Analytical Fluxes from Generic Schwarzschild Geodesics

Author(s)

Khalaf, Majed, Kavanagh, Chris, Telem, Ofri

Abstract

We present an analytic method for computing gravitational-wave fluxes from bound Schwarzschild geodesics with arbitrary eccentricity. Our approach systematically expands the Fourier coefficients of the emitted radiation in a Chebyshev basis, allowing them to be reduced to sums of Keplerian-like Fourier coefficients previously derived in the Quantum Spectral Method. Because the construction does not rely on a small-eccentricity expansion, it applies to a broad range of bound eccentric orbits. As an illustration, we implement the method using a $15$PN-expanded input and find that it reproduces the total flux for the case $(p,e)=(12.5,0.5)$ to relative accuracy $10^{-5}$, while for the stronger-field case $(p,e)=(10,0.8)$ it yields weighted mode-by-mode errors below $10^{-6}$ for the selected dominant modes analyzed. These results provide an analytic route to frequency-domain flux calculations relevant to EMRI modeling.

Figures

Radiation Flux per $(l,m)$ mode, emitted from a Schwarzschild geodesic. We plot the weighted relative error between our analytical result, computed with a 15PN Green's function, and the numerically computed fluxes from the Black Hole Perturbation Toolkit. The horizontal axis is the mode number $n$. Top: $p=12.5, e=0.5$, Bottom: $p=10, e=0.8$. In the top (bottom) panel, we only include modes whose relative contribution to the total flux exceeds $10^{-6}$ ($10^{-7}$).
Caption Radiation Flux per $(l,m)$ mode, emitted from a Schwarzschild geodesic. We plot the weighted relative error between our analytical result, computed with a 15PN Green's function, and the numerically computed fluxes from the Black Hole Perturbation Toolkit. The horizontal axis is the mode number $n$. Top: $p=12.5, e=0.5$, Bottom: $p=10, e=0.8$. In the top (bottom) panel, we only include modes whose relative contribution to the total flux exceeds $10^{-6}$ ($10^{-7}$).
Radiation Flux per $(l,m)$ mode, emitted from a Schwarzschild geodesic. We plot the weighted relative error between our analytical result, computed with a 15PN Green's function, and the numerically computed fluxes from the Black Hole Perturbation Toolkit. The horizontal axis is the mode number $n$. Top: $p=12.5, e=0.5$, Bottom: $p=10, e=0.8$. In the top (bottom) panel, we only include modes whose relative contribution to the total flux exceeds $10^{-6}$ ($10^{-7}$).
Caption Radiation Flux per $(l,m)$ mode, emitted from a Schwarzschild geodesic. We plot the weighted relative error between our analytical result, computed with a 15PN Green's function, and the numerically computed fluxes from the Black Hole Perturbation Toolkit. The horizontal axis is the mode number $n$. Top: $p=12.5, e=0.5$, Bottom: $p=10, e=0.8$. In the top (bottom) panel, we only include modes whose relative contribution to the total flux exceeds $10^{-6}$ ($10^{-7}$).
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  • [36] C. Whittall, L. Barack, and O. Long, Frequency-domain self-force calculations using Gegenbauer reconstruction, Phys. Rev. D 112, 124045 (2025), arXiv:2509.19439 [grqc]. S1 Supplemental Material for: Analytical Fluxes from Generic Schwarzschild Geodesics In this Supplemental Material, we provide further details on definitions, derivations and computations omitted from the main text. In particular, in Section A we derive the general form of ∆Anm; in Section B we define Dr and explain how to treat the square-root term; and in Section C we present a toy calculation illustrating the main features of the algorithm introduced in this work. Scaling this calculation yields the benchmarks shown in Fig. 1 of the main text. A. GENERAL FORM OF ∆Anm(r) In this section, we study the form of ∆Anm and elaborate on (15) of the main text. To begin, note that ∆Anm (r) = AS nm − AK nm =  n + Ωφ S Ωr S m  Ωr StS (r) −
  • [36] C. Whittall, L. Barack, and O. Long, Frequency-domain self-force calculations using Gegenbauer reconstruction, Phys. Rev. D 112, 124045 (2025), arXiv:2509.19439 [grqc]. S1 Supplemental Material for: Analytical Fluxes from Generic Schwarzschild Geodesics In this Supplemental Material, we provide further details on definitions, derivations and computations omitted from the main text. In particular, in Section A we derive the general form of ∆Anm; in Section B we define Dr and explain how to treat the square-root term; and in Section C we present a toy calculation illustrating the main features of the algorithm introduced in this work. Scaling this calculation yields the benchmarks shown in Fig. 1 of the main text. A. GENERAL FORM OF ∆Anm(r) In this section, we study the form of ∆Anm and elaborate on (15) of the main text. To begin, note that ∆Anm (r) = AS nm − AK nm =  n + Ωφ S Ωr S m  Ωr StS (r) −