Carlos Conde-Ocazionez, Tuojin Yin, Jaquelyn Noronha-Hostler, Nicolás Yunes

Future X-ray missions, such as NICER and LOFT, together with gravitational-wave observations from ground-based detectors, will provide new insights into neutron stars. Interpreting accurate observations in the future will require accurate models of their gravitational fields. In this first paper of a two-part series, we construct the perturbation equations for slowly-rotating, isolated, and unmagnetized neutron stars, extending the Hartle-Thorne approximation to seventh order in a slow-rotation expansion. We obtain exact, closed-form, analytical solutions for the exterior metric at each order in spin. From these solutions, we derive expressions for the mass and mass-current scalar multipole moments, $M_{\ell}$ and $S_{\ell}$, respectively, up to seventh order in spin frequency, using two distinct methods. This high-order expansion allows us to calculate second-, fourth-, and sixth-order relative spin corrections to the observed mass and moment of inertia; second- and fourth-order relative spin corrections to the quadrupole and octopole moments; second-order relative spin corrections to the hexadecapole and dotriacontapole moments; and leading-order-in-spin expressions for the hexacontatetrapole and hectoicosaoctapole moments. Going to seventh order in the spin-frequency approximation will enable very precise calculations of X-ray pulse profiles, as well as the I-Love-Q and three-hair relations for slowly-rotating neutron stars. These results will be valuable for breaking parameter degeneracies in future multimessenger observations.