Limit on graviton mass from galaxy cluster Abell 1689
  To date, the only limit on graviton mass using galaxy clusters was obtained
by Goldhaber and Nieto in 1974, using the fact that the orbits of galaxy
clusters are bound and closed, and extend up to 580 kpc. From positing that
only a Newtonian potential gives rise to such stable bound orbits, a limit on
the graviton mass $m_g<10^{-29}$ eV was obtained (PRD 9,1119, 1974). Recently,
it has been shown that one can obtain closed bound orbits for Yukawa potential
(arXiv:1705.02444), thus invalidating the main \emph{ansatz} used in Goldhaber
and Nieto to obtain the graviton mass bound. In order to obtain a revised
estimate using galaxy clusters, we use dynamical mass models of the Abell 1689
(A1689) galaxy cluster to check their compatibility with a Yukawa gravitational
potential. We assume mass models for the gas, dark matter, and galaxies for
A1689 from arXiv:1703.10219 and arXiv:1610.01543, who used this cluster to test
various alternate gravity theories, which dispense with the need for dark
matter. We quantify the deviations in the acceleration profile using these mass
models assuming a Yukawa potential and that obtained assuming a Newtonian
potential by calculating the $\chi^2$ residuals between the two profiles. Our
estimated bound on the graviton mass ($m_g$) is thereby given by, $m_g < 1.37
\times 10^{-29}$ eV or in terms of the graviton Compton wavelength of,
$\lambda_g>9.1 \times 10^{19}$ km at 90\% confidence level.
