We show that a gauge singlet scalar S, with a coupling to the Higgs doublet of the form l S S y SH y H and with the S mass entirely generated by the Higgs expectation value, has a thermally generated relic density V S ഠ 0.3 if m S ഠ ͑2.9 10.5͒ ͑V S ͞0. [3,4]. Observationally, the density profile of galaxies in the inner few kiloparsecs appears to be much shallower than predicted by numerical simulations (the central density of dark matter halos being 50 times smaller than the CCDM prediction for dwarf galaxies and roughly independent of halo mass [2,5]), while the number of dwarf galaxies in the local group is an order of magnitude fewer than predicted [3,4]. In addition, the CCDM predictions for the Tully-Fisher relation [6,7] and the stability of galactic bars in high surface brightness spiral galaxies [8] are not in agreement with what is observed, indicating lower density galaxy cores than predicted by CCDM. Although there is at present considerable uncertainty regarding the interpretation of observations and simulations [9,10], it has nevertheless been argued that all the discrepancies between observations and simulations may be understood as indicating that dark matter halos in CCDM simulations are more centrally concentrated than observed [11].In order to overcome the possible deficiencies of CCDM halos, one suggestion has been that the cold dark matter particles have a nondissipative self-interaction [12,13], and it has been shown that such cold, nondissipative selfinteracting dark matter (SIDM) can be effective in alleviating the various problems of CCDM [11]. Scattering of dark matter particles stops gravitational accretion at the center of the halo and so allows a smooth core to form. Simulations with SIDM [11,14] are able to simultaneously account for the observed density profiles of galactic halos and the number of subhalos [11]. In the future SIDM may be strongly constrained by gravitational lensing observations of the shape of cluster halos [15][16][17] and by the formation of massive black holes at the centers of galaxies, which is enhanced by self-interactions of dark matter particles [18].In order to be able to account for the observed properties of dark matter halos, the requirement on the mass M and