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The Atacama Cosmology Telescope: A Test of the Gravitational Force Law on Cosmological Scales Using the Kinematic Sunyaev-Zeldovich Effect

Using the kinematic Sunyaev-Zeldovich effect with data from the Atacama Cosmology Telescope and the Sloan Digital Sky Survey, this study confirms that the gravitational force between massive halos follows an inverse-square law ( $n \approx 2$ ) on cosmological scales, consistent with the standard $\Lambda$ CDM model.

Original authors: Patricio A. Gallardo, Kris Pardo, Oliver H. E. Philcox, Nicholas Battaglia, Elia S. Battistelli, Rachel Bean, Erminia Calabrese, Steve K. Choi, Rolando Dünner, Mark Devlin, Joanna Dunkley, Simone Ferr

Published 2026-04-17

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CC BY 4.0

Original authors: Patricio A. Gallardo, Kris Pardo, Oliver H. E. Philcox, Nicholas Battaglia, Elia S. Battistelli, Rachel Bean, Erminia Calabrese, Steve K. Choi, Rolando Dünner, Mark Devlin, Joanna Dunkley, Simone Ferraro, Yilun Guan, Erin Healy, Carlos Hervías-Caimapo, Matt Hilton, Adam D. Hincks, John C. Hood II, Arthur Kosowsky, Adrien La Posta, Thibaut Louis, Mathew S. Madhavacheril, Jeff McMahon, Kavilan Moodley, Tony Mroczkowski, Sigurd Naess, Laura Newburgh, Michael D. Niemack, Lyman A. Page, Bruce Partridge, Roberto Puddu, Emmanuel Schaan, Neelima Sehgal, Cristóbal Sifón, David N. Spergel, Suzanne T. Staggs, Alexander van Engelen, Cristian Vargas, Eve M. Vavagiakis, Kasey Wagoner, Edward J. Wollack

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 dance floor. On this floor, massive groups of stars and gas (called "galaxy clusters") are swirling around. Usually, we think of gravity as the invisible hand that pulls these dancers together. For decades, physicists have assumed this hand follows a very specific rule: the further apart two dancers are, the weaker the pull becomes, dropping off very quickly (like the square of the distance). This is the standard rule of Newton and Einstein, known as $\Lambda$ CDM.

However, the universe is expanding, and things are moving faster than expected in some places. This has led some scientists to wonder: Is the rulebook for gravity wrong on the biggest scales? Maybe the "hand" of gravity doesn't fade away as fast as we think, or maybe it follows a completely different rule.

This paper is like a cosmic detective story where the authors use a special tool to test the rules of the dance floor.

The Detective Tool: The "Cosmic Doppler Shift" (kSZ)

To test gravity, you need to measure how fast these galaxy clusters are moving toward or away from each other. But they are too far away to see moving with a telescope.

Instead, the authors used the Kinematic Sunyaev-Zeldovich (kSZ) effect. Here is a simple analogy:
Imagine the galaxy clusters are like giant, hot fans spinning in a room filled with a fine mist (the Cosmic Microwave Background, or the leftover heat from the Big Bang). As the fans spin, they push the mist. If you stand on the side where the fan is blowing toward you, the mist feels slightly warmer (compressed). If it's blowing away, the mist feels slightly cooler (stretched).

By looking at the Atacama Cosmology Telescope (ACT) data, the authors measured these tiny temperature shifts in the cosmic mist caused by the "fans" (galaxy clusters). This told them exactly how fast the clusters were moving relative to each other.

The Experiment: Testing the "Gravity Formula"

The authors took these speed measurements and compared them to the positions of the clusters. They asked: "If gravity works the way we think it does, do the speeds match the distances?"

They tested two main ideas:

The Standard Rule ( $\Lambda$ CDM): Gravity gets weaker very fast as distance increases (like $1/r^2$ ).
The Alternative Rule (MOND): A theory called Modified Newtonian Dynamics suggests that on huge scales, gravity gets weaker much more slowly (like $1/r$ ). This theory tries to explain cosmic motion without needing "Dark Matter."

The Verdict: The Standard Rule Wins

The results were clear. The "dance" of the galaxy clusters matched the Standard Rule perfectly.

The data showed that gravity fades away exactly as the standard model predicts.
The alternative rule (MOND), which suggests gravity stays stronger over long distances, did not fit the data at all. It was like trying to fit a square peg into a round hole; the math just didn't work.

The authors calculated that the "power law" of gravity is 2.1, which is incredibly close to the standard 2.0. This means the force of gravity behaves exactly as Newton and Einstein predicted, even across distances of hundreds of millions of light-years.

Why This Matters

Think of this like testing the laws of physics in a new, extreme environment. We know how gravity works in our solar system (like the Earth orbiting the Sun), but we didn't know for sure if those same rules applied to the entire universe.

This paper confirms that the universe is consistent. The same laws that govern a falling apple also govern the motion of galaxy clusters billions of light-years away. It also deals a significant blow to theories that try to get rid of Dark Matter by simply changing the rules of gravity.

The Future

The authors are optimistic. They say that with even better telescopes coming online (like the Simons Observatory), they will be able to test these rules with such precision that they could rule out alternative theories with 10 times more confidence than before.

In a nutshell: The universe is a well-ordered dance floor. The gravity pulling the dancers together follows the classic rules we've known for centuries, and the "new rules" proposed by some alternative theories just don't fit the music.

1. Problem Statement

The standard cosmological model ( $\Lambda$ CDM) assumes that General Relativity (and its Newtonian limit, $1/r^2$ ) accurately describes gravity on all scales. However, the observed cosmic acceleration and the nature of dark energy/dark matter have led to alternative theories, such as Modified Newtonian Dynamics (MOND), which propose that the gravitational force law deviates from the inverse-square law at large scales or low accelerations.

While gravity is well-tested within the solar system, direct measurements of the gravitational force law on cosmological scales (tens to hundreds of megaparsecs) are lacking. Existing constraints often rely on assumptions about the expansion history of the universe, which limits the space of testable theories. This paper aims to provide a direct, model-independent test of the gravitational force law by measuring the mean pairwise velocity of galaxy halos, which is driven directly by the gravitational acceleration field.

2. Methodology

Data Sources

The authors combine two major datasets:

CMB Intensity Maps: Data from the Atacama Cosmology Telescope (ACT) Data Release 6 (DR6), specifically the 150 GHz map combined with Planck 143 GHz data.
Galaxy Catalog: A sample of red luminous galaxies from the Sloan Digital Sky Survey (SDSS) DR15, restricted to a narrow redshift slice ( $0.44 < z < 0.66$ ) to minimize evolution effects. The sample contains $\sim$ 343,000 galaxies.

The Kinematic Sunyaev-Zeldovich (kSZ) Effect

The study utilizes the pairwise kSZ effect. As galaxy clusters move relative to the Cosmic Microwave Background (CMB), they scatter CMB photons, causing a Doppler shift. By measuring the temperature difference between pairs of galaxies along the line of sight, the authors can infer the mean pairwise momentum ( $\hat{p}$ ), which is proportional to the mean pairwise velocity ( $V$ ).
$\hat{p}(r) = -\frac{T_{CMB}}{c} \bar{\tau} V(r)$
where $\bar{\tau}$ is the effective mass-averaged optical depth.

Theoretical Framework

The authors develop a formalism to relate the observed pairwise velocity to the gravitational force law without assuming a specific growth history of structure.

Generalized Force Law: They parametrize the gravitational acceleration $g$ as a power law:
$g(r) \propto \frac{1}{r^n}$
where $n=2$ corresponds to standard Newtonian gravity ( $\Lambda$ CDM) and $n=1$ corresponds to the MOND regime (in the deep MOND limit).
Derivation: Starting from the equations of motion for halos and the conservation of pair numbers, they derive an expression for the mean pairwise velocity $V(r)$ as a function of the galaxy correlation function $\xi_{gg}(r)$ and the force law index $n$ :
$V(r) = A \left( \frac{4\pi G \rho_b r / \bar{a}}{1 + \xi_{gg}(r)} \bar{\xi}_{gg}(r) \right)^{n/2}$
Here, $A$ is an amplitude term degenerate with the optical depth $\bar{\tau}$ .
Analysis Strategy:
- They use the measured galaxy correlation function from SDSS to predict the theoretical pairwise velocity curve for different values of $n$ .
- They compare these predictions against the measured pairwise kSZ signal from ACT.
- They perform a likelihood analysis to constrain $n$ and the optical depth $\bar{\tau}$ simultaneously.

3. Key Contributions

Direct Test of Gravity: This is one of the first direct measurements of the gravitational force law on scales of 30–230 Mpc, independent of assumptions regarding the expansion history (Hubble flow) or the specific growth rate of structure.
Formalism Development: The paper provides a generalized derivation of the pairwise velocity equation that allows for arbitrary power-law modifications to gravity ( $g \propto 1/r^n$ ), bridging the gap between standard cosmology and modified gravity theories like MOND.
MOND Constraint: It provides a rigorous test of MOND on cosmological scales, a regime where MOND has historically struggled to explain cluster dynamics without dark matter.

4. Results

Force Law Index ( $n$ ): The analysis yields a constraint on the power-law index of the gravitational force law:
$n = 2.1 \pm 0.3 \quad (68\% \text{ C.I.})$
This result is consistent with the standard Newtonian inverse-square law ( $n=2$ ) within $0.4\sigma$ .
Exclusion of MOND: The MOND prediction ( $n=1$ ) is ruled out at $3.3\sigma$ significance. The probability to exceed ($PTE$) for the MOND fit is $3.65 \times 10^{-3}$ , indicating a poor fit to the data.
Goodness of Fit: The $\Lambda$ CDM model ( $n=2$ ) fits the data well with a $PTE = 0.17$. The best-fit curve for $\Lambda$ CDM matches previous ACT results and the theoretical prediction derived from the Planck cosmology.
Optical Depth: The analysis also constrains the average optical depth $\bar{\tau}$ , finding it consistent with previous estimates when normalized to the $\Lambda$ CDM model.

5. Significance and Future Outlook

Validation of $\Lambda$ CDM: The results strongly support the standard cosmological model, confirming that gravity follows an inverse-square law ( $1/r^2$ ) even on the largest scales probed by galaxy clusters.
Ruling Out Alternatives: The study demonstrates that modified gravity theories which rely on a $1/r$ force law (like the deep MOND limit) cannot simultaneously explain the observed galaxy correlation function and the pairwise velocities of clusters.
Future Potential: The authors project that upcoming surveys (e.g., DESI, LSST) combined with next-generation CMB experiments (e.g., Simons Observatory) will increase the sample size significantly. They forecast that with $\sim$ 4 million galaxies, the test could rule out $n=1$ at $10\sigma$ significance, providing an even more stringent test of gravity.
Methodological Advancement: The paper establishes the pairwise kSZ effect as a powerful, independent tool for testing gravity, complementing other probes like gravitational lensing and redshift-space distortions.

In conclusion, this paper uses the kinematic Sunyaev-Zeldovich effect to place the tightest constraints to date on the form of the gravitational force law at cosmological scales, finding strong evidence for standard Newtonian gravity and ruling out simple MOND-like modifications.