The comparison of posteriors shows that while the flux-limited method is less precise than the idealized complete method, it remains a viable statistical tool, with precision scaling favorably as the number of GW events increases.

5. Significance and Future Outlook

• Proof-of-Concept for Dark Sirens: The study validates that $H_0$ can be constrained using BBH mergers (dark sirens) without EM counterparts by leveraging the spatial correlation with flux-limited galaxy catalogs. This supplements BNS measurements and helps resolve the $H_0$ tension.
• Real-World Applicability: By accounting for flux limits and redshift-dependent bias, the paper bridges the gap between theoretical proposals and the application of real observational data (e.g., from LSST or Euclid).
• Limitations and Future Work:
  • The current study assumes GW events are unbiased tracers of matter. In reality, GW events may have their own redshift-dependent bias due to selection effects or halo occupation distribution.
  • The study assumes a uniform survey depth, whereas real catalogs have varying depths in different directions.
  • Future work must parameterize and marginalize over these additional biases and imperfections before applying the method to actual observational data.

Conclusion: The paper successfully demonstrates a robust Bayesian framework for inferring the Hubble constant using dark sirens and realistic, flux-limited galaxy catalogs, achieving a $\sim 9\%$ precision with 300 events, thereby offering a promising pathway for future cosmological constraints.