  The exterior gravitational field of a slowly-rotating neutron star can be
characterized by its multipole moments, the first few being the neutron star
mass, moment of inertia, and quadrupole moment to quadratic order in spin. In
principle, all of these quantities depend on the neutron star's internal
structure, and thus, on unknown nuclear physics at supra-nuclear energy
densities. We here find relations between the moment of inertia, the Love
numbers and the quadrupole moment (I-Love-Q relations) that do not depend
sensitively on the neutron star's internal structure. Three important
consequences derive from these I-Love-Q relations. On an observational
astrophysics front, the measurement of a single member of the I-Love-Q trio
would automatically provide information about the other two, even when the
latter may not be observationally accessible. On a gravitational wave front,
the I-Love-Q relations break the degeneracy between the quadrupole moment and
the neutron-star spins in binary inspiral waveforms, allowing second-generation
ground-based detectors to determine the (dimensionless) averaged spin to
$\mathcal{O}(10)%$, given a sufficiently large signal-to-noise ratio detection.
On a fundamental physics front, the I-Love-Q relations allow for tests of
General Relativity in the neutron-star strong-field that are both theory- and
internal structure-independent. As an example, by combining gravitational-wave
and electromagnetic observations, one may constrain dynamical Chern-Simons
gravity in the future by more than 6 orders of magnitude more stringently than
Solar System and table-top constraints.
