The particle that carries gravity may weigh a tiny amount.
dRGT Massive Gravity
The graviton carries a tiny mass, which gives gravity a finite range and acts like a cosmological constant.
Placeholder for a 3D visualisation of Large-Distance Modified Gravity. The interactive scene will land in Phase 3. This family changes gravity itself at the largest distances. Near stars and galaxies, gravity behaves as general relativity predicts. At distances comparable to the observable universe, it either leaks into an extra dimension or acts as if the graviton has a tiny mass. Either change can make the expansion accelerate with no dark energy, a result physicists call self-acceleration. The program began with the 2000 braneworld model of Dvali, Gabadadze and Porrati. Its four-dimensional limit led to the Galileon, and a 2014 review derives ghost-free massive gravity from the same higher-dimensional starting point. The three models here use different mechanisms, and all of them rely on screening, usually the Vainshtein mechanism, to hide their extra force near the Sun. The flagship self-accelerating solutions have since met ghost instabilities, poor fits to cosmological data and the 2017 measurement of the speed of gravity.
§1 · The claim, in one sentence
De Rham, Gabadadze and Tolley gave the graviton a tiny mass without the Boulware-Deser ghost that sank earlier attempts, but the theory's simplest cosmological solutions are still unstable.
§2 · Why it might be true
In general relativity the graviton, the particle that carries gravity, has no mass. Giving it a small mass would give gravity a finite range.
Van Dam and Veltman found in 1970 that a massive graviton, however light, conflicts with the orbit of Mercury. Vainshtein showed in 1972 that non-linear effects can restore the massless result near a source. Boulware and Deser, also in 1972, found a sixth mode in the full theory, with energy that has no lower bound.
De Rham, Gabadadze and Tolley built a non-linear theory in 2011 that is free of this ghost in its decoupling limit, to all orders. Hassan and Rosen showed in 2012 that the ghost is absent in the full theory as well.
A graviton mass acts on the expansion like a cosmological constant set by that mass. Gumrukcuoglu, Lin and Mukohyama found self-accelerating open universes on this basis in 2011. The acceleration then comes from gravity itself, with no dark energy.
The family stance
These theories try to explain cosmic acceleration by changing gravity at distances comparable to the observable universe, instead of adding dark energy. There gravity either spreads into an extra dimension or behaves as if its carrier particle, the graviton, has a tiny mass. Near stars and planets the extra effects switch off, so solar-system tests still agree with general relativity. The best-known self-accelerating versions face serious stability and observational problems.
§2.5 · Evidence
- Hassan and Rosen (2012) proved the Boulware-Deser ghost absent at the full non-linear level, which they describe as the first such demonstration in four dimensions.
- De Rham and Gabadadze (2010) found that the construction naturally generates the cubic, quartic and quintic Galileon interactions, tying it to the sibling variants.
- D'Amico and colleagues (2011) found approximately homogeneous patches, the size of the inverse graviton mass, that can describe the past and present universe.
§3 · What you'd need to test it
- Gravity has a finite range set by the graviton mass, with a Yukawa-type fall-off at the largest distances
- Gravitational waves of different frequencies travel at slightly different speeds, because massive gravitons disperse
- On some self-accelerating branches, an effective cosmological constant proportional to the graviton mass squared drives late-time acceleration
- Near massive bodies, the Vainshtein mechanism hides the extra modes and recovers the predictions of general relativity
§4 · Where it breaks
- D'Amico and colleagues (2011) showed that the constraint removing the ghost also forbids exactly homogeneous and isotropic universes.
- De Felice, Gumrukcuoglu and Mukohyama (2012) found every homogeneous and isotropic solution unstable, with at least one ghost.
- The self-accelerating open universes have no exactly flat limit, so the theory jumps discontinuously as spatial curvature goes to zero.
Go deeper
De Rham and Gabadadze generalised the Fierz-Pauli action by covariantly adding a mass term and polynomial interactions with arbitrary coefficients. In the decoupling limit, special choices of those coefficients remove the ghost-like pathologies of the helicity-0 mode. A local change of variables absorbs the helicity-0 and helicity-2 mixing and generates the Galileon terms. The 2011 dRGT paper resums all these non-linear terms and keeps the Hamiltonian constraint at least to quartic order.
Hassan and Rosen analysed the full two-parameter family of actions in the ADM formalism. The Hamiltonian constraint survives at the complete non-linear level, and they argue for a secondary constraint, which together remove the Boulware-Deser ghost. De Rham's 2014 Living Reviews article derives DGP, cascading gravity and ghost-free massive gravity from a higher-dimensional theory of general relativity.
In open FRW backgrounds the Stückelberg scalars have three branches. Two of them give an effective cosmological constant Λ± = c± m_g^2, where c± depends on the two dimensionless parameters of the theory. D'Amico and colleagues estimated that the graviton mass must lie roughly an order of magnitude below today's Hubble rate. They also noted that a mass arising from an extra condensing scalar field would allow homogeneous cosmologies.
GW150914 gave a 90% lower bound of 10^13 km on the graviton Compton wavelength. GWTC-3 tightened the mass bound to m_g ≤ 2.42 x 10^-23 eV/c^2, and the GWTC-4.0 preprint to 1.92 x 10^-23 eV/c^2, both at 90% credibility. GW170817 separately bounds the graviton mass in bimetric theories (Baker et al. 2017).
▸§5 · Who built it, and when(14 sources, 13 established, 1 needs verification)
- Establishedde Rham, Gabadadze & Tolley (2011). 'Resummation of Massive Gravity.' Phys. Rev. Lett. 106, 231101
- Establishedde Rham & Gabadadze (2010). 'Generalization of the Fierz-Pauli Action.' Phys. Rev. D 82, 044020
- EstablishedHassan & Rosen (2012). 'Resolving the Ghost Problem in non-Linear Massive Gravity.' Phys. Rev. Lett. 108, 041101
- Establishedvan Dam & Veltman (1970). 'Massive and massless Yang-Mills and gravitational fields.' Nucl. Phys. B 22, 397
- EstablishedVainshtein (1972). 'To the problem of nonvanishing gravitation mass.' Phys. Lett. B 39, 393
- EstablishedBoulware & Deser (1972). 'Can gravitation have a finite range?' Phys. Rev. D 6, 3368
- EstablishedGumrukcuoglu, Lin & Mukohyama (2011). 'Open FRW universes and self-acceleration from nonlinear massive gravity.' JCAP 11, 030
- EstablishedD'Amico, de Rham, Dubovsky, Gabadadze, Pirtskhalava & Tolley (2011). 'Massive Cosmologies.' Phys. Rev. D 84, 124046
- EstablishedDe Felice, Gumrukcuoglu & Mukohyama (2012). 'Massive gravity: nonlinear instability of the homogeneous and isotropic universe.' Phys. Rev. Lett. 109, 171101
- Establishedde Rham (2014). 'Massive Gravity.' Living Rev. Rel. 17, 7 (review)
- Establishedde Rham, Deskins, Tolley & Zhou (2017). 'Graviton Mass Bounds.' Rev. Mod. Phys. 89, 025004 (review)
- EstablishedLIGO Scientific & Virgo Collaborations (2016). 'Tests of general relativity with GW150914.' Phys. Rev. Lett. 116, 221101
- EstablishedLIGO Scientific, Virgo & KAGRA Collaborations (2025). 'Tests of General Relativity with GWTC-3.' Phys. Rev. D 112, 084080
- Needs verificationLIGO Scientific, Virgo & KAGRA Collaborations (2026). 'GWTC-4.0: Tests of General Relativity. II. Parameterized Tests.' Preprint
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