Assessment of universal relations among second-order moments of relativistic stars via reformulated perturbation equations

We assess the universal relations among second-order moments of relativistic stars, namely the moment of inertia, tidal deformability, and spin-induced quadrupole moment, via reformulated perturbation equations. After constructing the spherical background configuration by solving two ordinary differential equations as usual, these three moments are obtained by solving four additional ordinary differential equations. They are solved numerically from the stellar center to the surface, and we do not need to derive homogeneous solutions for obtaining the quadrupole moment. This small number of ordinary differential equations to be solved enables us to identify the primary variable for each second-order moment. Investigating the profile of these variables in the star, we speculate that their nonmonotonic behavior, enhanced typically for soft equations of state and/or high compactnesses, introduces the variety to the relations among these second-order moments unless the black-hole limit is approached. Because realistic relativistic stars are widely believed to be characterized by stiff equations of state, they enjoy the universal relation to a great extent.