PaperlayerView exact revision on arXiv

Source-linked AI summary

A recipe for echoes from exotic compact objects

Zachary Mark, Aaron Zimmerman, Song Ming Du, Yanbei Chen

arXiv:1706.06155v1gr-qc

TL;DR

The paper asks how gravitational-wave observations can distinguish black holes from highly compact ECOs whose early ringdowns may look nearly identical. It uses Green’s functions and a parametrized reflecting boundary to reprocess black-hole near-horizon radiation into ECO waveforms, finding that black-hole quasinormal modes shape the main burst and first echoes while echoes and ECO modes encode the boundary properties.

  • Problem

    Highly compact ECOs can have early ringdowns nearly identical to black holes, making standard quasinormal-mode tests insufficient for distinguishing them.

  • Method

    The paper derives a Green’s-function relation using a frequency-dependent reflecting boundary to transform black-hole waveforms and horizon radiation into corresponding ECO waveforms.

  • Results

    Black-hole quasinormal modes shape the ECO main burst and individual echoes, while the echo pulses and ECO mode frequencies encode the boundary reflectivity and location.

  • Takeaways & Limitations

    ECO waveforms can be analyzed as either echo-pulse sequences or mode superpositions, providing two connected descriptions of information about the compact object.

Abstract

from arXiv · show

Gravitational wave astronomy provides an unprecedented opportunity to test the nature of black holes and search for exotic, compact alternatives. Recent studies have shown that exotic compact objects (ECOs) can ring down in a manner similar to black holes, but can also produce a sequence of distinct pulses resembling the initial ringdown. These "echoes" would provide definite evidence for the existence of ECOs. In this work we study the generation of these echoes in a generic, parametrized model for the ECO, using Green's functions. We show how to reprocess radiation in the near-horizon region of a Schwarzschild black hole into the asymptotic radiation from the corresponding source in an ECO spacetime. Our methods allow us to understand the connection between distinct echoes and ringing at the resonant frequencies of the compact object. We find that the quasinormal mode ringing in the black hole spacetime plays a central role in determining the shape of the first few echoes. We use this observation to develop a simple template for echo waveforms. This template preforms well over a variety of ECO parameters, and with improvements may prove useful in the analysis of gravitational waves.

I. INTRODUCTION

The paper addresses how gravitational-wave observations can distinguish black holes from highly compact ECOs, whose early ringdown may resemble black-hole behavior but whose later echoes encode their reflective structure. It develops a generic reflecting-boundary and Green’s-function framework linking black-hole and ECO waveforms, including their echo and mode interpretations.

  • Motivation: Highly compact ECOs can initially produce ringdowns nearly identical to black holes, potentially fooling quasinormal-mode tests.Later echo pulses can instead reveal differences associated with the ECO’s compactness and reflective properties.
  • Motivation: The work targets a gap in prior studies by relating black-hole waveforms to ECO waveforms without restricting analysis to one specific ECO model or source orbit.The authors use test scalar fields as a proxy for gravitational perturbations.
  • Method: The Green’s-function formalism constructs the ECO waveform from the black-hole waveform plus radiation reprocessed by reflection from the near-horizon boundary.This allows wave propagation with the usual black-hole horizon condition to be compared with propagation using an ECO reflecting boundary.
  • Interpretation: ECO differences can be interpreted either as distinct echo pulses or as ringing associated with the ECO’s resonant modes.Echo timing depends on compactness, while pulse decay and shape encode reflective properties; less compact objects can instead show ringdowns consistent with ECO resonant frequencies.
  • Model: A static, spherically symmetric ECO is modeled by an exterior Schwarzschild spacetime joined to an interior at radius r0, with a frequency-dependent reflectivity at the compact-object surface.For very compact ECOs and sources in the Schwarzschild region, the interior boundary can be replaced by a reflecting boundary near the horizon.

B. Generating ECO waveforms from BH waveforms

The paper constructs ECO waveforms by modifying Schwarzschild black-hole Green’s functions with a reflecting boundary. The resulting distant signal combines ordinary black-hole emission with reprocessed horizon radiation that can appear as delayed echoes or resonant-cavity ringing.

  • B. Generating ECO waveforms from BH waveforms: The reflecting-boundary Green’s function is obtained by adding a homogeneous outgoing solution to the black-hole Green’s function and enforcing the ECO boundary condition.The construction applies to sources outside the reflecting boundary.
  • B. Generating ECO waveforms from BH waveforms: Wave propagation with the reflecting barrier equals black-hole propagation plus an additional component controlled entirely by the transfer function K̃ and the reflectivity R̃.The transfer function contains the dependence on the boundary reflectivity.
  • B. Generating ECO waveforms from BH waveforms: The distant ECO waveform is the usual black-hole emission plus K̃Z_H, allowing the asymptotic ECO signal to be computed from black-hole waveforms near infinity and the horizon.The calculation requires a chosen reflectivity R̃ and boundary location x0.
  • B. Generating ECO waveforms from BH waveforms: Expanding K̃ as a geometric series yields successive terms that reprocess boundary-impinging radiation and can produce distinct echo pulses.The first reflection acquires the boundary reflectivity, a round-trip phase delay, and black-hole transmission through the potential barrier.
  • B. Generating ECO waveforms from BH waveforms: The transfer function also has resonances, providing a complementary description in which incident waves excite resonant modes of the cavity between the boundary and potential barrier.The echo delay depends on propagation phases, while the delay between echoes is constant starting with the second echo.

III. EXAMPLES OF ECHOES

The examples construct echo waveforms by convolving black-hole response functions and show how the potential barrier shapes their delays, amplitudes, and ringing. Numerical results find low-frequency reflection, black-hole-QNM ringdown in the response functions, and echo-specific time-domain behavior.

  • III. EXAMPLES OF ECHOES: The echo waveform is decomposed into transfer functions K(n), with each term representing one echo contribution.The transfer functions are built from response functions for reflection and transmission.
  • III. EXAMPLES OF ECHOES: Frequency-domain multiplication of response functions becomes time-domain convolution, enabling numerical construction of individual echoes.The first echo and successive terms are obtained by convolving the relevant response functions and shifting them by propagation delays.
  • III. EXAMPLES OF ECHOES: Successive echo terms acquire repeated black-hole reflection factors and propagation shifts, with the echo delay becoming constant from the second echo onward.The boundary position x0 contributes the phase-related shifts shown in the time-domain construction.
  • III. EXAMPLES OF ECHOES: At low frequencies, waves incident on the ℓ=2 potential barrier are completely reflected.The condition is (Mω)^2 ≪ Vp, where Vp is the scale of the potential peak.
  • III. EXAMPLES OF ECHOES: Both black-hole reflection and transmission response functions ring down at the black-hole quasinormal-mode frequency Ω.The time-domain transmission response also contains a δ(t) singularity at t = 0 that is subtracted in the figure.

B. Frequency Independent Reflectivity

For frequency-independent reflectivity, the ECO reflectivity controls echo amplitudes while the boundary location controls most of their delay. Rescaled response functions and ISCO-plunge examples show early echoes dominated by black-hole ringing, with later echoes changing more slowly.

  • B. Frequency Independent Reflectivity: Frequency-independent reflectivity factors out of K(n), controlling each echo’s size without adding phase factors.Most of the inter-echo delay then comes from the round trip between the potential peak and boundary at x0.
  • B. Frequency Independent Reflectivity: The second and third response functions occupy progressively shifted, narrower frequency windows, while the tenth and eleventh differ very little in absolute value.As the window shifts to lower frequencies, |R_BH| approaches one and changes in the transfer functions slow.
  • B. Frequency Independent Reflectivity: The rescaled response functions become decaying sinusoids after an initial zero period, with frequencies near the fundamental QNM frequency for early echoes.For later echoes, the decay time lengthens and the oscillation frequency decreases slightly.
  • B. Frequency Independent Reflectivity: The reprocessing formalism cannot capture emission in an actual ECO spacetime after the particle passes the boundary x0.A particular ECO model could add this radiation directly, leaving only a small remaining inaccuracy from suppressed emission between x0 and the horizon.
  • B. Frequency Independent Reflectivity: The ISCO-plunge source produces (ℓ, m) = (2, 2) echoes whose early oscillations are highly suppressed and whose late-time behavior approaches decaying sinusoids.The late-time sinusoid has a complex frequency with the same qualitative behavior described for the response functions.
  • B. Frequency Independent Reflectivity: After rescaling by R~^n and shifting in time, the tenth and eleventh echoes change only slightly in duration and amplitude.The figures compare the second and third echoes with the tenth and eleventh for an ISCO-plunge source.

C. Wormhole

The reflecting-boundary formalism places wormhole echoes within the ECO framework: each wormhole echo corresponds to every second constant-reflectivity echo, producing longer delays and often more distinct pulses. The full waveform ranges from separated echo pulses to overlapping late-time behavior or a single decaying sinusoid, depending on boundary location and reflectivity.

  • C. Wormhole: Wormhole echoes correspond to the 2n-th echoes of the frequency-independent ˜R = 1 case.The mapping follows from the doubled propagation path in the wormhole spacetime.
  • C. Wormhole: When echo spacing is small relative to echo duration, the pulses overlap and the total waveform can resemble a single decaying sinusoid rather than distinct echoes.For ˜R = 1 and x0 = −3M, the waveform initially follows the BH waveform before transitioning to this sinusoid; the approximation is pushed near its limit.
  • C. Wormhole: The wormhole round-trip delay is twice that of the ˜R = 1 boundary, with shifts of ∆u = 4n|x0| versus ∆u = 2n|x0|.The longer spacing makes early wormhole echoes more distinct.
  • C. Wormhole: For x0 = −50M, roughly three to four distinct echoes appear after about |2x0|, whereas x0 = −20M yields only two before interference begins.Later echoes decay more slowly and overlap when their duration becomes comparable to the spacing.
  • C. Wormhole: At x0 = −50M, decreasing ˜R to 0.1 produces many rapidly decaying pulses, while larger reflectivities show only three to four visible echoes.The examples use an ISCO plunge waveform and vary ˜R from 0.01 to 1.

A. New Modes

The ECO response function introduces resonant modes determined by the reflecting boundary and the black-hole scattering properties. Their spacing is set by the cavity travel time, while reflectivity controls their decay and width; for a compact boundary, a single mode can dominate the echo waveform.

  • A. New Modes: The ECO modes arise from poles of the response function ˜K(ω), satisfying both the reflecting boundary condition at x0 and the outgoing condition at infinity.Peaks of |˜K| represent transfer-function resonances.
  • A. New Modes: For x0 = −3M, a single mode appears near the fundamental black-hole QNM for ˜R = 1 and the wormhole, while the ˜R = 0.5 peak is less visible.The resonance remains near the same frequency but is weaker for lower reflectivity.
  • A. New Modes: For x0 = −50M, constant reflectivity produces mode spacing 2π/(2|x0|), while wormholes produce spacing 2π/(4|x0|).These spacings correspond approximately to the light travel time from the potential peak to the boundary and back.
  • A. New Modes: The new modes decay when |˜R| < 1, and their frequency spacing is ωFSR.This agrees with the transfer-function structure discussed for the cavity-like resonances.
  • A. New Modes: The resonance width is controlled by the decay rate: ˜R = 0.5 produces broader resonances than ˜R = 1, while wormhole widths are similar to ˜R = 1.For frequency-dependent reflectivity, the spacing approximation incurs O(ωFSR/δω) errors under the stated scale-separation conditions.
  • B. Single Mode Excitation: For ˜R = 1 and x0 = −3M, the echo waveform inherits a resonance near the fundamental BH QNM but decays much more slowly.The resulting waveform appears as a single decaying sinusoid, consistent with excitation of one resonant mode.

C. Echoes from Interference of Modes

The echo waveform can be understood both as a sequence of pulses and as interference among resonant modes. The black-hole ringdown strongly shapes the first echoes, motivating a compact template that matches individual and complete echo waveforms.

  • Mode interpretation: For large |x0|, distinct time-domain echoes correspond to additional ECO resonances with spacing set by the echo delay.The pulse spacing is approximately T = 2|x0|, or 4|x0| for the wormhole case.
  • Mode interpretation: The frequency-domain echo amplitude contains the resonances of the response function because the horizon amplitude is substantial at those frequencies.This behavior appears for reflectivities ˜R = 1, 0.5, and the wormhole case.
  • Echo morphology: The first few echoes are shaped mainly by the black-hole ringdown, whose fundamental QNM frequency transmits more easily through the potential barrier than lower-frequency inspiral content.Early horizon-waveform components mostly reflect inside the barrier and contribute less to the first echo.
  • Echo morphology: Later echoes depend more intricately on the early horizon waveform after repeated scattering suppresses power near the reflectivity frequency.The low-frequency suppression makes the ringdown portion most important for the first several echoes.
  • Template: The proposed template models the horizon waveform with decaying sinusoids near the positive and negative QNM frequencies and transfers it to individual echoes and the echo sum.Its parameters are two complex amplitudes, a central start time, and a frequency width.
  • Template: Individual-echo overlaps remain approximately 0.96–0.97 and asymptote with echo number, while the first-echo example reaches ρ = 0.969.Parameters fitted to the first echo produce reasonably good overlaps for later echoes, but performance degrades for ˜R ≥ 0.99 at large x0 because narrow low-frequency resonances gain power.

C. Energy in the echoes

The formalism relates ECO echo energy to black-hole horizon and asymptotic energies. Echo energy decreases with reflectivity and, for very compact ECOs, becomes independent of boundary location.

  • Energy relation: For very compact ECOs, the formalism gives a simple relationship between the ECO waveform energy and the corresponding black-hole horizon and infinity energies.The derivation uses the echo energy together with correlations between the ECO and black-hole waveforms.
  • Energy relation: When echoes are temporally separated, their correlations can be neglected and the total echo energy approximated by the sum of individual echo energies.This applies when |x0| is much larger than the duration of each echo.
  • Reflectivity dependence: For |˜R| < 1, the echo energy is less than the black-hole horizon-waveform energy and tends to zero as ˜R → 0.For perfectly reflecting boundaries, the echo energy equals the horizon energy in the stated limit.
  • Reflectivity dependence: For perfectly reflecting, extremely compact ECOs with x0 > 20M, more than 97% of the horizon-waveform energy is radiated in the echoes.The ratio also becomes independent of x0 as x0 → ∞ for smaller reflectivities.

VI. CONCLUSIONS

The paper derives a Green’s-function mapping from black-hole to ECO waveforms with reflecting near-horizon boundaries. It interprets the added radiation as echoes or ECO modes, explains the black-hole QNM imprint, and develops a template that performs well for most tested parameters.

  • Formalism: A generic ECO exterior is modeled by replacing the black-hole inner boundary with a location and frequency-dependent reflectivity near the horizon.This parameterizes the ECO boundary through reflecting conditions in the black-hole exterior spacetime.
  • Formalism: The ECO waveform at infinity equals the black-hole waveform plus radiation produced by reflecting and reprocessing the black-hole horizon waveform with a transfer function.The additional radiation is the difference between the ECO and black-hole waveforms.
  • Interpretation: The additional waveform can be represented either as echo pulses or as modes associated with poles of the ECO Green’s function, encoding boundary reflectivity and location.These two descriptions connect time-domain echoes with resonant behavior of the compact object.
  • QNM imprint: The main ECO burst rings at black-hole QNM frequencies, while individual echo frequency content is largely determined by the horizon waveform near those frequencies.The black-hole QNM poles cancel between Green’s-function pieces in the full ECO expression, so they are not ECO Green’s-function poles.
  • Template performance: For an ISCO-plunge test charge, the echo template achieves normalized overlaps ρ > 0.95 for most boundary locations and frequency-independent reflectivities.The template is designed from numerical results and analytic observations of the echo structure.
  • Scope: Extending the formalism to Kerr perturbations and comparable-mass binaries remains future work.The Kerr extension faces additional algebraic complexity, the absence of Birkhoff’s theorem, and no simple boundary-condition parameterization.

Appendix A: Calculation of the reflection and transmission coefficients

The reflection and transmission coefficients are obtained from characteristic initial-value problems or frequency-domain boundary-value problems. The calculations extract the coefficients near the future horizon and future null infinity, with numerical checks on resolution and extraction choices.

  • Definitions: The reflection and transmission coefficients are defined from frequency-domain solutions and equivalently from time-domain characteristic initial-value problems.The appendix establishes equivalence between the two definitions.
  • Boundary matching: The near-horizon and far-field regions are treated where the potential is approximately zero, but determining the horizon and infinity fields requires all initial data.The matching uses superpositions of outward- and inward-traveling waves in these regions.
  • Frequency-domain calculation: Fourier transforming the time-domain solution and comparing with the frequency-domain definitions identifies the corresponding frequency-domain reflection and transmission coefficients.Direct frequency-domain calculations provide an independent check of the time-domain methods.
  • Time-domain calculation: In the time-domain calculation, delta-function initial data are posed near the horizon, and the transmitted and reflected fields are extracted near future null infinity and the future horizon.The transmission coefficient is evaluated along a ray near I+, while the reflection coefficient is evaluated along a ray near H+.
  • Frequency-domain calculation: At fixed frequency, the coefficients are obtained by solving the homogeneous wave equation with a boundary condition and matching the solution and derivative near the opposite boundary.The numerical integration proceeds outward from the horizon and matches an asymptotic expansion at r = 1000M.
  • Green’s functions: The Green’s-function construction specializes the scalar-field solutions to point-particle sources observed at future null infinity and the future horizon.This supplies the waveform ingredients used in the coefficient and echo calculations.

1. Green’s Function solution

The paper constructs black-hole horizon and asymptotic waveforms from retarded Green’s functions and source trajectories. These Green’s functions describe responses on the horizon and at infinity, enabling radiation calculations for the ISCO plunge.

  • Retarded Green’s functions and spherical-harmonic charge-density components are used to construct the scalar field in the black-hole spacetime.
  • Asymptotic Green’s functions separately describe the response on the horizon and at infinity.The horizon and infinity limits are taken with advanced and retarded times held fixed, respectively.
  • The source is specialized to ingoing and outgoing coordinates so horizon and asymptotic waveforms can be obtained from the particle trajectory.
  • Causality truncates the waveform integrals after the particle crosses the horizon and at the relevant retarded time.
  • The calculations focus on radiation from a test charge following the ISCO plunge orbit, whose early-time behavior approaches r = 6M and E_ISCO = 2.

2. Characteristic Initial Value Problem for the Green’s function

The Green’s function is computed as a characteristic initial-value problem in null coordinates using a distributional ansatz and finite-difference evolution. The resulting horizon and infinity responses are extracted numerically, while the ISCO waveform is windowed before echo calculations.

  • A distributional ansatz enforces the Green’s function’s causal singularity structure and reduces the smooth component to a homogeneous wave equation.
  • The homogeneous equation is solved numerically with a finite-difference characteristic code on a rectangular null-coordinate grid.
  • The code is second-order convergent, while the cell-stepping algorithm is derived with O(h^4) accuracy.
  • The asymptotic Green’s function is extrapolated to future null infinity using the field’s 1/r expansion and Richardson extrapolation.
  • The horizon Green’s function is extracted near the horizon and checked for convergence as the extraction ray approaches H+.
  • Because the exact ISCO plunge waveform has arbitrarily early oscillations, the echo calculations use a one-sided Planck-Taper window.The selected parameters retain 3 early oscillations and turn on smoothly over two oscillations.

Appendix C: Wormhole Reflectivity

For the wormhole model, the reflectivity is obtained by matching outgoing solutions across the throat and can be represented as a reflecting boundary at x0. The appendix also relates the result to black-hole scattering conventions and derives a Fourier-transform form for a Dirac comb.

  • Appendix C: Wormhole Reflectivity: A coordinate y covers the full wormhole, where scalar waves encounter a continuous but non-differentiable potential at the matching radius.
  • Appendix C: Wormhole Reflectivity: The reflectivity is defined as the ratio R̃W = C_in/C_out and is obtained by matching the outgoing solution across the throat.
  • Appendix C: Wormhole Reflectivity: The wormhole boundary condition matches waves of the form ψ ∝ e^-iωy + R̃e^iωy when R̃ equals the wormhole reflection coefficient.
  • Appendix C: Wormhole Reflectivity: The appendix relates the wormhole result to black-hole scattering coefficients through a shift in the tortoise-coordinate origin.
  • Appendix C: Wormhole Reflectivity: The wormhole can be treated as a reflecting boundary at x0.
  • The Fourier transform of the echo waveform is derived in two equivalent ways using the Dirac comb’s direct delta-function representation and Fourier-series expansion.
Loading 1706.06155v1…