The parametrized black hole quasinormal ringdown formalism is useful to compute quasinormal mode (QNM) frequencies if a master equation for the gravitational perturbation around a black hole has a small deviation from the Regge-Wheeler or Zerilli equation. In this formalism, the deviation of QNM frequency from general relativity can be calculated by small deviation parameters and model independent coefficients. In this paper, we derive recursion relations for the model independent coefficients. Using these relations, the higher order coefficients are written only by the lower order coefficients. Thus, we only need the lower order coefficients when we numerically compute the model independent coefficients.
Ringdown gravitational waves of compact object binaries observed by ground-based gravitational-wave detectors encapsulate rich information to understand remnant objects after the merger and to test general relativity in the strong field. In this work, we investigate the ringdown gravitational waves in detail to better understand their property, assuming that the remnant objects are black holes. For this purpose, we perform numerical simulations of post-merger phase of binary black holes by using the black hole perturbation scheme with the initial data given under the close-limit approximation,…
We provide an in-depth investigation of quasinormal-mode oscillations of Kerr black holes with nearly extremal angular momenta. We first discuss in greater detail the two distinct types of quasinormal mode frequencies presented in a recent paper (arXiv:1212.3271). One set of modes, that we call "zero-damping modes", has vanishing imaginary part in the extremal limit, and exists for all corotating perturbations (i.e., modes with azimuthal index m being nonnegative). The other set (the "damped modes") retains a finite decay rate even for extremal Kerr black holes, and exists only for a subset of…
Modifying black hole horizon can drastically change the spectrum of quasinormal modes. But if the modification is close enough to the horizon the ringdown signal remains almost unaltered, and well described by the quasinormal modes of the original GR solution. I show how the original quasinormal modes emerge in the sum over the new modes.
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 shi…
It is known that approximating the Regge-Wheeler Potential with step functions significantly modifies the Schwarzschild black hole quasinormal mode spectrum. Surprisingly, this change in the spectrum has little impact on the ringdown waveform. We examine whether this issue is caused by the jump discontinuities and/or the piecewise constant nature of step functions. We show that replacing the step functions with a continuous piecewise linear function does not qualitatively change the results. However, in contrast to previously published results, we discover that the ringdown waveform can be app…
Branch-dependent ringdown in black-bounce spacetimes: imprints of matter-source ambiguity on quasinormal modes
School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS（杭州高等研究院基础物理与数学学院，中国科学院大学）；Department of Physics, Yantai University（烟台大学物理学院）
We study the generation and propagation of gravitational waves in scalar-tensor gravity using numerical relativity simulations of scalar field collapses beyond spherical symmetry. This allows us to compare the tensor and additional massive scalar waves that are excited. As shown in previous work in spherical symmetry, massive propagating scalar waves decay faster than 1/r and disperse, resulting in an inverse chirp. These effects obscure the ringdown in any extracted signal by mixing it with the transient responses of the collapse during propagation. In this paper we present a simple method to…
It is commonly believed that the ringdown signal from a binary coalescence provides a conclusive proof for the formation of an event horizon after the merger. This expectation is based on the assumption that the ringdown waveform at intermediate times is dominated by the quasinormal modes of the final object. We point out that this assumption should be taken with great care, and that very compact objects with a light ring will display a similar ringdown stage, even when their quasinormal-mode spectrum is completely different from that of a black hole. In other words, universal ringdown wavefor…
We investigate the nonlinear evolution of black hole ringdown in the framework of higher-order metric perturbation theory. By solving the initial-value problem of a simplified nonlinear field model analytically as well as numerically, we find that (i) second-order quasinormal modes (QNMs) are indeed excited at frequencies different from those of first-order QNMs, as predicted recently. We also find serendipitously that (ii) late-time evolution is dominated by a new type of power-law tail. This ``second-order power-law tail'' decays more slowly than any late-time tails known in the first-order …
The motion of photons around black holes determines the shape of shadow and match the ringdown properties of a perturbed black hole. Observations of shadows and ringdown waveforms will reveal the nature of black holes. In this paper, we study the motion of photons in a general parametrized metric beyond the Kerr hypothesis. We investigated the radius and frequency of the photon circular orbits on the equatorial plane and obtained fitted formula with varied parameters. The Lyapunov exponent which connects to the decay rate of the ringdown amplitude is also calculated. We also analyzed the shape…
It is known that a quasinormal mode (QNM) of a remnant black hole dominates a ringdown gravitational wave (GW) in a binary black hole (BBH) merger. To study properties of the QNMs, it is important to determine the time when the QNMs appear in a GW signal as well as to calculate its frequency and amplitude. In this paper, we propose a new method of estimating the starting time of the QNM and calculating the QNM frequency and amplitude of BBH GWs. We apply it to simulated merger waveforms by numerical relativity and the observed data of GW150914. The results show that the obtained QNM frequencie…
Black-hole spectroscopy is arguably the most promising tool to test gravity in extreme regimes and to probe the ultimate nature of black holes with unparalleled precision. These tests are currently limited by the lack of a ringdown parametrization that is both robust and accurate. We develop an observable-based parametrization of the ringdown of spinning black holes beyond general relativity, which we dub ParSpec (Parametrized Ringdown Spin Expansion Coefficients). This approach is perturbative in the spin, but it can be made arbitrarily precise (at least in principle) through a high-order exp…
Exceptional Points in Quasinormal Spectra of Hairy Black Holes
College of Physics, Sichuan University（四川大学物理学院）；School of Data Science, Shanghai University of Finance and Economics Zhejiang College（上海财经大学浙江学院数据科学学院）
The general parametrization of spherically symmetric and asymptotically flat black-hole spacetimes in arbitrary metric theories of gravity was suggested in [3]. The parametrization is based on the continued fraction expansion in terms of the compact radial coordinate and has superior convergence and strict hierarchy of parameters. It is known that some observable quantities, related to particle motion around the black hole, such as the eikonal quasinormal modes, radius of the shadow, frequency at the innermost stable circular orbit, and others, depend mostly on only a few of the lowest coeffic…
Quasinormal Modes of Self-Dual Loop Quantum Gravity Corrected Kerr Black Holes
School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院）；School of Aeronautics and Astronautics, Shanghai Jiao Tong University（上海交通大学航空航天学院）
The waveforms from binary black hole mergers include inspiral, merger, and ringdown parts. Usually, the inspiral waveform can be obtained by calibrating from post-Newtonian approximation; The merger and ringdown ones can be gotten from the quasinormal modes with black hole perturbation theory. However, for more general black holes, the calculation of the quasinormal modes is not trivial. In this paper we use the photon sphere to get the quasinormal modes of spinning black holes. Then we connect the ringdown wave with the inspiral part to get full waveforms and compare with the ones from numeri…
We propose two frequency-domain filters to analyze ringdown signals of binary black hole mergers. The first rational filter is constructed based on a set of (arbitrary) quasi-normal modes (QNMs) of the remnant black holes, whereas the second full filter comes from the transmissivity of the remnant black holes. The two filters can remove corresponding QNMs from original time-domain ringdowns, while changing early inspiral signals in a trivial way - merely a time and phase shift. After filtering out dominant QNMs, we can visualize the existence of various subdominant effects. For example, by app…
Parametrized quasinormal modes, greybody factors and their correspondence
CENTRA, Departamento de Física, Instituto Superior Técnico - IST, Universidade de Lisboa - UL（里斯本大学高等技术学院物理系CENTRA）；Dipartimento di Fisica, “Sapienza” Universitá di Roma（罗马第一大学物理系）；Sezione INFN Roma1（意大利国家核物理研究所罗马一分部）
We consider a wave equation with a potential on the half-line as a model problem for wave propagation close to an extremal horizon, or the asymptotically flat end of a black hole spacetime. We propose a definition of quasinormal frequencies (QNFs) as eigenvalues of the generator of time translations for a null foliation, acting on an appropriate (Gevrey based) Hilbert space. We show that this QNF spectrum is discrete in a subset of $\mathbb{C}$ which includes the region $\{$Re$(s) >-b$, $|$Im $(s)|> K\}$ for any $b>0$ and some $K=K(b) \gg 1$. As a corollary we establish the meromorphicity of t…
Quasinormal modes describe the return to equilibrium of a perturbed system, in particular the ringdown phase of a black hole merger. But as globally-defined quantities, the quasinormal spectrum can be highly sensitive to global structure, including distant small perturbations to the potential. In what sense are quasinormal modes a property of the resulting black hole? We explore this question for the linearized perturbation equation with two potentials having disjoint bounded support. We give a composition law for the Wronskian that determines the quasinormal frequencies of the combined system…
Deep conceptual problems associated with classical black holes can be addressed in string theory by the ``fuzzball'' paradigm, which provides a microscopic description of a black hole in terms of a thermodynamically large number of regular, horizonless, geometries with much less symmetry than the corresponding black hole. Motivated by the tantalizing possibility to observe quantum gravity signatures near astrophysical compact objects in this scenario, we perform the first $3+1$ numerical simulations of a scalar field propagating on a large class of multicenter geometries with no spatial isomet…
Spectral instability of parametrized black hole quasinormal modes in the high-overtone limit via the exact WKB analysis
Osaka Metropolitan University（大阪公立大学）；Osaka Central Advanced Mathematical Institute (OCAMI)（大阪中央先进数学研究所）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学中心交叉理论与数学科学中心）；Research Center for the Early Universe (RESCEU), Graduate School of Science, The University of Tokyo（东京大学理学研究科早期宇宙研究中心）；Department of Physics , Kyoto University（京都大学物理系）；Center for Gravitational Physics and Quantum Information,Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理中心引力物理与量子信息）；The Hakubi Center for Advanced Research, Kyoto University（京都大学白兔高级研究中心）
The relaxation of a distorted black hole to its final state provides important tests of general relativity within the reach of current and upcoming gravitational wave facilities. In black hole perturbation theory, this phase consists of a simple linear superposition of exponentially damped sinusoids (the quasinormal modes) and of a power-law tail. How many quasinormal modes are necessary to describe waveforms with a prescribed precision? What error do we incur by only including quasinormal modes, and not tails? What other systematic effects are present in current state-of-the-art numerical wav…
We develop an analytical ringdown waveform model, including both the fundamental and the overtone quasinormal modes, for charged black holes and show that it is precise enough to analyze current gravitational wave data. Applying this waveform model to GW150914, the charge to mass ratio of the remnant black hole ($λ_f$) has been constrained with the sole ringdown gravitational wave data for the first time and the $90\%$ upper limit is $λ_f\leq 0.38$. Correspondingly, the deviation parameter of the scalar-tensor vector gravity ($α_{\rm s}$) is limited to be $α_{\rm s}\leq 0.17$. Our approach can…
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…
Quasinormal modes and their excitation beyond general relativity. II: isospectrality loss in gravitational waveforms
The Grainger College of Engineering, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校格兰杰工程学院）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（爱因斯坦研究所））；ETH Zürich（苏黎世联邦理工学院）；Universidad de Murcia（穆尔西亚大学）；University of Tübingen（蒂宾根大学）；University of Virginia（弗吉尼亚大学）
The spectrum of frequencies and characteristic times that compose the ringdown phase of gravitational waves emitted by neutron stars carries information about the matter content (the equation of state) and the underlying theory of gravity. Typically, modified theories of gravity introduce additional degrees of freedom/fields, such as scalars, which result in new families of modes composing the ringdown spectrum. Simple but physically promising candidates are scalar-tensor theories, which effectively introduce an additional massive scalar field (i.e. an ultra-light boson) that couples non-minim…
Linear perturbations of spherically symmetric spacetimes in general relativity are described by radial wave equations, with potentials that depend on the spin of the perturbing field. In previous work we studied the quasinormal mode spectrum of spacetimes for which the radial potentials are slightly modified from their general relativistic form, writing generic small modifications as a power-series expansion in the radial coordinate. We assumed that the perturbations in the quasinormal frequencies are linear in some perturbative parameter, and that there is no coupling between the perturbation…
We reanalyze the GW150914 post-merger data searching for quasinormal modes beyond the fundamental, quadrupolar mode. There is currently an ongoing disagreement in the literature about whether, and to what extent, the data contains evidence for a quasinormal mode overtone. We use a frequency-domain approach to ringdown data analysis that was recently proposed by the authors. Our analysis has several advantages compared to other analyses performed mainly in the time domain; in particular, the source sky position and the ringdown start time are marginalized over (as opposed to simply being fixed)…
