Looking to the future, there are a number of open questions remaining. First, in order to complete the inspiral model, accurate to less than order unity in the phase evolution, it will be necessary to include the dissipative part of the second-order-in-the-mass-ratio perturbations [60]. Currently there are no calculations of the second-order self-force, but the necessary formalism has been laid out [66] and computational tech- niques are emerging [70, 71]. Once the second-order orbit-averaged dissipative self-force can be computed, the results are easily added to this self-force interpolation scheme and inspiral model.
It will also be important to quantify the effects of the “geodesic self-force approximation.” The true self-force is a functional of the entire past history of the particle’s motion but in this work the self-force is taken at each instance to be that of a particle whose past history is motion along the tangent geodesic to the inspiralling worldline. This approximation introduces a small error which is important to quantify. Initial investigations made by comparing a self-consistent evolution with a geodesic self-force evolution in the scalar field case suggest this error is very small (with the phase error smaller than the error-bars from either evolution [57–59]). Once self-consistent evolutions can be made in the gravitational case the results of this work can be used for comparison to quantify the error from the geodesic self-force approximation.
This work concentrates primarily on inspirals into a Schwarzschild black hole, but it is expected that astrophysical black holes will generally be rotating. Thus it is important to extend inspiral models to motion around a Kerr black hole. There has been much progress recently on computing self-forces in Kerr spacetime [38, 54, 122, 123] and these results can be used to compute inspirals in much the same way as done here. The techniques developed in Chapter 6 of this work may be a key first step to computing the self-force for generic bound orbits in Kerr spacetime. The self-force in Kerr spacetime is generally computed in a radiation gauge. Thus, even in Schwarzschild spacetime, it would be interesting to compare an evolution computed using a radiation gauge self-force with these evolutions computed using a Lorenz-gauge self-force. Whilst the coordinate descriptions of the two evolutions might differ, the gauge invariant evolution of the waveform phase should be the same.
Finally, although inspirals can be rapidly computed in a matter of minutes, this is probably still not quick enough for use in practical matched filtering searches. A similar problem is encountered when evaluating the time-domain effective-one-body models for use in gravitational-wave searches with LIGO data. One successful technique that has been applied in that case is the use of Reduced Order Modeling [111] that allows
for interpolation and rapid evaluation of the effective-one-body waveforms. No doubt a similar approach would be beneficial for more extreme mass ratios as well.
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