Graviton Mass Constraints

• Graviton Mass Constraints is the study of establishing limits on a finite graviton rest mass, impacting gravitational interactions from solar system scales to cosmology.
• Multiple observations—including solar system dynamics, S-star monitoring, gravitational-wave phase shifts, and CMB B-mode analyses—yield bounds ranging from 10⁻²³ to sub–10⁻³⁴ eV.
• Future enhancements from advanced GW detectors, precise astrometric methods, and cosmological surveys are expected to further tighten these constraints and refine modified gravity theories.

The graviton mass constraint is a critical interface between gravitational theory and experiment, delineating the permissible parameter space for modifications of General Relativity (GR) involving a finite graviton rest mass. A finite graviton mass $m_g$ induces Yukawa-type modifications in the gravitational potential at large distances, engenders frequency-dependent dispersion in gravitational waves, and affects the orbital and wave propagation phenomenology across astrophysical, cosmological, and laboratory systems. Stringent empirical and theoretical bounds on $m_g$ arise from a diverse suite of observations, including Solar System dynamics, S-star astrometry at the Galactic Center, gravitational-wave phasing, strong gravitational lensing time delays, cosmic microwave background (CMB) B-mode polarization, and large-scale structure, as well as high-SNR black hole ringdown signals. The current landscape features constraints ranging from model-independent upper limits in the $10^{-21}$–$10^{-23}\ \text{eV}$ regime (solar system, GW, S-star, multi-messenger lensing) to ultra-stringent sub–$10^{-33}$ eV limits from cosmological and CMB tensor-mode analyses, with the latter requiring strong theoretical model assumptions.

1. Theoretical Foundations for Graviton Mass Constraints

Theoretical consideration of a finite graviton mass leads to essential changes in gravitational interaction at both static and dynamical levels. In the static, weak-field regime, a massive graviton mediates a Yukawa-potential

$$V(r) = -\frac{GM}{r}\exp\left(-\frac{r}{\lambda_g}\right)\,, \qquad \lambda_g = \frac{\hbar}{m_gc}\,,$$

which suppresses gravity on scales $r \gtrsim \lambda_g$. For the interaction to be operative throughout the visible universe ($L_0 = c/H_0\sim10^{26}$ m), the “Yukawa” upper bound is $m_g < \hbar / (c L_0) \sim 10^{-32}\ \text{eV}$ (Ali et al., 2016). Field-theoretic consistency in de Sitter backgrounds (Higuchi bound) enforces $m_g^2 \gtrsim \hbar^2/(c^2L_0^2)$, yielding $m_g = \mathcal{O}(10^{-32})$ eV as a rough theoretical floor (no gap allowed).

In radiative and dynamical regimes, the graviton’s non-zero mass $m_g$ introduces a Lorentz-invariant dispersion relation

$E^2 = p^2c^2 + m_g^2c^4\,,$

and the group velocity for gravitational waves of frequency $f$ is

$$v_g(f) = c\sqrt{1 - \left(\frac{m_gc^2}{hf}\right)^2} \approx c\left[1 - \frac{m_g^2c^4}{2(hf)^2}\right]\,.$$

This implies a characteristic frequency-dependent time delay and phase shift in GW signals, which forms the basis of strong experimental upper bounds.

In massive (bi-)gravity and other modified gravity models, additional mechanisms (e.g., Vainshtein screening with $r_V\propto (GM/m_g^2)^{1/5}$) restore agreement with local tests at sub-solar scales (Zakharov et al., 2017), constraining not only $m_g$ but also model-specific parameters and cosmic evolution.

2. Solar System and S-Star Constraints

Classical tests of Newtonian gravity modified by a Yukawa potential enable the most direct, model-independent bounds in the weak-field, static limit. Solar system orbital fits, notably with planet ephemerides such as INPOP17b, include the leading correction to the acceleration,

$$\delta\mathbf{a}_A = \sum_{B\neq A} \frac{G M_B}{2\lambda_g^2} \frac{\mathbf{r}_B-\mathbf{r}_A}{|\mathbf{r}_B-\mathbf{r}_A|}\,,$$

fitted globally to radio-science and ranging data to obtain (Bernus et al., 2019, Fienga et al., 2022): $$\lambda_g > 8.3 \times 10^{13}\ \mathrm{km}\,, \quad m_g < 1.51 \times 10^{-23}\ \mathrm{eV}/c^2 \quad \text{(90\% C.L., with BepiColombo simulation)}$$ with systematics dominated by correlations with fit parameters and the range accuracy of the Cassini spacecraft at Saturn and Mercury (Fienga et al., 2022).

Astrometric monitoring of S-stars near Sgr A*, exploiting the detection of Schwarzschild precession, yields independent astrophysical limits. For the bright S2 star ($P=16$ yr, $e=0.88$) and recent GRAVITY Collaboration results, matching the observed precession to the combination of GR and Yukawa potentials yields (Jovanović et al., 2024, Jovanović et al., 2023): $$m_g \lesssim (1.5\pm0.8)\times 10^{-22}\ \text{eV} \text{ (68\% C.L.)}\,,$$ with the constraint expected to improve to the few-$10^{-23}$ eV regime over several decades and for more circular, longer-period orbits (Zakharov et al., 2018, Jovanović et al., 2023).

Table: Representative Weak-Field (Static) Constraints

Probe	Bound on $m_g$	Compton Wavelength
Solar system	$<1.5\times10^{-23}$ eV	$>8.3\times10^{13}$ km
S2 star (Sgr A*)	$<1.5\times10^{-22}$ eV	$>8.2\times10^{12}$ km

These bounds are entirely model-independent in the sense that they rely solely on the parametrization of gravity by a Yukawa potential and do not require assumptions about the generation or propagation of gravitational radiation.

3. Gravitational Wave Constraints

Propagation of massive gravitons causes dispersive phase shifts in compact binary coalescence signals, forming the basis of the tightest direct experimental constraints. The frequency-dependent phase shift in the inspiral waveform enters at $-1$PN order,

$$\Delta\Psi(f) = -\frac{\pi D}{\lambda_g^2 (1+z)} f^{-1}\,,$$

where $D$ is the dispersion distance (Keppel et al., 2010, Gao, 2022). Standard Fisher-matrix or Bayesian likelihood analyses over GW event catalogs (GWTC-1, -3) yield (Zakharov et al., 2017, Keppel et al., 2010, Gao, 2022): $$m_g < 1.27 \times 10^{-23}\ \text{eV} \quad (\text{GWTC-3}; 90\%\ \mathrm{C.L.})$$ with best current bounds dominated by long-baseline, lower-frequency sources (binary black holes and future LISA massive binaries). Third-generation (Cosmic Explorer, ET) and LISA-class detectors are projected to reach $m_g < 10^{-24}$–$10^{-27}$ eV depending on source class (Keppel et al., 2010, Gao, 2022). Inclusion of ringdown (QNM) frequencies and amplitude modification has also been considered, with ringdown-alone limits much weaker ($m_g \lesssim 10^{-15}$ eV) (Chung et al., 2018), highlighting the dominance of the phasing channel.

Lensed multi-messenger events offer new constraints through measurement of the differential time-delay between EM and GW images. The time-delay difference is sensitive directly and model-independently to the graviton mass: $$\Delta t_g = (1+\frac{m_g^2}{2\omega^2})\Delta t_\gamma\,,$$ where $\omega$ is GW angular frequency (Colangeli et al., 3 Sep 2025). For a lensed system with $\Delta t_\gamma\sim 1000$ days and GW time-delay accuracy $\sigma_t \ll 1$ s (LISA-class), the inferred limit is

$m_g < 3\times 10^{-23}\ \text{eV}/c^2$

independent of lens model, waveform model, or cosmology—establishing a robust, rapidly scalable multi-messenger bound.

Table: Representative Radiative Limits

Observation	$m_g$ Upper Bound
LIGO–Virgo GWTC-3 (O1–O3)	$1.27\times 10^{-23}$ eV
Lensed GW–EM time-delay (prospective)	$3\times 10^{-23}$ eV
LISA-class (Fisher forecast)	$2.1\times 10^{-27}$ eV

4. Cosmological and CMB B-mode Constraints

Cosmological tensor modes (primordial GWs) are modified by a graviton mass, as apparent in the wave equation for the Fourier modes: $$h_k'' + 2\mathcal{H} h_k' + [k^2 + a^2 m_g^2] h_k = 0$$ causing scale- and time-dependent damping and phase evolution of primordial GWs. The most striking observational effect is the emergence of a plateau in the CMB B-mode power spectrum at low multipoles $\ell\lesssim 100$ for $$m_g \gtrsim (10\,\mathrm{Mpc})^{-1} \sim 10^{-30}\ \mathrm{eV}$$ (0907.1658, Malsawmtluangi et al., 2017). Absence of such a feature in current BICEP/Keck and Planck data imposes extremely tight constraints: $m_g \lesssim 10^{-30}\ \mathrm{eV}$ with a systematic uncertainty arising from B-mode delensing and primordial tensor amplitude uncertainties (Malsawmtluangi et al., 2017).

Minimal Theory of Massive Gravity (MTMG) and its extensions, when constrained cosmologically via Planck CMB, BAO, SNe, and weak lensing, enforce: $$m_g < 6.6\times10^{-34}\ \mathrm{eV}\ (95\%\,\mathrm{C.L.})\ [2311.10530],\qquad m_g < 8.4\times10^{-34}\ \mathrm{eV}\ (95\%\,\mathrm{C.L.})\ [2110.01237]$$ assuming the normal branch and absence of background deviations from $\Lambda$CDM. These bounds are orders of magnitude below any weak-field or astrophysical test, but rest on specific model assumptions and cosmological parameter choices.

5. Astrophysical and Model-Dependent Constraints

Constraints on $m_g$ from galactic and cluster dynamics, pulsar timing, and large-scale structure are by nature highly model-dependent, sensitive to the assumed form of the modified gravitational potential and to the screening mechanisms operative in nonlinear massive gravity theories. Dynamical analyses of galaxy clusters extend the reach to

$m_g\lesssim 10^{-29}\ \mathrm{eV}$

but are contingent on the assumed scaling and universality of the dispersion relation and the absence of environmental screening (Zakharov et al., 2017).

Pulsar timing arrays are forecast to approach

$m_g \lesssim 3\times 10^{-23}\ \mathrm{eV}$

in future campaigns (Zakharov et al., 2017), complementing GW–based and CMB–based methods.

Mass-variation and screening models (e.g., MVMG) allow $m_g$ to be Hubble-scale in the cosmological background but enhanced near black holes. There, signal detection in ringdown QNMs and late–time GW echoes can set direct constraints on $m_g(r_h)\lesssim 10^{-10}\ \mathrm{eV}$, orders of magnitude weaker than for the cosmological graviton mass but directly testing the strong–field regime (Zhang et al., 2017).

6. Implications for Gravity Theories

Measured bounds on $m_g$ delimit the parameter space for a wide class of massive gravity models:

• dRGT/bigravity theories require $m_g\ll 10^{-32}$ eV to recover GR behavior and cosmological consistency (Ali et al., 2016).
• $f(R)$ models possess a residual “scalaron” graviton mass $m_g$ tied to the second derivative $f''(R)$; GW and Solar System constraints carve out allowed parameter brackets, e.g. $10^6<\mu<3\times10^{22}$ for Hu–Sawicki $f(R)$ (Vainio et al., 2016).
• Screening mechanisms (Vainshtein, chameleon) impact the effective mass probed, with current GW detectors insensitive to the screening regime unless $m_g\to H_0$ (Perkins et al., 2018).

The persistent gap of $\sim 10$ orders of magnitude between the most stringent weak–field experimental bounds ($m_g\sim10^{-23}$ eV) and the theoretical upper limit from universe size ($m_g\sim10^{-32}$ eV) remains a principal driver for ongoing precision tests, next–generation GW detectors, CMB–B mode missions, and deeper model analyses.

7. Prospects for Improvement and Future Observational Pathways

Future improvements in $m_g$ constraints are anticipated from:

• Third-generation GW detectors (ET, CE) and space-based GW observatories (LISA, Taiji, TianQin), which will access lower GW frequencies and longer baselines, improving sensitivity to $m_g$ by factors of $\sim10$–$10^3$ (Keppel et al., 2010, Gao, 2022).
• Extended S-star monitoring at the Galactic Center (VLTI/GRAVITY, E-ELT, TMT), reaching sub-10% precision in Schwarzschild precession, and possibly discovering closer-in stellar or pulsar companions (Zakharov et al., 2018, Jovanović et al., 2024).
• Lensed GW–EM events, where single “golden” events with ms–level delay contrast can provide cosmology–independent constraints competitive with (and complementary to) GW phasing bounds (Colangeli et al., 3 Sep 2025).
• Precision CMB B-mode and weak lensing surveys (LSST, Euclid, Roman, CMB-S4), which will push cosmological bounds further below $10^{-33}$ eV if cosmological gravity matches $\Lambda$CDM at high accuracy (Felice et al., 2021, Felice et al., 2023).
• Stacking of GW events, lowering statistical uncertainties as $1/\sqrt{N}$, may approach the screening regime relevant for some massive gravity models (Perkins et al., 2018).

Remaining systematic limitations include parameter degeneracies in planetary ephemerides, astrometric reference frame uncertainties in S-star observations, and modeling uncertainties in CMB and lensing analyses. The interplay between model-independent and model-dependent bounds, as well as between different astrophysical and cosmological scales, remains essential for robustly constraining (or revealing) a finite graviton mass.

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Summary Table: Key Experimental Upper Limits on $m_g$

Technique/Domain	Bound on $m_g$ (eV)	Reference
Solar system ephemerides	$<1.5 \times 10^{-23}$	(Fienga et al., 2022)
S2 star precession	$<1.5 \times 10^{-22}$	(Jovanović et al., 2024)
GW propagation (GWTC-3)	$<1.27 \times 10^{-23}$	(Zakharov et al., 2017)
Lensed GW-EM time delays	$<3\times 10^{-23}$	(Colangeli et al., 3 Sep 2025)
CMB B-mode (BICEP/Planck)	$<1\times 10^{-30}$	(0907.1658)
Cosmic structure/MTMG	$<8.4\times10^{-34}$	(Felice et al., 2021)

All current data are consistent with $m_g=0$, with no evidence for deviations from GR or finite graviton mass. Further constraints are expected from upcoming observational and experimental programs at all scales.