Effective no-hair relations for neutron stars and quark stars: Relativistic results

Astrophysical charge-free black holes are known to satisfy no-hair relations through which all multipole moments can be specified in terms of just their mass and spin angular momentum. We here investigate the possible existence of no-hair-like relations among multipole moments for neutron stars and quark stars that are independent of their equation of state. We calculate the multipole moments of these stars up to hexadecapole order by constructing uniformly rotating and unmagnetized stellar solutions to the Einstein equations. For slowly rotating stars, we construct stellar solutions to quartic order in spin in a slow-rotation expansion, while for rapidly rotating stars, we solve the Einstein equations numerically with the LORENE and RNS codes. We find that the multipole moments extracted from these numerical solutions are consistent with each other and agree with the quartic-order slow-rotation approximation for spin frequencies below roughly 500 Hz. We also confirm that the current dipole is related to the mass quadrupole in an approximately equation-of-state-independent fashion, which does not break for rapidly rotating neutron stars or quark stars. We further find that the current-octupole and the mass-hexadecapole moments are related to the mass quadrupole in an approximately equation-of-state-independent way to roughly $\mathcal{O}(10%)$, worsening in the hexadecapole case. All of our findings are in good agreement with previous work that considered stellar solutions to leading order in a weak-field, Newtonian expansion. In fact, the hexadecapole-quadrupole relation agrees with the Newtonian one quite well even in moderately relativistic regimes. The quartic in spin, slowly rotating solutions found here allows us to estimate the systematic errors in the measurement of the neutron star's mass and radius with future x-ray observations, such as Neutron star Interior Composition ExploreR (NICER) and Large Observatory for X-ray Timing (LOFT). We find that the effect of these quartic-in-spin terms on the quadrupole and hexadecapole moments and stellar eccentricity may dominate the error budget for very rapidly rotating neutron stars. The new universal relations found here should help to reduce such systematic errors.