Numerical Kludge Scheme for EMRI Waveforms

• Numerical kludge scheme is a hybrid method that integrates multipolar post-Minkowskian, post-Newtonian, and black hole perturbation theories to approximate gravitational waveforms from extreme-mass-ratio inspirals.
• The framework employs osculating Kerr geodesics and local self-force computations to simulate gravitational radiation reaction and capture transient resonance effects.
• It maps trajectories from Boyer–Lindquist to harmonic coordinates, enabling efficient waveform synthesis for advanced gravitational wave observatory data analysis.

The numerical kludge scheme is a hybrid framework devised to construct approximate gravitational waveforms for extreme-mass-ratio inspirals (EMRIs), particularly those comprising a stellar-mass compact object orbiting a spinning supermassive black hole. This method integrates disparate approximation techniques from general relativity: multipolar post-Minkowskian expansions for metric perturbations and self-force, post-Newtonian expansions for source moments, and black hole perturbation theory for particle trajectories. The core objective is to emulate the generic inspiral—dynamical evolution dictated by gravitational radiation reaction and yielding waveforms with sufficient fidelity for data analysis in future space and ground-based detectors such as LISA. The framework is characterized by strong locality of its self-force prescription, enabling studies of transient resonances in non-adiabatic regimes (Sopuerta et al., 2012, Sopuerta et al., 2011).

1. Hybrid Approximation Methodology

The numerical kludge scheme synthesizes three principal approximation schemes:

• Multipolar post-Minkowskian (MPM) expansion: Utilized for both distant-zone metric perturbations (far-zone waveform structure) and the local radiation reaction force. In the far zone, one obtains the gravitational waveform as a multipole sum; locally, radiation reaction is encoded by potentials constructed from higher-order time derivatives of the source multipole moments.
• Post-Newtonian (PN) trajectory encoding: PN expansions specify the source multipole moments $M_L(t)$ and $S_L(t)$ in terms of the compact object's (SCO) trajectory in harmonic coordinates. At leading order, moments reduce to Newtonian forms (e.g., mass quadrupole $M_{ij}(t)=\eta\,m\,z_{\langle i}z_{j\rangle}$), while higher-order PN corrections systematically refine accuracy.
• Black hole perturbation theory: The SCO is treated as moving along Kerr geodesics whose constants of motion $\{E, L_z, Q\}$ evolve secularly due to the local MPM self-force. The trajectory is constructed as a sequence of osculating, time-dependent geodesic fragments.

This hybridization provides a closed set of algebraic and ODE updates that integrate the inspiral and self-consistently generate corresponding gravitational waveforms (Sopuerta et al., 2012, Sopuerta et al., 2011).

2. Metric Perturbation and Far-Zone Waveform Construction

In the far zone, the metric perturbation in a transverse-traceless (TT) gauge is expressed as

$h_{ij}(t, \mathbf{n}) = \frac{2}{r}\, \ddot{M}_{ij}(t_r) + \frac{2}{3 r}\left[\dddot{M}_{ijk}(t_r) n^k + 4 \epsilon_{kl(i}\ddot{S}_{j)k}(t_r) n^l\right] + \frac{1}{6 r}\left[\ddddot{M}_{ijkl}(t_r) n^k n^l + 6 \epsilon_{kl(i}\dddot{S}_{j)km}(t_r) n^l n^m\right]$

with $n^i = x^i/r$, $t_r = t - r$, and symmetrization over indices. This sum is truncated at mass hexadecapole and current octopole order for practical EMRI waveform modeling. The plus and cross GW polarizations are constructed through projections using polarization tensors (Sopuerta et al., 2011).

3. Source Multipole Moments and Harmonic Trajectories

PN expansions define the source multipoles from the SCO trajectory $z^i(t)$ in harmonic coordinates:

• Mass Quadrupole: $$M_{ij}(t) = \eta\,m\,z_{\langle i}(t) z_{j\rangle}(t)$$
• Mass Octopole: $$M_{ijk}(t) = \eta\,\delta m\,z_{\langle i}z_j z_{k\rangle}(t)$$
• Current Quadrupole: $$S_{ij}(t) = \eta\,\delta m\,\epsilon_{kl\langle i} z^k(t) \dot{z}^l(t) z_{j\rangle}(t)$$
• Mass Hexadecapole: $$M_{ijkl}(t) = \eta\,m\,z_{\langle i}z_jz_kz_{l\rangle}(t)$$
• Current Octopole: $$S_{ijk}(t) = \eta\,m\,\epsilon_{lm\langle i} z^l(t) z_j(t) z_{k\rangle}(t) \dot{z}^m(t)$$

where $\eta = mM/(m+M)^2$, $\delta m = M - m$, and STF (symmetric-tracefree) projection is applied. Further PN corrections may be incorporated for accuracy as required (Sopuerta et al., 2012, Sopuerta et al., 2011).

4. Osculating Elements and Geodesic Evolution

To model the non-geodesic inspiral, the scheme uses osculating Kerr geodesics. Each geodesic segment, described by $\{E, L_z, Q, C\}$, evolves on the radiation-reaction timescale due to the MPM self-force:

$$\frac{dE}{d\tau} = -\zeta^{(t)}_\alpha\, a^\alpha,\quad \frac{dL_z}{d\tau} = \zeta^{(\phi)}_\alpha a^\alpha,\quad \frac{dQ}{d\tau} = 2\,\xi_{\alpha\beta} u^\alpha a^\beta$$

where $\zeta^{(t)}$ and $\zeta^{(\phi)}$ are Kerr's timelike and axial Killing vectors, $\xi_{\alpha\beta}$ is the Killing tensor, and $a^\alpha$ is the self-acceleration derived from the dissipative piece of $h_{\mu\nu}$.

One expresses the geodesic motion in BL coordinates $\{t, r, \theta, \phi\}$ through standard relations, and relates $\{E, L_z, C\}$ to osculating orbital elements $(p, e, \iota)$ using algebraic inversion of geodesic turning-point equations (Sopuerta et al., 2012, Sopuerta et al., 2011).

5. Boyer–Lindquist to Harmonic Coordinate Mapping

For waveform generation, all relevant physical quantities are mapped to harmonic coordinates using the explicit Abe–Ichinose–Nakanishi transformation:

$$t_H = t_{BL},\quad x_H + i y_H = \sqrt{(r_{BL} - M)^2 + a^2}\,\sin\theta\, e^{i [\phi - \Phi(r_{BL})]},\quad z_H = (r_{BL} - M) \cos\theta$$

$$\Phi(r) = \frac{\pi}{2} - \arctan \left\{ \frac{(r-M)/a + \Omega(r)}{1 - [(r-M)/a] \Omega(r)} \right\}$$

$$\Omega(r) = \tan \left[ \frac{a}{2\sqrt{M^2 - a^2}\ln \frac{r-r_-}{r-r_+}} \right]$$

Inverse mappings allow conversion in both directions and operational implementation for trajectory calculations and waveform evaluation in harmonic coordinates (Sopuerta et al., 2012, Sopuerta et al., 2011).

6. Algorithmic Implementation Workflow

The numerical kludge algorithm proceeds through:

• Initializing orbital elements $(p, e, \iota, \phi_0)$ and their mapping to $(E, L_z, C)$ with chosen initial phases.
• Integration of geodesic ODEs using angle variables $r(t) = p M / (1 + e \cos\psi(t))$, $\cos^2\theta(t) = \cos^2\theta_{\min}\cos^2\chi(t)$ to mitigate stiffness at turning points.
• Mapping BL trajectories to harmonic coordinates.
• Construction of source multipoles $M_L(t), S_L(t)$ from $x_H^i(t), v_H^i(t)$ and fitting to discrete multi-Fourier series $$f(t) = \sum_{k, m, n} f_{kmn} e^{-i(k\Omega_r+m\Omega_\theta+n\Omega_\phi)t}$$.
• Evaluation of radiation-reaction potentials $V, V^i$ and self-acceleration $a^\alpha$ at the instantaneous SCO location.
• Update of orbital constants via $dE/dt, dL_z/dt, dQ/dt$ and inversion to $(p, e, \iota)$.
• Iteration over time steps $\Delta t \ll T_\text{orbital}$.
• Assembly of the far-zone waveform $h_{ij}(t)$ and projection onto GW polarizations and detector response functions (Sopuerta et al., 2012, Sopuerta et al., 2011).

7. Local Self-Force, Transient Resonances, and Extensions

A salient feature is the strictly local-in-time character of the MPM self-force, ensuring the osculating-elements evolution exhibits small-scale oscillations at the system's orbital frequencies. This property is essential for capturing transient resonances, which arise when two orbital frequencies become commensurate $\Omega_r/\Omega_\theta = p/q$, resulting in instantaneous jumps in the evolution of action variables and GW phase. No special treatment is prescribed—the self-force responds automatically to frequency commensurability.

The framework admits systematic improvement by inclusion of higher-PN multipole corrections, conservative self-force contributions (via the time-symmetric $h_{\mu\nu}$ piece), and chunk-averaging of the self-force for resonance studies. The algorithm and all its ingredients are intended for the simulation of generic inspirals in systems from extreme to intermediate mass ratios, targeting the requirements of space-based and advanced ground gravitational-wave observatories (Sopuerta et al., 2012, Sopuerta et al., 2011).