#### graviton MASS

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
• • We do not use the following data for averages, fits, limits, etc. • •
$<1.76 \times 10^{-23}$ 1
 2021
DISP LIGO Virgo catalog GWTC-2
$<3.2 \times 10^{-23}$ 2
 2020
YUKA Planetary ephemeris INPOP19a
$<2 \times 10^{-28}$ 3
 2020
$<4.7 \times 10^{-23}$ 4
 2019
DISP LIGO Virgo catalog GWTC-1
$<7 \times 10^{-23}$ 5
 2019
YUKA Planetary ephemeris INPOP17b
$<3.1 \times 10^{-20}$ 6
 2019
DISP Binary pulsar orbital decay rate
$<1.4 \times 10^{-29}$ 7
 2018
YUKA Gal cluster Abell 1689
$<5 \times 10^{-30}$ 8
 2018
YUKA Using SPT-SZ
$<3 \times 10^{-30}$ 8
 2018
YUKA Using Planck all-sky SZ
$<1.3 \times 10^{-29}$ 8
 2018
YUKA Using redMaPPer SDSS-DR8
$<6 \times 10^{-30}$ 9
 2018
YUKA Weak lensing in massive clusters
$<8 \times 10^{-30}$ 10
 2018
YUKA SZ effect in massive clusters
$<1.0 \times 10^{-23}$ 11
 2018
$<7 \times 10^{-23}$ 4
 2017
DISP Combined dispersion limit from three BH mergers
$<1.2 \times 10^{-22}$ 4
 2016
DISP Combined dispersion limit from two BH mergers
$<2.9 \times 10^{-21}$ 12
 2016
YUKA S2 star orbit
$<5 \times 10^{-23}$ 13
 2013
MGRV Spinning black holes bounds
$<6 \times 10^{-32}$ 14
 2005
MGRV Solar System observations
$<6 \times 10^{-32}$ 15
 2004
YUKA Weak gravitational lensing
$<9.0 \times 10^{-34}$ 16
 2004
MGRV From $\Omega _{tot}$ value assuming RTG
$<8 \times 10^{-20}$ 17, 18
 2002
DISP Binary pulsar orbital period decrease
$<7 \times 10^{-23}$
 1988
YUKA Solar system planetary astrometric data
$<1.3 \times 10^{-29}$ 19
 1974
YUKA Rich clusters
$<7 \times 10^{-28}$
 1973
YUKA Galaxy
$<8 \times 10^{4}$
 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 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.
 3 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.
 4 ABBOTT 2019 , ABBOTT 2017 , and ABBOTT 2016 assumed modified gravitational waves dispersion to establish limits on graviton mass.
 5 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.
 6 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(${\mathit m}_{{{\mathit g}}}$).
 7 DESAI 2018 limit based on dynamical mass models of galaxy cluster Abell 1689.
 8 GUPTA 2018 obtains graviton mass limits using stacked clusters from 3 disparate surveys.
 9 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.
 10 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.
 11 WILL 2018 limit from perihelion advances of the planets, notably Earth, Mars, and Saturn. Alternate analysis yields $<$ $6 \times 10^{-24}$.
 12 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.
 13 BRITO 2013 explore massive graviton (spin-2) fluctuations around rotating black holes.
 14 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.
 15 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.
 16 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.
 17 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.
 18 As of 2020, limits on dP/dt are now about 0.1$\%$ (see T. Damour, Experimental tests of gravitational theory,'' in this $\mathit Review$).
 19 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
 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