Fast frequency-domain algorithm for gravitational self-force: Circular orbits in Schwarzschild spacetime

Fast, reliable orbital evolutions of compact objects around massive black holes will be needed as input for gravitational wave search algorithms in the data stream generated by the planned Laser Interferometer Space Antenna (LISA). Currently, the state of the art is a time domain code by [Phys. Rev. D 81, 084021 (2010)] that computes the gravitational self-force on a point particle in an eccentric orbit around a Schwarzschild black hole. Existing time domain codes take up to a few days to compute just one point in parameter space. In a series of articles, we advocate the use of a frequency domain approach to the problem of gravitational self-force (GSF) with the ultimate goal of orbital evolution in mind. Here, we compute the GSF for a particle in a circular orbit in Schwarzschild spacetime. We solve the linearized Einstein equations for the metric perturbation in Lorenz gauge. Our frequency domain code reproduces the time-domain results for the GSF up to $\sim 1000$ times faster for small orbital radii. In forthcoming companion papers, we will generalize our frequency domain computations of the GSF to include bound (eccentric) orbits in Schwarzschild spacetimes, where we will employ the method of extended homogeneous solutions [Phys. Rev. D 78, 084021 (2008)]. We will eventually extend our methods to attempt a frequency domain computation of the GSF in Kerr spacetime.

Figure 1

The runtimes for the ${10}^{-4}$, ${10}^{-6}$, ${10}^{-7}$ overall fractional error runs. Panels (a), (b) and (c) display plots of runtime (in minutes) versus orbital radius $r_0$ at which we compute the GSF. ${\Delta }_{\mathrm{GSF}}$ denotes the overall fractional error in our numerical computation of the GSF. This error is what we refer to as our (relative) “accuracy”. As can be seen in panel (a), at an accuracy of ${10}^{-4}$, our code takes less than two minutes to compute the GSF for $r_0\lesssim 15M$. This grows nearly to a day as $r_0$ approaches $100M$. Panel (b) shows that an accuracy of ${10}^{-6}$ increases the runtimes by a factor of 2 to three for $r_0\lesssim 10M$, but the runtimes are still $\lesssim 10$ minutes for $r_0\lesssim 20M$. However, beyond $r_0=50M$, this accuracy becomes unattainable. As panel (c) shows, an accuracy of ${10}^{-7}$ is achievable for $r_0\lesssim 30M$ and the overall runtimes do not change much for these strong field GSF computations. Interestingly enough, for $r_0\lesssim 20M$, the $r_0\le 8M$ runs seem to take more time than $r_0\ge 9M$ runs. This is a result of our having to compute more modes to obtain the GSF for the $r_0\le 8M$ runs because the large-$l$ tail could not be computed to the desired accuracy of ${10}^{-6}$ or ${10}^{-7}$ using just 17 scalar modes, which is what we had done for the $r_0\ge 9M$ runs. We think the reason for this is that the magnitudes of the individual $l$ modes of the GSF are large enough for $r_0\le 8M$ that more modes are needed in order for the tail to be fit correctly. Finally in panel (d), we present the runtimes for a few $r_0\le 20M$ run for all three accuracies. As expected, the runtimes increase with demand for higher accuracy (except for the $10M$ run). Most importantly, the figure shows that all $r_0\le 20M$ runs take less than 15 minutes up to an accuracy of ${10}^{-7}$.