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Constraining gravity with the decay rate of cosmological gravitational potential

By utilizing a recent measurement of the decay rate of the cosmological gravitational potential derived from DESI and Planck data, this study constrains four one-parameter modified gravity models and finds them fully consistent with General Relativity, with future full-sky surveys expected to double the current constraining power.

Original authors: Xinyi Zhao, Pengjie Zhang, Fuyu Dong

Published 2026-07-14

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CC BY 4.0

Original authors: Xinyi Zhao, Pengjie Zhang, Fuyu Dong

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, stretching trampoline. In the middle of this trampoline, we have heavy bowling balls (galaxies and clusters of matter) that create dips. According to the rules of General Relativity (GR)—the "gold standard" of gravity written by Einstein—these dips should stay pretty stable as the trampoline stretches out. But if the universe is accelerating (stretching faster and faster), something is pushing back. Is it a mysterious "Dark Energy" field, or are the rules of gravity themselves broken and needing a rewrite (Modified Gravity)?

A team of astronomers led by Xinyi Zhao, Pengjie Zhang, and Fuyu Dong decided to test these rules by watching how fast those gravity dips "decay" or flatten out over time. They call this the Decay Rate (DR).

The Cosmic Detective Work

To solve this mystery, the team acted like cosmic detectives combining two different clues:

The ISW Effect: This is like listening to the echo of a shout in a canyon. As light from the Cosmic Microwave Background (CMB) travels through the universe, it gains or loses energy depending on whether the gravity wells it passes through are growing or shrinking.
Weak Lensing: This is like looking at a funhouse mirror. The gravity of galaxies bends the light from background objects, telling us how deep the gravity wells are right now.

By combining these two, the team created a special measurement (the DR) that is immune to many of the usual "messy" problems in astronomy, like how biased our galaxy maps might be. They used data from the DESI imaging surveys (specifically the DR9 catalog) and Planck satellite maps, looking at galaxies between redshifts 0.2 ≤ z ≤ 1.4.

The Big Reveal: Einstein Still Wins

The team tested four different "what-if" scenarios where gravity might behave differently than Einstein predicted. They were looking for a specific number, called γ (gamma), which describes how fast structures grow.

The Prediction: General Relativity says this number should be about 0.55.
The Measurement: The team found γ = 0.47 ± 0.15 to 0.22 (specifically 0.47 +0.22 −0.15).

Here is the verdict: The measurement is fully consistent with Einstein's prediction. The "Modified Gravity" ideas they tested? They didn't pan out. The data suggests that gravity behaves exactly as General Relativity says it should, even on the massive scales of the universe.

They also tested a parameter called Σ (Sigma), which represents a change in the strength of gravity. They tried three different ways this strength could change over time:

ΣΛ: A constant-like change. Result: 0.018 ± 0.052.
Σ1: A change that grows linearly. Result: 0.020 ± 0.065.
Σ2: A change that grows with the square of time. Result: 0.027 ± 0.067.

In all three cases, the results are fully consistent with zero (which means no change, i.e., standard gravity). The paper explicitly rules out the idea that these specific modifications to gravity are happening, at least within the precision of their current data.

Why This Matters (And What It's Not)

You might wonder, "Is this a total proof?" The authors are careful. They say their result is consistent with General Relativity, but they don't claim to have "solved" the mystery of dark energy forever. They note that other recent studies using different methods (like counting galaxy clusters) have suggested a much higher gamma value (around 1.2), which would imply gravity is broken. However, this team's method, using the decay rate, suggests those other results might be due to specific data quirks rather than a fundamental flaw in gravity.

The team also checked if their results would change if they tweaked the amount of matter in the universe (Ωm) or if they ignored the oldest or youngest galaxies in their sample. The answer was a resounding "no." The results stayed robust, shifting by less than 0.3σ (a tiny statistical wiggle) even when they changed the inputs.

The Future

Right now, the team's measurement has a "signal-to-noise" ratio of about 3.1σ. It's a solid hint, but not a slam dunk. The authors are excited because they expect that upcoming full-sky galaxy surveys will improve this precision by a factor of 2. If those future surveys confirm these findings, we can be even more confident that Einstein's gravity is the correct rulebook for our expanding universe, and that the acceleration we see is likely driven by something else (like Dark Energy) rather than a broken law of physics.

So, for now, the trampoline rules are holding up. The gravity wells are decaying exactly as Einstein predicted, and the universe is playing by the old rules.

Technical Summary: Constraining Gravity with the Decay Rate of Cosmological Gravitational Potential

Problem Statement
A central challenge in modern cosmology is distinguishing between dark energy (DE) and modified gravity (MG) as the drivers of late-time cosmic acceleration. While expansion rate probes (e.g., supernovae, BAO) can differentiate some models, many require measurements of the structure growth rate to break degeneracies. Existing probes such as redshift-space distortions (RSD) and weak gravitational lensing are sensitive to galaxy bias and other astrophysical uncertainties. The Integrated Sachs-Wolfe (ISW) effect offers a unique probe of the time variation of the cosmological gravitational potential ( $\dot{\psi} + \dot{\phi}$ ), but its direct measurement is difficult due to low signal-to-noise ratios and dependence on galaxy bias.

Methodology
This study utilizes the Decay Rate (DR) of the cosmological gravitational potential, a quantity proposed by P. Zhang (2006) that combines ISW-galaxy and weak lensing-galaxy cross-correlations. The DR is defined as:
$DR(z) = \left( -\frac{d \ln D_{\psi+\phi}}{d \ln a} \right) \left( \frac{aH(z)/c}{WL(z)} \right)$
where $D_{\psi+\phi}$ is the linear growth factor of the (Weyl) potential and $WL(z)$ is the lensing kernel. Crucially, DR is independent of galaxy bias, the matter clustering amplitude ( $\sigma_8$ ), the slope ( $n_s$ ), and the Hubble constant ( $H_0$ ), depending primarily on the matter density $\Omega_m$ and gravity parameters.

The analysis employs DR measurements from F. Dong et al. (2025), derived from the DESI imaging survey DR9 galaxy catalog and Planck CMB maps, covering the redshift range $0.2 \le z \le 1.4$ . The total significance of these measurements is $3.1\sigma$ . The authors restrict their analysis to one-parameter extensions of the flat $\Lambda$ CDM cosmology, fixing $\Omega_m = 0.3169$ (based on CMB, ACT, and SPT data) to isolate gravity parameters.

Four specific MG parameterizations are tested:

Growth Index ( $\gamma$ ): Defined via $f = \Omega_m^\gamma(a)$ , where $f$ is the growth rate. General Relativity (GR) predicts $\gamma \simeq 0.55$ .
Effective Gravitational Constant ( $\Sigma$ ): Assuming the gravitational slip parameter $\eta = \phi/\psi = 1$ $η = ϕ / ψ = 1$ , the authors test variations in $G_{\text{eff}}/G = 1 + \Sigma(a)$ $G_{eff} / G = 1 + Σ (a)$ using three functional forms:
- $\Sigma(a) = \Sigma_\Lambda \Omega_\Lambda(a)/\Omega_\Lambda$
- $\Sigma(a) = \Sigma_1 a$
- $\Sigma(a) = \Sigma_2 a^2$

Key Results
The study yields the following constraints, all consistent with GR predictions within $1\sigma$ (or $\sim 0.5\sigma$ for $\Sigma$ ):

Growth Index: $\gamma = 0.47^{+0.22}_{-0.15}$ . This is consistent with the GR prediction of $\gamma \simeq 0.55$ . The authors note this result provides an independent perspective on recent controversies where other datasets (e.g., X-ray cluster counts or specific CMB lensing combinations) suggested higher values ( $\gamma \sim 1.2$ ).
Modified Gravity Parameters:
- $\Sigma_\Lambda = 0.018^{+0.052}_{-0.053}$
- $\Sigma_1 = 0.020^{+0.065}_{-0.062}$
- $\Sigma_2 = 0.027^{+0.067}_{-0.069}$
  All three are consistent with zero (the GR case) regardless of the parameterization of $\Sigma(a)$ .

Significance and Robustness
The paper claims that despite the relatively low signal-to-noise ratio ( $\sim 3$ ) of the current DR measurement, the constraints on MG are already competitive with those derived from combining full-shape galaxy clustering, CMB, weak lensing, and supernovae (e.g., M. Ishak et al. 2025).

The authors conducted several robustness checks:

$\Omega_m$ Sensitivity: Varying $\Omega_m$ within the tension range between BAO, CMB, and SNe Ia ($0.2975$ to $0.352$) shifts the best-fit $\gamma$ and $\Sigma_\Lambda$ by less than $0.3\sigma$ , confirming that current uncertainties in $\Omega_m$ do not invalidate the GR consistency.
Redshift Calibration: The impact of potential redshift calibration biases ( $|\delta z| \sim 0.02$ ) on DR is calculated to be less than 3%, well below statistical uncertainties.
Bin Removal: Excluding the lowest or highest redshift bins yields results fully consistent with the fiducial analysis.

Conclusion
The paper concludes that the DR measurement supports General Relativity across multiple MG parameterizations. It highlights that DR is a powerful, bias-independent probe of gravity. The authors project that upcoming full-sky galaxy surveys will improve DR measurement precision by a factor of 2, and the combination of DR with other Large Scale Structure (LSS) probes will further tighten constraints on both the effective gravitational constant ( $G_{\text{eff}}$ ) and the gravitational slip ( $\eta$ ).

The Cosmic Detective Work

The Big Reveal: Einstein Still Wins

Why This Matters (And What It's Not)

The Future

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