  Self-interacting dark matter (SIDM) was introduced by Spergel & Steinhardt to
address possible discrepancies between collisionless dark matter simulations
and observations on scales of less than 1 Mpc. We examine the case in which
dark matter particles not only have strong self-interactions but also have
strong interactions with baryons. The presence of such interactions will have
direct implications for nuclear and particle astrophysics. Among these are a
change in the predicted abundances from big bang nucleosynthesis (BBN) and the
flux of gamma-rays produced by the decay of neutral pions which originate in
collisions between dark matter and Galactic cosmic rays (CR). From these
effects we constrain the strength of the baryon--dark matter interactions
through the ratio of baryon - dark matter interaction cross section to dark
matter mass, $s$. We find that BBN places a weak upper limit to this ratio $<
10^8 cm^2/g$. CR-SIDM interactions, however, limit the possible DM-baryon cross
section to $< 5 \times 10^{-3} cm^2/g$; this rules out an energy-independent
interaction, but not one which falls with center-of-mass velocity as $s \propto
1/v$ or steeper.
