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Gravitational waves and galaxies cross-correlations: a forecast on GW biases for future detectors

This paper forecasts that cross-correlating gravitational waves from future third-generation detectors with galaxy surveys will enable precise measurements of binary black hole clustering bias and magnification lensing effects, provided that degeneracies between bias parameters are resolved through optimal redshift selection or external constraints.

Original authors: Stefano Zazzera, José Fonseca, Tessa Baker, Chris Clarkson

Published 2026-07-27

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CC BY 4.0

Original authors: Stefano Zazzera, José Fonseca, Tessa Baker, Chris Clarkson

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 as a giant, invisible ocean of dark matter. We can't see this ocean directly, but we know it's there because it acts like a cosmic scaffold, holding everything else together. For decades, astronomers have used galaxies as their "buoys" to map this hidden structure. By counting how many galaxies clump together in certain areas, they can infer the shape of the dark matter beneath them. But recently, a new kind of cosmic messenger has arrived: gravitational waves. These are ripples in the fabric of space-time, created when massive objects like black holes crash into each other. Since their first detection in 2015, these waves have opened a new window into the cosmos. However, unlike galaxies, which glow with light, gravitational waves are silent and invisible to our eyes; we only "hear" them. The big question scientists are asking is: Do these black hole mergers live in the same neighborhoods as the galaxies we see? If they do, they can act as new buoys to help us map the dark matter ocean even more precisely. If they don't, it might mean black holes are wandering alone in the dark, or perhaps even formed in a completely different way than we thought.

This paper is a forecast, a kind of "crystal ball" study, looking ahead to the future of gravitational wave astronomy. The authors, Stefano Zazzera and his team, are asking: "If we build the next generation of super-sensitive gravitational wave detectors, like the Einstein Telescope (ET) and Cosmic Explorer (CE), how well will we be able to measure the 'clustering bias' of black holes?" In simple terms, "clustering bias" is a number that tells us how much more likely black holes are to hang out in crowded areas compared to the average dark matter. A bias of 1 means they follow the crowd perfectly; a higher number means they are very picky and only live in the densest party zones. The team also wants to see if we can spot subtle effects where the gravity of other objects bends the gravitational waves (like a lens bending light) or changes how we count them as they move through time.

The researchers used a mathematical tool called the "Fisher formalism" to simulate what would happen if we combined data from these future gravitational wave detectors with massive galaxy surveys like the Vera Rubin Observatory (LSST) and Euclid. They didn't just look at one scenario; they tested different combinations of current and future detectors against various galaxy surveys to see which team-up would give the best results.

Here is what they found. If we stick with our current detectors (like the ones running the O4 and O5 observing runs), the results are a bit fuzzy. The team predicts that with current technology, the error in measuring how black holes cluster could be as high as 50%. That's like trying to guess the weight of a person while wearing thick, blurry glasses. However, the picture changes dramatically with the third-generation detectors. When they simulated cross-correlating data from the Einstein Telescope with the LSST galaxy survey, the precision skyrocketed. They forecast that these future instruments could measure the clustering bias with an error of just 2.5%. That's like switching from blurry glasses to a high-definition microscope.

The study also looked at other tricky effects. They found that with the Einstein Telescope and LSST, we could measure the "magnification lensing" effect on gravitational waves with about 3% precision. This is exciting because it would allow us to test the laws of gravity on a cosmic scale in a whole new way. However, there is a catch. The team discovered a "degeneracy," which is a fancy word for a confusing mix-up. The math shows that the effect of magnification and the effect of how the black hole population evolves over time are tangled together. It's like trying to figure out how much of a cake's sweetness comes from sugar and how much comes from honey when they are mixed together perfectly; you can't easily separate them. Because of this, measuring both at the same time is very hard. The authors suggest that if we assume we already know one of these values, or if we focus our observations on a specific "sweet spot" of the universe (between redshifts 1 and 2.5), we can untangle the mess and get precise measurements.

In short, this paper suggests that while our current tools are a bit too rough to give us a clear picture of how black holes cluster, the future is incredibly bright. By the time the Einstein Telescope comes online around 2035, we should be able to map the dark matter distribution with black holes as our guides with incredible precision, potentially revealing secrets about how these cosmic giants are born and where they choose to live.

Technical Summary: Gravitational waves and galaxies cross-correlations: a forecast on GW biases for future detectors

Problem Statement
Since the first detection of gravitational waves (GWs) in 2015, they have emerged as powerful cosmological probes. However, GW signals from binary black hole (BBH) mergers without electromagnetic counterparts provide distance information only in the form of luminosity distance ( $D_L$ ), rather than redshift ( $z$ ). This necessitates the construction of number count fluctuations in luminosity distance space (LDS), which differs significantly from the standard redshift space (RS) analysis used for galaxy surveys.

In LDS, the observed number count fluctuation depends on three distinct bias parameters: clustering bias ( $b$ ), magnification bias ( $s$ ), and evolution bias ( $b_e$ ). While galaxy surveys have well-characterized these biases, the clustering bias of GWs remains largely unconstrained. This parameter is crucial as it links BBHs to the underlying dark matter distribution, helping to distinguish between astrophysical black holes residing in dark matter halos and other potential populations (e.g., primordial black holes). Furthermore, the magnification and evolution biases encode information about the BBH population's merger rate and chirp mass distribution. The challenge lies in measuring these parameters with sufficient precision, particularly given the limited number of events expected from current (LVK O4/O5) and near-future detectors compared to the millions expected from third-generation (3G) instruments.

Methodology
The authors employ a Fisher matrix formalism to forecast the precision of measuring GW bias parameters through cross-correlations between GW events and galaxy surveys.

Theoretical Framework: The study utilizes the number count fluctuation equations for tracers in both redshift space (galaxies) and luminosity distance space (GWs). The GW fluctuation $\Delta_{GW}$ includes terms for matter density contrast, Doppler effects, Luminosity Distance Space Distortions (LSD), and lensing magnification. The amplitudes of these terms depend on the clustering, magnification, and evolution biases.
Survey Modeling:
- GWs: The authors model the BBH merger rate using a standard Madau-Dickinson rate and a Power-Law + Peak chirp mass distribution. They consider current/future LVK runs (O4, O5) and third-generation detectors (Einstein Telescope - ET, Cosmic Explorer - CE). The magnification and evolution biases are derived from the detector's signal-to-noise ratio (SNR) threshold and the merger rate.
- Galaxies: Four galaxy survey types are modeled: DESI Bright Galaxy Sample (BGS), Euclid spectroscopic, LSST photometric, and the futuristic Megamapper. Each survey has specific redshift ranges and sky coverages.
Cross-Correlation Analysis: The authors compute the angular power spectra ( $C_\ell$ ) for auto-correlations (GW-GW, Galaxy-Galaxy) and cross-correlations (GW-Galaxy). They account for shot noise, sky fraction, and the beam smearing effect due to GW sky localization uncertainty (assumed to be $5^\circ$ ).
Fisher Forecast: The Fisher matrix is constructed to constrain the clustering bias amplitude ( $B$ ) and slope ( $\alpha$ ) of the model $b = B(1+z)^\alpha$ . In the primary analysis, galaxy biases are fixed to known values to isolate GW constraints. The analysis also marginalizes over standard cosmological parameters ( $H_0, \Omega_b, \Omega_{cdm}, A_s, n_s$ ).

Key Contributions

Multi-Tracer Forecast: The paper provides a comprehensive forecast of GW bias constraints using a multi-tracer approach, explicitly comparing current (O4/O5) and future (ET) detectors against various galaxy surveys (DESI, Euclid, LSST, Megamapper).
Relativistic Effects: The study includes a detailed forecast for the detectability of relativistic corrections in the GW number counts, specifically the magnification lensing effect, LSD, and Doppler terms.
Bias Degeneracy Analysis: The authors investigate the degeneracy between magnification and evolution biases, demonstrating how fixing one parameter can significantly improve the constraints on the other.
Redshift Dependence: The analysis maps the precision of bias measurements as a function of redshift, identifying optimal redshift ranges for future observations.

Results

Clustering Bias Precision:
- Current Detectors: Cross-correlations involving current or near-future detectors (e.g., O4 $\times$ DESI BGS) yield limited precision, with measurement errors on the clustering bias amplitude as high as $\sim50\%$ . Even O5 $\times$ DESI BGS only reaches $\sim30\%$ precision.
- Third-Generation Detectors: The precision improves dramatically with 3G detectors. An ET-like experiment cross-correlated with an LSST-like survey achieves a measurement error of $\sim2.5\%$ on the clustering bias amplitude after 5 years of observation. An ET $\times$ Megamapper combination yields percent-level precision but with slightly higher errors due to smaller redshift overlap.
- Redshift Evolution: For an ET $\times$ LSST combination, the uncertainty on the clustering bias is lowest just before $z=1$ (due to high source counts) and remains below 5% ( $1\sigma$ ) up to $z < 1.5$ even with only 1 year of data.
Relativistic Effects:
- Lensing: The magnification lensing effect is highly detectable. With 1 year of ET $\times$ LSST data, the lensing amplitude can be measured with $\sim10\%$ precision, improving to $\sim3\%$ with 5 years of data.
- LSD and Doppler: The LSD effect is measurable but with lower precision ( $\sim10\%$ error after 5 years). The Doppler and other subdominant GR corrections are likely unmeasurable due to their small amplitude in the power spectrum.
Magnification and Evolution Biases:
- When marginalizing over both magnification ( $s_{GW}$ ) and evolution ( $b_{GW}^e$ ) biases simultaneously, large uncertainties persist due to strong degeneracy.
- However, if one bias is assumed to be known (fixed), the other can be measured with percent-level precision, particularly in the redshift range $1 < z < 2.5$ . This optimal range balances the increasing lensing signal at higher $z$ against the rising shot noise at $z > 2$ .

Significance
The paper concludes that cross-correlating GWs in luminosity distance space with galaxies in redshift space offers a powerful new avenue for cosmology. While current detectors lack the statistical power to constrain GW biases precisely, the advent of third-generation detectors like the Einstein Telescope will enable percent-level measurements of the clustering bias. This will allow for a direct probe of how BBHs populate dark matter halos, potentially distinguishing between astrophysical and primordial origins. Furthermore, the ability to measure the lensing magnification effect on GWs to $\sim3\%$ precision opens new opportunities to test General Relativity and modified gravity theories on large scales. The study highlights that addressing the degeneracy between magnification and evolution biases is critical for fully exploiting these future datasets, suggesting that optimal constraints will be found in the $1 < z < 2.5$ redshift range.

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