Quasi-Normal Modes of Stars and Black Holes

Black Hole Formation-Null Geodesic Correspondence

We provide evidence for a correspondence between the formation of black holes and the stability of circular null geodesics around the collapsing perturbation. We first show that the critical threshold of the compaction function to form a black hole in radiation is well approximated by the critical threshold for the appearance of the first unstable circular orbit in a spherically symmetric background. We also show that the critical exponent in the scaling law of the primordial black hole mass close to the threshold is set by the inverse of the Lyapunov coefficient of the unstable orbits when a self-similar stage is developed close to criticality.

Nonlinear Treatment of a Black Hole Mimicker Ringdown

We perform the first nonlinear and self-consistent study of the merger and ringdown of a black hole mimicking object with stable light rings. To that end, we numerically solve the full Einstein-Klein-Gordon equations governing the head-on collisions of a series of binary boson stars in the large-mass-ratio regime resulting in spinning horizonless remnants with stable light rings. We broadly confirm the appearance of features in the extracted gravitational waveforms expected based on perturbative methods: the signal from the prompt response of the remnants approaches that of a Kerr black hole in the large-compactness limit, and the subsequent emissions contain periodically appearing bursts akin to so-called gravitational wave echoes. However, these bursts occur at high frequencies and are sourced by perturbations of the remnant's internal degrees of freedom. Furthermore, the emitted waveforms also contain a large-amplitude and long-lived component comparable in frequency to black hole quasinormal modes. We further characterize the emissions, obtain basic scaling relations of relevant timescales, and compute the energy emitted in gravitational waves.

Ringdown Gravitational Waves from Close Scattering of Two Black Holes

We have numerically investigated close scattering processes of two black holes (BHs). Our careful analysis shows for the first time a nonmerging ringdown gravitational wave induced by dynamical tidal deformations of individual BHs during their close encounter. The ringdown wave frequencies turn out to agree well with the quasinormal ones of a single BH in perturbation theory, despite its distinctive physical context from the merging case. Our study shows a new type of gravitational waveform and opens up a new exploration of strong gravitational interactions using BH encounters.

Revisiting the Cosmic String Origin of GW190521

For the first time we analyze gravitational-wave strain data using waveforms constructed from strong gravity simulations of cosmic string loops collapsing to Schwarzschild black holes; a previously unconsidered source. Since the expected signal is dominated by a black-hole ringdown, it can mimic the observed gravitational waves from high-mass binary black hole mergers. To illustrate this, we consider GW190521, a short duration gravitational-wave event observed in the third LIGO-Virgo-KAGRA observing run. We show that describing this event as a collapsing cosmic string loop is favored over previous cosmic string analyses by an approximate log Bayes factor of 22. The binary black hole hypothesis is still preferred, mostly because the cosmic string remnant is nonspinning. It remains an open question whether a spinning remnant could form from loops with angular momentum, but if possible, it would likely bring into contention the binary black hole preference. Finally, we suggest that searches for ringdown-only waveforms would be a viable approach for identifying collapsing cosmic string events and estimating their event rate. This Letter opens up an important new direction for the cosmic-string and gravitational-wave communities.

Hyperboloidal approach for static spherically symmetric spacetimes: a didactical introduction and applications in black-hole physics

This work offers a didactical introduction to the calculations and geometrical properties of a static, spherically symmetric spacetime foliated by hyperboloidal time surfaces. We discuss the various degrees of freedom involved, namely the height function, responsible for introducing the hyperboloidal time coordinate, and a radial compactification function. A central outcome is the expression of the Trautman-Bondi mass in terms of the hyperboloidal metric functions. Moreover, we apply this formalism to a class of wave equations commonly used in black-hole perturbation theory. Additionally, we provide a comprehensive derivation of the hyperboloidal minimal gauge, introducing two alternative approaches within this conceptual framework: the in-out and out-in strategies. Specifically, we demonstrate that the height function in the in-out strategy follows from the well-known tortoise coordinate by changing the sign of the terms that become singular at future null infinity. Similarly, for the out-in strategy, a sign change also occurs in the tortoise coordinate's regular terms. We apply the methodology to the following spacetimes: Singularity-approaching slices in Schwarzschild, higher-dimensional black holes, black hole with matter halo, and Reissner-Nordström-de Sitter. From this heuristic study, we conjecture that the out-in strategy is best adapted for black hole geometries that account for environmental or effective quantum effects. This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'.

Relativistic Perturbation Theory for Black-Hole Boson Clouds

We develop a relativistic perturbation theory for scalar clouds around rotating black holes. We first introduce a relativistic product and corresponding orthogonality relation between modes, extending a recent result for gravitational perturbations. We then derive the analog of time-dependent perturbation theory in quantum mechanics, and apply it to calculate self-gravitational frequency shifts. This approach supersedes the nonrelativistic "gravitational atom" approximation, brings close agreement with numerical relativity, and has practical applications for gravitational-wave astronomy.

Quasinormal Modes of Optical Solitons

Quasinormal modes (QNMs) are essential for understanding the stability and resonances of open systems, with increasing prominence in black hole physics. We present here the first study of QNMs of optical potentials. We show that solitons can support QNMs, deriving a soliton perturbation equation and giving exact analytical expressions for the QNMs of fiber solitons. We discuss the boundary conditions in this intrinsically dispersive system and identify novel signatures of dispersion. From here, we discover a new analogy with black holes and describe a regime in which the soliton is a robust black hole simulator for light-ring phenomena. Our results invite a range of applications, from the description of optical pulse propagation with QNMs to the use of state-of-the-art technology from fiber optics to address questions in black hole physics, such as QNM spectral instabilities and the role of nonlinearities in ringdown.

Geodesics and gravitational waves in chaotic extreme-mass-ratio inspirals: the curious case of Zipoy-Voorhees black-hole mimickers

Due to the growing capacity of gravitational-wave astronomy and black-hole imaging, we will soon be able to emphatically decide if astrophysical dark objects lurking in galactic centers are black holes. Sgr A*, one of the most prolific astronomical radio sources in our galaxy, is the focal point for tests of general relativity. Current mass and spin constraints predict that the central object of the Milky Way is supermassive and slowly rotating, thus can be conservatively modeled as a Schwarzschild black hole. Nevertheless, the well-established presence of accretion disks and astrophysical environments around supermassive compact objects can significantly deform their geometry and complicate their observational scientific yield. Here, we study extreme-mass-ratio binaries comprised of a minuscule secondary object inspiraling onto a supermassive Zipoy-Voorhees compact object; the simplest exact solution of general relativity that describes a static, spheroidal deformation of Schwarzschild spacetime. We examine geodesics of prolate and oblate deformations for generic orbits and reevaluate the non-integrability of Zipoy-Voorhees spacetime through the existence of resonant islands in the orbital phase space. By including radiation loss with post-Newtonian techniques, we evolve stellar-mass secondary objects around a supermassive Zipoy-Voorhees primary and find clear imprints of non-integrability in these systems. The peculiar structure of the primary, allows for, not only typical single crossings of transient resonant islands, that are well-known for non-Kerr objects, but also inspirals that transverse through several islands, in a brief period of time, that lead to multiple glitches in the gravitational-wave frequency evolution of the binary. The detectability of glitches with future spaceborne detectors can, therefore, narrow down the parameter space of exotic solutions that, otherwise, can cast identical shadows with black holes.

Black Hole Spectroscopy by Mode Cleaning

We formulate a Bayesian framework to analyze ringdown gravitational waves from colliding binary black holes and test the no-hair theorem. The idea hinges on mode cleaning-revealing subdominant oscillation modes by removing dominant ones using newly proposed "rational filters." By incorporating the filter into Bayesian inference, we construct a likelihood function that depends only on the mass and spin of the remnant black hole (no dependence on mode amplitudes and phases) and implement an efficient pipeline to constrain the remnant mass and spin without Markov chain Monte Carlo. We test ringdown models by cleaning combinations of different modes and evaluating the consistency between the residual data and pure noise. The model evidence and Bayes factor are used to demonstrate the presence of a particular mode and to infer the mode starting time. In addition, we design a hybrid approach to estimate the remnant black hole properties exclusively from a single mode using Markov chain Monte Carlo after mode cleaning. We apply the framework to GW150914 and demonstrate more definitive evidence of the first overtone by cleaning the fundamental mode. This new framework provides a powerful tool for black hole spectroscopy in future gravitational-wave events.

Nonlinear Effects in Black Hole Ringdown

We report evidence for nonlinear modes in the ringdown stage of the gravitational waveform produced by the merger of two comparable-mass black holes. We consider both the coalescence of black hole binaries in quasicircular orbits and high-energy, head-on black hole collisions. The presence of nonlinear modes in the numerical simulations confirms that general-relativistic nonlinearities are important and must be considered in gravitational-wave data analysis.

Novel Ringdown Amplitude-Phase Consistency Test

The ringdown signal emitted during a binary black hole coalescence can be modeled as a linear superposition of the characteristic damped modes of the remnant black hole that get excited during the merger phase. While checking the consistency of the measured frequencies and damping times against the Kerr BH spectrum predicted by general relativity (GR) is a cornerstone of strong-field tests of gravity, the consistency of measured excitation amplitudes and phases have been largely left unexplored. For a nonprecessing, quasicircular binary black hole merger, we find that GR predicts a narrow region in the space of mode amplitude ratio and phase difference, independently of the spin of the binary components. Using this unexpected result, we develop a new null test of strong-field gravity which demands that the measured amplitudes and phases of different ringdown modes should lie within this narrow region predicted by GR. We call this the amplitude-phase consistency test and introduce a procedure for performing it using information from the ringdown signal. Lastly, we apply this test to the GW190521 event, using the multimodal ringdown parameters inferred by Capano et al. [arXiv:2105.05238]. While ringdown measurements errors for this event are large, we show that GW190521 is consistent with the amplitude-phase consistency test. Our test is particularly well suited for accommodating multiple loud ringdown detections as those expected in the near future, and can be used complementarily to standard black-hole spectroscopy as a proxy for modified gravity, compact objects other than black holes, binary precession and eccentricity.

Analysis of Ringdown Overtones in GW150914

We analyze GW150914 postmerger data to understand if ringdown overtone detection claims are robust. We find no evidence in favor of an overtone in the data after the waveform peak. Around the peak, the Bayes factor does not indicate the presence of an overtone, while the support for a nonzero amplitude is sensitive to changes in the starting time much smaller than the overtone damping time. This suggests that claims of an overtone detection are noise dominated. We perform GW150914-like injections in neighboring segments of the real detector noise, and we show that noise can indeed induce artificial evidence for an overtone.

Temperature Effects on Core g-Modes of Neutron Stars

Neutron stars provide a unique physical laboratory in which to study the properties of matter at high density and temperature. We study a diagnostic of the composition of high-density matter, namely, g-mode oscillations, which are driven by buoyancy forces. These oscillations can be excited by tidal forces and couple to gravitational waves. We extend prior results for the g-mode spectrum of cold neutron star matter to high temperatures that are expected to be achieved in neutron star mergers using a parameterization for finite-temperature effects on equations of state recently proposed by Raithel, Özel and Psaltis. We find that the g-modes of canonical mass neutron stars (≈1.4M⊙) are suppressed at high temperatures, and core g-modes are supported only in the most massive (≥2M⊙) of hot neutron stars.

Gravitational Wave Signatures of Black Hole Quasinormal Mode Instability

Black hole (BH) spectroscopy has emerged as a powerful approach to extracting spacetime information from gravitational wave (GW) observed signals. Yet, quasinormal mode (QNM) spectral instability under small scale perturbations has been recently shown to be a common classical general relativistic phenomenon [J. L. Jaramillo et al., Phys. Rev. X 11, 031003 (2021)PRXHAE2160-330810.1103/PhysRevX.11.031003]. This requires assessing its impact on the BH QNM spectrum, in particular on BH QNM overtone frequencies. We conclude (i) perturbed BH QNM overtones are indeed potentially observable in the GW waveform, providing information on small-scale environment BH physics, and (ii) their detection poses a challenging data analysis problem of singular interest for LISA astrophysics. We adopt a twofold approach, combining theoretical results from scattering theory with a fine-tuned data analysis on a highly accurate numerical GW ringdown signal. The former introduces a set of effective parameters (partially relying on a BH Weyl law) to characterize QNM instability physics. The latter provides a proof of principle demonstrating that the QNM spectral instability is indeed accessible in the time-domain GW waveform, though certainly requiring large signal-to-noise ratios. Particular attention is devoted to discussing the patterns of isospectrality loss under QNM instability, since the disentanglement between axial and polar GW parities may already occur within the near-future detection range.

Destabilizing the Fundamental Mode of Black Holes: The Elephant and the Flea

Recent work applying the notion of pseudospectrum to gravitational physics showed that the quasinormal mode spectrum of black holes is unstable, with the possible exception of the longest-lived (fundamental) mode. The fundamental mode dominates the expected signal in gravitational wave astronomy, and there is no reason why it should have privileged status. We compute the quasinormal mode spectrum of two model problems where the Schwarzschild potential is perturbed by a small "bump" consisting of either a Pöschl-Teller potential or a Gaussian, and we show that the fundamental mode is destabilized under generic perturbations. We present phase diagrams and study a simple double-barrier toy problem to clarify the conditions under which the spectral instability occurs.

Compact Objects in Alternative Gravities

We address neutron stars and black holes in alternative gravities, after recalling their basic properties in General Relativity. Among the plethora of interesting alternative gravities we here focus on an interesting set of scalar-tensor theories. We discuss the phenomenon of spontaneous scalarization, that is, matter induced for neutron stars and curvature induced for black holes. Along with other relevant physical properties, we address the quasi-normal modes of these compact objects. In particular, we consider universal relations of neutron stars to largely reduce the dependence on the equation of state, and we briefly address the shadow of black holes.

Normalization, orthogonality, and completeness of quasinormal modes of open systems: the case of electromagnetism [Invited]

The scattering of electromagnetic waves by resonant systems is determined by the excitation of the quasinormal modes (QNMs), i.e. the eigenmodes, of the system. This Review addresses three fundamental concepts in relation to the representation of the scattered field as a superposition of the excited QNMs: normalization, orthogonality, and completeness. Orthogonality and normalization enable a straightforward assessment of the QNM excitation strength for any incident wave. Completeness guarantees that the scattered field can be faithfully expanded into the complete QNM basis. These concepts are not trivial for non-conservative (non-Hermitian) systems and have driven many theoretical developments since initial studies in the 70's. Yet, they are not easy to grasp from the extensive and scattered literature, especially for newcomers in the field. After recalling fundamental results obtained in initial studies on the completeness of the QNM basis for simple resonant systems, we review recent achievements and the debate on the normalization, clarify under which circumstances the QNM basis is complete, and highlight the concept of QNM regularization with complex coordinate transforms.

The effect of mission duration on LISA science objectives

The science objectives of the LISA mission have been defined under the implicit assumption of a 4-years continuous data stream. Based on the performance of LISA Pathfinder, it is now expected that LISA will have a duty cycle of ≈ 0.75 , which would reduce the effective span of usable data to 3 years. This paper reports the results of a study by the LISA Science Group, which was charged with assessing the additional science return of increasing the mission lifetime. We explore various observational scenarios to assess the impact of mission duration on the main science objectives of the mission. We find that the science investigations most affected by mission duration concern the search for seed black holes at cosmic dawn, as well as the study of stellar-origin black holes and of their formation channels via multi-band and multi-messenger observations. We conclude that an extension to 6 years of mission operations is recommended.

Black Hole Quasinormal Modes and Seiberg-Witten Theory

We present new analytic results on black hole perturbation theory. Our results are based on a novel relation to four-dimensional N = 2 supersymmetric gauge theories. We propose an exact version of Bohr-Sommerfeld quantization conditions on quasinormal mode frequencies in terms of the Nekrasov partition function in a particular phase of the Ω -background. Our quantization conditions also enable us to find exact expressions of eigenvalues of spin-weighted spheroidal harmonics. We test the validity of our conjecture by comparing against known numerical results for Kerr black holes as well as for Schwarzschild black holes. Some extensions are also discussed.

Testing General Relativity with Gravitational Waves: An Overview

The detections of gravitational-wave (GW) signals from compact binary coalescence by ground-based detectors have opened up the era of GW astronomy. These observations provide opportunities to test Einstein’s general theory of relativity at the strong-field regime. Here we give a brief overview of the various GW-based tests of General Relativity (GR) performed by the LIGO-Virgo collaboration on the detected GW events to date. After providing details for the tests performed in four categories, we discuss the prospects for each test in the context of future GW detectors. The four categories of tests include the consistency tests, parametrized tests for GW generation and propagation, tests for the merger remnant properties, and GW polarization tests.

Spectral Problems for Quasinormal Modes of Black Holes

This is an unconventional review article on spectral problems in black hole perturbation theory. Our purpose is to explain how to apply various known techniques in quantum mechanics to such spectral problems. The article includes analytical/numerical treatments, semiclassical perturbation theory, the (uniform) WKB method and useful mathematical tools: Borel summations, Padé approximants, and so forth. The article is not comprehensive, but rather looks into a few examples from various points of view. The techniques in this article are widely applicable to many other examples.

Modal approximation for plasmonic resonators in the time domain: the scalar case

We study the electromagnetic field scattered by a metallic nanoparticle with dispersive material parameters in a resonant regime. We consider the particle placed in a homogeneous medium in a low-frequency regime. We define modes for the non-Hermitian problem as perturbations of electro-static modes, and obtain a modal approximation of the scattered field in the frequency domain. The poles of the expansion correspond to the eigenvalues of a singular boundary integral operator and are shown to lie in a bounded region near the origin of the lower-half complex plane. Finally, we show that this modal representation gives a very good approximation of the field in the time domain. We present numerical simulations in two dimensions to corroborate our results.

Constraining Dense Matter Physics Using f-Mode Oscillations in Neutron Stars

In this paper, an investigation of the role of nuclear saturation parameters on f-mode oscillations in neutron stars is performed within the Cowling approximation. It is found that the uncertainty in the effective nucleon mass plays a dominant role in controlling the f-mode frequencies. The effect of the uncertainties in saturation parameters on previously-proposed empirical relations of the frequencies with astrophysical observables relevant for asteroseismology are also investigated. These results can serve as an important tool for constraining the nuclear parameter space and understand the behaviour of dense nuclear matter from the future detection of f-modes.

Bekenstein-Hod Universal Bound on Information Emission Rate Is Obeyed by LIGO-Virgo Binary Black Hole Remnants

Causality and the generalized laws of black hole thermodynamics imply a bound, known as the Bekenstein-Hod universal bound, on the information emission rate of a perturbed system. Using a time-domain ringdown analysis, we investigate whether remnant black holes produced by the coalescences observed by Advanced LIGO and Advanced Virgo obey this bound. We find that the bound is verified by the astrophysical black hole population with 94% probability, providing a first confirmation of the Bekenstein-Hod bound from black hole systems.

Perturbations of the Gravitational Energy in the TEGR: Quasinormal Modes of the Schwarzschild Black Hole

We calculate the gravitational energy spectrum of the perturbations of a Schwarzschild black hole described by quasinormal modes, in the framework of the teleparallel equivalent of general relativity (TEGR). We obtain a general formula for the gravitational energy enclosed by a large surface of constant radius r, in the region m<<r<<∞, where m is the mass of the black hole. Considering the usual asymptotic expression for the perturbed metric components, we arrive at finite values for the energy spectrum. The perturbed energy depends on the two integers n and l that describe the quasinormal modes. In this sense, the energy perturbations are discretized. We also obtain a simple expression for the decrease of the flux of gravitational radiation of the perturbations.

Probing Crust Meltdown in Inspiraling Binary Neutron Stars

Thanks to recent measurements of tidal deformability and radius, the nuclear equation of state and structure of neutron stars are now better understood. Here, we show that through resonant tidal excitations in a binary inspiral, the neutron crust generically undergoes elastic-to-plastic transition, which leads to crust heating and eventually meltdown. This process could induce ∼O(0.1) phase shift in the gravitational waveform. Detecting the timing and induced phase shift of this crust meltdown will shed light on the crust structure, such as the core-crust transition density, which previous measurements are insensitive to. A direct search using GW170817 data has not found this signal, possibly due to limited signal-to-noise ratio. We predict that such a signal may be observable with Advanced LIGO Plus and more likely with third-generation gravitational-wave detectors such as the Einstein Telescope and Cosmic Explorer.

Quasinormal Mode Oscillations in an Analogue Black Hole Experiment

The late stages of the relaxation process of a black hole are expected to depend only on its mass and angular momentum and not on the details of its formation process. Inspired by recent analogue gravity experiments, which demonstrate that certain black hole processes take place in gravitational and hydrodynamical systems alike, we conduct an experiment to search for quasinormal mode oscillations of the free surface of a hydrodynamical vortex flow. Our results demonstrate the occurrence and hint at the ubiquity of quasinormal ringing in nonequilibrium analog black hole experiments.