Multipole Love Relations

Abstract: Gravitational-wave observations in the near future may allow us to measure tidal deformabilities of neutron stars, which leads us to the understanding of physics at nuclear density. In principle, the gravitational waveform depends on various tidal parameters, which correlate strongly. Therefore, it would be useful if one can express such tidal parameters with a single parameter. Here, we report on universal relations among various $\ell$-th (dimensionless) electric, magnetic and shape tidal deformabilities in neutron stars and quark stars that do not depend sensitively on the equation of state. Such relations allow us to break the degeneracy among the tidal parameters. In this paper, we focus on gravitational waves from non-spinning neutron-star binary inspirals. We first derive the leading contribution of the $\ell$-th electric and $\ell=2$ magnetic tidal deformabilities to the gravitational-wave phase, which enters at $2\ell +1$ and $6$ post-Newtonian orders relative to the leading Newtonian one respectively. We then calculate the useful number of gravitational-wave cycles and show that not only the $\ell=2$ but also $\ell=3$ electric tidal deformabilities are important for parameter estimation with third-generation gravitational-wave detectors such as LIGO III and Einstein Telescope. Although the correlation between the $\ell=2$ and $\ell=3$ electric tidal deformabilities deteriorate the measurement accuracy of the former deformability parameter, one can increase its measurement accuracy significantly by using the universal relation. We provide a fitting formula for the LIGO III noise curve in the Appendix.