  The emergence of novel differential rotation laws that can reproduce the
rotational profile of binary neutron star merger remnants has opened the way
for the construction of equilibrium models with properties that resemble those
of remnants in numerical simulations. We construct models of merger remnants,
using the 4-parameter differential rotation law by Uryu et al. (2017) and three
tabulated, zero-temperature equations of state. The models have angular momenta
that are determined by empirical relations, constructed through numerical
simulations. After a systematic exploration of the parameter space of merger
remnant equilibrium sequences, which includes the determination of turning
points along constant-angular-momentum sequences, we find that a particular
rotation law can reproduce the threshold mass to prompt collapse to a black
hole with a relative difference of only 1% with respect to numerical
simulations, in all cases considered. Furthermore, our results indicate a
possible correlation between the compactness of equilibrium models of remnants
at the threshold mass and the compactness of maximum-mass nonrotating models.
Another key prediction of binary neutron star merger simulations is a
relatively slowly rotating inner region, where the angular velocity $\Omega$
(as measured by an observer at infinity) is mostly due to the frame dragging
angular velocity $\omega$. In our investigation of the parameter space of the
Uryu+ rotation law, we naturally find quasi-spherical (Type A) remnant models
with this property. Our investigation clarifies the impact of the differential
rotation law and of the equation of state on key properties of binary neutron
star remnants and lays the groundwork for including thermal effects in future
studies.
