Rotation-dependent $I$-Love-$Q$-$δM$ relations in perturbation theory

TL;DR

This work derives rotation-dependent I-Love-Q-δM relations for slowly rotating stars by combining perturbative (Hartle–Thorne) theory with extended normalization schemes. It introduces tilde quantities based on the observable mass $M_S$ and shows that a rotation-dependent extension based on $\delta M$ can recover $M_0$ and thus produce more accurate multipolar moments than the standard approach. The authors present two complementary routes—the extended analytical framework and a numerical perturbative approach—demonstrating close agreement (percent-level) with fully numerical results for modest spins and highlighting significant improvements over the traditional method. These results enhance the viability of gravitational-wave inferences of neutron-star structure by removing dependence on the unobservable static mass and providing robust, rotation-aware universal relations. The findings invite extensions to higher multipoles and broader EoS coverage, potentially improving parameter estimation in GW observations.

Abstract

The so-called $I$-Love-$Q$ relations link some normalized versions of the moment of inertia, the Love number, and the quadrupole moment of a star. These relations, in principle, enable the inference of two of the quantities given the third. However, their use has been limited because the normalized versions of the multipole moments rely on the static mass derived from the Tolman-Oppenheimer-Volkoff equation, which is not directly observable. In this work, using perturbation theory, we find that the $I$-Love-$Q$ relations can also be formulated in terms of an alternative set of normalized quantities that do not depend on the static mass, but on the actual (observable) mass.

Figures (4)

• Figure 1: Different relations of the ${\widetilde{I}}$--Love--${\widetilde{Q}}(\chi_S)$ set. The dots in each plot represent the results from the individual EoS's, while the surfaces are the polynomial functions from \ref{['lnIbarrotTL']}-\ref{['lnIbarrotQbarT']}. Panel a) shows $\ln{\color{Black}{{\widetilde{I}}\lvert_{(\lambda_S,\chi_S)}^{{\color{Black}{\textup{NP}}}}\xspace}}$\ref{['lnIbarrotTL']}, b) $\ln{\color{Black}{{\widetilde{Q}}\lvert_{(\lambda_S,\chi_S)}^{{\color{Black}{\textup{NP}}}}\xspace}}$\ref{['lnQbarrotTL']}, c) $\ln{\color{Black}{{\widetilde{I}}\lvert_{({\widetilde{Q}},\chi_S)}^{{\color{Black}{\textup{NP}}}}\xspace}}$\ref{['lnIbarrotQbarrotT']}, and d) represents $\ln{\color{Black}{{\widetilde{I}}\lvert_{({\overline{Q}},\chi_S)}^{{\color{Black}{\textup{NP}}}}\xspace}}$\ref{['lnIbarrotQbarT']}.
• Figure 2: Panel a) represents $\mathcal{E}_{{\widetilde{I}}(\lambda_S,\chi_S)}^{{{\color{Black}{\textup{NP}}}}\xspace/{{\color{Black}{\textup{ext}}}}\xspace}$\ref{['errorILTX']}, b) $\mathcal{E}_{{\widetilde{Q}}(\lambda_S,\chi_S)}^{{{\color{Black}{\textup{NP}}}}\xspace/{{\color{Black}{\textup{ext}}}}\xspace}$\ref{['errorQLTX']}, and c) $\mathcal{E}_{{\widetilde{I}}({\overline{Q}},\chi_S)}^{{{\color{Black}{\textup{NP}}}}\xspace/{{\color{Black}{\textup{ext}}}}\xspace}$\ref{['errorIQTX']}. The color bars show that the relative errors are below $\sim0.2\%$ in all cases.
• Figure 3: Panel a) represents $\ln{\color{Black}{{\widetilde{I}}\lvert_{({\widetilde{Q}},\chi_S)}^{{\color{Black}{\textup{NP}}}}\xspace}}$\ref{['lnIbarrotQbarrotT']} (green surface) and $\ln{\color{Black}{{\widetilde{I}}}\lvert^{{{\color{Black}{\textup{RNS}}}}\xspace}_{({\widetilde{Q}},\chi_S)}}$\ref{['lnIbarrotchakra']} (blue surface), which is the restriction of $\ln{\color{Black}{{\color{Black}{{\widetilde{I}}{}^{\color{Black}{\textup{full}}}\xspace}}{}\lvert^{{\color{Black}{\textup{RNS}}}}\xspace_{({\color{Black}{{\widetilde{Q}}{}^{\color{Black}{\textup{full}}}\xspace}},{\color{Black}{\chi_S^{\color{Black}{\textup{full}}}\xspace{}}})}}}$ to second order in $\Omega_S$. Panel b) shows the relative error between both surfaces as defined in \ref{['errorIQchakra']}.
• Figure 4: All possible relations involving the quantity ${\widetilde{\delta M}}$. The dots in each plot represent the results from all the EoS's, while the surfaces are the polynomial functions from \ref{['lndMbarrotLT']}-\ref{['lndMbarrotQbarrotT']}. Panel a) shows $\ln{\color{Black}{{\widetilde{\delta M}}\lvert_{(\lambda_S,\chi_S)}^{{\color{Black}{\textup{NP}}}}\xspace}}$\ref{['lndMbarrotLT']}, b) $\ln{\color{Black}{{\widetilde{\delta M}}\lvert_{({\overline{I}},\chi_S)}^{{\color{Black}{\textup{NP}}}}\xspace}}$\ref{['lndMbarrotIbarT']}, c) $\ln{\color{Black}{{\widetilde{\delta M}}\lvert_{({\widetilde{I}},\chi_S)}^{{\color{Black}{\textup{NP}}}}\xspace}}$\ref{['lndMbarrotIbarrotT']}, d) $\ln{\color{Black}{{\widetilde{\delta M}}\lvert_{({\overline{Q}},\chi_S)}^{{\color{Black}{\textup{NP}}}}\xspace}}$\ref{['lndMbarrotQbarT']}, and e) represents $\ln{\color{Black}{{\widetilde{\delta M}}\lvert_{({\widetilde{Q}},\chi_S)}^{{\color{Black}{\textup{NP}}}}\xspace}}$\ref{['lndMbarrotQbarrotT']}.