Improving the inference of the stellar quantities using the extended $I$-Love-$Q$-$δM$ relations

TL;DR

The paper addresses the limitation that the $I$-Love-$Q$ relations rely on normalising by the background mass $M_0$, which is not directly observable, making direct application challenging for moderately to rapidly rotating stars. It contrasts the standard approach, which fixes $M_0^{std}=M_S$ and is valid only at very slow rotation, with the extended $I$-Love-$Q$-$\delta M$ framework that infers $M_0$ from the mass-perturbation term $\delta M$ via $\overline{\delta M} = \frac{M_S - M_0}{\Omega_S^2 M_0^3 \overline{I}^2} = \frac{M_S - M_0}{\chi_S^2 M_0}$, leading to $M_0^{\text{ext}} = \frac{M_S}{\overline{\delta M}\chi_S^2 + 1}$. Once $M_0^{\text{ext}}$ is obtained, the standard $I$-Love and $Q$-Love relations provide $\overline{I}$ and $\overline{Q}$, yielding $I_S^{\text{ext}} = (M_0^{\text{ext}})^3\,\overline{I}$ and $Q_S^{\text{ext}} = \chi_S^2 (M_0^{\text{ext}})^3\,\overline{Q}$. In a polytropic test, the extended method consistently reduces relative errors in $M_0$, $I_S$, and $Q_S$, especially at higher spins, enhancing the observational applicability of the universal relations to faster-rotating neutron/ quark stars and improving inferences from gravitational-wave data.

Abstract

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 $χ_S<0.1$). The more accurate alternative approach, based on the $I$-Love-$Q$-$δM$ set of relations, circumvents this limitation by enabling the inference of $M_0$. Here we review both approaches and provide numerical comparisons.

Figures (1)

• Figure 1: Relative errors of $M_0$ (left column), $I_S$ (middle column) and $Q_S$ (right column) as computed using perturbation theory, for the extended approach (upper row) and standard approach (lower row) as functions of $P_c$ and for different values of $\chi_S$. In order to compare the results from both approaches in a more visual way, a dashed line marks the highest error of the extended approach in every column.