The I-Love-Q Relations for Superfluid Neutron Stars

The I-Love-Q relations are approximate equation-of-state independent relations that connect the moment of inertia, the spin-induced quadrupole moment, and the tidal deformability of neutron stars. In this paper, we study the I-Love-Q relations for superfluid neutron stars for a general relativistic two-fluid model: one fluid being the neutron superfluid and the other a conglomerate of all charged components. We study to what extent the two-fluid dynamics might affect the robustness of the I-Love-Q relations by using a simple two-component polytropic model and a relativistic mean field model with entrainment for the equation-of-state. Our results depend crucially on the spin ratio $\Omega_{\rm n}/\Omega_{\rm p}$ between the angular velocities of the neutron superfluid and the normal component. We find that the I-Love-Q relations can still be satisfied to high accuracy for superfluid neutron stars as long as the two fluids are nearly co-rotating $\Omega_{\rm n}/\Omega_{\rm p} \approx 1$. However, the deviations from the I-Love-Q relations increase as the spin ratio deviates from unity. In particular, the deviation of the Q-Love relation can be as large as $O(10\%)$ if $\Omega_{\rm n}/\Omega_{\rm p}$ differ from unity by a few tens of percent. As $\Omega_{\rm n}/\Omega_{\rm p} \approx 1$ is expected for realistic neutron stars, our results suggest that the two-fluid dynamics should not affect the accuracy of any gravitational waveform models for neutron star binaries that employ the relation to connect the spin-induced quadrupole moment and the tidal deformability.