We explore the possible cross sections for the elastic scattering of
neutralinos chi on nucleons p,n in the minimal supersymmetric extension of the
standard model (MSSM). Universality of the soft supersymmetry-breaking scalar
masses for the Higgs multiplets is not assumed, but the MSSM parameters are
nevertheless required to lead consistently to an electroweak vacuum. We explore
systematically the region of MSSM parameter space where LEP and other
accelerator constraints are respected, and the relic neutralino density lies in
the range 0.1 < Omega_chi h^2 < 0.3 preferred by cosmology. We also discuss
models with Omega_chi h^2 < 0.1, in which case we scale the density of
supersymmetric dark matter in our galactic halo by Omega_chi h^2 / 0.1,
allowing for the possible existence of some complementary form of cold dark
matter. We find values of the cross sections that are considerably lower than
the present experimental sensitivities. At low neutralino masses, m_chi < 100
GeV, the cross sections may be somewhat higher than in the constrained MSSM
with universal soft Higgs masses, though they are generally lower. In the case
of large m_chi, the cross sections we find may be considerably larger than in
the constrained model, but still well below the present experimental
sensitivity.Comment: 25 pages LaTeX, 7 eps figure
