Fundamental Quasinormal Mode Research Articles

Correspondence between grey-body factors and quasinormal modes

Quasinormal modes and grey-body factors are spectral characteristics corresponding to different boundary conditions: the former imply purely outgoing waves to the event horizon and infinity, while the latter allow for an incoming wave from the horizon, thus describing a scattering problem. Nevertheless, we show that there is a link between these two characteristics. We establish an approximate correspondence between the quasinormal modes and grey-body factors, which becomes exact in the high-frequency (eikonal) regime. We show that, in the eikonal regime, the grey-body factors of spherically symmetric black holes can be remarkably simply expressed via the fundamental quasinormal mode, while at smaller ℓ, the correction terms include values of the overtones. This might be interesting in the context of the recently observed connection between grey-body factors and the amplitudes of gravitational waves from black holes. The correspondence might explain why grey-body factors are more stable, i.e. less sensitive, than higher overtones to small deformation of the effective potential.

Quasinormal modes of Einstein–scalar–Gauss–Bonnet black holes

In this paper, we investigate quasinormal modes of scalar and electromagnetic fields in the background of Einstein–scalar–Gauss–Bonnet (EsGB) black holes. Using the scalar and electromagnetic field equations in the vicinity of the EsGB black hole, we study nature of the effective potentials. The dependence of real and imaginary parts of the fundamental quasinormal modes on parameter p (which is related to the Gauss–Bonnet coupling parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}) for different values of multipole numbers l are studied. We analyzed the effects of massive scalar fields on the EsGB black hole, which tells us the existence of quasi-resonances. In the eikonal regime, we find the analytical expression for the quasinormal frequency and show that the correspondence between the eikonal quasinormal modes and null geodesics is valid in the EsGB theory for the test fields. Finally, we study grey-body factors of the electromagnetic fields for different multipole numbers l, which deviates from Schwarzschild’s black hole.

Quasinormal modes and grey-body factors of regular black holes with a scalar hair from the Effective Field Theory

The Effective Field Theory (EFT) of perturbations on an arbitrary background geometry with a timelike scalar profile has been recently constructed in the context of scalar-tensor theories. Unlike General Relativity, the regular Hayward metric is realized as an exact background metric in the Effective Field Theory with timelike scalar profile without resorting to special matter field, such as nonlinear electrodynamics. The fundamental quasinormal mode for axial graviational perturbations of this black hole has been considered recently with the help of various methods. Here we make a further step in this direction and find that, unlike the fundamental mode, a few first overtones deviate from their Schwarzschild limit at a much higher rate. This outburst of overtones occurs because the overtones are extremely sensitive to the least change of the near- horizon geometry. The analytical formula for quasinormal modes is obtained in the eikonal regime. In addition, we calculated grey-body factors and showed that the regular Hayward black hole with a scalar hair has a smaller grey-body factor than the Schwarzschild one. Integration of the wave-like equation in the time-domain shows that the power-law tails, following the ring-down phase, are indistinguishable from the Schwarzschild ones at late times.

Quasinormal modes in higher-derivative gravity: Testing the black hole parametrization and sensitivity of overtones

The fundamental quasinormal modes of black holes in higher-derivative gravity given by the Einstein-Weyl action are known to be moderately corrected by the Weyl term. Here we will show that the first several overtones are highly sensitive to even a relatively small Weyl correction, which might be important when representing the earlier stage of the black hole ringdown. In addition, we have solved the problem related to the analytical parametrized approximation of the numerical black hole solution in the Einstein-Weyl theory: In some range of parameters the approximation for the metric developed up to the third order leads to the unusual highly nonmonotonic behavior of the frequencies. We have shown that this problem can be solved via the extension of the parametrization of the metric to higher orders until reaching the regime when the frequencies do not change with further increasing of the order.

Quasinormal modes of Schwarzschild black holes on the real axis

We study the scattering of gravitational waves by a Schwarzschild black hole and its perturbed siblings to investigate influences of proposed spectral instability of quasinormal modes on the ringdown signal. Our results indicate that information of dominant ringdown signals, which are ascribed to the fundamental (i.e., least damping) quasinormal mode of unperturbed Schwarzschild black holes, is imprinted in the phase shift defined from the transmission amplitude (1/A_{in} in our notation). This approximately parallels the fact that the resonance of quantum systems is imprinted in the phase shift of the S-matrix. The phase shift around the oscillation frequency of the fundamental mode is modified only perturbatively even if the quasinormal-mode spectrum is destabilized by a perturbative bump at a distant location, signifying the stability of the ringdown signal. At the same time, the phase shift at low frequencies is modulated substantially reflecting the late-time excitation of echo signals associated with the quasinormal-mode spectrum after destabilization.

Multipole moments on the common horizon in a binary-black-hole simulation

We construct the covariantly defined multipole moments on the common horizon of an equal-mass, nonspinning, quasicircular binary-black-hole system. We see a strong correlation between these multipole moments and the gravitational waveform. We find that the multipole moments are well described by the fundamental quasinormal modes at sufficiently late times. For each nonzero multipole moment with $\ensuremath{\ell}\ensuremath{\le}6$, at least two fundamental quasinormal modes of different $\ensuremath{\ell}$ are detectable in the best model. These models provide faithful estimates of the true mass and spin of the remnant black hole. We also show that by including overtones, the $\ensuremath{\ell}=m=2$ mass multipole moment admits an excellent quasinormal-mode description at all times after the merger. This demonstrates the perhaps surprising power of perturbation theory near the merger.

Stability of the fundamental quasinormal mode in time-domain observations against small perturbations

Black hole spectroscopy with gravitational waves is an important tool to measure the mass and spin of astrophysical black holes and to test their Kerr nature. Next-generation ground- and space-based detectors will observe binary black hole mergers with large signal-to-noise ratios and perform spectroscopy routinely. It was recently shown that small perturbations due, e.g., to environmental effects (the "flea") to the effective potential governing gravitational-wave generation and propagation in black hole exteriors (the "elephant") can lead to arbitrarily large changes in the black hole's quasinormal spectrum, including the fundamental mode, which is expected to dominate the observed signal. This raises an important question: is the black hole spectroscopy program robust against perturbations? We clarify the physical behavior of time-domain signals under small perturbations in the potential, and we show that changes in the amplitude of the fundamental mode in the prompt ringdown signal are parametrically small. This implies that the fundamental quasinormal mode extracted from the observable time-domain signal is stable against small perturbations. The stability of overtones deserves further investigation.

Theory-agnostic reconstruction of potential and couplings from quasinormal modes

In this work, we use a parametrized theory-agnostic approach that connects the observation of black hole quasi-normal modes with the underlying perturbation equations, with the goal of reconstructing the potential and the coupling functions appearing in the latter. The fundamental quasi-normal mode frequency and its first two overtones are modeled through a second order expansion in the deviations from general relativity, which are assumed to be small but otherwise generic. By using a principal component analysis, we demonstrate that percent-level measurements of the fundamental mode and its overtones can be used to constrain the effective potential of tensor perturbations and the coupling functions between tensor modes and ones of different helicity, without assuming an underlying theory. We also apply our theory-agnostic reconstruction framework to analyze simulated quasi-normal mode data produced within specific theories extending general relativity, such as Chern-Simons gravity.

Gravitational ringing of rotating black holes in higher-derivative gravity

We study gravitational perturbations of slowly rotating black holes in a general effective-field-theory extension of general relativity that includes up to eight-derivative terms. We show that two Schr\"odinger-like equations with spin-dependent effective potentials govern the odd- and even-parity master variables. These equations are coupled for parity-violating corrections, and this coupling affects the quasinormal modes even at linear order in the higher-derivative corrections, due to their isospectrality in general relativity. We provide results for the shifts in the fundamental quasinormal mode frequencies at linear order in the spin, which we expect to be valuable for high-precision phenomenology through future gravitational wave observations.

Scalar and Dirac perturbations of the Reissner-Nordström black hole and Painlevé transcendents

We investigate spin-$0$ and spin-$\frac{1}{2}$ perturbations for non-extremal and extremal Reissner-Nordstr\"om backgrounds using the isomonodromic method. We calculate the fundamental quasinormal modes (QNMs) associated with each perturbation as a function of the electromagnetic coupling $qQ$ and ratio $Q/M$. After corroborating the literature values for generic $qQ$ and $Q/M$, we turn to study the near-extremal limit. In parallel with the study of QNMs for the Kerr geometry, we find the existence of "non-damping" modes for $qQ$ above a spin-dependent critical value, and locate the bifurcation point in the $q-Q$ parameter space.

Constraint on nuclear symmetry energy imposed by f-mode oscillation of neutron stars

Due to improvements in the sensitivity of gravitational wave (GW) detectors, the detection of GWs originating from the fundamental quasi-normal mode (f-mode) of neutron stars has become possible. The future detection of GWs originating from the f-mode of neutron stars will provide a potential way to improve our understanding of the nature of nuclear matter inside neutron stars. In this work, we investigate the constraint imposed by the f-mode oscillation of neutron stars on the symmetry energy of nuclear matter using Bayesian analysis and parametric EOS. It is shown that if the frequency of the f-mode of a neutron star of known mass is observed precisely, the symmetry energy at twice the saturation density (E sym(2ρ 0)) of nuclear matter can be constrained within a relatively narrow range. For example, when all the following parameters are within the given intervals: 220 ≤ K 0 ≤ 260 MeV, 28 ≤ E sym(ρ 0) ≤ 36 MeV, 30 ≤ L ≤ 90 MeV, −800 ≤ J 0 ≤ 400 MeV, − 400 ≤ K sym ≤ 100 MeV, −200 ≤ J sym ≤ 800 MeV, E sym(2ρ 0) will be constrained to within MeV if the f-mode frequency of a canonical neutron star (1.4 M⊙) is observed to be 1.720 kHz with a 1% relative error. Furthermore, if only f-mode frequency detection is available, i.e. there is no stellar mass measurement, a precisely detected f-mode frequency can also impose an accurate constraint on the symmetry energy. For example, given the same parameter space and the same assumed observed f-mode frequency mentioned above, and assuming that the stellar mass is in the range of 1.2–2.0 M⊙, E sym(2ρ 0) will be constrained to within . In addition, it is shown that a higher slope of 69 ≤ L ≤ 143 MeV will give a higher posterior distribution of E sym(2ρ 0), .

Eikonal quasinormal modes of black holes beyond general relativity. III. Scalar Gauss-Bonnet gravity

In a recent series of papers, we have shown how the eikonal/geometrical optics approximation can be used to calculate analytically the fundamental quasinormal mode frequencies associated with coupled systems of wave equations, which arise, for instance, in the study of perturbations of black holes in gravity theories beyond General Relativity. As a continuation to this series, we focus here on the quasinormal modes of nonrotating black holes in scalar Gauss-Bonnet gravity assuming a small-coupling expansion. We show that the axial perturbations are purely tensorial and are described by a modified Regge-Wheeler equation, while the polar perturbations are of mixed scalar-tensor character and are described by a system of two coupled wave equations. When applied to these equations, the eikonal machinery leads to axial quasinormal modes that deviate from the general relativistic results at quadratic order in the Gauss-Bonnet coupling constant. We show that this result is in agreement with an analysis of unstable circular null orbits around black holes in this theory, allowing us to establish the geometrical optics--null geodesic correspondence for the axial quasinormal modes. For the polar quasinormal modes, the small-coupling approximation forces us to consider the ordering between eikonal and small-coupling perturbative parameters, one of which we show, by explicit comparison against numerical data, yields the correct identification of the quasinormal modes of the scalar-tensor coupled system of wave equations. These corrections lift the general relativistic degeneracy between scalar and tensorial eikonal quasinormal modes at quadratic order in Gauss-Bonnet coupling in a way reminiscent of the Zeeman effect. In general, our analytic, eikonal quasinormal mode frequencies (normalized to the General Relativity ones) agree with numerical results with an error of $\mathcal{O}(10%)$ in the regime of small coupling constant. Finally, we find that the analytical expressions for the quasinormal modes are common to a broad class of scalar-Gauss-Bonnet theories to leading eikonal order, showing a degeneracy between the quasinormal modes of nonrotating black holes in particular scalar-Gauss-Bonnet theories in the geometrical optics limit.

Spin effects on neutron star fundamental-mode dynamical tides: Phenomenology and comparison to numerical simulations

Gravitational waves from neutron star binary inspirals contain information on strongly-interacting matter in unexplored, extreme regimes. Extracting this requires robust theoretical models of the signatures of matter in the gravitational-wave signals due to spin and tidal effects. In fact, spins can have a significant impact on the tidal excitation of the quasi-normal modes of a neutron star, which is not included in current state-of-the-art waveform models. We develop a simple approximate description that accounts for the Coriolis effect of spin on the tidal excitation of the neutron star's quadrupolar and octupolar fundamental quasi-normal modes and incorporate it in the SEOBNRv4T waveform model. We show that the Coriolis effect introduces only one new interaction term in an effective action in the co-rotating frame of the star, and fix the coefficient by considering the spin-induced shift in the resonance frequencies that has been computed numerically for the mode frequencies of rotating neutron stars in the literature. We investigate the impact of relativistic corrections due to the gravitational redshift and frame-dragging effects, and identify important directions where more detailed theoretical developments are needed in the future. Comparisons of our new model to numerical relativity simulations of double neutron star and neutron star-black hole binaries show improved consistency in the agreement compared to current models used in data analysis

Gravitational wave signatures of highly magnetized neutron stars

Motivated by the recent gravitational wave detection by the LIGO–VIRGO observatories, we study the Love number and dimensionless tidal polarizability of highly magnetized stars. We also investigate the fundamental quasi-normal mode of neutron stars subject to high magnetic fields. To perform our calculations we use the chaotic field approximation and consider both nucleonic and hyperonic stars. As far as the fundamental mode is concerned, we conclude that the role played by the constitution of the stars is far more relevant than the intensity of the magnetic field, and if massive stars are considered, the ones constituted by nucleons only present frequencies somewhat lower than the ones with hyperonic cores. This feature that can be used to point out the real internal structure of neutron stars. Moreover, our studies clearly indicate that strong magnetic fields play a crucial role in the deformability of low mass neutron stars, with possible consequences on the interpretation of the detected gravitational waves signatures.

Eikonal quasinormal modes of black holes beyond general relativity. II. Generalized scalar-tensor perturbations

Black hole `spectroscopy', i.e. the identification of quasinormal mode frequencies via gravitational wave observations, is a powerful technique for testing the general relativistic nature of black holes. In theories of gravity beyond general relativity perturbed black holes are typically described by a set of coupled wave equations for the tensorial field and the extra scalar/vector degrees of freedom, thus leading to a theory-specific quasinormal mode spectrum. In this paper we use the eikonal/geometric optics approximation to obtain analytic formulae for the frequency and damping rate of the fundamental quasinormal mode of a generalised, theory-agnostic system of equations describing coupled scalar-tensor perturbations of spherically symmetric black holes. Representing an extension of our recent work, the present model includes a massive scalar field, couplings through the field derivatives and first-order frame dragging rotational corrections. Moving away from spherical symmetry, we consider the simple model of the scalar wave equation in a general stationary-axisymmetric spacetime and use the eikonal approximation to compute the quasinormal modes associated with equatorial and nonequatorial photon rings.

Black Hole Ringdown: The Importance of Overtones

It is possible to infer the mass and spin of the remnant black hole from binary black hole mergers by comparing the ringdown gravitational wave signal to results from studies of perturbed Kerr spacetimes. Typically these studies are based on the fundamental quasinormal mode of the dominant $\ell=m=2$ harmonic. By modeling the ringdown of accurate numerical relativity simulations, we find that the fundamental mode alone is insufficient to recover the true underlying mass and spin, unless the analysis is started very late in the ringdown. Including higher overtones associated with this $\ell=m=2$ harmonic resolves this issue, and provides an unbiased estimate of the true remnant parameters. Further, including overtones allows for the modeling of the ringdown signal for all times beyond the peak strain amplitude, indicating that the linear quasinormal regime starts much sooner than previously expected. A model for the ringdown beginning at the peak strain amplitude can exploit the higher signal-to-noise ratio in detectors, reducing uncertainties in the extracted remnant quantities. Tests of the no-hair theorem should consider incorporating overtones in the analysis.

Testing the No-Hair Theorem with GW150914.

We analyze gravitational-wave data from the first LIGO detection of a binary black-hole merger (GW150914) in search of the ringdown of the remnant black hole. Using observations beginning at the peak of the signal, we find evidence of the fundamental quasinormal mode and at least one overtone, both associated with the dominant angular mode (ℓ=m=2), with 3.6σ confidence. A ringdown model including overtones allows us to measure the final mass and spin magnitude of the remnant exclusively from postinspiral data, obtaining an estimate in agreement with the values inferred from the full signal. The mass and spin values we measure from the ringdown agree with those obtained using solely the fundamental mode at a later time, but have smaller uncertainties. Agreement between the postinspiral measurements of mass and spin and those using the full waveform supports the hypothesis that the GW150914 merger produced a Kerr black hole, as predicted by general relativity, and provides a test of the no-hair theorem at the ∼10% level. An independent measurement of the frequency of the first overtone yields agreement with the no-hair hypothesis at the ∼20% level. As the detector sensitivity improves and the detected population of black-hole mergers grows, we can expect that using overtones will provide even stronger tests.

Eikonal quasinormal modes of black holes beyond general relativity

Much of our physical intuition about black hole quasinormal modes in general relativity comes from the eikonal/geometric optics approximation. According to the well-established eikonal model, the fundamental quasinormal mode represents wavepackets orbiting in the vicinity of the black hole's geodesic photon ring, slowly peeling off towards the event horizon and infinity. Besides its strength as a "visualisation" tool, the eikonal approximation also provides a simple quantitative method for calculating the mode frequency, in close agreement with rigorous numerical results. In this paper we move away from Einstein's theory and its garden-variety black holes and go on to consider spherically symmetric black holes in modified theories of gravity through the lens of the eikonal approximation. The quasinormal modes of such black holes are typically described by a set of coupled wave equations for the various field degrees of freedom. Considering a general, theory-agnostic, system of two equations for two perturbed fields, we derive eikonal formulae for the complex fundamental quasinormal mode frequency. In addition we show that the eikonal modes can be related to the extremum of an effective potential and its associated "photon ring". As an application of our results we consider a specific example of a modified theory of gravity with known black hole quasinormal modes and find that these are well approximated by the eikonal formulae.

Kerr–(anti–)de Sitter black holes: Perturbations and quasinormal modes in the slow rotation limit

We study the perturbations of scalar, vector, and tensor fields in a slowly rotating Kerr-(Anti-)de Sitter black hole spacetime, presenting new and existing Schr\"odinger style master equations for each type of perturbation up to linear order in black hole spin $a$. For each type of field we calculate analytical expressions for the fundamental quasi-normal mode frequencies. These frequencies are compared to existing results for Schwarzschild-de Sitter, slowly rotating Kerr, and slowly rotating Kerr-de Sitter black holes. In all cases good agreement is found between the analytic expressions and those frequencies calculated numerically. In addition, the axial and polar gravitational frequencies are shown to be isospectral to linear order in $a$ for all cases other than for both non-zero $a$ and $\Lambda$.

Post-Kerr black hole spectroscopy

One of the central goals of the newborn field of gravitational wave astronomy is to test gravity in the highly nonlinear, strong field regime characterizing the spacetime of black holes. In particular, "black hole spectroscopy" (the observation and identification of black hole quasinormal mode frequencies in the gravitational wave signal) is expected to become one of the main tools for probing the structure and dynamics of Kerr black holes. In this paper we take a significant step towards that goal by constructing a "post-Kerr" quasinormal mode formalism. The formalism incorporates a parametrized but general perturbative deviation from the Kerr metric and exploits the well-established connection between the properties of the spacetime's circular null geodesics and the fundamental quasinormal mode to provide approximate, eikonal limit formulae for the modes' complex frequencies. The resulting algebraic toolkit can be used in waveform templates for ringing black holes with the purpose of measuring deviations from the Kerr metric. As a first illustrative application of our framework, we consider the Johannsen-Psaltis deformed Kerr metric and compute the resulting deviation in the quasinormal mode frequency relative to the known Kerr result.