Post-Newtonian theory provides theoretical predictions for the dynamics and gravitational waves emitted by compact object inspiralling binary systems, the most promising sources for the upcoming generation of ground-based detectors LIGO/Virgo. The faintness of the signal and the challenge posed by the observations has driven a lot of effort to improve the modeling of the expected signals to very high accuracy.
In this talk, we will address in particular the question of the modelling of spin effects in these systems within the post-Newtonian approach. From astrophysical observations, black holes spins are expected to be generically close to maximal, and they play an important role by affecting the phasing and causing orbital plane precession.
We present an extension of a Lagrangian formalism for point particle with spins, where finite size effects are represented by an additional multipolar structure. When applied to the case of a spin-induced octupole, the formalism allows for the computation of the cubic-in-spin effects that enter at the order 3.5PN. We also report on recent results obtained for quadratic-in-spin effects at the next-to-leading 3PN order, and for linear-in-spin effects at the next-to-next-to-leading 3.5PN order and at 4PN, where hereditary effects are present. We recover existing results for the dynamics, and derive for the first time the gravitational wave energy flux and orbital phasing. These results can be directly used for the data analysis of the detectors of gravitational waves.