On the proof of dark matter, the law of gravity, and the mass of neutrinos

We develop a new method to predict the density associated with weak-lensing maps of (un) relaxed clusters in a range of theories interpolating between general relativity (GR) and modified Newtonian dynamics (MOND). We apply it to fit the lensing map of the Bullet merging cluster 1E 0657 - 56, in order to constrain more robustly the nature and amount of collisionless matter in clusters beyond the usual assumption of spherical equilibrium ( Pointecouteau & Silk) and the validity of GR on cluster scales (Clowe et al.). Strengthening the proposal of previous authors, we show that the Bullet Cluster is dominated by a collisionless - most probably nonbaryonic component in GR as well as in MOND, a result consistent with the dynamics of many X-ray clusters. Our findings add to the number of known pathologies for a purely baryonic MOND, including its inability to fit the latest data from the Wilkinson Microwave Anisotropy Probe. A plausible resolution of all these issues and standard issues of cold dark matter (CDM) with galaxy rotation curves is the "marriage" of MOND with ordinary hot neutrinos of 2 eV. This prediction is just within the GR-independent maximum of neutrino mass from current beta-decay experiments and will be falsifiable by the Karlsruhe Tritium Neutrino (KATRIN) experiment by 2009. Issues of consistency with strong-lensing arcs and the large relative velocity of the two clusters comprising the Bullet Cluster are also addressed.