Analytical results for binary dynamics at the first post-Newtonian order in Einstein-Cartan theory with the Weyssenhoff fluid

The quantum spin effects inside matter can be modeled via the Weyssenhoff fluid, which permits us to unearth a formal analogy between general relativity and Einstein-Cartan theory at the first post-Newtonian order. In this framework, we provide some analytical formulas pertaining to the dynamics of binary systems having the spins aligned perpendicular to the orbital plane. We derive the expressions of the relative orbit and the coordinate time, which in turn allow us to determine the gravitational waveform, and the energy- and angular-momentum fluxes. The potentialities of our results are presented in two astrophysical applications, where we compute (i) the quantum spin contributions to the energy flux and gravitational waveform during the inspiral phase and (ii) the macroscopic angular momentum of one of the bodies starting from the time-averaged energy flux and the knowledge of a few timing parameters.