  Self-interacting dark matter (SIDM), especially bosonic, has been considered
a promising candidate to replace cold dark matter (CDM) as it resolves some of
the problems associated with CDM. Here, we rule out the possibility that dark
matter is a repulsive boson in thermal equilibrium. We develop the model first
proposed by Goodman (2000) and derive the equation of state at finite
temperature. Isothermal spherical halo models indicate a Bose-Einstein
condensed core surrounded by a non-degenerate envelope, with an abrupt density
drop marking the boundary between the two phases. Comparing this feature with
observed rotation curves constrains the interaction strength of our model's DM
particle, and Bullet Cluster measurements constrain the scattering cross
section. Both ultimately can be cast as constraints on the particle's mass. We
find these two constraints cannot be satisfied simultaneously in any realistic
halo model---and hence dark matter cannot be a repulsive boson in thermal
equilibrium. It is still left open that DM may be a repulsive boson provided it
is not in thermal equilibrium; this requires that the mass of the particle be
significantly less than a millivolt.
