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The Redshift-Space Cluster-Galaxy Cross-Correlation as a Probe of Gravity on Cosmic Scales

This paper presents a Fisher matrix forecast demonstrating that the Dark Energy Spectroscopic Instrument (DESI) Bright Galaxy Survey can constrain departures from General Relativity to a precision of $\sigma(F_{\rm G}) = 0.033$ at $z \sim 0.3$ by analyzing the redshift-space cluster-galaxy cross-correlation function, a capability sufficient to distinguish modified gravity models like $f(R)$ and Galileon from GR at the $\gtrsim 4\sigma$ level.

Original authors: Nicole Marcelina Gountanis, David H. Weinberg, Andrés N. Salcedo, Claire Lamman

Published 2026-10-30

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CC BY 4.0

Original authors: Nicole Marcelina Gountanis, David H. Weinberg, Andrés N. Salcedo, Claire Lamman

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

For decades, the most pressing mystery in cosmology has been the accelerating expansion of the universe. Scientists know the cosmos is stretching out faster and faster, but they disagree on why. One possibility is that the universe is filled with a mysterious, repulsive energy that pushes space apart. The other is that our understanding of gravity itself is incomplete. Albert Einstein's theory of General Relativity has passed every test thrown at it, from the motion of planets to the collision of black holes, but it might fail on the vast scales of the cosmos. If gravity behaves differently on these huge scales, it would rewrite the laws of physics. To find the answer, researchers need to watch gravity in action, specifically how it pulls matter together to form the massive structures that dot the universe.

The universe is not a smooth, empty void; it is a web of filaments and clumps. At the intersections of this web sit galaxy clusters, the most massive structures in existence, containing hundreds or thousands of galaxies bound together by gravity. As galaxies fall toward these massive clusters, they speed up, creating a flow of matter known as infall. In the standard theory of gravity, the speed of this infall is precisely determined by the mass of the cluster. However, if gravity is stronger than Einstein predicted in certain environments, these galaxies would fall faster. By measuring how fast galaxies are moving toward clusters, astronomers can test whether gravity is behaving exactly as expected or if it is showing signs of a "fifth force" that modifies the rules of the game.

A team of researchers has now used powerful computer simulations to forecast how well a new, massive telescope survey will be able to perform this test. The study focuses on the Dark Energy Spectroscopic Instrument, or DESI, which is currently mapping the positions and velocities of millions of galaxies. The researchers built a detailed virtual universe using the AbacusSummit simulations, a suite of supercomputer models that track the motion of dark matter and galaxies. They populated this virtual world with fake galaxies and clusters, mimicking the specific types of objects that DESI will observe. Their goal was to see if the data from DESI would be precise enough to detect even tiny deviations in the speed of infalling galaxies, which would signal a breakdown of General Relativity.

The researchers focused on a specific way of looking at the data: the redshift-space cross-correlation between galaxies and clusters. In astronomy, "redshift" is a measure of how fast an object is moving away from us due to the expansion of the universe, but it is also affected by the object's own motion through space. When galaxies fall toward a cluster, their motion distorts the apparent shape of the cluster in the data, stretching or squashing it depending on the direction of the flow. This distortion is a direct fingerprint of the gravitational pull at work. The team simulated this effect, creating a virtual map of how galaxies and clusters are arranged and how their motions warp that arrangement. They then tested how sensitive this map would be to changes in the strength of gravity, represented by a single factor that could speed up or slow down the infall.

The results of the forecast are promising. The study suggests that the DESI survey will be able to measure the strength of gravity in these cluster infall regions with a precision of about 3.3 percent. This level of accuracy is a significant leap forward from previous attempts using older surveys, which were limited by statistical noise and could not rule out many alternative theories of gravity. With this new precision, the researchers found that DESI could distinguish between standard General Relativity and several popular theories of modified gravity, such as f(R) gravity and Galileon gravity, with a high degree of confidence. Specifically, the data would be strong enough to reject these alternative models if they were true, or confirm them if the universe is indeed behaving differently than Einstein predicted.

A major challenge in this type of research is separating the signal of gravity from the messy details of how galaxies form and live inside dark matter halos. The researchers had to account for many uncertainties, such as how galaxies are distributed within clusters and how accurately we can estimate the mass of a cluster based on what we see. They found that while these uncertainties complicate the picture, they do not hide the signal of gravity. The most significant obstacle turned out to be the uncertainty in the relationship between a cluster's visible properties and its true mass. If this relationship is not well understood, it can mimic the effect of modified gravity. However, by combining the measurement of galaxy motions with the total number of clusters observed, the researchers showed that this confusion can be resolved, allowing the true gravitational signal to emerge clearly.

The study also looked at how well the survey would measure the fundamental properties of the universe, such as the amount of matter it contains and how clumpy that matter is. The forecast indicates that the same data used to test gravity will also provide very tight constraints on these cosmological parameters, with an accuracy of about 1 percent. Crucially, the measurements of gravity and the measurements of the universe's composition are not tangled together; they can be determined independently. This means that scientists will not have to guess the amount of matter in the universe to test gravity, nor will they have to assume a specific theory of gravity to measure the universe's contents.

Looking ahead, the researchers emphasize that this forecast is a simulation of what will happen, not a measurement of what has already happened. The next step is to apply these methods to the real data that DESI is currently collecting. The team plans to build more sophisticated tools to interpret the data, including better models for how clusters are selected and how galaxies move within them. They also intend to test these ideas across different distances in the universe, checking if the strength of gravity changes over time. If the predictions hold up, the DESI survey will provide one of the most direct and powerful tests of gravity ever conducted, probing the invisible forces that shape the cosmos on scales where Einstein's theory might finally show its limits.

Problem Statement
Testing General Relativity (GR) on cosmological scales requires probes sensitive to gravitational dynamics, particularly in regimes where modified gravity theories predict deviations from GR. While the origin of cosmic acceleration remains unknown, distinguishing between a cosmological constant and modified gravity often relies on measuring the growth of structure and the behavior of gravitational potentials. Previous analyses using the Sloan Digital Sky Survey (SDSS) attempted to measure the cluster infall velocity field via the redshift-space cluster–galaxy cross-correlation function, $\xi^s_{cg}(r_p, r_\pi)$ . However, these studies were limited by statistical uncertainties and the inability to model the observable directly within a parameter inference framework, preventing definitive tests of modified gravity models like $f(R)$ and Galileon gravity. The Dark Energy Spectroscopic Instrument (DESI) Bright Galaxy Survey (BGS) offers a significantly larger volume and higher galaxy number density, presenting an opportunity to revisit this analysis with reduced statistical errors and more sophisticated modeling.

Methodology
The authors perform a Fisher matrix forecast to estimate the precision with which the DESI BGS can constrain departures from GR. The analysis utilizes the following components:

Simulations and Mock Catalogs: Mock galaxy and cluster catalogs are constructed from the AbacusSummit N-body simulation suite. Galaxies are populated into halos using a Halo Occupation Distribution (HOD) framework, while clusters are selected via a log-normal mass–observable relation to simulate realistic observational scatter between proxy masses (e.g., richness) and true halo mass.
Parameterization of Modified Gravity: Instead of simulating specific modified gravity theories, the authors adopt a phenomenological approach. Departures from GR are parameterized by a single multiplicative factor, $F_G$ , which rescales the mean infall velocities of halos relative to the GR prediction ( $v_{rad} \to F_G v^{\text{GR}}_{rad}$ ). In standard modified gravity scenarios, $F_G > 1$ .
Observable: The primary observable is the redshift-space cluster–galaxy cross-correlation function, $\xi^s_{cg}(r_p, r_\pi)$ , measured in cylindrical coordinates. The data vector also includes the galaxy number density ( $\log n_{gal}$ ) and the cluster (halo) number density ( $\log n_{halo}$ ).
Fisher Forecast: Derivatives of the observables with respect to model parameters (HOD, mass–observable relation, cosmological parameters $\Omega_m$ and $\sigma_8$ , and $F_G$ ) are computed using centered finite differences. These derivatives, combined with a covariance matrix estimated from simulation subvolumes, are used to construct the Fisher information matrix. The analysis marginalizes simultaneously over all nuisance parameters, including HOD parameters, mass–observable scatter, and cosmological parameters.

Key Contributions and Results

Constraint on Modified Gravity: The forecast indicates that the DESI BGS can constrain the modified gravity parameter to $\sigma(F_G) = 0.033$ at redshift $z \sim 0.3$ . This precision is sufficient to distinguish representative $f(R)$ and Galileon gravity models from GR at a significance level of $\gtrsim 4\sigma$ , representing a substantial improvement over the $\sim 10–30\%$ uncertainties of previous SDSS-based analyses.
Degeneracies and Limitations: The strongest limitation on the $F_G$ constraint is a degeneracy with the scatter in the cluster mass–observable relation ( $\sigma_{\log M_c}$ ). This degeneracy arises because both parameters modulate the effective infall velocity signal. However, this degeneracy is partially broken by the independent constraint on cluster abundance provided by $\log n_{halo}$ . The analysis finds no strong degeneracy between $F_G$ and individual HOD parameters, demonstrating that the infall velocity signal can be isolated from uncertainties in the galaxy–halo connection.
Cosmological Constraints: The same data vector yields tight constraints on cosmological parameters: $\sigma(\Omega_m) = 0.010$ and $\sigma(\sigma_8) = 0.010$ . The correlation between these cosmological parameters and $F_G$ is moderate, suggesting that modified gravity constraints can be obtained without significant mutual degradation.
Information Content: The study compares the constraining power of the full redshift-space correlation function against a direct measurement of the mean radial infall velocity profile, $v_{rad}(r)$ . The results show that $\xi^s_{cg}$ recovers most of the information encoded in the infall profile; replacing $\xi^s_{cg}$ with $v_{rad}(r)$ improves the $F_G$ constraint only marginally (from 0.033 to 0.029) but significantly weakens constraints on cluster selection parameters.
Role of Number Densities: Including the number densities in the data vector is critical. While $\log n_{gal}$ provides moderate leverage, $\log n_{halo}$ is the dominant factor in tightening constraints on cluster selection parameters and mitigating the $F_G$ – $\sigma_{\log M_c}$ degeneracy.

Significance
The paper establishes the potential of DESI BGS cluster–galaxy redshift-space correlations to test gravitational physics on quasi-linear scales. The cluster infall regime occupies a physically distinct region in gravity tests: it lies beyond the screened, virialized cluster interior where fifth forces are suppressed, yet within scales where coherent infall flows are strong. In this environment, departures from GR are predicted to be most pronounced.

The authors argue that this method complements existing probes of modified gravity and weak-lensing measurements of cluster mass profiles. By combining $\xi^s_{cg}(r_p, r_\pi)$ measurements with weak gravitational lensing around the same cluster sample, future work could perform a "gravitational slip" test in the cluster infall regime. This would probe the ratio of the two scalar metric potentials governing light deflection and matter acceleration, offering a direct test of whether these potentials are equal as GR demands. The results position $\xi^s_{cg}(r_p, r_\pi)$ as a compelling tool for distinguishing between General Relativity and specific modified gravity theories using upcoming spectroscopic survey data.

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