# Testing the multipole structure and conservative dynamics of compact binaries using gravitational wave observations: The spinning case

## TL;DR

This paper extends a parametrized gravitational wave model to include spins in compact binary systems, enabling tests of general relativity's multipole structure and dynamics with future gravitational wave detectors.

## Contribution

It introduces a spin-inclusive parametrized phasing formula and assesses the measurement accuracy of multipole and energy parameters with upcoming detectors.

## Key findings

- Third-generation detectors and LISA can similarly constrain binary dynamics.
- The parametrized waveform allows for upper limits on deviations from general relativity.
- Enhanced models facilitate tests of fundamental physics with future observations.

## Abstract

In an earlier work [S. Kastha et al., PRD {\bf 98}, 124033 (2018)], we developed the {\it parametrized multipolar gravitational wave phasing formula} to test general relativity, for the non-spinning compact binaries in quasi-circular orbit. In this paper, we extend the method and include the important effect of spins in the inspiral dynamics. Furthermore, we consider parametric scaling of PN coefficients of the conserved energy for the compact binary, resulting in the parametrized phasing formula for non-precessing spinning compact binaries in quasi-circular orbit. We also compute the projected accuracies with which the second and third generation ground-based gravitational wave detector networks as well as the planned space-based detector LISA will be able to measure the multipole deformation parameters and the binding energy parameters. Based on different source configurations, we find that a network of third-generation detectors would have comparable ability to that of LISA in constraining the conservative and dissipative dynamics of the compact binary systems. This parametrized multipolar waveform would be extremely useful not only in deriving the first upper limits on any deviations of the multipole and the binding energy coefficients from general relativity using the gravitational wave detections, but also for science case studies of next generation gravitational wave detectors.