Weighing the Graviton: Stellar Orbits vs. Quantum Gravity Mass ✛ ✦
Context
Understanding gravity's nature remains a central task of physics. The classical general relativity of Einstein has triumphed in every test, yet provides no answer to the puzzles of dark matter and dark energy. Moreover, it fundamentally resists quantization, creating a rift with the rest of physics. Massive gravity theories, where the field quantum—the graviton—has a non-zero mass, offer an elegant bypass by modifying gravity's law on large scales. The most sensitive tests of such theories are possible in extreme conditions—near the event horizons of supermassive black holes or when analyzing gravitational waves.
Methods
Two complementary approaches were used to constrain the graviton mass, based on archival astrometric measurements of guide stars S2 and S38 in the Galactic center, obtained with Keck telescopes and the VLTI/GRAVITY interferometer. In the first method, a Yukawa addition exp(-r/λ) was introduced into the Newtonian potential, where λ is the graviton's Compton wavelength (according to the law of universal gravitation in the weak-field limit). Taking into account Kepler's third law, which links the period and semi-major axis for nearly Keplerian orbits, the orbital precession, parameterized by the $f_{SP}$ factor from GRAVITY collaboration data, was compared with a modified expression that includes contributions from the Yukawa potential and GR corrections at the first post-Newtonian order. In the second, numerical modeling of orbits was performed by minimizing χ² over all orbital parameters, including the central black hole mass, distance, and eccentricity, using an evolutionary algorithm.
Results
Both methods yielded consistent upper limits on the graviton mass: for star S2, mg < (124.9 ± 120.2) × 10⁻²⁴ eV/c² from the precession analysis and mg ≈ (142.9 ± 24.1) × 10⁻²⁴ eV/c² from direct fitting; for S38, mg < (76.9 ± 73.3) × 10⁻²⁴ eV/c². These values are roughly an order of magnitude weaker than the best constraints from gravitational-wave events by the LIGO-Virgo collaboration (mg < 2.4 × 10⁻²³ eV/c²), but they are obtained by a completely independent method in the strong-field regime. This complements earlier observations of gravitational redshift in the spectrum of S2, which confirmed GR. Remarkably, the graviton's Compton wavelength for the best fit to S2 reaches 5.8 × 10⁴ AU, confirming the colossal sensitivity of astrometry. The deviation from Schwarzschild precession (first described by Karl Schwarzschild) is proportional to the square of the ratio of the semi-major axis to λ, making the method particularly effective for stars with large orbits. Interestingly, the Yukawa correction always increases precession, so constraints only arise when an excess pericenter shift relative to GR is observed.
Implications
The results reinforce the status of stellar dynamics near supermassive black holes as a precision laboratory for testing fundamental physics. They leave an imprint on the interpretation of hidden mass—if the graviton has mass, the need for some dark matter vanishes. Agreement with LIGO limits underscores the complementarity of electromagnetic and gravitational-wave astronomy. Moreover, lower bounds on the gravity interaction length rule out a whole class of theories with small Compton wavelengths, such as some f(R) variants, and support dRGT-type models where the graviton could be ultralight.
Future development
Key improvements are expected from the GRAVITY+ interferometer, whose recent activation promises to boost astrometric precision to tens of microarcseconds. This will allow more stars, including tight orbits, to be incorporated into the analysis, possibly uncovering systematic deviations hinting at new physics. In parallel, theory must advance in accounting for nonlinear Vainshtein screening effects that could mask massive graviton signatures in local tests. Merging data with future gravitational-wave catalogs (ET, LISA) will create a multichannel test.
Impact
The methodology will directly influence data analysis from next-generation missions like THEIA and MICADO@ELT, aimed at mapping the Galactic center. It also stimulates dialogue between experts in black hole mergers and stellar dynamics, uniting efforts in the search for GR deviations.
Next steps
In the near term—refinement of the f_SP factor using new GRAVITY Collaboration data (2024 update) for four S-stars simultaneously. Combining with planetary ephemerides, where record constraints have already been achieved, also looks promising.
Key open problems
The question of the graviton mass is inextricably linked to dark energy—perhaps the cosmological constant is a direct consequence of the gravitational field's massiveness. Moreover, it is key to the problem of quantizing gravity: a non-zero mass would make the theory renormalizable, bringing us closer to a unified picture. And, of course, it is a central element in solving the dark matter mystery, first formulated by Fritz Zwicky in the 1930s.
The knowledge machine's view
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What the work does
You have placed constraints on the graviton mass using the orbits of stars S2 and S38 at the Galactic center, employing two methods: analysis of precession with a Yukawa addition and direct orbit fitting with χ² minimization. Upper limits of order 10⁻²³ eV/c² were obtained, independent of gravitational-wave estimates.
Where its strength is
This work demonstrates how astrometry in a strong-field regime can provide competitive constraints on fundamental parameters of gravity theories, complementing interferometric observations of mergers.
Where it could go next
- It might be interesting to see how astrometric constraints combine with GR tests from 168 mergers: there is a nearby work that checked the consistency of signals with Einstein's predictions—this would provide a cross-check in different regimes.based on: Gravity Passes Another Test: 168 Mergers Confirm Einstein arXiv:2607.19293
- To extend the method to other modified gravity theories, one can rely on an approach where gravitational waves are used to test a new theory—there, a parameterization of deviations is discussed that can be adapted to orbit analysis.based on: Gravitational Waves to Put New Theory to the Test arXiv:2606.15295
Worth trying
- Try applying Bayesian analysis of orbits with several alternative potentials, as is done for gravitational waves in [2606.15295].
- Check whether the constraints improve when jointly fitting the orbits of S2 and S38 with a consistency condition on the central black hole parameters.
Why this may matter
Such independent constraints are important for narrowing down the class of viable massive-gravity theories and could point to systematic effects if different methods yield divergent results.
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Original abstract
as written by the authors arXiv:2607.11764
Along with a whole range of alternative theories of gravity, variants of the massive theory of gravity, i.e. the theory of gravity, in which the graviton has mass, have been actively discussed in recent years. Theorists have proposed versions of massive gravity theories that address the shortcomings of early versions of such theories. Astrophysicists and experimental physicists have been discussing limitations on the graviton mass from various astronomical observations. In particular, in the first LIGO paper, where the discovery of gravitational waves from binary black holes was reported, a limit on the mass of the graviton was obtained from the analysis of the profile of the gravitational wave signal. In the paper graviton mass constraints were obtained by analyzing the trajectory of a bright star in the vicinity of the center of our Galaxy, using observations from the GRAVITY and Keck groups. Briefly other astronomical ways to limit a graviton mass were discussed.
The wavelength of a graviton with mass 10⁻²² eV/c² is comparable to the distance to the nearest stars, such a ‘smeared’ quantum field is hard to detect in the lab, but it leaves a noticeable imprint on stellar orbits.
In Greg Egan's novel ‘Schild's Ladder’, physicists explore a vacuum with altered laws, echoing the idea of a massive graviton changing the structure of spacetime on galactic scales.
Key numbers
- graviton mass (upper limit, S2, precession method): 1.25 × 10⁻²² eV/c²
- graviton mass (upper limit, LIGO-Virgo): 2.4 × 10⁻²³ eV/c²
- graviton Compton wavelength (S2, best-fit): 5.8 × 10⁴ AU
- orbital period of S2: 16.05 years
- orbital eccentricity of S2: 0.88