MOND plus classical neutrinos are not enough for cluster lensing

Clusters of galaxies offer a robust test bed for probing the nature of dark matter that is insensitive to the assumption of the gravity theories. Both Modified Newtonian Dynamics (MOND) and General Relativity (GR) would require similar amounts of non-baryonic matter in clusters as MOND boosts the gravity only mildly on cluster scales. Gravitational lensing allows us to estimate the enclosed mass in clusters on small (∼ 20 − 50 kpc) and large (∼ several 100 kpc) scales independent of the assumptions of equilibrium. Here we show for the first time that a combination of strong and weak gravitational lensing effects can set interesting limits on the phase space density of dark matter in the centers of clusters. The phase space densities derived from lensing observations are inconsistent with neutrino masses ranging from 2 7 eV, and hence do not support the 2 eV-range particles required by MOND. To survive, the plausible modifications for MOND may be either an additional degree of dynamical freedom in a co-variant incarnation or mass-varying theories of neutrinos.