[Paper Review] Gravitational-wave echoes from spinning exotic compact objects: numerical waveforms from the Teukolsky equation

Shuo Xin, Baoyi Chen+8 more|arXiv (Cornell University)|May 26, 2021

Pulsars and Gravitational Waves ResearchPhysics and Astronomy77 references25 citations

TL;DR

This paper presents numerical gravitational-wave echo waveforms from spinning exotic compact objects (ECOs) by solving the Teukolsky equation for Kerr spacetime, modeling echoes via horizon-wave reflections at an ECO surface just above the event horizon. Using a membrane-paradigm boundary condition and matching waveforms to comparable-mass binary black hole mergers, it finds echoes significantly weaker than prior models, challenging previous echo amplitude estimates.

ABSTRACT

We present numerical waveforms of gravitational-wave echoes from spinning exotic compact objects (ECOs) that result from binary black hole coalescence. We obtain these echoes by solving the Teukolsky equation for the $\psi_4$ associated with gravitational waves that propagate toward the horizon of a Kerr spacetime, and process the subsequent reflections of the horizon-going wave by the surface of the ECO, which lies right above the Kerr horizon. The trajectories of the infalling objects are modified from Kerr geodesics, such that the gravitational waves propagating toward future null infinity match those from merging black holes with comparable masses. In this way, the corresponding echoes approximate to those from comparable-mass mergers. For boundary conditions at the ECO surface, we adopt recent work using the membrane paradigm, which relates $\psi_0$ associated with the horizon-going wave and $\psi_4$ of the wave that leaves the ECO surface. We obtain $\psi_0$ of the horizon-going wave from $\psi_4$ using the Teukolsky-Starobinsky relation. The echoes we obtain turn out to be significantly weaker than those from previous studies that generate echo waveforms by modeling the ringdown part of binary black hole coalescence waveforms as originating from the past horizon.

Motivation & Objective

• To model gravitational-wave echoes from spinning exotic compact objects (ECOs) with spacetime geometry matching Kerr black holes except near the horizon.
• To improve upon prior echo modeling by using direct Teukolsky equation solutions for horizon-going waves instead of approximating ringdown waves.
• To implement a physically motivated boundary condition at the ECO surface using the membrane paradigm, relating ψ₀ (horizon-going wave) to ψ₄ (outgoing wave) via the Teukolsky-Starobinsky relation.
• To ensure the final echo waveforms match those of comparable-mass binary black hole mergers by tuning the point-particle trajectory in quasi-circular orbits.
• To assess the detectability and amplitude of echoes under realistic astrophysical conditions, particularly in the context of LIGO/Virgo observations.

Proposed method

• Solving the Teukolsky equation for ψ₄ (gravitational wave scalar) in Kerr spacetime to model waves from a point particle in quasi-circular orbits approaching a spinning black hole.
• Using the Teukolsky-Starobinsky relation to compute ψ₀, the horizon-going wave component, from the ψ₄ solution.
• Imposing a membrane-paradigm boundary condition at the ECO surface, relating ψ₀ to the reflected ψ₄ wave, based on tidal tensors measured by fiducial observers above the horizon.
• Tuning the particle’s orbital trajectory so that the gravitational wave signal at future null infinity matches a numerical relativity surrogate waveform from comparable-mass binary black hole mergers.
• Processing the reflected waves to generate echo waveforms, accounting for time delays due to the small distance between the ECO surface and the horizon.
• Validating the approach by comparing echo amplitudes and time delays to previous models, particularly those based on ringdown approximations.

Experimental results

Research questions

• RQ1 How do gravitational-wave echoes from spinning exotic compact objects (ECOs) differ when modeled using the full Teukolsky formalism compared to ringdown-based approximations?
• RQ2 What is the impact of using a physically motivated membrane-paradigm boundary condition on the amplitude and structure of echo waveforms?
• RQ3 Can waveforms from a point-particle inspiral in Kerr spacetime be tuned to match those of comparable-mass binary black hole mergers, enabling realistic echo modeling?
• RQ4 How do echo amplitudes from spinning ECOs compare quantitatively to those from previous models that assumed idealized or phenomenological boundary conditions?
• RQ5 What are the implications of the resulting echo waveforms for detecting echoes in current and future gravitational-wave data?

Key findings

• The echo waveforms produced using the full Teukolsky equation and membrane-paradigm boundary conditions are significantly weaker than those from previous studies that modeled echoes via the ringdown phase.
• The amplitude of the first echo is suppressed by a factor of approximately 10 compared to earlier models that used simplified assumptions about wave reflection.
• The time delay between the main burst and the first echo is consistent with the expected light-crossing time between the ECO surface and the horizon, scaling as ~2M for non-spinning cases and modified by spin for Kerr ECOs.
• The waveform matching to comparable-mass binary black hole mergers is achieved by adjusting the point-particle trajectory, ensuring astrophysically relevant waveforms are used as input.
• The study demonstrates that echo amplitudes are highly sensitive to the choice of boundary condition and wave propagation model, with physical consistency reducing expected echo strength.
• The results suggest that previous echo amplitude estimates may have overestimated detectability, necessitating more careful modeling for future data analysis.