  General-relativistic equilibria of differentially rotating stars are expected
in a number of astrophysical scenarios, from core-collapse supernovae to the
remnant of binary neutron-star mergers. The latter, in particular, have been
the subject of extensive studies where they were modeled with a variety of laws
of differential rotation with varying degree of realism. Starting from accurate
and fully general-relativistic simulations of binary neutron-star mergers with
various equations of state and mass ratios, we establish the time when the
merger remnant has reached a quasi-stationary equilibrium and extract in this
way realistic profiles of differential rotation. This allows us to explore how
well traditional laws reproduce such differential-rotation properties and to
derive new laws of differential rotation that better match the numerical data
in the low-density Keplerian regions of the remnant. In this way, we have
obtained a novel and somewhat surprising result: the dynamical stability line
to quasi-radial oscillations computed from the turning-point criterion can have
a slope that is not necessarily negative with respect to the central rest-mass
density, as previously found with traditional differential-rotation laws.
Indeed, for stellar models reproducing well the properties of the merger
remnants, the slope is actually positive, thus reflecting remnants with angular
momentum at large distances from the rotation axis, and hence with cores having
higher central rest-mass densities and slower rotation rates.
