  We evaluate the neutrino fluxes to be expected from neutralino LSP
annihilations inside the Sun, within the minimal supersymmetric extension of
the Standard Model with supersymmetry-breaking scalar and gaugino masses
constrained to be universal at the GUT scale (the CMSSM). We find that there
are large regions of typical CMSSM $(m_{1/2}, m_0)$ planes where the LSP
density inside the Sun is not in equilibrium, so that the annihilation rate may
be far below the capture rate. We show that neutrino fluxes are dependent on
the solar model at the 20% level, and adopt the AGSS09 model of Serenelli et
al. for our detailed studies. We find that there are large regions of the CMSSM
$(m_{1/2}, m_0)$ planes where the capture rate is not dominated by
spin-dependent LSP-proton scattering, e.g., at large $m_{1/2}$ along the CMSSM
coannihilation strip. We calculate neutrino fluxes above various threshold
energies for points along the coannihilation/rapid-annihilation and focus-point
strips where the CMSSM yields the correct cosmological relic density for
tan(beta) = 10 and 55 for $\mu$ > 0, exploring their sensitivities to
uncertainties in the spin-dependent and -independent scattering matrix
elements. We also present detailed neutrino spectra for four benchmark models
that illustrate generic possibilities within the CMSSM. Scanning the
cosmologically-favored parts of the parameter space of the CMSSM, we find that
the IceCube/DeepCore detector can probe at best only parts of this parameter
space, notably the focus-point region and possibly also at the low-mass tip of
the coannihilation strip.
