Abstract

The main observable for the detection of gravitational waves generated by compact binary systems is the phase of the wave. Within the post-Newtonian approximation, where the objects are assumed to be well-separated and to have small velocities, one can obtain analytical expressions for the phase as a series in (v/c)^2. Currently, the phase (as well as the complete waveform) is known up to 3.5 post-Newtonian (PN) order. In this talk, I will review the history of the longstanding project of obtaining the phase at 4.5PN, as well as the gravitational waves modes at 4PN order. I will then focus on recent advances, in particular the non-linear contribution of the tails-of-memory. This is a hybrid, higher-order effect that combines the tails, i.e. linear gravitational waves that scatter against the curved background, and the memory effect, due to the re-radiation of gravitational waves by gravitational waves themselves. Finally, I will discuss the last steps we are performing for the soon-to-come expression of the 4.5PN phase.