We recently put out a key letter from our second-order self-force framework. In this work we compute the gravitational wave (GW) flux through second-order in the mass ratio for the first time. We do for quasi-circular orbits around an initially non-rotating black hole, but at second-order we can also account for slow rotating of the primary (large) black hole. Our flux results show remarkable agreement with numerical relativity simulations from the SXS collaboration for 10:1 mass ratios or smaller. Second-order self-force calculations are crucial to enable fundamental physics tests with extreme mass ratio inspirals (EMRIs) and our work shows they will also be useful to modelling intermediate mass ratio inspirals (IMRIs) as well. Our result are summarized by the following plot for a binary with mass ratio 10:1 \( (q=10)\).

The figure shows the Newtonian-normalized flux for the the (2,2)-mode as a function of the (inverse) orbital separation, \(\bar{x}\) which is computed from the frequency, \(\varpi\) extracted from the waveform. Weak field orbits are to the left of the plot and strong field orbits are on the right. The location of the innermost stable circular orbit (ISCO) for geodesic motion is shown by the dashed vertical line. On the figure we plot the (2,2)-mode flux computed using various methods.

The gold standard reference is the numerical relativity result (NR) which is taken from SXS:BBH:1107. We estimate the error bars in the NR data using the different extrapolation orders they use to compute the waveform at null infinity (shown by the shaded region between the two blue curves). The oscillations in the NR result is likely due to a combination of residual eccentricity and motion of the centre of mass of the binary influencing the mode decomposition. The plot also shows the PN result which is valid in the weak field but diverges strongly from the NR result in the strong field.

The dashed, green curve is the first-order self-force result that could have been plotted since the 60s when people first solve the Regge-Wheeler equation. Our new second-order (2SF) result is the solid, red curve with triangular markers. Over much of the frequency range where we have NR data our 2SF result is within the error bars of the NR result. Only towards the merger do we see a small discrepancy before our result breaks down as the ISCO is approached. Future work will improve this by attaching a transition to plunge.

The letter goes on to show 6 more figures that show (i) the agreement between NR and 2SF at \(q=1\) is still, unexpectedly, very good, (ii) the agreement is less good for subdominant modes, but we can resum the results to drastically improve the comparison, (iii) we also see good agreement for spin aligned binaries where the angular momentum of each black hole is small. For the secondary, so long as the mass ratio is small, this corresponds to any value of the Kerr spin parameter and for the primary this corresponds to small values of the Kerr spin parameter, and (iv) we also see agreement with all the known PN terms up to 3.5PN.

The remarkable agreement that we observe between NR and 2SF results at \(q=10\) strongly suggests that our 2SF results will be excellent models for binaries with, say, \(q=50\). Very recently we have been able the waveforms associated with the binary and at \( q=10 \) the comparison with NR looks amazing. Watch this space for more details soon.