Post-Newtonian theory enables us to predict the waveform of the gravitational waves emitted by a system of two compact objects coalescing in its inspiral phase. State-of-the-art works provide the phase of the expected signal up to 3.5PN (i.e. up to 1/c^7). Comparison with numerical relativity, as well as the promising evolution of gravitational wave detectors incite us to pursue this computation to a higher order. In our current attempt, we are reaching the phase of the signal at the 4.5PN order (i.e. 1/c^9). For this purpose, the flux emitted by such a system has to be known at 4.5PN. I will describe the so-called Blanchet-Damour-Iyer formalism that we're using, and show how it combines near-zone and far-zone results. I will focus on the 4.5PN coefficient in the flux for circular orbits due to specific non-local terms -called tails- that was computed recently [1].

[1] Marchand T., Blanchet L., Faye G. Class.Quant.Grav. 33 (2016) no.24, 244003