It is likely that the graviton is massless. More than fifty years ago Van Dam and Veltman (VANDAM 1970 ), Iwasaki (IWASAKI 1970 ), and Zakharov (ZAKHAROV 1970 ) almost simultaneously showed that in the linear approximation a theory with a finite graviton mass does not approach GR as the mass approaches zero. Attempts have been made to evade this "vDVZ discontinuity" by invoking modified gravity or nonlinear theory by De Rahm (DE-RHAM 2017 ) and others. More recently, the analysis of gravitational wave dispersion has led to bounds that are largely independent of the underlying model, even if not the strongest. We quote the best of these as our best limit.

Experimental limits have been set based on a Yukawa potential (YUKA), dispersion relation (DISP), or other modified gravity theories (MGRV).

The following conversions are useful: 1 eV = $1.783 \times 10^{-33}$ g = $1.957 \times 10^{-6}{\mathit m}_{{{\mathit e}}}$; $ƛ_{C}$ = ($1.973 \times 10^{-7}$ m)${\times }$(1 eV/${\mathit m}_{{{\mathit g}}}$).

VALUE (eV) DOCUMENT ID TECN  COMMENT $\bf{<1.76 \times 10^{-23}}$ 1 ABBOTT 2021 DISP LIGO Virgo catalog GWTC-2 • • We do not use the following data for averages, fits, limits, etc. • • $<8 \times 10^{-34}$ 2 DEFELICE 2021 MGRV Normal branch Minimal Theory of Massive Gravity $<3.2 \times 10^{-23}$ 3 BERNUS 2020 YUKA Planetary ephemeris INPOP19a $<2 \times 10^{-28}$ 4 SHAO 2020 DISP Binary pulsar Galileon radiation $<4.7 \times 10^{-23}$ 5 ABBOTT 2019 DISP LIGO Virgo catalog GWTC-1 $<7 \times 10^{-23}$ 6 BERNUS 2019 YUKA Planetary ephemeris INPOP17b $<3.1 \times 10^{-20}$ 7 MIAO 2019 DISP Binary pulsar orbital decay rate $<1.4 \times 10^{-29}$ 8 DESAI 2018 YUKA Gal cluster Abell 1689 $<5 \times 10^{-30}$ 9 GUPTA 2018 YUKA Using SPT-SZ $<3 \times 10^{-30}$ 9 GUPTA 2018 YUKA Using Planck all-sky SZ $<1.3 \times 10^{-29}$ 9 GUPTA 2018 YUKA Using redMaPPer SDSS-DR8 $<6 \times 10^{-30}$ 10 RANA 2018 YUKA Weak lensing in massive clusters $<8 \times 10^{-30}$ 11 RANA 2018 YUKA SZ effect in massive clusters $<1.0 \times 10^{-23}$ 12 WILL 2018 YUKA Perihelion advances of planets $<7 \times 10^{-23}$ 5 ABBOTT 2017 DISP Combined dispersion limit from three BH mergers $<1.2 \times 10^{-22}$ 5 ABBOTT 2016 DISP Combined dispersion limit from two BH mergers $<2.9 \times 10^{-21}$ 13 ZAKHAROV 2016 YUKA S2 star orbit $<5 \times 10^{-23}$ 14 BRITO 2013 MGRV Spinning black holes bounds $<6 \times 10^{-32}$ 15 GRUZINOV 2005 MGRV Solar System observations $<6 \times 10^{-32}$ 16 CHOUDHURY 2004 YUKA Weak gravitational lensing $<9.0 \times 10^{-34}$ 17 GERSHTEIN 2004 MGRV From $\Omega _{tot}$ value assuming RTG $<8 \times 10^{-20}$ 18, 19 FINN 2002 DISP Binary pulsar orbital period decrease $<7 \times 10^{-23}$ TALMADGE 1988 YUKA Solar system planetary astrometric data $<1.3 \times 10^{-29}$ 20 GOLDHABER 1974 YUKA Rich clusters $<7 \times 10^{-28}$ HARE 1973 YUKA Galaxy $<8 \times 10^{4}$ HARE 1973 YUKA 2${{\mathit \gamma}}$ decay 1  ABBOTT 2021 assumed modified gravitational-wave dispersion to establish a limit on graviton mass, using LIGO-Virgo O1-O3a binary black hole (BBH) events. 2  DEFELICE 2021 studies the normal branch of the Minimal Theory of Massive Gravity (MTMG) to find that after five parameters are adjusted to obtain agreement with all presently available data, today's squared mass ${{\mathit m}_{{g}}^{2}}$ = ($2.5$ ${}^{+4.5}_{-4.8}$) $ \times 10^{-67}$ eV${}^{2}$ or ${{\mathit m}_{{g}}}$ $<$ $8.4 \times 10^{-33}$ eV, both at the 95$\%$ CL. 3  BERNUS 2020 use the latest solution of the ephemeris INPOP (19a) in order to improve the constraint in BERNUS 2019 on the existence of a Yukawa suppression to the Newtonian potential, generically associated to a gravitons mass. 4  SHAO 2020 sets limit, 95$\%$ CL, based on non-observation of excess gravitational radiation in 14 well-timed binary pulsars in the context of the cubic Galileon model. 5  ABBOTT 2019 , ABBOTT 2017 , and ABBOTT 2016 assumed modified gravitational waves dispersion to establish limits on graviton mass. 6  BERNUS 2019 use the planetary ephemeris INPOP 17b to constraint the existence of a Yukawa suppression to the Newtonian potential, generically associated to a gravitons mass. 7  MIAO 2019 90$\%$ CL limit is based on orbital period decay rates of 9 binary pulsars using a Bayesian prior uniform in graviton mass. Limit becomes $<$ $5.2 \times 10^{-21}$ eV for a prior uniform in ln(${\mathit m}_{{{\mathit g}}}$). 8  DESAI 2018 limit based on dynamical mass models of galaxy cluster Abell 1689. 9  GUPTA 2018 obtains graviton mass limits using stacked clusters from 3 disparate surveys. 10  RANA 2018 limit, 68$\%$ CL, obtained using weak lensing mass profiles out to the radius at which the cluster density falls to 200 times the critical density of the Universe. Limit is based on the fractional change between Newtonian and Yukawa accelerations for the 50 most massive galaxy clusters in the Local Cluster Substructure Survey. Limits for other CL's and other density cuts are also given. 11  RANA 2018 limit, 68$\%$ CL, obtained using mass measurements via the SZ effect out to the radius at which the cluster density falls to 500 times the critical density of the Universe for 182 optically confirmed galaxy clusters in an Altacama Cosmology Telescope survey. Limits for other CL's and other density cuts are also given. 12  WILL 2018 limit from perihelion advances of the planets, notably Earth, Mars, and Saturn. Alternate analysis yields $<$ $6 \times 10^{-24}$. 13  ZAKHAROV 2016 constrains range of Yukawa gravity interaction from S2 star orbit about black hole at Galactic center. The limit is $<$ $2.9 \times 10^{-21}$ eV for $\delta $ = 100. 14  BRITO 2013 explore massive graviton (spin-2) fluctuations around rotating black holes. 15  GRUZINOV 2005 uses the DGP model (DVALI 2000 ) showing that non-perturbative effects restore continuity with Einstein's equations as the gravition mass approaches zero, then bases his limit on Solar System observations. 16  CHOUDHURY 2004 concludes from a study of weak-lensing data that masses heavier than about the inverse of 100 Mpc seem to be ruled out if the gravitation field has the Yukawa form. 17  GERSHTEIN 2004 use non-Einstein field relativistic theory of gravity (RTG), with a massive graviton, to obtain the 95$\%$ CL mass limit implied by the value of $\Omega _{tot}$ = $1.02$ $\pm0.02$ current at the time of publication. 18  FINN 2002 analyze the orbital decay rates of PSR$~$B1913+16 and PSR$~$B1534+12 with a possible graviton mass as a parameter. The combined frequentist mass limit is at 90$\%$CL. 19  As of 2020, limits on dP/dt are now about 0.1$\%$ (see T. Damour, ``Experimental tests of gravitational theory,'' in this $\mathit Review$). 20  GOLDHABER 1974 establish this limit considering the binding of galactic clusters, corrected to Planck ${{\mathit h}_{{0}}}$ = 0.67.

References:

ABBOTT 2021 PR D103 122002 Tests of general relativity with binary black holes from the second LIGO-Virgo gravitational-wave transient catalog DEFELICE 2021 JCAP 2112 011 Minimal theory of massive gravity and constraints on the graviton mass BERNUS 2020 PR D102 021501 Constraint on the Yukawa suppression of the Newtonian potential from the planetary ephemeris INPOP19a SHAO 2020 PR D102 024069 New Graviton Mass Bound from Binary Pulsars ABBOTT 2019 PR D100 104036 Tests of General Relativity with the Binary Black Hole Signals from the LIGO-Virgo Catalog GWTC-1 BERNUS 2019 PRL 123 161103 Constraining the mass of the graviton with the planetary ephemeris INPOP MIAO 2019 PR D99 123015 Bounding the mass of graviton in a dynamic regime with binary pulsars DESAI 2018 PL B778 325 Limit on graviton mass from galaxy cluster Abell 1689 GUPTA 2018 ANP 399 85 Limit on graviton mass using stacked galaxy cluster catalogs from SPT-SZ, Planck-SZ and SDSS-redMaPPer RANA 2018 PL B781 220 Bounds on graviton mass using weak lensing and SZ effect in galaxy clusters WILL 2018 CQG 35 17LT01 Solar system versus gravitational-wave bounds on the graviton mass ABBOTT 2017 PRL 118 221101 GW170104: Observation of a 50-Solar-Mass Binary Black Hole Coalescence at Redshift 0.2 ABBOTT 2016 PRL 116 061102 Observation of Gravitational Waves from a Binary Black Hole Merger ZAKHAROV 2016 JCAP 1605 045 Constraining the range of Yukawa gravity interaction from S2 star orbits II: Bounds on graviton mass BRITO 2013 PR D88 023514 Massive Spin-2 Fields on Black Hole Spacetimes: Instability of the Schwarzschild and Kerr Solutions and Bounds on the Graviton Mass GRUZINOV 2005 NAST 10 311 On the Graviton Mass CHOUDHURY 2004 ASP 21 559 Probing Large Distance Higher Dimensional Gravity from Lensing Data GERSHTEIN 2004 PAN 67 1596 Graviton Mass, Quintessence and Oscillatory Character of the Universe Evolution FINN 2002 PR D65 044022 Bounding the Mass of the Graviton using Binary Pulsar Observations TALMADGE 1988 PRL 61 1159 Model Independent Constraints on Possible Modifications of Newtonian Gravity GOLDHABER 1974 PR D9 1119 Mass of the Graviton HARE 1973 CJP 51 431 Mass of the Graviton