arXiv subjects

José Luis Jaramillo

Publications and source records attributed to José Luis Jaramillo.

At least 19 recordsLinked to original sources

arXiv2026

Universality in the Transition from Inspiral to Plunge for Extreme-Mass-Ratio-Inspirals: High-Accuracy Analytic Solutions and Catastrophe Theory

We revisit the transition from inspiral to plunge for extreme mass-ratio inspirals on quasi-circular, inclined orbits in Kerr spacetime from the perspective of catastrophe theory. Our goal is to uncover the mathematical structures underlying the universality of the transition dynamics, which remains governed by the same Painlevé I differential equation as for equatorial inspirals despite the additional complexity. We first analyze the solution of the Painlevé I equation selected by the physical boundary conditions of slowly evolving quasi-circular inspiral at early times. We argue that these conditions uniquely select the tritronquée solution of Painlevé I. We then compare existing high-accuracy analytic approximations of the tritronquée solution with direct numerical integrations of the Painlevé I equation, finding comparable accuracy and improved stability under differentiation and integration for the analytic solution. In the second part of this work, we show that the equilibrium structure of the Kerr radial effective potential admits a natural interpretation in terms of catastrophe theory. Equatorial orbits are associated with the fold catastrophe, while inclined orbits are described by the cusp catastrophe. In both cases, the transition to plunge corresponds to slow evolution across fold lines of the catastrophe manifold, providing a geometric explanation for the universal appearance of the Painlevé I equation in the transition dynamics.

gr-qc↗

arXiv2026

Vishveshwara's waveform revisited: insights from a Keldysh quasinormal mode expansion

We revisit Vishveshwara's linear scattering on a Schwarzschild black hole to study the evolution of quasinormal mode activation along the full time-domain waveform. Specifically, by adopting a hyperboloidal approach we cast the scattering problem in a non-selfadjoint dynamics setting where a (Keldysh) resonant expansion in a bi-orthogonal system of quasinormal modes can be readily performed. The calculation reveals the neat correlation of the second peak in Vishveshwara's waveform to the fundamental mode and first overtones, closely following the ringdown waveform pattern of non-linear evolutions. The first peak is completely controlled, for even (Zerilli) perturbations, by the algebraically special mode and the nearby branch cut. This phenomenon is absent for odd (Regge-Wheeler) perturbations. These qualitative features are robust under change of initial data and, we argue, might provide insight into the mechanisms underlying the ringdown activation time in non-linear evolutions.

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arXiv2026

A geometric approach to QNMs in optics: application to pseudospectrum and structural stability

We develop a geometric--spectral framework for the computation and stability analysis of quasi-normal modes (QNMs) in open optical cavities of compact support. The hyperboloidal approach, transferred from gravitational physics to electromagnetism [1], incorporates outgoing boundary conditions directly into the formulation of the optical resonance problem. Dispersive and absorbing media are described by a Lorentz-model permittivity through an auxiliary-field formulation, and the resulting non-Hermitian spectral problem is solved using Chebyshev spectral discretization.Pseudospectrum analysis is then extended to the optical setting following [2] and used to study the stability of optical resonances under perturbations. The pseudospectrum provides information on the sensitivity of the resonances that is not contained in the QNM spectrum alone. Particular attention is given to the role of the norm used to define the pseudospectrum, by comparing the stability properties obtained with different choices of scalar product. The results show that the assessment of spectral stability is closely related to the functional setting in which perturbations are measured. Combining the hyperboloidal formulation with pseudospectrum analysis therefore makes it possible to compute optical QNMs while also studying their spectral stability. The approach extends techniques developed for resonance problems in gravitational physics to dispersive electromagnetism and provides a basis for studying stability and sensitivity in open optical systems.

physics.optics↗

arXiv2025

A Weyl's law for black holes

We discuss a Weyl's law for the quasi-normal modes of black holes that recovers the structural features of the standard Weyl's law for the eigenvalues of Laplacian-like operators in compact regions. Specifically, we propose that the asymptotics of the counting function $N(ω)$ of quasi-normal modes of $(d+1)$-dimensional black holes follows a power-law $N(ω)\sim \mathrm{Vol}_d^{\mathrm{eff}}ω^d$, with $\mathrm{Vol}_d^{\mathrm{eff}}$ an effective $d$-volume determined by the light-trapping properties of the black hole geometry. Concretely, the factorisation $\mathrm{Vol}_d^{\mathrm{eff}} \sim \left(8π/κ\right) \cdot \mathrm{Vol}^{\mathrm{trapped}}_{d-1}$ makes apparent the two underlying structural ingredients, namely the (local) redshift effect controlled by the surface gravity $κ$ and the volume $\mathrm{Vol}^{\mathrm{trapped}}_{d-1}$ of the (phase space) trapped set. In particular, this proposal extends the Weyl's law proved by Dyatlov & Zworski for the counting of slowest decaying quasi-normal modes, to include overtones. As an application, these Weyl's laws could provide a probe into the effective spacetime dimensionality, upon the counting of sufficiently many quasi-normal modes in the ringdown signal of binary black hole mergers.

gr-qc↗

arXiv2025

Asymptotics and Universality in Black Holes: from the quasinormal Weyl's law to the binary merger waveform

Current state-of-the-art approaches to black hole (BH) dynamics, encompassing several effective approximation schemes, offer a remarkable control of the quantitative aspects of strong gravity. They also provide key insights into some qualitative aspects of the problem. In spite of this, there remain blind spots that hinder the understanding of the mechanisms underlying some observed phenomena, in particular concerning simplicity and universality in BH spacetimes. Adopting an 'asymptotic reasoning' approach, by filtering non-essential degrees of freedom, can potentially unveil universality patterns by identifying key underlying structural stability mechanisms. We first illustrate such an asymptotic approach by focusing on a BH quasinormal (QNM) Weyl's law, that accounts for the universal asymptotics of the QNM "counting function". This permits to identify light-trapping and the (local) redshift effect as the underlying mechanisms, also offering a bridge to the universal patterns found in BH QNM spectral instability. As a by-product, Weyl's law universality formally opens an observational access to spacetime (effective) dimensionality. More heuristically, we sketch a program recently put forward to apply such 'asymptotic reasoning' to address the observed simplicity and universality patterns in binary BH merger dynamics. This program is built as a hierarchy of asymptotic models, potentially making contact with integrability theory in gravity, namely through the background sector in a "wave-mean flow" approach to BH binary dynamics.

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arXiv2025

Quasi-normal mode expansions of black hole perturbations: a hyperboloidal Keldysh's approach

We study quasinormal mode expansions by adopting a Keldysh scheme for the spectral construction of asymptotic resonant expansions. Quasinormal modes are first cast in terms of a non-selfadjoint problem by adopting, in a black hole perturbation setting, a spacetime hyperboloidal approach. Then the Keldysh expansion of the resolvent, built on bi-orthogonal systems, provides a spectral version of Lax-Phillips expansions on scattering resonances. We clarify the role of scalar product structures in the Keldysh setting, that prove non-necessary to construct the resonant expansions (in particular the quasinormal mode time-series at null infinity), but are required to define the (constant) excitation coefficients in the bulk resonant expansion. We demonstrate the efficiency and accuracy of the Keldysh spectral approach to (non-selfadjoint) dynamics, even beyond its limits of validity, in particular recovering Schwarzschild black hole late power-law tails. We also study early dynamics by exploring i) the existence of an earliest time of validity of the resonant expansion and ii) the interplay between overtones extracted with the Keldysh scheme and regularity. Specifically, we address convergence aspects of the series and, on the other hand, we implement non-modal analysis tools, namely assessing $H^p$-Sobolev dynamical transient growths and constructing $H^p$-pseudospectra. Finally, we apply the Keldysh scheme to calculate ''second-order'' quasinormal modes and complement the qualitative study of overtone distribution by presenting the Weyl law for the counting of quasinormal modes in black holes with different (flat, De Sitter, anti-De Sitter) spacetime asymptotics.

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arXiv2024

Integrability in Perturbed Black Holes: Background Hidden Structures

In this work we investigate the presence of integrable hidden structures in the dynamics of perturbed non-rotating black holes (BHs). This can also be considered as a first step in a wider program of an effective identification of ``slow'' and ``fast'' degrees of freedom (DoFs) in the (binary) BH dynamics, following a wave-mean flow perspective. The slow DoFs would be associated with a nonlinear integrable dynamics, on which the fast ones propagate following an effective linear dynamics. BH perturbation theory offers a natural ground to test these properties. Indeed, the decoupling of Einstein equations into wave master equations with a potential provides an instance of such splitting into (frozen) slow DoFs (background potential) over which the linear dynamics of the fast ones (perturbation master functions) evolve. It has been recently shown that these wave equations possess an infinite number of symmetries that correspond to the flow of the infinite hierarchy of Korteweg-de Vries (KdV) equations. Starting from these results, we systematically investigate the presence of integrable structures in BH perturbation theory. We first study them in Cauchy slices and then extend the analysis to hyperboloidal foliations. This second step introduces a splitting of the master equation into bulk and boundary contributions, unveiling an underlying structural relation with the slow and fast DoFs. This insight represents a first step to establish the integrable structures associated to the slow DoFs as bulk symmetries of the dynamics of perturbed BHs.

gr-qc↗

arXiv2024

Structural aspects of the anti-de Sitter black hole pseudospectrum

Black holes in anti-de Sitter spacetime provide an important testing ground for both gravitational and field-theoretic phenomena. In particular, the study of perturbations can be useful to further our understanding regarding certain physical processes, such as superradiance, or the dynamics of strongly coupled conformal field theories through the holographic principle. In this work we continue our systematic study of the ultraviolet instabilities of black-hole quasinormal modes, built on the characterization of the latter as eigenvalues of a (spectrally unstable) non-selfadjoint operator and using the pseudospectrum as a main analysis tool, extending our previous studies in the asymptotically flat setting to Anti-de Sitter asymptotics. Very importantly, this step provides a singularly well-suited probe into some of the key structural aspects of the pseudospectrum. This is a consequence of the specific features of the Schwarzschild-anti-de Sitter geometry, together with the existence of a sound characterization by Warnick of quasinormal modes as eigenvalues, that is still absent in asymptotic flatness. This work focuses on such structural aspects, with an emphasis on the convergence issues of the pseudospectrum and, in particular, the comparison between the hyperboloidal and null slicing cases. As a physical by-product of this structural analysis we assess, in particular, the spectral stability of purely imaginary ``hydrodynamic" modes, which appear for axial gravitational perturbations, that become dominant when the black-hole horizon is larger than the anti-de Sitter radius. We find that their spectral stability, under perturbations, depends on how close they are to the real axis, or conversely how distant they are from the first oscillatory overtone.

gr-qc↗

arXiv2023

Universality in Binary Black Hole Dynamics: An Integrability Conjecture

The waveform of a binary black hole coalescence appears to be both simple and universal. In this essay we argue that the dynamics should admit a separation into 'fast and slow' degrees of freedom, such that the latter are described by an integrable system of equations, accounting for the simplicity and universality of the waveform. Given that Painlevé transcendents are a smoking gun of integrable structures, we propose the Painlevé-II transcendent as the key structural element threading a hierarchy of asymptotic models aiming at capturing different (effective) layers in the dynamics. Ward's conjecture relating integrable and (anti)self-dual solutions can provide the avenue to encode background binary black hole data in (non-local) twistor structures.

gr-qc↗

arXiv2023

Pseudospectrum of horizonless compact objects: a bootstrap instability mechanism

Recent investigations of the pseudospectrum in black hole spacetimes have shown that quasinormal mode frequencies suffer from spectral instabilities. This phenomenon may severely affect gravitational-wave spectroscopy and limit precision tests of general relativity. We extend the pseudospectrum analysis to horizonless exotic compact objects which possess a reflective surface arbitrarily close to the Schwarzschild radius, and find that their quasinormal modes also suffer from an overall spectral instability. Even though all the modes themselves decay monotonically, the pseudospectrum contours of equal resonance magnitude around the fundamental mode and the lowest overtones can cross the real axis into the unstable regime of the complex plane, unveiling the existence of nonmodal pseudo-resonances. A pseudospectrum analysis further predicts that fluctuations to the system may destabilize the object when next to leading-order effects are considered, as the triggering of pseudo-resonant growth can break the order-expansion of black-hole perturbation theory.

gr-qc↗

arXiv2022

Painlevé-II approach to binary black hole merger dynamics: universality from integrability

The binary black hole merger waveform is both simple and universal. Adopting an effective asymptotic description of the dynamics, we aim at accounting for such universality in terms of underlying (effective) integrable structures. More specifically, under a ``wave-mean flow'' perspective, we propose that fast degrees of freedom corresponding to the observed waveform would be subject to effective linear dynamics, propagating on a slowly evolving background subject to (effective) non-linear integrable dynamics. The Painlevé property of the latter would be implemented in terms of the so-called Painlevé-II transcendent, providing a structural link between i) orbital (in particular, EMRI) dynamics in the inspiral phase, ii) self-similar solutions of non-linear dispersive Korteweg-de Vries-like equations (namely, the `modified Korteweg-de Vries' equation) through the merger and iii) the matching with the isospectral features of black hole quasi-normal modes in late ringdown dynamics. Moreover, the Painlevé-II equation provides also a `non-linear turning point' problem, extending the linear discussion in the recently introduced Airy approach to binary black hole merger waveforms. Under the proposed integrability perspective, the simplicity and universality of the binary black hole merger waveform would be accounted to by the `hidden symmetries' of the underlying integrable (effective) dynamics. In the spirit of asymptotic reasoning, and considering Ward's conjecture linking integrability and self-dual Yang-Mills structures, it is tantalizing to question if such universal patterns would reflect the actual full integrability of a (self-dual) sector of general relativity, ultimately responsible for the binary black hole waveform patterns.

gr-qc↗

arXiv2022

Pseudospectrum and binary black hole merger transients

The merger phase of binary black hole coalescences is a transient between an initial oscillating regime (inspiral) and a late exponentially damped phase (ringdown). In spite of the non-linear character of Einstein equations, the merger dynamics presents a surprisingly simple behaviour consistent with effective linearity. On the other hand, energy loss through the event horizon and by scattering to infinity renders the system non-conservative. Hence, the infinitesimal generator of the (effective) linear dynamics is a non-selfadjoint operator. Qualitative features of transients in linear dynamics driven by non-selfadjoint (in general, non-normal) operators are captured by the pseudospectrum of the time generator. We propose the pseudospectrum as a unifying framework to thread together the phases of binary black hole coalescences, from the inspiral-merger transition up to the late quasinormal mode ringdown.

gr-qc↗

arXiv2022

Airy-function approach to binary black hole merger waveforms: The fold-caustic diffraction model

From numerical simulations of the Einstein equations, and also from gravitational wave observations, the gravitational wave signal from a binary black hole merger is seen to be simple and to possess certain universal features. The simplicity is somewhat surprising given that non-linearities of general relativity are thought to play an important role at the merger. The universal features include an increasing amplitude as we approach the merger, where transition from an oscillatory to a damped regime occurs in a pattern apparently oblivious to the initial conditions. We propose an Airy-function pattern to model the binary black hole (BBH) merger waveform, focusing on accounting for its simplicity and universality. We postulate that the relevant universal features are controlled by a physical mechanism involving: i) a caustic phenomenon in a basic `geometric optics' approximation and, ii) a diffraction over the caustic regularizing its divergence. Universality of caustics and their diffraction patterns account for the observed universal features, as in optical phenomena such as rainbows. This postulate, if true, allows us to borrow mathematical techniques from Singularity (Catastrophe) Theory, in particular Arnol'd-Thom's theorem, and to understand binary mergers in terms of fold caustics. The diffraction pattern corresponding to the fold-caustic is given in terms of the Airy function, which leads (under a `uniform approximation') to the waveform model written in terms of a parameterized Airy function. The post-merger phase does not share the same features of simplicity and universality, and must be added separately. Nevertheless, our proposal allows a smooth matching of the inspiral and post-merger signals by using the known asymptotics of the Airy function.

gr-qc↗

arXiv2021

Pseudospectrum of Reissner-Nordström black holes: quasinormal mode instability and universality

Black hole spectroscopy is a powerful tool to probe the Kerr nature of astrophysical compact objects and their environment. The observation of multiple ringdown modes in gravitational waveforms could soon lead to high-precision gravitational spectroscopy, so it is critical to understand if the quasinormal mode spectrum is stable against perturbations. It was recently shown that the pseudospectrum can shed light on the spectral stability of black hole quasinormal modes. We study the pseudospectrum of Reissner-Nordström spacetimes and we find a spectral instability of scalar and gravitoelectric quasinormal modes in subextremal and extremal black holes, extending similar findings for the Schwarzschild spacetime. The asymptotic structure of pseudospectral contour levels is the same for scalar and gravitoelectric perturbations. By making different gauge choices in the hyperboloidal slicing of the spacetime, we find that the broad features of the pseudospectra are remarkably gauge-independent. The gravitational-led and electromagnetic-led quasinormal modes of extremal Reissner-Nordström black holes exhibit "strong" isospectrality: not only their spectrum coincides, but the whole pseudospectrum is the same for both classes of perturbations. We observe that a conformal duality between the extremal horizon and spacetime boundaries at infinity is responsible for such "strong" isospectrality property.

gr-qc↗

arXiv2021

Energy scales and black hole pseudospectra: the structural role of the scalar product

A pseudospectrum analysis has recently provided evidence of a potential generic instability of black hole (BH) quasinormal mode (QNM) overtones under high-frequency perturbations. Such instability analysis depends on the assessment of the size of perturbations. This is encoded in the scalar product and its choice is not unique. We address the impact of the scalar product choice, founding it on the physical energy scales of the problem. The article is organized in three parts: basics, applications and heuristic proposals. In the first part, we revisit the energy scalar product used in the hyperboloidal approach to QNMs, extending previous effective analyses and placing them on solid spacetime basis. The second part focuses on applications of the scalar product in the QNM problem: i) we demonstrate that the QNM instability is not an artifact of previous spectral numerical schemes, by implementing a finite elements calculation from a weak formulation; ii) using Keldysh's theorem, we provide QNM resonant expansions for the gravitational waveform, with explicit expressions of the expansion coefficients; iii) we propose the notion of "epsilon-dual QNM expansions" to exploit BH QNM instability in BH spectroscopy, exploiting both non-perturbed and perturbed QNMs, the former informing on large scales and the latter probing small scales. The third part enlarges the conceptual scope of BH QNM instability proposing: a) spiked perturbations are more efficient in triggering BH QNM instabilities than smooth ones, b) a general picture of the BH QNM instability problem is given, supporting the conjecture (built on Burnett's conjecture) that Nollert-Price branches converge universally to logarithmic Regge branches in the high-frequency limit and c) aiming at a fully geometric description of QNMs, BMS states are hinted as possible asymptotic/boundary degrees of freedom for an inverse scattering problem.

gr-qc↗

arXiv2021

Gravitational wave signatures of black hole quasi-normal mode instability

Black hole (BH) spectroscopy has emerged as a powerful approach to extract spacetime information from gravitational wave (GW) observed signals. Yet, quasinormal mode (QNM) spectral instability under high wave-number perturbations has been recently shown to be a common classical general relativistic phenomenon [1]. This requires to assess 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 two-fold 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 lying on a BH Weyl law) to characterise 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 discuss 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.

gr-qc↗

arXiv2021

Pseudospectrum and black hole quasi-normal mode (in)stability

We study the stability of quasinormal modes (QNM) in asymptotically flat black hole spacetimes by means of a pseudospectrum analysis. The construction of the Schwarzschild QNM pseudospectrum reveals the following: (i) the stability of the slowest-decaying QNM under perturbations respecting the asymptotic structure, reassessing the instability of the fundamental QNM discussed by Nollert [H. P. Nollert, About the Significance of Quasinormal Modes of Black Holes, Phys. Rev. D 53, 4397 (1996)] as an "infrared" effect; (ii) the instability of all overtones under small-scale ("ultraviolet") perturbations of sufficiently high frequency, which migrate towards universal QNM branches along pseudospectra boundaries, shedding light on Nollert's pioneer work and Nollert and Price's analysis [H. P. Nollert and R. H. Price, Quantifying Excitations of Quasinormal Mode Systems, J. Math. Phys. (N.Y.) 40, 980 (1999)]. Methodologically, a compactified hyperboloidal approach to QNMs is adopted to cast QNMs in terms of the spectral problem of a non-self-adjoint operator. In this setting, spectral (in)stability is naturally addressed through the pseudospectrum notion that we construct numerically via Chebyshev spectral methods and foster in gravitational physics. After illustrating the approach with the Pöschl-Teller potential, we address the Schwarzschild black hole case, where QNM (in)stabilities are physically relevant in the context of black hole spectroscopy in gravitational-wave physics and, conceivably, as probes into fundamental high-frequency spacetime fluctuations at the Planck scale.

gr-qc↗

arXiv2018

Hyperboloidal slicing approach to quasi-normal mode expansions: the Reissner-Nordström case

We study quasi-normal modes of black holes, with a focus on resonant (or quasi-normal mode) expansions, in a geometric frame based on the use of conformal compactifications together with hyperboloidal foliations of spacetime. Specifically, this work extends the previous study of Schwarzschild in this geometric approach to spherically symmetric asymptotically flat black hole spacetimes, in particular Reissner-Nordström. The discussion involves, first, the non-trivial technical developments needed to address the choice of appropriate hyperboloidal slices in the extended setting as well as the generalization of the algorithm determining the coefficients in the expansion of the solution in terms of the quasi-normal modes. In a second stage, we discuss how the adopted framework provides a geometric insight into the origin of regularization factors needed in Leaver's Cauchy based foliations, as well as into the discussion of quasi-normal modes in the extremal black hole limit.

gr-qc↗