Post-Newtonian expansion of fluxes from a scalar charge on an inclined-spherical orbit about a Kerr black hole

TL;DR

This paper derives analytic post-Newtonian expansions for scalar-field fluxes from a scalar charge on inclined-spherical Kerr orbits, achieving up to $12$PN order with exact dependence on spin $a$ and inclination $x$ using the MST method. It separates the scalar field into angular and radial parts, expands the spheroidal harmonics and Teukolsky-like radial functions, and assembles mode sums to obtain energy and angular momentum fluxes at infinity and the horizon, validating against high-precision numerical Teukolsky results. The results reveal a structured interplay between spin and inclination, including leading-spin terms in half-integer and integer PN orders and non-polynomial $a$-dependence at high PN order, with explicit $3.5$PN expressions and comprehensive $12$PN data available online. By using a scalar-field proxy for gravitational self-force problems, the work informs EMRI modeling in Kerr spacetime and sets the stage for extension to the conservative sector and to full gravitational perturbations, while highlighting near-extremal regime limitations and horizon-flux extensions.

Abstract

Efforts are underway to accurately model extreme-mass-ratio inspirals for binaries with a spinning (Kerr) primary. At lowest order the adiabatic evolution depends on the radiation fluxes. Fluxes and other self-force quantities can be expanded analytically in post-Newtionian (PN) series allowing the early evolutionary phase to be understood. When it comes to more complicated background geodesic orbits, it proves useful to use the scalar field model problem to guide development and testing of techniques. In this paper, we present analytical expressions for the scalar fluxes from a scalar point-charge in inclined-spherical orbit about a Kerr black hole up to 12PN relative order, with expressions that are exact in terms of the inclination parameter $x$ and black hole spin $a$. The expressions are constructed using the Mano, Suzuki, and Takasugi method of solving the scalar wave equation in a Kerr background. We compare the numerical evaluation of these flux expressions to full numerical ($s=0$) Teukolsky code results, examining their degree of utility as the strong-field region is approached.

Figures (2)

• Figure 1: Relative error in the infinity energy flux when comparing the numerically evaluated scalar infinity flux expression with numerical scalar fluxes computed from a Teukolsky code. Results are shown as a function of $p$ for $a = 0.9M$ and $x \in \{1/2,-1/2,1/4,-1/4\}$. For retrograde orbits, $p = 7$ is excluded as the innermost stable spherical orbit for $a = 0.9M$ is located at $p_{ISSO} \simeq 7.1029$SteiWarb19. The residuals plotted in this figure are estimated to have a relative PN-scaling of $(n+1/2)$PN, where $n$ is the relative PN order of the flux expansion used.
• Figure 2: Relative error in the horizon energy flux when comparing the numerically evaluated scalar horizon flux expression with numerical scalar fluxes computed from a Teukolsky code. Results are shown as a function of $p$ for $a \in \{0.3M,0.6M\}$ and $x \in \{-1/2,1/2\}$. The residuals plotted in this figure are estimated to have a relative PN-scaling of $(n+1/2)$PN, where $n$ is the relative PN order of the flux expansion used. We note that several downward spikes appear in the log-of-absolute-value curves presented, these are due to a zero-crossing, $\lim_{p\to p_c}\Delta \dot{E}_{\mathcal{H}} = 0$, at that specific order of approximation.