  The opportunity to study physics at supra-nuclear densities through X-ray
observations of neutron stars has led to in-depth investigations of certain
approximately universal relations that can remove degeneracies in pulse profile
models. One such set of relations determines all of the multipole moments of a
neutron star just from the first three (the mass monopole, the current dipole
and the mass quadrupole moment) approximately independently of the equation of
state. These three-hair relations were found to hold in neutron stars that
rotate rigidly, as is the case in old pulsars, but neutron stars can also
rotate differentially, as is the case for proto-neutron stars and hypermassive
transient remnants of binary mergers. We here extend the three-hair relations
to differentially rotating stars for the first time with a generic rotation law
using two approximations: a weak-field scheme (an expansion in powers of the
neutron star compactness) and a perturbative differential rotation scheme (an
expansion about rigid rotation). These approximations allow us to analytically
derive approximately universal relations that allow us to determine all of the
multipole moments of a (perturbative) differentially rotating star in terms of
only the first four moments. These new four-hair relations for differentially
rotating neutron stars are found to be approximately independent of the
equation of state to a higher degree than the three-hair relations for
uniformly rotating stars. Our results can be instrumental in the development of
four-hair relations for rapidly differentially rotating stars in full General
Relativity using numerical simulations.
