Gravitational radiation from Kerr black holes using the Sasaki-Nakamura formalism: waveforms and fluxes at infinity

TL;DR

The paper develops a new integration-by-parts scheme for the Sasaki–Nakamura formalism to compute gravitational waveforms and infinity fluxes from Kerr perturbations, removing the need for an extra radial integration when forming the SN source term. By introducing an auxiliary function $Y(r)$ and decomposing the SN source into tractable components, the method yields a nonoscillatory, source-independent path to $X^{\infty}_{\ell m \omega}$ and thus $h_+ - i h_\times$ for bound and unbound orbits. The approach is validated against established Teukolsky-based codes across generic bound orbits and radial infalls, achieving excellent agreement while maintaining comparable performance and enabling efficient generation of amplitude/flux data for extreme mass ratio inspiral modeling. These developments promise accelerated, cross-validated waveform generation for EMRIs, including eccentric and inclined configurations, with practical impact on space-based GW data analysis.

Abstract

In linear perturbation theory for Kerr black holes, there are two equivalent formalisms, namely the Teukolsky and the Sasaki-Nakamura (SN) formalism. Typically, one defaults to the Teukolsky formalism, especially when calculating extreme mass ratio inspiral waveforms, and uses the SN formalism when dealing with extended sources, as it offers superior convergence when employing the Green's function method for calculating the inhomogeneous solution. In this work, we present a new scheme for solving the inhomogeneous SN equation, based on integration by parts, that eliminates the extra radial integration step required in the standard formulation to construct the source term for convolution with the SN variable. Our approach enables efficient computations of gravitational waveforms within the SN formalism in all cases, from compact to extended sources. We validate our scheme and code implementation against the literature and find excellent agreement, achieving comparable performance without employing any special optimization techniques.

Figures (10)

• Figure 1: The $Y^{\rm in}$ solutions for Eq. \ref{['Eq.ODEforY']} with boundary conditions $Y^{\rm in}(r\to\infty)={Y^{\rm in}}'(r\to\infty)=0$ and $\ell=m=2$, $a/M=0.9$. From the top to the bottom, the frequency is set to $M\omega=1$, $0.5$, and $0.1$, respectively.
• Figure 2: The energy flux at infinity for $a=0.9M$, $p=6.0M$, $e=0.7$, $x=\cos\pi/4$. The mode indexes are $\ell=m=2$ and $\ell=m=4$ with polar index $k=0$ and radial index $n=0$ to $n=70$. The two approaches agree very well.
• Figure 3: The waveform snapshot for $a=0.9M$, $p=6.0M$, $e=0.7$, $x=\cos\pi/4$ viewing at $\theta=\pi/2$ and $\varphi=0$.
• Figure 4: The $\mathcal{E}=1$ case. Panel (a) illustrates the variation of $f_0$, $f_1$, and $f_2$ with $r$. As $r\to\infty$, $f_0$ converges at a rate of $1/r^{1/2}$, $f_1$ converges at a rate of $1/r^{3/2}$, and $f_2$ converges at a rate of $1/r^3$. Panel (b) shows the variation of the $\mathcal{W}(r_{*})$ function. It converges at the same rate as $f_0$, i.e. $1/r^{1/2}$, and its oscillation frequency increases with increasing $r_{*}$. Panel (c) presents the magnitudes of the integrands in the Green's function integrals for the and non- methods. The method defined in Eq. \ref{['Eq.I_SNIBP']} exhibits a convergence rate of $1/r^{1/2}$, while the non- (i.e., the original ) method defined in Eq. \ref{['Eq.X^Infty']} converges faster as $1/r^{7/2}$. Other parameters are $\ell=2$, $m=0$, $a/M=0.9$, and $M\omega=0.5$.
• Figure 5: The same as Fig. \ref{['fig:unbound_rest']}, but with $\mathcal{E}=3$.