Toward exponentially-convergent simulations of extreme-mass-ratio inspirals: A time-domain solver for the scalar Teukolsky equation with singular source terms

Abstract

Gravitational wave signals from extreme mass ratio inspirals are a key target for space-based gravitational wave detectors. These systems are typically modeled as a distributionally-forced Teukolsky equation, where the smaller black hole is treated as a Dirac delta distribution. Time-domain solvers often use regularization approaches that approximate the Dirac distribution that often introduce small length scales and are a source of systematic error, especially near the smaller black hole. We describe a multi-domain discontinuous Galerkin method for solving the distributionally-forced Teukolsky equation that describes scalar fields evolving on a Kerr spacetime. To handle the Dirac delta, we expand the solution in spherical harmonics and recast the sourced Teukolsky equation as a first-order, one-dimensional symmetric hyperbolic system. This allows us to derive the method's numerical flux to correctly account for the Dirac delta. As a result, our method achieves global spectral accuracy even at the source's location. To connect the near field to future null infinity, we use the hyperboloidal layer method, allowing us to supply outer boundary conditions and providing direct access to the far-field waveform. We document several numerical experiments where we test our method, including convergence tests against exact solutions, energy luminosities for circular orbits, the scheme's superconvergence properties at future null infinity, and the late-time tail behavior of the scalar field. We also compare two systems that arise from different choices of the first-order reduction variables, finding that certain choices are numerically problematic in practice. The methods developed here may be beneficial when computing gravitational self-force effects, where the regularization procedure has been developed for the spherical harmonic modes and high accuracy is needed at the Dirac delta's location.

arXiv commentVersion 2 includes proof that our first-order system is symmetric hyperbolic, scalar energy fluxes for circular orbits in Kerr, and a correction about our scheme's convergence rate for non-smooth hyperboloidal layers. Version 3 corrects typo in table two's caption and matches the published version

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Cited by in corpus (2)

• Hyperboloidal approach for static spherically symmetric spacetimes: a didactical introduction and applications in black-hole physics
• Superconvergent Discontinuous Galerkin Method for the Scalar Teukolsky Equation on Hyperboloidal Domains: Efficient Waveform and Self-Force Computation