[Paper Review] The landscape of massive black-hole spectroscopy with LISA and Einstein Telescope

S. Bhagwat, Costantino Pacilio+2 more|arXiv (Cornell University)|Dec 31, 2021

Pulsars and Gravitational Waves ResearchPhysics and Astronomy119 references43 citations

TL;DR

This paper investigates the prospects for massive black hole spectroscopy using the Laser Interferometer Space Antenna (LISA) and Einstein Telescope (ET), employing Fisher matrix analysis to forecast the precision of quasi-normal mode (QNM) parameter measurements across diverse astrophysical models. It finds that LISA can achieve sub-0.1% uncertainty on the dominant QNM frequency in heavy seed scenarios, enabling high-precision tests of general relativity, while ET can measure 3 QNM parameters with ~10% error in light seed scenarios.

ABSTRACT

Measuring the quasi-normal mode~(QNM) spectrum emitted by a perturbed black-hole~(BH) --~also known as BH spectroscopy~-- provides an excellent opportunity to test the predictions of general relativity in the strong-gravity regime. We investigate the prospects and precision of BH spectroscopy in massive binary black hole ringdowns, one of the primary science objectives of the future Laser Interferometric Space Antenna~(LISA) mission. We simulate various massive binary BH population models, featuring competing prescriptions for the Delays between galaxy and BH mergers, for the impact of supernova feedback on massive BH growth, and for the initial population of high redshift BH seeds (light versus heavy seeds). For each of these scenarios, we compute the average number of expected events for precision BH spectroscopy using a Fisher-matrix analysis. We find that, for any heavy seed scenario, LISA will measure the dominant mode frequency within ${\cal O}(0.1) \%$ relative uncertainty and will estimate at least 3 independent QNM parameters within $1 \%$ error. The most optimistic heavy seed scenarios produce $\mathcal{O}(100)$ events with $1 \%$ measurability for 3 or more QNM quantities during LISA's operational time. On the other hand, light seed scenarios produce lighter merger remnants, which ring at frequencies higher than LISA's sensitivity. Interestingly, the light seed models give rise to a fraction of mergers in the band of Einstein Telescope, allowing for the measurement of 3 QNM parameters with $\sim 10 \%$ relative errors in approximately a few to ten events/yr. More precise BH spectroscopy in the light seed scenarios would require instruments operating in the deciHertz band.

Motivation & Objective

• To assess the feasibility and precision of black hole spectroscopy using future gravitational wave detectors, specifically LISA and Einstein Telescope.
• To quantify the measurement uncertainty of quasi-normal mode (QNM) parameters in ringdown signals from massive binary black hole mergers.
• To evaluate how different astrophysical models—particularly heavy vs. light black hole seed scenarios—affect the detectability and precision of QNM measurements.
• To determine the number of observable events with sufficient signal-to-noise ratio for multi-parameter QNM testing within LISA’s operational lifetime.
• To compare the complementary roles of LISA (for high-mass, low-frequency ringdowns) and ET (for lighter, higher-frequency remnants) in strong-field gravity tests.

Proposed method

• Simulating 100 years of gravitational wave data containing analytical ringdown signals for 8 distinct massive binary black hole population models.
• Applying a numerical Fisher matrix formalism to estimate parameter estimation uncertainties for up to 5 QNM parameters (frequency, damping time, etc.) per event.
• Using the Fisher matrix to compute relative measurement uncertainties, defined as 'measurability', to assess the precision of QNM parameter estimation.
• Evaluating different astrophysical models with varying initial black hole seed masses (light vs. heavy), delays between galaxy and black hole mergers, and supernova feedback effects on black hole growth.
• Computing average event rates and expected number of events with 1% or better measurability for 3 or more QNM parameters over LISA’s mission lifetime.
• Assessing the sensitivity of LISA and ET to ringdown signals based on the remnant black hole mass and frequency, considering detector sensitivity curves.

Experimental results

Research questions

• RQ1 What is the expected number of massive binary black hole ringdown events detectable by LISA with sufficient signal-to-noise ratio for multi-parameter QNM spectroscopy?
• RQ2 How precisely can LISA measure the dominant quasi-normal mode frequency and other QNM parameters in the heavy seed scenario?
• RQ3 Can Einstein Telescope perform robust QNM spectroscopy on lighter remnant black holes that are outside LISA’s optimal sensitivity band?
• RQ4 How do uncertainties in astrophysical modeling—such as seed mass function, merger delay times, and feedback—impact the prospects for BH spectroscopy?
• RQ5 What is the minimum number of QNM parameters that must be measured with sub-1% uncertainty to perform a meaningful test of the no-hair theorem?

Key findings

• In the most optimistic heavy seed scenarios, LISA is expected to detect approximately 100 ringdown events with 1% or better measurability for 3 or more QNM parameters during its operational lifetime.
• For heavy seed models, LISA can measure the dominant QNM frequency with a relative uncertainty of order 0.1%, enabling high-precision tests of general relativity.
• LISA can estimate at least 3 independent QNM parameters with less than 1% relative error in the heavy seed scenario, satisfying the condition for model-independent no-hair theorem tests.
• Light seed models produce lighter remnant black holes whose ringdown frequencies lie above LISA’s optimal sensitivity band, limiting LISA’s performance.
• Einstein Telescope can measure 3 QNM parameters with approximately 10% relative error in a few to ten events per year for light seed scenarios, indicating viable but less precise spectroscopy.
• More precise BH spectroscopy in light seed scenarios would require future deciHertz-band detectors, as ET’s sensitivity is insufficient for sub-1% precision on higher-frequency modes.