Cosmology with the angular cross-correlation of gravitational-wave and galaxy catalogs: forecasts for next-generation interferometers and the Euclid survey

This paper forecasts that the angular cross-correlation of next-generation gravitational-wave catalogs with the Euclid galaxy survey can constrain the Hubble constant to percent-level precision and significantly improve cosmological constraints, provided multiple interferometers enable precise sky localization.

The Gist

The Big Idea: A Cosmic Game of "Where Are You?"

Imagine the universe is a giant, dark room filled with two types of objects: Galaxies (like glowing lanterns) and Gravitational Waves (like ripples from stones dropped in a pond).

For a long time, astronomers have only been able to map the "lanterns" (galaxies). They know exactly where these lanterns are in the sky and how far away they are based on how much their light has stretched (redshift). This has helped them build a map of the universe.

However, a new tool has arrived: Gravitational Waves (GWs). These are ripples in space-time caused by massive collisions (like black holes smashing together).

• The Problem: GWs tell us exactly how far away a collision happened (like a ruler), but they are terrible at telling us where in the sky it happened or what its "redshift" is.
• The Solution: This paper proposes a clever trick. Instead of trying to match one specific galaxy to one specific gravitational wave, the authors suggest looking at the patterns of both groups.

The Analogy: The "Ghostly" Overlap

Think of the universe as a layered cake.

1. Layer A (Galaxies): We have a very detailed map of this layer. We know exactly how many "lanterns" are in each slice of the cake.
2. Layer B (Gravitational Waves): We have a blurry map of this layer. We know the ripples are there, but the edges are fuzzy.

The authors ask: "If we stack these two maps on top of each other, do the patterns line up?"

Because both galaxies and the black holes that create gravitational waves live in the same "dark matter" neighborhoods, their patterns should match up perfectly only if we use the correct ruler to measure distance.

• The "Magic" Connection: If you guess the wrong distance for the gravitational waves, the patterns won't line up with the galaxy map. They will look like mismatched puzzle pieces.
• The Goal: By finding the distance ruler that makes the two maps fit together perfectly, the scientists can calculate the Hubble Constant ($H_0$). This is a number that tells us how fast the universe is expanding.

The Tools: Next-Generation "Ears" and "Eyes"

The paper looks at the future, specifically the next generation of tools:

• The "Eyes" (Euclid Survey): A powerful space telescope that will take pictures of billions of galaxies.
• The "Ears" (3G Detectors): Future gravitational wave detectors (like the Einstein Telescope and Cosmic Explorer) that will be so sensitive they can hear millions of black hole collisions, not just the few we hear today.

The authors used a computer simulation (a "Fisher matrix," which is like a statistical crystal ball) to predict how well these future tools would work together.

The Results: A Perfect Match

Here is what the paper found:

1. Super-Precision: By combining the galaxy map and the gravitational wave map, they can measure the expansion of the universe with 1% precision (or even better).
   • Analogy: If the universe were a 100-meter track, current methods might guess the length is between 95 and 105 meters. This new method narrows it down to 99 to 101 meters.
2. Better Together: Using just the galaxy map or just the gravitational wave map gives okay results. But putting them together is like having a superpower; it improves the accuracy by a factor of 10.
3. The "Sweet Spot": The method works best at a specific distance (redshift) where the gravitational wave detectors can pinpoint the location well enough, but the galaxies are still bright enough to be seen.

Why This Matters (According to the Paper)

Currently, there is a disagreement in the scientific community about how fast the universe is expanding (the "Hubble Tension"). Some methods say it's fast; others say it's slow.

This paper claims that by using this "cross-correlation" technique (matching the patterns of galaxies and gravitational waves), we can get a very accurate, independent answer to this mystery. It doesn't rely on guessing the properties of the black holes; it relies on the statistical "clumping" of the universe itself.

Summary in One Sentence

This paper predicts that by using next-generation telescopes and gravitational wave detectors to match the "fingerprint" of galaxy clusters with the "fingerprint" of black hole collisions, we can measure the speed of the universe's expansion with unprecedented accuracy, solving a major puzzle in modern cosmology.

Technical Summary

Technical Summary: Cosmology with the angular cross-correlation of gravitational-wave and galaxy catalogs

Problem and Motivation
The spatial clustering of galaxies has long served as a primary probe for cosmology, mapping the large-scale structure of the Universe. However, gravitational waves (GWs) from compact binary mergers offer a complementary tracer: they provide direct measurements of luminosity distance ($d_L$) but lack direct redshift ($z$) information. While galaxies are observed in redshift space, GWs are observed in distance space. The authors investigate the potential of cross-correlating these two distinct tracers to constrain the distance-redshift relation, specifically targeting the Hubble constant ($H_0$) and matter density ($\Omega_m$). This approach leverages the fact that the cross-correlation signal is maximized only when the distance bins of the GW catalog align with the redshift bins of the galaxy catalog according to the correct cosmological model.

Methodology
The study employs a Fisher matrix formalism to forecast the statistical power of third-generation (3G) GW detectors (Einstein Telescope and Cosmic Explorer) combined with the Euclid galaxy survey.

• Observables: The analysis utilizes the angular power spectra ($C_\ell$) for galaxy-galaxy auto-correlations, GW-GW auto-correlations, and GW-galaxy cross-correlations. These are computed in tomographic bins of redshift (for galaxies) and luminosity distance (for GWs).
• Signal Modeling: The theoretical framework follows perturbation theory in both redshift and luminosity distance spaces. The authors include dominant contributions: density fluctuations, redshift-space distortions (RSD) or luminosity-distance space distortions (LSD), and gravitational lensing. The signal is modeled using the Limber approximation.
• Noise and Systematics: The forecast incorporates shot noise and the limited angular resolution of GW detectors. The latter is modeled as an exponential damping of the power spectrum at high multipoles ($\ell$), determined by the sky localization uncertainty ($\Delta\Omega$).
• Parameter Marginalization: A key aspect of the methodology is the "agnostic" treatment of nuisance parameters. The authors marginalize over:
  • Tracer biases ($b_g$ for galaxies, $b_{GW}$ for GWs), allowing a free bias parameter per redshift/distance bin.
  • Primordial power spectrum parameters ($A_s, n_s$) and baryon density ($\Omega_b h^2$).
  • The fiducial cosmology is set to Planck 2018 values.
• Survey Specifications:
  • Galaxies: The Euclid photometric sample ($\sim 1.6 \times 10^9$ galaxies, $0 < z < 2$) is used as the fiducial case, divided into 13 equally populated redshift bins.
  • GWs: A network consisting of two L-shaped Einstein Telescope (ET) detectors and two Cosmic Explorer (CE) detectors is simulated over a 5-year observing period. The analysis uses realistic compact binary coalescence population models and error forecasts based on the gwfast code.
  • Binning: Distance bins for GWs are derived from the galaxy redshift bins using the fiducial cosmology, with additional high-distance bins added to cover the full GW detection horizon.

Key Contributions

1. Comprehensive Nuisance Treatment: Unlike previous studies that often fixed tracer biases or cosmological parameters, this work explicitly marginalizes over a wide range of astrophysical and cosmological nuisance parameters, including a fully free bias model per bin. This demonstrates the robustness of the technique under realistic uncertainty.
2. Synergy Quantification: The paper quantifies the information gain from combining galaxy auto-correlations with GW-galaxy cross-correlations, showing how the different degeneracy directions in the $H_0$-$\Omega_m$ plane complement each other.
3. Binning and Network Analysis: The study systematically investigates the impact of different binning strategies and compares various 3G network configurations (e.g., ET 2L + 2CE) and observing times (1, 5, and 10 years).
4. Detection Bias: The authors re-evaluate the detectability of GW clustering bias under model-independent assumptions, confirming its potential measurability.

Results

• Signal-to-Noise Ratio (SNR): The cross-correlation between the Euclid photometric sample and a 3G GW network (2 ET + 2 CE) over 5 years yields a total SNR of approximately 21.2. The GW-GW auto-correlation alone has a much lower SNR (~1.8), indicating that the cross-correlation is the primary driver of cosmological constraints in this setup.
• Cosmological Constraints:
  • The GW-galaxy cross-correlation alone constrains $H_0$ to a precision of $\sim 6\%$.
  • The galaxy-galaxy auto-correlation alone constrains $H_0$ to $\sim 15\%$ (under the conservative multipole cuts used to exclude non-linear scales).
  • Combined: Combining both probes improves the constraint on $H_0$ to $\sim 1\%$ (percent or sub-percent precision), representing an improvement of a factor of $\sim 6$ over the cross-correlation alone and $\sim 10$ over either probe in isolation.
• Dependence on Configuration: The precision depends on the binning strategy, the specific detector network configuration, and the observing time. The analysis highlights that multiple interferometers with accurate sky localization are essential for this performance.
• Spectroscopic vs. Photometric: While the photometric sample is preferred for its sheer number of sources (reducing shot noise), the paper notes that spectroscopic samples (like Euclid's or SKA Phase II) offer different trade-offs, which are discussed in the context of bias modeling.

Significance and Claims
The paper claims that the angular cross-correlation of GW and galaxy catalogs is a viable and competitive method for measuring cosmic expansion in the era of 3G detectors. The technique is shown to deliver robust constraints on $H_0$ and $\Omega_m$ even when marginalizing over significant astrophysical uncertainties (such as tracer bias) and cosmological nuisance parameters. The authors emphasize that this method is distinct from "dark siren" approaches (which rely on identifying host galaxies for individual events) and offers a complementary systematic error profile. The results suggest that 3G detectors, in synergy with large-scale galaxy surveys like Euclid, can provide independent, high-precision measurements of the Hubble constant, potentially helping to resolve current tensions in cosmological parameters.