  With positive signals from multiple direct detection experiments it will, in
principle, be possible to measure the mass and cross sections of
weakly-interacting massive particle (WIMP) dark matter. Recent work has shown
that, with a polynomial parameterisation of the WIMP speed distribution, it is
possible to make an unbiased measurement of the WIMP mass, without making any
astrophysical assumptions. However, direct detection experiments are not
sensitive to low-speed WIMPs and, therefore, any model-independent approach
will lead to a bias in the cross section. This problem can be solved with the
addition of measurements of the flux of neutrinos from the Sun. This is because
the flux of neutrinos produced from the annihilation of WIMPs which have been
gravitationally captured in the Sun is sensitive to low-speed WIMPs. Using mock
data from next-generation direct detection experiments and from the IceCube
neutrino telescope, we show that the complementary information from IceCube on
low-speed WIMPs breaks the degeneracy between the cross section and the speed
distribution. This allows unbiased determinations of the WIMP mass and
spin-independent and spin-dependent cross sections to be made, and the speed
distribution to be reconstructed. We use two parameterisations of the speed
distribution: binned and polynomial. While the polynomial parameterisation can
encompass a wider range of speed distributions, this leads to larger
uncertainties in the particle physics parameters.
