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Toward testing gravity with LSST using $E_G$

This paper updates the theoretical framework for the gravity test statistic $E_G$ to incorporate LSST and DESI data, proposing a method to consistently handle uncertainties in matter density that enables a model-agnostic test capable of potentially rejecting General Relativity under certain modified gravity scenarios.

Original authors: C. D. Leonard, S. Alam, R. Mandelbaum, M. M. Rau, S. Singh, C. M. A. Zanoletti, the LSST Dark Energy Science Collaboration

Published 2026-03-24

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CC BY 4.0

Original authors: C. D. Leonard, S. Alam, R. Mandelbaum, M. M. Rau, S. Singh, C. M. A. Zanoletti, the LSST Dark Energy Science Collaboration

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 trampoline. In our everyday experience, we know that if you place a heavy bowling ball (a galaxy) on that trampoline, it curves the fabric, causing smaller marbles (other galaxies) to roll toward it. This is how gravity works according to Einstein's General Relativity (GR).

But what if the trampoline fabric behaves differently on a cosmic scale? What if there's a hidden force or a different set of rules making the universe expand faster or pull things together in weird ways? This is the big question cosmologists are trying to answer.

This paper is a "forecast" (a prediction of the future) about how a massive new telescope project, LSST (the Rubin Observatory), combined with a galaxy-mapping robot, DESI, will test these rules. Specifically, they are using a clever statistical tool called $E_G$ (pronounced "E-G").

Here is the breakdown of their work using simple analogies:

1. The Detective's Tool: What is $E_G$ ?

Imagine you are a detective trying to figure out if a crime was committed by a known suspect (General Relativity) or a new, unknown criminal (Modified Gravity).

Usually, detectives look at one clue at a time. But $E_G$ is a super-clue. It combines two different types of evidence:

Galaxy Clustering: How galaxies are grouped together (like seeing how people cluster at a party).
Weak Lensing: How light bends around those galaxies (like seeing how a funhouse mirror distorts the reflection of the people).

By mixing these two clues together, $E_G$ creates a measurement that is immune to certain "noise."

The Noise Problem: Usually, when we look at galaxies, we don't know exactly how "clumpy" they are naturally. It's like trying to measure the weight of a person, but you don't know if they are wearing a heavy coat or a light t-shirt. This "coat" is called galaxy bias.
The $E_G$ Magic: The beauty of $E_G$ is that it mathematically cancels out the "coat." It isolates the pure gravity signal, allowing us to test the laws of physics without being confused by how galaxies behave.

2. The New Challenge: The "Scale" Problem

In the past, scientists only looked at very large, smooth distances (like looking at a forest from a helicopter). At those distances, the rules are simple.
But the new LSST telescope is so powerful it will see billions of galaxies. This means they can look at smaller, messier distances (like walking through the trees).

The Issue: On these smaller scales, galaxies get messy. They interact, merge, and form complex shapes. This is "non-linear" physics.
The Solution in the Paper: The authors realized that if they use the $E_G$ tool correctly, the messiness of the galaxies actually cancels itself out again! It's like if the "coat" on the left side of the equation is exactly the same as the "coat" on the right side; they cancel, leaving the pure gravity signal clean. They proved that even with the messy data LSST will provide, $E_G$ remains a clean test of gravity.

3. The Hidden Trap: The "Omega" Problem

Here is the most critical part of the paper, and it's a bit like a recipe for a cake.

To test if the gravity rules are broken, you need to compare your measurement to a "perfect cake" recipe (the prediction of General Relativity).
But the recipe depends on one ingredient: $\Omega_M$ (the amount of matter in the universe).
The Trap: If you don't know exactly how much "flour" ( $\Omega_M$ ) is in the universe, you can't be sure if your cake is wrong because the recipe (gravity) is bad, or just because you used the wrong amount of flour.

In the past, we didn't know the amount of flour very well. But now, with better data, the uncertainty in the flour measurement is almost as big as the uncertainty in the cake itself. If we aren't careful, we might think gravity is broken when it's actually just our estimate of the matter density that's slightly off.

4. The Proposed Fix: The "Posterior Predictive Test"

The authors propose a new way to handle this uncertainty, which they call a Posterior Predictive Test.

The Analogy: Imagine you are baking a cake to test a recipe. Instead of just baking one cake and comparing it to the picture, you bake 1,000 cakes.
In each of these 1,000 cakes, you slightly vary the amount of flour (based on what we think we know about $\Omega_M$ ).
Then, you look at your real measurement. Does it look like one of the 1,000 cakes you baked?
- Yes: The recipe (General Relativity) is likely correct.
- No: Even after accounting for all the possible flour amounts, your real cake looks nothing like the ones from the recipe. The recipe is wrong! Gravity is modified.

5. The Big Conclusion: It's All About the "Flour"

The authors ran simulations to see how well LSST and DESI would work with this new method. Their findings were surprising:

The Telescope isn't the Limit: Having more data (LSST Year 10 vs. Year 1) helps, but not as much as you'd think.
The "Flour" Knowledge is King: The most important factor is how well we know the amount of matter in the universe ( $\Omega_M$ $Ω_{M}$ ) before we start.
- If we use a "wobbly" estimate of the flour (from older surveys), we will likely fail to detect new gravity theories, even if they exist. We'll just think our flour measurement was wrong.
- If we use a "precise" estimate of the flour (from the Planck satellite's cosmic microwave background data), we have a very high chance of detecting if gravity is broken.

Summary

This paper is a guidebook for the future. It tells us:

$E_G$ is a powerful tool that can test gravity even with messy, small-scale data.
We have a new method to handle the uncertainty of how much matter is in the universe.
The Catch: To successfully prove that Einstein's gravity is wrong (or right) using this new telescope, we must have a very precise, independent measurement of the universe's matter density from other sources (like the Cosmic Microwave Background). Without that precise "flour" measurement, the most powerful telescope in the world might not be enough to solve the mystery.

1. Problem Statement

The paper addresses the challenge of testing General Relativity (GR) on cosmological scales using the upcoming LSST (Rubin Observatory) and DESI (Dark Energy Spectroscopic Instrument) surveys. Specifically, it focuses on the $E_G$ statistic, a model-independent summary statistic designed to test gravity by combining galaxy-galaxy weak lensing and galaxy clustering.

Three primary challenges are identified:

Covariance Modeling: Previous Stage III surveys often assumed independence between the constituent probes of $E_G$ (lensing, clustering, and redshift-space distortions). However, with the high statistical power of Stage IV data (LSST+DESI), these components share the same lens sample, leading to non-negligible cross-covariances that must be accurately modeled.
Nonlinear Effects: Standard approaches often restrict analysis to linear scales to avoid uncertainties in nonlinear structure formation and galaxy bias. However, LSST's statistical power makes linear-only cuts suboptimal. The paper investigates whether $E_G$ remains robust against nonlinear effects and how to handle nonlinear galaxy bias.
Theoretical Dependence on $\Omega_M$ : The GR prediction for $E_G$ depends on the present-day matter density parameter, $\Omega_M$ . As measurement uncertainties shrink, the theoretical uncertainty due to the prior knowledge of $\Omega_M$ becomes a limiting factor. A consistent statistical framework is needed to incorporate this uncertainty without biasing the test.

2. Methodology

The authors employ a forecasting approach using simulations and analytic methods to evaluate the power of LSST+DESI to detect deviations from GR.

Survey Configuration:
- Lenses: DESI Luminous Red Galaxies (LRGs) in the redshift range $z \in [0.4, 0.8]$ .
- Sources: LSST weak lensing source galaxies (simulating Year 1 and Year 10 data).
- Overlap: 5000 deg $^2$ .
Theoretical Framework:
- Gravity Models: The study forecasts performance against two specific modified gravity theories: Hu-Sawicki $f(R)$ gravity and normal-branch Dvali-Gabadadze-Porrati (nDGP) gravity.
- Covariance Estimation:
  - Analytic: Derived for auto- and cross-covariances of the constituent observables ( $\Upsilon_{gm}$ , $\Upsilon_{gg}$ , and $\beta$ ).
  - Simulation-based: Used N-body simulations (MedRes) with Halo Occupation Distribution (HOD) modeling to generate mock catalogs. A jackknife resampling method was used to estimate the full covariance matrix, including cross-terms between lensing and clustering. Shape noise was added analytically to the simulation results.
- Nonlinear Bias Handling: The authors tested the impact of nonlinear galaxy bias on scale cuts. They proposed a correction factor, $C_b(r_p)$ , based on the ratio of clustering to lensing signals, to correct the measured $E_G$ for nonlinear bias effects, allowing the inclusion of smaller scales.
Statistical Test (Posterior Predictive Test):
- To address the $\Omega_M$ dependence, the authors introduced a Bayesian posterior predictive test.
- Procedure:
  1. Generate simulated data realizations from an alternative gravity theory.
  2. Fit a constant $E_G$ model (GR prediction) to this data, treating $\Omega_M$ as a free parameter with a specific prior (either Stage III 3 $\times$ 2pt or Planck CMB).
  3. Generate "replicated data" from the posterior distribution of the fitted $\Omega_M$ .
  4. Compare the original simulated data against the distribution of replicated data. If the original data falls outside the 95% confidence region, GR is rejected.

3. Key Contributions

Updated Covariance Framework: The paper provides the first comprehensive covariance modeling for $E_G$ in a Stage IV context, demonstrating that cross-covariances between $\Upsilon_{gm}$ , $\Upsilon_{gg}$ , and $\beta$ are non-negligible and must be included.
Robustness to Nonlinearity: The study shows that while nonlinear galaxy bias significantly affects small-scale $E_G$ measurements, the proposed correction factor ( $C_b$ ) effectively recovers the linear signal, allowing for the inclusion of more data points without introducing bias.
Posterior Predictive Framework for $\Omega_M$ : The authors propose and implement a rigorous statistical method to consistently incorporate uncertainty in $\Omega_M$ into gravity tests, moving beyond simple heuristic comparisons.
Quantification of Prior Sensitivity: The work quantifies how the choice of prior on $\Omega_M$ dictates the sensitivity of $E_G$ tests, a critical insight for future survey planning.

4. Key Results

Covariance Structure: The correlation matrices reveal significant off-diagonal terms between the constituent probes, particularly between $\Upsilon_{gm}$ and $\beta$ , and $\Upsilon_{gg}$ and $\beta$ . This contradicts the common Stage III assumption of independence.
Scale Cuts:
- Without bias correction, nonlinear galaxy bias forces the removal of small-scale data points ( $r_p < 30$ Mpc/h) to maintain linearity.
- With the proposed nonlinear bias correction, the authors can retain the full data vector (down to $r_p \approx 1.4$ Mpc/h) without violating linearity assumptions, significantly increasing statistical power.
Testing Power (Rejection Rates):
- Prior Dependence: The ability to reject GR is overwhelmingly determined by the precision of the $\Omega_M$ $Ω_{M}$ prior, not the survey depth (Year 1 vs. Year 10).
  - Weak Prior (DES Y3 3 $\times$ 2pt): Rejection rates for modified gravity scenarios are extremely low ( $\sim 1-2\%$ ), even for large deviations. The data can be "explained" by shifting $\Omega_M$ within the wide prior.
  - Strong Prior (Planck CMB): Rejection rates increase dramatically. For the most extreme $f(R)$ case ( $f_{R0} = 10^{-4}$ ), the rejection rate jumps to >92%. For nDGP, rates reach $\sim 45-55\%$ .
- Survey Depth: Moving from LSST Year 1 to Year 10 yields only marginal improvements in rejection rates compared to the impact of tightening the $\Omega_M$ prior.
Specific Models:
- The study finds that for the considered theories (which are scale-independent in $E_G$ ), the test is most sensitive to large deviations (e.g., $f_{R0} = 10^{-4}$ ).
- The results suggest that for realistic deviations from GR, the current methodology may struggle to reject GR unless the $\Omega_M$ prior is extremely tight.

5. Significance

This paper is a critical "reality check" for the community planning to use $E_G$ with LSST and DESI. Its main significance lies in shifting the focus from simply collecting more data to improving theoretical priors.

Paradigm Shift: It demonstrates that for consistency tests like $E_G$ , the limiting factor is no longer just statistical noise from the survey, but the theoretical uncertainty in cosmological parameters (specifically $\Omega_M$ ).
Methodological Standard: The proposed posterior predictive framework provides a robust, generalizable tool for testing gravity that accounts for parameter degeneracies, applicable beyond just LSST+DESI.
Future Directions: The authors conclude that to maximize the power of $E_G$ $E_{G}$ as a consistency check, future analyses must either:
1. Utilize extremely tight priors on $\Omega_M$ (e.g., from CMB or future 21cm surveys).
2. Develop methods that are less dependent on specific cosmological parameters.
3. Consider that if the true deviation from GR is scale-dependent, $E_G$ might be even more powerful than the scale-independent scenarios tested here.

In summary, while LSST+DESI will provide unprecedented data, the paper argues that the success of $E_G$ as a gravity test hinges on the precision of our knowledge of the background cosmology ( $\Omega_M$ ) and the rigorous application of bias corrections and covariance modeling.

1. The Detective's Tool: What is EGE_GEG​?