Dark Matter And $B_s \to μ^+ μ^-$ With Minimal $SO_{10}$ Soft SUSY Breaking

CMSSM boundary conditions are usually used when calculating cosmological dark matter densities. In this paper we calculate the cosmological density of dark matter in the MSSM using minimal $SO_{10}$ soft SUSY breaking boundary conditions. These boundary conditions incorporate several attractive features: they are consistent with $SO_{10}$ Yukawa unification, they result in a "natural" inverted scalar mass hierarchy and they reduce the dimension 5 operator contribution to the proton decay rate. With regards to dark matter, on the other hand, this is to a large extent an unexplored territory with large squark and slepton masses $m_{16}$, large $A_0$ and small $ \{\mu, M_{1/2} \} $. We find that in most regions of parameter space the cosmological density of dark matter is considerably less than required by the data. However there is a well--defined, narrow region of parameter space which provides the observed relic density of dark matter, as well as a good fit to precision electroweak data, including top, bottom and tau masses, and acceptable bounds on the branching fraction of $B_s \to \mu^+ \mu^-$. We present predictions for Higgs and SUSY spectra, the dark matter detection cross section and the branching ratio ${\rm BR}(B_s\to \mu^+ \mu^-)$ in this region of parameter space.