  (Abridged) Modified Newtonian Dynamics (MOND) can fit a broad range of galaxy
kinematic data, but struggles with clusters of galaxies. MONDian clusters need
dark matter, and here we test the 11 eV sterile neutrino - used to fit the
first three acoustic peaks of the cosmic microwave background - by
investigating their equilibrium distributions in 30 groups and clusters over a
wide range of temperatures. Knowing both the sterile neutrino density and
velocity dispersion tells us the Tremaine-Gunn phase-space limit at all radii.
We find that all 30 systems serendipitously reach the Tremaine-Gunn limit by
the centre, which means a portion of the dynamical mass must always be covered
by the brightest cluster galaxy. Interestingly, the typical fitted K-band
mass-to-light ratio is unity and at most 1.2, which is very consistent -
although leaving no margin for error - with stellar population synthesis
models. Amidst the sample there are several special cases including the Coma
cluster (for which dark matter was first proposed), NGC 720 (where geometrical
evidence for dark matter was found) and the bullet cluster (where dark matter -
of some kind - in clusters was directly proven to exist). We demonstrate that
11 eV sterile neutrinos are unlikely to influence spiral galaxy rotation
curves, as they don't influence even some very massive early-types (NGC 4125
and NGC 6482). Finally, we conclude that it is intriguing that the minimum mass
of sterile neutrino particle that can match the cosmic microwave background is
identical to the minimum mass found here to be consistent with equilibrium
configurations of MONDian clusters of galaxies.
