  We construct three new Hartree-Fock-Bogoliubov (HFB) mass models, labeled
HFB-19, HFB-20, and HFB-21, with unconventional Skyrme forces containing $t_4$
and $t_5$ terms, i.e., density-dependent generalizations of the usual $t_1$ and
$t_2$ terms, respectively. The new forces underlying these models are fitted
respectively to three different realistic equations of state of neutron matter
for which the density dependence of the symmetry energy ranges from the very
soft to the very stiff, reflecting thereby our present lack of complete
knowledge of the high-density behavior of nuclear matter. All unphysical
instabilities of nuclear matter, including the transition to a polarized state
in neutron-star matter, are eliminated with the new forces. At the same time
the new models fit essentially all the available mass data with rms deviations
of 0.58 MeV and give the same high quality fits to measured charge radii that
we obtained in earlier models with conventional Skyrme forces. Being
constrained by neutron matter, these new mass models, which all give similar
extrapolations out to the neutron drip line, are highly appropriate for studies
of the $r$-process and the outer crust of neutron stars. Moreover, the
underlying forces, labeled BSk19, BSk20 and BSk21, respectively, are well
adapted to the study of the inner crust and core of neutron stars. The new
family of Skyrme forces thus opens the way to a unified description of all
regions of neutron stars.
