Black Hole Perturbation Research Articles

Nonlinearities in Black Hole Ringdowns.

The gravitational wave strain emitted by a perturbed black hole (BH) ringing down is typically modeled analytically using first-order BH perturbation theory. In this Letter, we show that second-order effects are necessary for modeling ringdowns from BH merger simulations. Focusing on the strain's (ℓ,m)=(4,4) angular harmonic, we show the presence of a quadratic effect across a range of binary BH mass ratios that agrees with theoretical expectations. We find that the quadratic (4,4) mode's amplitude exhibits quadratic scaling with the fundamental (2,2) mode-its parent mode. The nonlinear mode's amplitude is comparable to or even larger than that of the linear (4,4) mode. Therefore, correctly modeling the ringdown of higher harmonics-improving mode mismatches by up to 2 orders of magnitude-requires the inclusion of nonlinear effects.

Constraining modifications of black hole perturbation potentials near the light ring with quasinormal modes

In modified theories of gravity, the potentials appearing in the Schr\"odinger-like equations that describe perturbations of nonrotating black holes are also modified. In this paper we ask: can these modifications be constrained with high-precision gravitational-wave measurements of the black hole's quasinormal mode frequencies? We expand the modifications in a small perturbative parameter regulating the deviation from the general-relativistic potential, and in powers of $M/r$. We compute the quasinormal modes of the modified potential up to quadratic order in the perturbative parameter. Then we use Markov-chain Monte Carlo methods to recover the coefficients in the $M/r$ expansion in an ``optimistic'' scenario where we vary them one at a time, and in a ``pessimistic'' scenario where we vary them all simultaneously. In both cases, we find that the bounds on the individual parameters are not robust. Because quasinormal mode frequencies are related to the behavior of the perturbation potential near the light ring, we propose a different strategy. Inspired by Wentzel-Kramers-Brillouin (WKB) theory, we demonstrate that the value of the potential and of its second derivative at the light ring can be robustly constrained. These constraints allow for a more direct comparison between tests based on black hole spectroscopy and observations of black hole ``shadows'' by the Event Horizon Telescope and future instruments.

Shadow, lensing, quasinormal modes, greybody bounds and neutrino propagation by dyonic ModMax black holes

Motivated by recent work on the Modified Maxwell (ModMax) black holes [Phys Lett B 10.1016/j.physletb.2020.136011], which are invariant in duality rotations and conformal transformations founded in [Phys Rev D 10.1103/PhysRevD.102.121703], we probe its effects on the shadow cast, weak field gravitational lensing, and neutrino propagation in its vicinity. Using the EHT data for the shadow diameter of Sgr. A* and M87*, and LIGO/VIRGO experiments for the dyonic ModMax black hole perturbations, we find constraints for ModMax parameters such as Q_text {m} and the screening factor gamma . We also analyze how the shadow radius behaves as perceived by a static observer and one that is comoving with the cosmic expansion. The effect of the ModMax parameters is constant for a static observer, and we found That it varies when the observer is comoving with cosmic expansion. We also analyzed its effect on the weak deflection angle by exploiting the Gauss–Bonnet theorem and its application to Einstein ring formation. We also consider the finite distance effect and massive particle deflection. Our results indicate that the far approximation of massive particle gives the largest deflection angle and amplifies the effect of Q_text {m} and gamma . Then we also calculate the quasinormal modes and greybody bounds which encode unique characteristic features of the dyonic ModMax black hole. With the advent of improving space technology, we reported that it is possible to detect the deviation caused through the shadow cast, Einstein rings, quasinormal modes, and neutrino oscillations.

Surrogate model for gravitational wave signals from nonspinning, comparable-to large-mass-ratio black hole binaries built on black hole perturbation theory waveforms calibrated to numerical relativity

We present a reduced-order surrogate model of gravitational waveforms from non-spinning binary black hole systems with comparable to large mass-ratio configurations. This surrogate model, \texttt{BHPTNRSur1dq1e4}, is trained on waveform data generated by point-particle black hole perturbation theory (ppBHPT) with mass ratios varying from 2.5 to 10,000. \texttt{BHPTNRSur1dq1e4} extends an earlier waveform model, \texttt{EMRISur1dq1e4}, by using an updated transition-to-plunge model, covering longer durations up to 30,500 $m_1$ (where $m_1$ is the mass of the primary black hole), includes several more spherical harmonic modes up to $\ell=10$, and calibrates subdominant modes to numerical relativity (NR) data. In the comparable mass-ratio regime, including mass ratios as low as $2.5$, the gravitational waveforms generated through ppBHPT agree surprisingly well with those from NR after this simple calibration step. We also compare our model to recent SXS and RIT NR simulations at mass ratios ranging from $15$ to $32$, and find the dominant quadrupolar modes agree to better than $\approx 10^{-3}$. We expect our model to be useful to study intermediate-mass-ratio binary systems in current and future gravitational-wave detectors.

Approach to computing spectral shifts for black holes beyond Kerr

Recent measurements of gravitational-wave ringdown following the merger of binary black holes raise the prospect of precision black hole spectroscopy in the near future. To perform the most sensitive tests of the nature of black holes using ringdown measurements, it is critical to compute the deviations to the spectrum of black holes in particular extensions of relativity. These spectral shifts are also needed to interpret any violations of the predictions of relativity that may be detected during ringdown. Here we present a first step towards computing the shifts to the spectrum of Kerr black holes with arbitrary spins, by deriving a modified Teukolsky equation governing the perturbations of black holes in theories beyond GR. Our approach applies to a class of theories which includes dynamical Chern-Simons gravity and shift-symmetric scalar Gauss-Bonnet gravity, in the case where the deviations from relativity are small. This allows for a perturbative approach to solving the equations of motion. Further, we show how to use the modified equation to compute the leading-order spectral shifts of Kerr black holes, using eigenvalue perturbation methods. Our formalism provides a practical approach to predicting ringdown for black holes in a range of promising extensions to relativity, enabling future precision searches for their signatures in black hole ringdown.

Gravitational waveform model based on photon motion for spinning black holes

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 numerical relativity. We find that they match with each other very well. In principle this method can be extended to some general compact objects. As an example, the ringdown waveforms from parametrized axisymmetric black holes are obtained. We also use this method to get the ringdown signals of the accelerating final black hole; this is due to gravitational recoil during the merger. Even for the extreme cases, the acceleration due to the recoil cannot produce detectable effects.

Spacetime Metrics and Ringdown Waveforms for Galactic Black Holes Surrounded by a Dark Matter Spike

Theoretical models suggest the existence of a dark matter spike surrounding the supermassive black holes at the core of galaxies. The spike density is thought to obey a power law that starts at a few times the black hole horizon radius and extends to a distance, R sp, of the order of a kiloparsec. We use the Tolman–Oppenheimer–Volkoff equations to construct the spacetime metric representing a black hole surrounded by such a dark matter spike. We consider the dark matter to be a perfect fluid, but make no other assumption about its nature. The assumed power-law density provides in principle three parameters with which to work: the power-law exponent γ sp, the external radius R sp, and the spike density at R sp. These in turn determine the total mass of the spike. We focus on Sagittarius A* and M87 for which some theoretical and observational bounds exist on the spike parameters. Using these bounds in conjunction with the metric obtained from the Tolman–Oppenheimer–Volkoff equations, we investigate the possibility of detecting the dark matter spikes surrounding these black holes via the gravitational waves emitted at the ringdown phase of black hole perturbations. Our results suggest that if the spike to black hole mass ratio is roughly constant, greater mass black holes require relatively smaller spike densities to yield potentially observable signals. We find that is unlikely for the spike in M87 to be detected via the ringdown waveform with currently available techniques unless its mass is roughly an order of magnitude larger than existing observational estimates. However, given that the signal increases with black hole mass, dark matter spikes might be observable for more massive galactic black holes in the not too distant future.

Dynamical perturbations around an extreme mass ratio inspiral near resonance

Extreme mass ratio inspirals (EMRIs) -- systems with a compact object orbiting a much more massive (e.g., galactic center) black hole -- are of interest both as a new probe of the environments of galactic nuclei, and their waveforms are a precision test of the Kerr metric. This work focuses on the effects of an external perturbation due to a third body around an EMRI system. This perturbation will affect the orbit most significantly when the inner body crosses a resonance with the outer body, and result in a change of the conserved quantities (energy, angular momentum, and Carter constant) or equivalently of the actions, which results in a subsequent phase shift of the waveform that builds up over time. We present a general method for calculating the changes in action during a resonance crossing, valid for generic orbits in the Kerr spacetime. We show that these changes are related to the gravitational waveforms emitted by the two bodies (quantified by the amplitudes of the Weyl scalar $\psi_4$ at the horizon and at $\infty$) at the frequency corresponding to the resonance. This allows us to compute changes in the action variables for each body, without directly computing the explicit metric perturbations, and therefore we can carry out the computation by calling an existing black hole perturbation theory code. We show that our calculation can probe resonant interactions in both the static and dynamical limit. We plan to use this technique for future investigations of third-body effects in EMRIs and their potential impact on waveforms for LISA.

Love symmetry

Perturbations of massless fields in the Kerr-Newman black hole background enjoy a (“Love”) SL(2, ℝ) symmetry in the suitably defined near zone approximation. We present a detailed study of this symmetry and show how the intricate behavior of black hole responses in four and higher dimensions can be understood from the SL(2, ℝ) representation theory. In particular, static perturbations of four-dimensional black holes belong to highest weight SL(2, ℝ) representations. It is this highest weight properety that forces the static Love numbers to vanish. We find that the Love symmetry is tightly connected to the enhanced isometries of extremal black holes. This relation is simplest for extremal charged spherically symmetric (Reissner-Nordström) solutions, where the Love symmetry exactly reduces to the isometry of the near horizon AdS2 throat. For rotating (Kerr-Newman) black holes one is lead to consider an infinite-dimensional SL(2, ℝ) ⋉ hat{textrm{U}}{(1)}_{mathcal{V}} extension of the Love symmetry. It contains three physically distinct subalgebras: the Love algebra, the Starobinsky near zone algebra, and the near horizon algebra that becomes the Bardeen-Horowitz isometry in the extremal limit. We also discuss other aspects of the Love symmetry, such as the geometric meaning of its generators for spin weighted fields, connection to the no-hair theorems, non-renormalization of Love numbers, its relation to (non-extremal) Kerr/CFT correspondence and prospects of its existence in modified theories of gravity.

Fixing the BMS frame of numerical relativity waveforms with BMS charges

The Bondi-van der Burg-Metzner-Sachs (BMS) group, which uniquely describes the symmetries of asymptotic infinity and therefore of the gravitational waves that propagate there, has become increasingly important for accurate modeling of waveforms. In particular, waveform models, such as post-Newtonian (PN) expressions, numerical relativity (NR), and black hole perturbation theory, produce results that are in different BMS frames. Consequently, to build a model for the waveforms produced during the merging of compact objects, which ideally would be a hybridization of PN, NR, and black hole perturbation theory, one needs a fast and robust method for fixing the BMS freedoms. In this work, we present the first means of fixing the entire BMS freedom of NR waveforms to match the frame of either PN waveforms or black hole perturbation theory. We achieve this by finding the BMS transformations that change certain charges in a prescribed way -- e.g., finding the center-of-mass transformation that maps the center-of-mass charge to a mean of zero. We find that this new method is 20 times faster, and more correct when mapping to the superrest frame, than previous methods that relied on optimization algorithms. Furthermore, in the course of developing this charge-based frame fixing method, we compute the PN expression for the Moreschi supermomentum to 3PN order without spins and 2PN order with spins. This Moreschi supermomentum is effectively equivalent to the energy flux or the null memory contribution at future null infinity $\mathscr{I}^{+}$. From this PN calculation, we also compute oscillatory ($m\not=0$ modes) and spin-dependent memory terms that have not been identified previously or have been missing from strain expressions in the post-Newtonian literature.

Generalized Regge-Wheeler equation from Effective Field Theory of black hole perturbations with a timelike scalar profile

Recently, the Effective Field Theory (EFT) of perturbations on an arbitrary background metric with a timelike scalar profile was formulated in the context of scalar-tensor theories. Here, we generalize the dictionary between the coefficients in the EFT action and those in covariant theories to accommodate shift- and reflection-symmetric quadratic higher-order scalar-tensor theories, including DHOST as well as U-DHOST. We then use the EFT action to study the dynamics of odd-parity perturbations on a static and spherically symmetric black hole background with a timelike scalar profile. Finally, we obtain the generalized Regge-Wheeler equation that can be used, e.g., to determine the spectrum of quasinormal modes and tidal Love numbers.

Effective field theory of black hole perturbations with timelike scalar profile: formulation

We formulate the Effective Field Theory (EFT) of perturbations within scalar-tensor theories on an inhomogeneous background. The EFT is constructed while keeping a background of a scalar field to be timelike, which spontaneously breaks the time diffeomorphism. We find a set of consistency relations that are imposed by the invariance of the EFT under the 3d spatial diffeomorphism. This EFT can be generically applied to any inhomogeneous background metric as long as the scalar profile is everywhere timelike. For completeness, we report a dictionary between our EFT parameters to those of Horndeski theories. Finally, we compute background equations for a class of spherically symmetric, static black hole backgrounds, including a stealth Schwarzschild-de Sitter solution.

Post-Newtonian expansion of the spin-precession invariant for eccentric-orbit nonspinning extreme-mass-ratio inspirals to 9PN and e16

We calculate the eccentricity dependence of the high-order post-Newtonian (PN) expansion of the spin-precession invariant $\psi$ for eccentric-orbit extreme-mass-ratio inspirals with a Schwarzschild primary. The series is calculated in first-order black hole perturbation theory through direct analytic expansion of solutions in the Regge-Wheeler-Zerilli formalism, using a code written in \textsc{Mathematica}. Modes with small values of $l$ are found via the Mano-Suzuki-Takasugi (MST) analytic function expansion formalism for solutions to the Regge-Wheeler equation. Large-$l$ solutions are found by applying a PN expansion ansatz to the Regge-Wheeler equation. Previous work has given $\psi$ to 9.5PN order and to order $e^2$ (i.e., the near circular orbit limit). We calculate the expansion to 9PN but to $e^{16}$ in eccentricity. It proves possible to find a few terms that have closed-form expressions, all of which are associated with logarithmic terms in the PN expansion. We also compare the numerical evaluation of our PN expansion to prior numerical calculations of $\psi$ in close orbits to assess its radius of convergence. We find that the series is not as rapidly convergent as the one for the redshift invariant at $r \simeq 10M$ but still yielding $\sim 1\%$ accuracy for eccentricities $e \lesssim 0.25$.

Black hole perturbations in higher-order scalar–tensor theories: initial value problem and dynamical stability

We propose a physically sensible formulation of initial value problem for black hole perturbations in higher-order scalar–tensor theories. As a first application, we study monopole perturbations around stealth Schwarzschild solutions in a shift- and reflection-symmetric subclass of degenerate higher-order scalar–tensor (DHOST) theories. In particular, we investigate the time evolution of the monopole perturbations by solving a two-dimensional wave equation and analyze the Vishveshwara’s classical scattering experiment, i.e., the time evolution of a Gaussian wave packet. As a result, we confirm that stealth Schwarzschild solutions in the DHOST theory are dynamically stable against the monopole perturbations with the wavelength comparable or shorter than the size of the black hole horizon. We also find that the damped oscillations at the late time do not show up unlike the ringdown phase in the standard case of general relativity. Moreover, we investigate the characteristic curves of the monopole perturbations as well as a static spherically symmetric solution with monopole hair.

High-order post-Newtonian expansion of the redshift invariant for eccentric-orbit nonspinning extreme-mass-ratio inspirals

We calculate the eccentricity dependence of the high-order post-Newtonian (PN) series for the generalized redshift invariant $⟨{u}^{t}{⟩}_{\ensuremath{\tau}}$ for eccentric-orbit extreme-mass-ratio inspirals on a Schwarzschild background. These results are calculated within first-order black hole perturbation theory using Regge-Wheeler-Zerilli (RWZ) gauge. Our Mathematica code is based on a familiar procedure, using PN expansion of the Mano-Suzuki-Takasugi analytic function formalism for $l$ modes up to a certain maximum and then using a direct general-$l$ PN expansion of the RWZ equation for arbitrarily high $l$. We calculate dual expansions in PN order and in powers of eccentricity, reaching 10PN relative order and ${e}^{20}$. Detailed knowledge of the eccentricity expansion at each PN order allows us to find within the eccentricity dependence numerous closed-form expressions and multiple infinite series with known coefficients. We find leading logarithm sequences in the PN expansion of the redshift invariant that reflect a similar behavior in the PN expansion of the energy flux to infinity. A set of flux terms and special functions that appear in the energy flux, like the Peters-Mathews flux itself, are shown to reappear in the redshift PN expansion.

Quasinormal modes of NUT-charged black branes in the AdS/CFT correspondence

We study the scalar, electromagnetic and gravitational perturbations of planar AdS4 black holes with NUT charge. In the context of the AdS/CFT correspondence, these solutions describe a thermal quantum field theory embedded in a Gödel-type Universe with closed time-like curves. For a given temperature and NUT charge, two different planar Taub–NUT solutions exist, but we show that only the one with a positive specific heat contributes to the Euclidean saddle point in the path integral. By using the Newman–Penrose formalism, we then derive the master equations satisfied by scalar, electromagnetic and gravitational perturbations in this background, and show that the corresponding equations are separable. Interestingly, the solutions pile up in the form of Landau levels, and hence are characterized by a single quantum number q. We determine the appropriate boundary conditions satisfied by the master variables and using these we compute the quasinormal modes of scalar and gravitational perturbations. On the other hand, electromagnetic perturbations depend on a free parameter whose determination is problematic. We find that all the scalar and gravitational QNM frequencies lie in the lower half of the complex plane, indicating that these Taub–NUT spacetimes are stable. We discuss the implications of these results in the light of the AdS/CFT correspondence.

Six-dimensional non-extremal Reissner-Nordstrom black hole, charged massive scalar perturbation and black hole bomb

The superradiant stability of higher dimensional non-extremal Reissner-Nordstrom black hole under charged massive scalar perturbation is analytically studied. We extend our previous studies of four- and five-dimensional non-extremal Reissner-Nordstrom black hole cases to six-dimensional case. By analyzing the derivative of the effective potential with an analytical method, we find that no potential well exists outside the outer horizon of the black hole for the superradiant scalar modes. This means that there is no black hole bomb for the system consisting of six-dimensional Reissner-Nordstrom black hole and charged massive scalar perturbation and the system is superradiantly stable.

Detailed properties of gravitational-wave mergers from flyby perturbations of wide binary black holes in the field

ABSTRACT Wide black hole binaries (wide-BBHs; ≥103 au) in the field can be perturbed by random stellar flybys that excite their eccentricities. Once a wide binary is driven to a sufficiently small pericentre approach, gravitational wave (GW) emission becomes significant, and the binary inspirals and merges. In our previous study, using simplified models for wide-BBHs, we found that successive flybys lead to significant merger fractions of wide-BBHs in less than Hubble time, making the flyby perturbation mechanism a relevant contributor to the production rate of GW-sources. However, the exact rates and detailed properties of the resulting GW sources depend on the wide binary progenitors. In this paper, we use detailed population synthesis models for the initial wide-BBH population, considering several populations corresponding to different natal-kick models and metallicities, and then follow the wide-BBHs evolution due to flyby perturbations and GW-emission. We show that the cumulative effect of flybys is conducive for the production of GW sources in non-negligible rates of 1−20 Gpc−3 yr−1, which are sensitive to the natal kicks model. Such rates are relevant to the observationally inferred rate. Our models, now derived from detailed population of binaries, provide detailed properties of the produced GW-sources, including mass-functions and delay times. The observational signatures from this channel include circular orbits in aLIGO band; preference for high-velocity dispersion host galaxies (in particular ellipticals); relatively uniform delay-time distribution; and, likely, mild – i.e. less than isolated evolution channels and more than dynamical channels – spin–spin and spin–orbit alignments.

Quasinormal modes of slowly-rotating black holes in dynamical Chern-Simons gravity

The detection of gravitational waves from compact binary mergers by the LIGO/Virgo Collaboration has, for the first time, allowed us to test relativistic gravity in its strong, dynamical, and nonlinear regime, thus opening a new arena to confront general relativity (and modifications thereof) against observations. We consider a theory which modifies general relativity by introducing a scalar field coupled to a parity-violating curvature term known as dynamical Chern-Simons gravity. In this theory, spinning black holes are different from their general relativistic counterparts and can thus serve as probes to this theory. We study linear gravito-scalar perturbations of black holes in dynamical Chern-Simons gravity at leading order in spin and (i) obtain the perturbed field equations describing the evolution of the perturbed gravitational and scalar fields, (ii) numerically solve these equations by direct integration to calculate the quasinormal mode frequencies for the dominant and higher multipoles and tabulate them, (iii) find strong evidence that these rotating black holes are linearly stable, and (iv) present general fitting functions for different multipoles for gravitational and scalar quasinormal mode frequencies in terms of spin and Chern-Simons coupling parameter. Our results can be used to validate the ringdown of small-spin remnants of numerical relativity simulations of black hole binaries in dynamical Chern-Simons gravity and pave the way towards future tests of this theory with gravitational wave ringdown observations.

Elementary integral series for Heun functions: Application to black-hole perturbation theory

Heun differential equations are the most general second order Fuchsian equations with four regular singularities. An explicit integral series representation of Heun functions involving only elementary integrands has hitherto been unknown and noted as an important open problem in a recent review. We provide such representations of the solutions of all equations of the Heun class: general, confluent, bi-confluent, doubly confluent, and triconfluent. All the series are illustrated with concrete examples of use, and Python implementations are available for download. We demonstrate the utility of the integral series by providing the first representation of the solution to the Teukolsky radial equation governing the metric perturbations of rotating black holes that is convergent everywhere from the black hole horizon up to spatial infinity.