arXiv subjects

Gregory B. Cook

Publications and source records attributed to Gregory B. Cook.

At least 19 recordsLinked to original sources

arXiv2026

Choosing the phase for the spin-weighted spheroidal functions

The spin-weighted spheroidal functions are the eigenfunctions of the angular Teukolsky equation. They are a generalization of the widely used spin-weighted spherical functions, and are extremely important in the area of black-hole perturbation theory. Like other special functions, they have an inherent phase ambiguity and need to be phase fixed to be uniquely defined. Clearly, such a phase choice does not have a direct physical impact. But, without specifying phase choices, meaningful comparison between results reported in various works becomes more difficult. To date, possible phase choices for the spin-weighted spheroidal functions have received little attention. Here, we clearly define and extensively explore two useful phase fixing schemes, and we propose that the spherical-limit phase-fixing scheme be adopted as the default phase-fixing scheme for the spin-weighted spheroidal functions.

gr-qc↗

arXiv2025

Black hole spectroscopy: from theory to experiment

The "ringdown" radiation emitted by oscillating black holes has great scientific potential. By carefully predicting the frequencies and amplitudes of black hole quasinormal modes and comparing them with gravitational-wave data from compact binary mergers we can advance our understanding of the two-body problem in general relativity, verify the predictions of the theory in the regime of strong and dynamical gravitational fields, and search for physics beyond the Standard Model or new gravitational degrees of freedom. We summarize the state of the art in our understanding of black hole quasinormal modes in general relativity and modified gravity, their excitation, and the modeling of ringdown waveforms. We also review the status of LIGO-Virgo-KAGRA ringdown observations, data analysis techniques, and the bright prospects of the field in the era of LISA and next-generation ground-based gravitational-wave detectors.

gr-qc↗

arXiv2025

Robustness of extracting quasinormal mode information from black hole merger simulations

In linear perturbation theory, the ringdown of a gravitational wave (GW) signal is described by a linear combination of quasinormal modes (QNMs). Detecting QNMs from GW signals is a promising way to test GR, central to the developing field of black-hole spectroscopy. More robust black-hole spectroscopy tests could also consider the ringdown amplitude-phase consistency. That requires an accurate understanding of the excitation and stability of the QNM expansion coefficients. In this paper, we investigate the robustness of the extracted $m=2$ QNM coefficients obtained from a high-accuracy numerical relativity waveform. We explore a framework to assess the robustness of QNM coefficients. Within this framework, we not only consider the traditional criterion related to the constancy of a QNM's expansion coefficients over a window in time, but also emphasize the importance of consistency among fitting models. In addition, we implement an iterative greedy approach within which we fix certain QNM coefficients. We apply this approach to linear fitting, and to nonlinear fitting where the properties of the remnant black hole are treated as unknown variables. We find that the robustness of overtone coefficients is enhanced by our greedy approach, particularly for the $(2,2,2,+)$ overtone. Based on our robustness criteria applied to the $m=2$ signal modes, we find the $(2\!\sim\!4,2,0,+)$ and $(2,2,1\!\sim\!2,+)$ modes are robust, while the $(3,2,1,+)$ subdominant mode is only marginally robust. After we subtract the contributions of the $(2\!\sim\!4,2,0,+)$ and $(2\!\sim\!3,2,1,+)$ QNMs from signal mode $(4,2)$, we also find evidence for the quadratic QNM $(2,1,0,+)\times(2,1,0,+)$.

gr-qc↗

arXiv2025

Multimode ringdown modelling with $\texttt{qnmfits}$ and $\texttt{KerrRingdown}$

In the last decade, the ringdown community has made large strides in understanding the aftermath of binary black hole mergers through the study of numerical simulations. In this note, we introduce two flavors of fitting algorithms, that have been verified against each other, for the extraction of quasinormal mode amplitudes from ringdown waveforms - $\texttt{qnmfits}$ in Python and $\texttt{KerrRingdown}$ in Mathematica.

gr-qc↗

arXiv2022

New total transmission modes of the Kerr geometry with Schwarzschild limit frequencies at complex infinity

In addition to the well-known quasinormal modes, the gravitational modes of the Kerr geometry also include sets of total-transmission modes. Each mode can be considered as an element of a sequence of modes parameterized by the angular momentum of the black hole. One family of gravitational total-transmission modes of Kerr have been known for some time. Modes in this family connect to a Schwarzschild limit where the mode frequency is finite and purely imaginary. Recently, what was thought to be an additional branch of this original family of modes was discovered. However, this new branch is actually a part of one of two entirely new families of total-transmission modes. Modes in these new families, surprisingly, connect to a Schwarzschild limit where the mode frequencies exist at complex infinity. We have numerically constructed full sets of sequences of gravitational total-transmission modes for harmonic indices $\ell=[2,8]$. Using these numerical sequences, we have been able to construct analytic asymptotic expansions for the mode frequencies and their associated separation constants. The asymptotic expansion for the separation constant used in constructing the total-transmission modes seems to be valid for general complex values of the oblateness parameter.

gr-qc↗

arXiv2022

Understanding solutions of the angular Teukolsky equation in the prolate asymptotic limit

Solutions to the Angular Teukolsky Equation have been used to solve various applied problems in physics and are extremely important to black-hole physics, particularly in computing quasinormal modes and in the extreme-mass-ratio inspiral problem. The eigenfunctions of this equation, known as spin-weighted spheroidal functions, are essentially generalizations of both the spin-weighted spherical harmonics and the scalar spheroidal harmonics. While the latter functions are quite well understood analytically, the spin-weighted spheroidal harmonics are only known analytically in the spherical and oblate asymptotic limits. Attempts to understand them in the prolate asymptotic limit have met limited success. Here, we make use of a high-accuracy numerical solution scheme to extensively explore the space of possible prolate solutions and extract analytic asymptotic expansions for the eigenvalues in the prolate asymptotic limit. Somewhat surprisingly, we find two classes of asymptotic behavior. The behavior of one class, referred to as "normal", is in agreement with the leading-order behavior derived analytically in prior work. The second class of solutions was not previously predicted, but solutions in this class are responsible for unexplained behavior seen in previous numerical prolate solutions during the transition to asymptotic behavior. The behavior of solutions in this "anomalous" class is more complicated than that of solutions in the normal class, with the anomalous class separating into different types based on the behavior of the eigenvalues at different asymptotic orders. We explore the question of when anomalous solutions appear and find necessary, but not sufficient conditions for their existence. It is our hope that this extensive numerical investigation of the prolate solutions will inspire and inform new analytic investigations into these important functions.

gr-qc↗

arXiv2020

Aspects of multimode Kerr ring-down fitting

A black hole that is ringing down to quiescence emits gravitational radiation of a very specific nature that can inform us of its mass and angular momentum, test the no-hair theorem for black holes, and perhaps even give us additional information about its progenitor system. This paper provides a detailed description of, and investigation into the behavior of, multimode fitting of the ring-down signal provided by numerical simulations. We find that there are at least three well-motivated multimode fitting schemes that can be used. These methods are tested against a specific numerical simulation to allow for comparison to prior work.

gr-qc↗

arXiv2019

An unknown branch of the total-transmission modes for the Kerr-geometry

The gravitational modes of the Kerr geometry include both quasinormal modes and total-transmission modes. Sequences of these modes are parameterized by the angular momentum of the black hole. The quasinormal and total-transmission modes are usually distinct, having mode frequencies that are different at any given value of the angular momentum. But a discrete and countably infinite subset of the left-total-transmission modes are simultaneously quasinormal modes. Most of these special modes exist along previously unknown branches of the gravitational total-transmission modes. In this paper, we give detailed plots of the total-transmission modes for harmonic indices $\ell=[2,7]$, with special emphasis given to the $m=0$ modes which all contain previously unknown branches. All of these unknown branches have purely imaginary mode frequencies. We find that as we approach the Schwarzschild limit along these new branches, the mode frequencies approach $-i\infty$ in stark contrast to the finite mode frequency obtained in the Schwarzschild limit along the previously known branches. We explain when and why, at certain frequencies, the left-total-transmission modes are simultaneously quasinormal modes. At these same frequencies, the right-total-transmission modes are missing. We also derive analytic expressions for the asymptotic behavior of the total-transmission mode frequencies, and for the values of the angular momentum at which the modes are simultaneously quasinormal modes.

gr-qc↗

arXiv2016

Modes of the Kerr geometry with purely imaginary frequencies

In this paper, we examine the behavior of modes of the Kerr geometry when the mode's frequency is purely imaginary. We demonstrate that quasinormal modes must be polynomial in nature if their frequency is purely imaginary, and present a method for computing such modes. The nature of these modes, however, is not always easy to determine. Some of the polynomial modes we compute are quasinormal modes. However, some are simultaneously quasinormal modes and total transmission modes, while others fail to satisfy the requisite boundary conditions for either. This analysis is, in part, an extension of the results known for Schwarzschild black holes, but clarifies misconceptions for the behavior of modes when the black hole has angular momentum. We also show that the algebraically special modes of Kerr with m=0 have an additional branch of solutions not seen before in the literature. All of these results are in precise agreement with new numerical solutions for sequences of gravitational quasinormal modes of Kerr. However, we show that some prior numerical and analytic results concerning the existence of quasinormal modes of Kerr with purely imaginary frequencies were incorrect.

gr-qc↗

arXiv2016

Purely imaginary quasinormal modes of the Kerr geometry

We present a method for determining the purely imaginary quasinormal modes of the Kerr geometry. Such modes have previously been explored, but we show that prior results are incorrect. The method we present, based on the theory of Heun polynomials, is very general and can be applied to a broad class of problems, making it potentially useful to all branches of physics. Furthermore, our application provides an example where the method of matched asymptotic expansions seems to have failed. A deeper understanding of why it fails in this case may provide useful insights for other situations.

gr-qc↗

arXiv2014

Gravitational perturbations of the Kerr geometry: High-accuracy study

We present results from a new code for computing gravitational perturbations of the Kerr geometry. This new code carefully maintains high precision to allow us to obtain high-accuracy solutions for the gravitational quasinormal modes of the Kerr space-time. Part of this new code is an implementation of a spectral method for solving the angular Teukolsky equation that, to our knowledge, has not been used before for determining quasinormal modes. We focus our attention on two main areas. First, we explore the behavior of these quasinormal modes in the extreme limit of Kerr, where the frequency of certain modes approaches accumulation points on the real axis. We compare our results with recent analytic predictions of the behavior of these modes near the accumulation points and find good agreement. Second, we explore the behavior of solutions of modes that approach the special frequency $Mω=-2i$ in the Schwarzschild limit. Our high-accuracy methods allow us to more closely approach the Schwarzschild limit than was possible with previous numerical studies. Unlike previous work, we find excellent agreement with analytic predictions of the behavior near this special frequency. We include a detailed description of our methods, and make use of the theory of confluent Heun differential equations throughout. In particular, we make use of confluent Heun polynomials to help shed some light on the controversy of the existence, or not, of quasinormal and total-transmission modes at certain special frequencies in the Schwarzschild limit.

gr-qc↗

arXiv2008

Excision boundary conditions for the conformal metric

Shibata, Uryū and Friedman recently suggested a new decomposition of Einstein's equations that is useful for constructing initial data. In contrast to previous decompositions, the conformal metric is no longer treated as a freely-specifiable variable, but rather is determined as a solution to the field equations. The new set of freely-specifiable variables includes only time-derivatives of metric quantities, which makes this decomposition very attractive for the construction of quasiequilibrium solutions. To date, this new formalism has only been used for binary neutron stars. Applications involving black holes require new boundary conditions for the conformal metric on the domain boundaries. In this paper we demonstrate how these boundary conditions follow naturally from the conformal geometry of the boundary surfaces and the inherent gauge freedom of the conformal metric.

gr-qc↗

arXiv2008

Study of the preheating phase of chaotic inflation

Particle production and its effects on the inflaton field are investigated during the preheating phase of chaotic inflation using a model consisting of a massive scalar inflaton field coupled to N massless quantum scalar fields. The effects of spacetime curvature and interactions between the quantum fields are ignored. A large N expansion is used to obtain a coupled set of equations including a backreaction equation for the classical inflaton field. Previous studies of preheating using these equations have been done. Here the first numerical solutions to the full set of equations are obtained for various values of the coupling constant and the initial amplitude of the inflaton field. States are chosen so that initially the backreaction effects on the inflaton field are small and the mode equations for the quantum fields take the form of Mathieu equations. Potential problems relating to the parametric amplification of certain modes of the quantum fields are identified and resolved. A detailed study of the damping of the inflaton field is undertaken. Some predictions of previous studies are verified and some new results are obtained.

hep-ph↗

arXiv2008

High-accuracy comparison of numerical relativity simulations with post-Newtonian expansions

Numerical simulations of 15 orbits of an equal-mass binary black hole system are presented. Gravitational waveforms from these simulations, covering more than 30 cycles and ending about 1.5 cycles before merger, are compared with those from quasi-circular zero-spin post-Newtonian (PN) formulae. The cumulative phase uncertainty of these comparisons is about 0.05 radians, dominated by effects arising from the small residual spins of the black holes and the small residual orbital eccentricity in the simulations. Matching numerical results to PN waveforms early in the run yields excellent agreement (within 0.05 radians) over the first $\sim 15$ cycles, thus validating the numerical simulation and establishing a regime where PN theory is accurate. In the last 15 cycles to merger, however, {\em generic} time-domain Taylor approximants build up phase differences of several radians. But, apparently by coincidence, one specific post-Newtonian approximant, TaylorT4 at 3.5PN order, agrees much better with the numerical simulations, with accumulated phase differences of less than 0.05 radians over the 30-cycle waveform. Gravitational-wave amplitude comparisons are also done between numerical simulations and post-Newtonian, and the agreement depends on the post-Newtonian order of the amplitude expansion: the amplitude difference is about 6--7% for zeroth order and becomes smaller for increasing order. A newly derived 3.0PN amplitude correction improves agreement significantly ($<1%$ amplitude difference throughout most of the run, increasing to 4% near merger) over the previously known 2.5PN amplitude terms.

gr-qc↗

arXiv2007

Measuring eccentricity in binary black-hole initial data

Initial data for evolving black-hole binaries can be constructed via many techniques, and can represent a wide range of physical scenarios. However, because of the way that different schemes parameterize the physical aspects of a configuration, it is not alway clear what a given set of initial data actually represents. This is especially important for quasiequilibrium data constructed using the conformal thin-sandwich approach. Most initial-data studies have focused on identifying data sets that represent binaries in quasi-circular orbits. In this paper, we consider initial-data sets representing equal-mass black holes binaries in eccentric orbits. We will show that effective-potential techniques can be used to calibrate initial data for black-hole binaries in eccentric orbits. We will also examine several different approaches, including post-Newtonian diagnostics, for measuring the eccentricity of an orbit. Finally, we propose the use of the ``Komar-mass difference'' as a useful, invariant means of parameterizing the eccentricity of relativistic orbits.

gr-qc↗

arXiv2007

Inspiral, merger and ring-down of equal-mass black-hole binaries

We investigate the dynamics and gravitational-wave (GW) emission in the binary merger of equal-mass black holes as obtained from numerical relativity simulations. Results from the evolution of three sets of initial data are explored in detail, corresponding to different initial separations of the black holes. We find that to a good approximation the inspiral phase of the evolution is quasi-circular, followed by a "blurred, quasi-circular plunge", then merger and ring down. We present first-order comparisons between analytical models of the various stages of the merger and the numerical results. We provide comparisons between the numerical results and analytical predictions based on the adiabatic Newtonain, post-Newtonian (PN), and non-adiabatic resummed-PN models. From the ring-down portion of the GW we extract the fundamental quasi-normal mode and several of the overtones. Finally, we estimate the optimal signal-to-noise ratio for typical binaries detectable by GW experiments.

gr-qc↗

arXiv2007

Approximate Killing Vectors on S^2

We present a new method for computing the best approximation to a Killing vector on closed 2-surfaces that are topologically S^2. When solutions of Killing's equation do not exist, this method is shown to yield results superior to those produced by existing methods. In addition, this method appears to provide a new tool for studying the horizon geometry of distorted black holes.

gr-qc↗

arXiv2006

Circular orbits and spin in black-hole initial data

The construction of initial data for black-hole binaries usually involves the choice of free parameters that define the spins of the black holes and essentially the eccentricity of the orbit. Such parameters must be chosen carefully to yield initial data with the desired physical properties. In this paper, we examine these choices in detail for the quasiequilibrium method coupled to apparent-horizon/quasiequilibrium boundary conditions. First, we compare two independent criteria for choosing the orbital frequency, the "Komar-mass condition" and the "effective-potential method," and find excellent agreement. Second, we implement quasi-local measures of the spin of the individual holes, calibrate these with corotating binaries, and revisit the construction of non-spinning black hole binaries. Higher-order effects, beyond those considered in earlier work, turn out to be important. Without those, supposedly non-spinning black holes have appreciable quasi-local spin; furthermore, the Komar-mass condition and effective potential method agree only when these higher-order effects are taken into account. We compute a new sequence of quasi-circular orbits for non-spinning black-hole binaries, and determine the innermost stable circular orbit of this sequence.

gr-qc↗