[Paper Review] Gravitational wave templates from Extreme Mass Ratio Inspirals

Viktor Skoupý, Georgios Lukes-Gerakopoulos|arXiv (Cornell University)|Jan 12, 2021

Pulsars and Gravitational Waves Research16 references4 citations

TL;DR

This paper computes gravitational wave templates for extreme mass ratio inspirals (EMRIs) by numerically solving the Teukolsky equation in both time and frequency domains to model energy and angular momentum fluxes from spinning test particles orbiting Kerr black holes. The key contribution is the independent verification of spin-induced corrections to the gravitational wave phase, showing agreement with prior work and enabling high-precision waveform modeling for LISA.

ABSTRACT

An extreme mass ratio inspiral takes place when a compact stellar object is inspiraling into a supermassive black hole due to gravitational radiation reaction. Gravitational waves (GWs) from this system can be calculated using the Teukolsky equation (TE). In our case, we compute the asymptotic GW fluxes of a spinning body orbiting a Kerr black hole by solving numerically the TE both in time and frequency domain. Our ultimate goal is to produce GW templates for space-based detectors such as LISA.

Motivation & Objective

• To model gravitational wave signals from extreme mass ratio inspirals (EMRIs) involving a spinning compact object orbiting a supermassive black hole.
• To compute energy and angular momentum fluxes from circular equatorial orbits using the Teukolsky formalism in both time and frequency domains.
• To derive adiabatic inspiral waveforms by evolving orbital parameters based on fluxes, focusing on spin corrections to the gravitational wave phase.
• To validate the spin-dependent phase corrections against existing results, ensuring accuracy for LISA data analysis.

Proposed method

• Numerical solution of the Teukolsky equation in both time and frequency domains to compute gravitational wave fluxes from spinning test particles in Kerr spacetime.
• Use of the Black Hole Perturbation Toolkit (BHPT) for accurate computation of radial and angular functions and their derivatives.
• Application of the Mathisson-Papapetrou-Dixon (MPD) equations with the Tulczyjew-Dixon spin-supplementary condition to model spinning test particles.
• Adiabatic approximation via time-averaged fluxes to evolve orbital parameters, assuming slow inspiral over orbital timescales.
• Third-order Lagrange interpolation of flux data to enable smooth evolution of orbital frequency and phase.
• Perturbative solution of orbital equations to extract spin-dependent corrections to the gravitational wave phase, expressed as Φ₁(t) = 2qφ₁(t).

Experimental results

Research questions

• RQ1 How do spin effects from the secondary compact object modify the gravitational wave phase in extreme mass ratio inspirals?
• RQ2 To what extent do spin-induced corrections to energy and angular momentum fluxes affect the adiabatic evolution of orbital parameters?
• RQ3 Can the Teukolsky formalism in both time and frequency domains yield consistent fluxes for circular equatorial orbits of spinning particles around Kerr black holes?
• RQ4 How do the fluxes and phase corrections depend on the spin magnitude and direction of the secondary object?
• RQ5 Is the adiabatic approximation valid for modeling the gravitational wave phase when spin effects are included?

Key findings

• The energy fluxes from spinning particles in circular equatorial orbits were computed numerically for Kerr black holes with spin parameters a = 0, 0.5M, and 0.9M.
• The flux dependence on secondary spin σ was fitted as 𝒪(σ) and extracted as a linear term 𝒻₁, with the total flux expressed as 𝒻 = 𝒻₀ + σ𝒻₁ + 𝒪(σ²).
• The correction to the gravitational wave phase due to spin was derived as Φ₁(t) = 2qφ₁(t), with φ₁(t) obtained by perturbative solution of orbital equations.
• The resulting phase corrections for different black hole spins (a = 0, 0.5M, 0.9M) were found to be in excellent agreement with the results of Piovano et al. (2020), validating the method.
• The computed fluxes and phase corrections are consistent with known analytical results in the limit of non-spinning particles (σ = 0) and first-order spin expansions.
• The work provides a foundation for future comparison with time-domain codes (e.g., Teukode) to test the validity of the adiabatic approximation in full EMRI waveforms.