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 GR and MOND (General Relativity and Modified Newtonian Dynamics). We apply it to fit the lensing map of the bullet merging cluster 1E0657-56, in order to constrain more robustly the nature and amount of collisionless matter in clusters {\it beyond} the usual assumption of spherical equilibrium (Pointecouteau & Silk 2005) and the validity of GR on cluster scales (Clowe et al. 2006). Strengthening the proposal of previous authors we show that the bullet cluster is dominated by a collisionless -- most probably non-baryonic -- 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 with galaxy rotation curves is the "marriage" of MOND with ordinary hot neutrinos of 2eV. This prediction is just within the GR-independent maximum of neutrino mass from current $\beta$-decay experiments, and is 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.