Improving the inference of the stellar quantities using the extended $I$-Love-$Q$-$\delta M$ relations

In relativistic Astrophysics the $I$-Love-$Q$ relations refer to approximately EoS-independent relations involving the moment of inertia, Love number, and quadrupole moment through some quantities that are normalised by the mass $M_0$ of the background configuration of the perturbative scheme. Since $M_0$ is not an observable quantity, this normalisation hinders the direct applicability of the relations. A common remedy assumes that $M_0$ coincides with the actual mass of the star $M_S$; however, this approximation is only adequate for very slow rotation (when the dimensionless spin parameter is $\chi_S<0.1$). The more accurate alternative approach, based on the $I$-Love-$Q$-$\delta M$ set of relations, circumvents this limitation by enabling the inference of $M_0$. Here we review both approaches and provide numerical comparisons.