Frequency-domain algorithm for the Lorenz-gauge gravitational self-force

State-of-the-art computations of the gravitational self-force (GSF) on massive particles in black hole spacetimes involve numerical evolution of the metric perturbation equations in the time-domain, which is computationally very costly. We present here a new strategy, based on a frequency-domain treatment of the perturbation equations, which offers considerable computational saving. The essential ingredients of our method are (i) a Fourier-harmonic decomposition of the Lorenz-gauge metric perturbation equations and a numerical solution of the resulting coupled set of ordinary equations with suitable boundary conditions; (ii) a generalized version of the method of extended homogeneous solutions [Phys. Rev. D {\bf 78}, 084021 (2008)] used to circumvent the Gibbs phenomenon that would otherwise hamper the convergence of the Fourier mode-sum at the particle's location; and (iii) standard mode-sum regularization, which finally yields the physical GSF as a sum over regularized modal contributions. We present a working code that implements this strategy to calculate the Lorenz-gauge GSF along eccentric geodesic orbits around a Schwarzschild black hole. The code is far more efficient than existing time-domain methods; the gain in computation speed (at a given precision) is about an order of magnitude at an eccentricity of 0.2, and up to three orders of magnitude for circular or nearly circular orbits. This increased efficiency was crucial in enabling the recently reported calculation of the long-term orbital evolution of an extreme mass ratio inspiral [Phys. Rev. D {\bf 85}, 061501(R) (2012)]. Here we provide full technical details of our method to complement the above report.Comment: 27 pages, 4 figure

Numerical computation of the EOB potential q using self-force results

The effective-one-body theory (EOB) describes the conservative dynamics of compact binary systems in terms of an effective Hamiltonian approach. The Hamiltonian for moderately eccentric motion of two non-spinning compact objects in the extreme mass-ratio limit is given in terms of three potentials: $a(v), \bar{d}(v), q(v)$. By generalizing the first law of mechanics for (non-spinning) black hole binaries to eccentric orbits, [\prd{\bf92}, 084021 (2015)] recently obtained new expressions for $\bar{d}(v)$ and $q(v)$ in terms of quantities that can be readily computed using the gravitational self-force approach. Using these expressions we present a new computation of the EOB potential $q(v)$ by combining results from two independent numerical self-force codes. We determine $q(v)$ for inverse binary separations in the range $1/1200 \le v \lesssim 1/6$. Our computation thus provides the first-ever strong-field results for $q(v)$. We also obtain $\bar{d}(v)$ in our entire domain to a fractional accuracy of $\gtrsim 10^{-8}$. We find to our results are compatible with the known post-Newtonian expansions for $\bar{d}(v)$ and $q(v)$ in the weak field, and agree with previous (less accurate) numerical results for $\bar{d}(v)$ in the strong field.Comment: 4 figures, numerical data at the end. Fixed the typos, added the journal referenc

Critical phenomena at the threshold of immediate merger in binary black hole systems: the extreme mass ratio case

In numerical simulations of black hole binaries, Pretorius and Khurana [Class. Quant. Grav. {\bf 24}, S83 (2007)] have observed critical behaviour at the threshold between scattering and immediate merger. The number of orbits scales as $n\simeq -\gamma\ln|p-p_*|$ along any one-parameter family of initial data such that the threshold is at $p=p_*$. Hence they conjecture that in ultrarelavistic collisions almost all the kinetic energy can be converted into gravitational waves if the impact parameter is fine-tuned to the threshold. As a toy model for the binary, they consider the geodesic motion of a test particle in a Kerr black hole spacetime, where the unstable circular geodesics play the role of critical solutions, and calculate the critical exponent $\gamma$. Here, we incorporate radiation reaction into this model using the self-force approximation. The critical solution now evolves adiabatically along a sequence of unstable circular geodesic orbits under the effect of the self-force. We confirm that almost all the initial energy and angular momentum are radiated on the critical solution. Our calculation suggests that, even for infinite initial energy, this happens over a finite number of orbits given by $n_\infty\simeq 0.41/\eta$, where $\eta$ is the (small) mass ratio. We derive expressions for the time spent on the critical solution, number of orbits and radiated energy as functions of the initial energy and impact parameter.Comment: Version published in PR

A Fast Frequency-Domain Algorithm for Gravitational Self-Force: I, 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{\bf 81}, 084021, (2010)] that computes the gravitational self-force on a point-particle in an eccentric orbit around a Schwarzschild black hole. Currently, 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 methods to include bound (eccentric) orbits in Schwarzschild and (eventually) Kerr spacetimes for computing the GSF, where we will employ the method of extended homogeneous solutions [Phys. Rev. D {\bf 78}, 084021 (2008)].Comment: 26 pages, 4 figures, with minor typos now fixe

A hybrid post-Newtonian -- effective-one-body scheme for spin-precessing compact-binary waveforms

We introduce \texttt{TEOBResumSP}: an efficient, accurate hybrid scheme for generating gravitational waveforms from spin-precessing compact binaries. The precessing waveforms are generated via the established technique of Euler rotating the non-precessing \texttt{TEOBResumS} waveforms from a precessing frame to an inertial frame. We obtain the Euler angles by solving the post-Newtonian precession equations expanded to second post-Newtonian order. Current version of \texttt{TEOBResumSP} produces precessing waveforms through the inspiral phase up to the onset of the merger. We compare \texttt{TEOBResumSP} to current state-of-the-art precessing approximants \texttt{NRSur7dq4}, \texttt{SEOBNRv4PHM}, and \texttt{IMRPhenomPv3HM} for 200 cases of precessing compact binary inspirals with orbital inclinations up to 90 degrees, mass ratios up to four, and the effective precession parameter $\chi_p$ up to 0.75. We further provide an extended comparison with \texttt{SEOBNRv4PHM} involving 1030 more inspirals with $\chi_p\le 1$ and mass ratios up to 10. We find that 91\% of the \texttt{TEOBResumSP}-\texttt{NRSur7dq4} matches, 85\% of the \texttt{TEOBResumSP}-\texttt{SEOBNRv4PHM} matches, and 77\% of the \texttt{TEOBResumSP}-\texttt{IMRPhenomPv3HM} matches are greater than $0.965$. Most disagreements occur for large mass ratios and $\chi_p \gtrsim 0.6$. We identify the mismatch of the \emph{non}-precessing $(2,1)$ mode as one of the leading causes of disagreements. We also introduce a new parameter, $\chi_{\perp,\text{max}}$, to measure the strength of precession and hint that the mismatch between the above approximants shows an exponential dependence on $\chi_{\perp,\text{max}}$ though this requires further study. Our results indicate that \texttt{TEOBResumSP} is on its way to becoming a robust precessing approximant to be employed in the parameter estimation of generic-spin compact binaries.Comment: 23 pages, 14 figures. Matches the accepted Phys. Rev. D version modulo abridged abstract, American vs. British English, and some figure placement

Comparison Between Self-Force and Post-Newtonian Dynamics: Beyond Circular Orbits

The gravitational self-force (GSF) and post-Newtonian (PN) schemes are complementary approximation methods for modelling the dynamics of compact binary systems. Comparison of their results in an overlapping domain of validity provides a crucial test for both methods, and can be used to enhance their accuracy, e.g. via the determination of previously unknown PN parameters. Here, for the first time, we extend such comparisons to noncircular orbits---specifically, to a system of two nonspinning objects in a bound (eccentric) orbit. To enable the comparison we use a certain orbital-averaged quantity $\langle U \rangle$ that generalizes Detweiler's redshift invariant. The functional relationship $\langle U \rangle(\Omega_r,\Omega_\phi)$, where $\Omega_r$ and $\Omega_\phi$ are the frequencies of the radial and azimuthal motions, is an invariant characteristic of the conservative dynamics. We compute $\langle U \rangle(\Omega_r,\Omega_\phi)$ numerically through linear order in the mass ratio $q$, using a GSF code which is based on a frequency-domain treatment of the linearized Einstein equations in the Lorenz gauge. We also derive $\langle U \rangle(\Omega_r,\Omega_\phi)$ analytically through 3PN order, for an arbitrary $q$, using the known near-zone 3PN metric and the generalized quasi-Keplerian representation of the motion. We demonstrate that the $\mathcal{O}(q)$ piece of the analytical PN prediction is perfectly consistent with the numerical GSF results, and we use the latter to estimate yet unknown pieces of the 4PN expression at $\mathcal{O}(q)$.Comment: 44 pages, 2 figures, 4 table