Gravitational Waves from a Particle in Circular Orbits around a Schwarzschild Black Hole to the 22nd Post-Newtonian Order

TL;DR

The paper derives 22PN expressions for gravitational waves from a test particle in circular orbits around a Schwarzschild black hole, using the MST formalism within the Teukolsky framework. It demonstrates that the 22PN energy flux matches high-precision numerical results with relative errors around 10^−5 near ISCO and shows exceptional phase accuracy over a two-year inspiral, supporting their suitability for LISA-like EMRI data analysis. A Kerr-hybrid flux is proposed to extend the approach to rotating black holes, revealing spin-dependent improvements and highlighting the need for even higher PN orders or more modes for large spins. Collectively, the work confirms the value of very high PN orders for precise gravitational-wave modeling in extreme mass-ratio inspirals and lays groundwork for Kerr generalizations and EFT-based calibration.

Abstract

We extend our previous results of the 14th post-Newtonian (PN) order expansion of gravitational waves for a test particle in circular orbits around a Schwarzschild black hole to the 22PN order, i.e. $v^{44}$ beyond the leading Newtonian approximation where $v$ is the orbital velocity of a test particle. Comparing our 22PN formula for the energy flux with high precision numerical results, we find that the relative error of the 22PN flux at the innermost stable circular orbit is about $10^{-5}$. We also estimate the phase difference between the 22PN waveforms and numerical waveforms after a two-year inspiral. We find that the dephase is about $10^{-9}$ for $μ/M=10^{-4}$ and $10^{-2}$ for $μ/M=10^{-5}$ where $μ$ is the mass of the compact object and $M$ the mass of the central supermassive black hole. Finally, we construct a hybrid formula of the energy flux by supplementing the 4PN formula of the energy flux for circular and equatorial orbits around a Kerr black hole with all the present 22PN terms for the case of a Schwarzschild black hole. Comparing the hybrid formula with the the full numerical results, we examine the performance of the hybrid formula for the case of Kerr black hole.

Figures (4)

• Figure 1: Absolute values of the difference of the energy flux to infinity between numerical results and the PN approximation as a function of the orbital velocity. Note that the relative error of the energy flux between 22PN and the numerical calculation is an order of magnitude smaller than the one between 14PN and the numerical calculation even around ISCO, $v=1/\sqrt{6}=0.40825$.
• Figure 2: Absolute values of the dephasing during two-year inspiral between the PN and the numerical waveforms for the dominant $\ell=m=2$ mode as a function of time in month. The left (right) panel represents the early (late) inspiral phase in the LISA band. The left panel shows the dephase for System-I of masses $(M,\mu)=(10^5,10)M_{\odot}$, which evolves from $r_0\simeq 29M$ to $r_0\simeq 16M$ with associated frequencies $f_{\rm GW}\in [4\times 10^{-3},10^{-2}]$Hz. The right panel shows the dephase for System-II of masses $(M,\mu)=(10^6,10)M_{\odot}$, which explores orbital radius in a region $r_0/M\in [6.0,11]$ and frequencies in a range $f_{\rm GW}\in [1.8\times 10^{-3},4.4\times 10^{-3}]$Hz. Note that the dephase between the 18PN (22PN) waveforms and numerical waveforms for System-I due to the two-year inspiral is about $8\times 10^{-9}$ ($10^{-9}$) rads, which is below the lowest value of the dephase in the left panel. The dephase between the 22PN waveforms and numerical waveforms for System-II after the two-year inspiral is about $10^{-2}$ rads, which may suggest that the 22PN waveforms will provide the data analysis accuracy comparable to the one using numerical waveforms.
• Figure 3: Absolute values of the difference of energy flux to infinity between numerical results and PN approximation as a function of orbital velocity, $v=(M/r_0)^{1/2}[1+q\,(M/r_0)^{3/2}]^{-1/3}$, up to ISCO for $q=0.01,\,0.05,\,0.1,\,0.3,\,0.5$ and $0.9$. The energy flux at $n$-PN combines $n$-PN energy flux for a Schwarzschild black hole and spin dependent terms of the 4PN energy flux for a Kerr black hole ref:TSTS. For $q<0.1$, higher PN order terms for a non-spinning black hole improve the relative accuracy of the energy flux. For $q\ge 0.1$, however, higher PN order terms for a non-spinning black hole do not always improve the accuracy although one can find the accuracy of some PN order is an order of magnitude better than that of the 4PN energy flux.
• Figure 4: Same as Fig. \ref{['fig:flux_kerr']} but for $q=-0.01,\,-0.05,\,-0.1,\,-0.3,\,-0.5$ and $q=-0.9$. For $|q|<0.1$, adding higher post-Newtonian order terms for a non-spinning black hole achieves about an order of magnitude better accuracy than using the 4PN energy flux for a spinning black hole. For $q<-0.1$ the improvement becomes small when $|q|$ large.