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Tidal Love numbers of a slowly spinning neutron star

By extending our recent framework to describe the tidal deformations of a spinning compact object, we compute for the first time the tidal Love numbers of a spinning neutron star to linear order in the angular momentum. The spin of the object introduces couplings between electric and magnetic distortions and new classes of spin-induced ("rotational") tidal Love numbers emerge. We focus on stationary tidal fields, which induce axisymmetric perturbations. We present the perturbation equations for both electric-led and magnetic-led rotational Love numbers for generic multipoles and explicitly solve them for various tabulated equations of state and for a tidal field with an electric (even parity) and magnetic (odd parity) component with $\ell=2,3,4$. For a binary system close to the merger, various components of the tidal field become relevant. In this case we find that an octupolar magnetic tidal field can significantly modify the mass quadrupole moment of a neutron star. Preliminary estimates, assuming a spin parameter $χ\approx0.05$, show modifications $\gtrsim10\%$ relative to the static case, at an orbital distance of five stellar radii. Furthermore, the rotational Love numbers as functions of the moment of inertia are much more sensitive to the equation of state than in the static case, where approximate universal relations at the percent level exist. For a neutron-star binary approaching the merger, we estimate that the approximate universality of the induced mass quadrupole moment deteriorates from $1\%$ in the static case to roughly $6\%$ when $χ\approx0.05$. Our results suggest that spin-tidal couplings can introduce important corrections to the gravitational waveforms of spinning neutron-star binaries approaching the merger.