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Catalogue of gravitational-wave echo waveforms
A repository of templates for GW echoes from exotic compact objects
(page under development)

All material is free for use, please make reference to this webpage and to the relevant papers.

If you are interested to post other GW echo waveforms on this page please contact us.

Mathematica notebooks:
  • echo_template_spinning.nb (analytical template in the frequency domain using low-frequency approximation of Teukolsky's equation) [9]
  • echo_template_nonspinning.nb (analytical template in the frequency domain, using Poschl-Teller approximation) [1]
Picture
References:
  1. Adriano Testa & Paolo Pani, An analytical template for echoes, arXiv:1806.04253, Phys. Rev. D 98, 044018 (2018)
  2. Adriano Testa, Master thesis, Sapienza University of Rome (2018)
  3. Vitor Cardoso & Paolo Pani, The observational evidence for horizons: from echoes to precision gravitational-wave physics  - arXiv:1707.03021, Nat.Astron. 1 (2017) no.9, 586-591​
  4. Vitor Cardoso, Seth Hopper, Caio F. B. Macedo, Carlos Palenzuela, Paolo Pani, Echoes of ECOs: gravitational-wave signatures of exotic compact objects and of quantum corrections at the horizon scale - Phys. Rev. D 94, 084031 (2016)
  5. Vitor Cardoso, Edgardo Franzin, Paolo Pani, Is the gravitational-wave ringdown a probe of the event horizon? - Phys. Rev. Lett. 116, 171101 (2016)
  6. Zachary Mark, Aaron Zimmerman, Song Ming Du, and Yanbei Chen, A recipe for echoes from exotic compact objects, Phys. Rev. D 96, 084002 (2017)
  7. Gaurav Khanna, http://gravity.umassd.wikispaces.net/Echoes 
  8. Raposo, Pani,  Bezares, Palenzuela, Cardoso, "Anisotropic stars as ultracompact objects in General Relativity" Phys. Rev. D 99, 104072 (2019)
  9. Maggio, Testa, Bhagwat, Pani, "Analytical model for gravitational-wave echoes from spinning remnants" (arXiv:1907.03091, 2019)
​
​Reviews: 
  • Vitor Cardoso & Paolo Pani, Testing the nature of dark compact objects: a status report, (arXiv:1904.05363, Liv. Rev. Rel, in press)
  • Vitor Cardoso & Paolo Pani, The observational evidence for horizons: from echoes to precision gravitational-wave physics  (arXiv:1707.03021, Nat.Astron. 1 (2017) no.9, 586-591)

Datafiles 

​In the tables below, M is the total mass of the object (G=c=1 units adopted).
echoes from spinning remnants [NEW!]
Ringdown is always fundamental l=m=2, linearly polarized
Echoes might have both polarizations (format of datafiles: t/M, h_+, h_x)
Prompt signal
Reflection coefficient (R)
Compactness parameter
​(delta/M)
Spin
​(chi=a/M)
​​Data
Reference
Ringdown
(grav., l=m=2)
1
1e-7
0.0
echo.dat
[9]
Ringdown
(grav., l=m=2)
1
1e-7
0.7
echo.dat
[9]
Ringdown
(grav., l=m=2)
1
1e-7
0.9
echo.dat
[9]
Ringdown
(grav., l=m=2)
0.5
1e-7
0
echo.dat
[9]
Ringdown
(grav., l=m=2)
0.5
1e-7
0.7
echo.dat
[9]
Ringdown
(grav., l=m=2)
0.5
1e-7
0.9
echo.dat
[9]
Ringdown
(grav., l=m=2)
-1
1e-7
0
echo.dat
[9]
Ringdown
(grav., l=m=2)
-1
1e-7
0.7
echo.dat
[9]
Ringdown
(grav., l=m=2)
-1
1e-7
0.9
echo.dat
​[9]
Ringdown
(grav., l=m=2)
Exp[i*pi/3]
1e-7
​0
echo.dat
​[9]
Ringdown
(grav., l=m=2)
Exp[i*pi/3]
1e-7
0.7
echo.dat 
[9]
Ringdown
(grav., l=m=2)
Exp[i*pi/3]
​1e-7
0.9
echo.dat
[9]
ECHOES FROM NON-SPINNING REMNANTS
Source localized near the surface (nonspinning remnant):
Prompt signal
Reflection coefficient (R)
Width of the cavity
​(d/M)
Data/Audio
​Reference
Ringdown
​(grav. polar, l=2)
1
100
echo.dat / echo.wav
[1,2]
​Ringdown
(grav. polar, l=2)
1
75
echo.dat / echo.wav
[1,2]
Ringdown
(grav. polar, l=2)
1
50
echo.dat / echo.wav
[1,2]
Ringdown
(grav. polar, l=2)
0.75
100
echo.dat / echo.wav
​[1,2]
Ringdown
(grav. polar, l=2)
0.75
75
echo.dat / echo.wav
[1,2]
​Ringdown
(grav. polar, l=2)
0.75
50
echo.dat / echo.wav
[1,2]
Ringdown
(grav. polar, l=2)
0.5
100
echo.dat / echo.wav
[1,2]
​Ringdown
(grav. polar, l=2)
0.5
75
echo.dat / echo.wav
[1,2]
Ringdown
(grav. polar, l=2)
0.5
50
echo.dat / echo.wav
​[1,2]
Ringdown
(scalar, l=2)
1
100
echo.dat / echo.wav
[1,2]
Ringdown
(scalar, l=2)
1
75
echo.dat / echo.wav
[1,2]
Ringdown
(scalar, l=2)
1
50
echo.dat / echo.wav
[1,2]
​Ringdown
(scalar, l=2)
0.75
100
echo.dat / echo.wav
[1,2]
Ringdown
(scalar, l=2)
0.75
75
echo.dat / echo.wav
[1,2]
​Ringdown
(scalar, l=2)
0.75
50
echo.dat / echo.wav
[1,2]
Ringdown
(scalar, l=2)
0.5
100
echo.dat / echo.wav
[1,2]
Ringdown
(scalar, l=2)
0.5
75
echo.dat / echo.wav
[1,2]
​Ringdown
(scalar, l=2)
0.5
50
echo.dat / echo.wav
​[1,2]

​Point-particle source (nonspinning remnant):

(gamma is the Lorentz factor of the particle at infinity)
Prompt signal
Reflection coefficient (R)
Width of the cavity
​(d/M)
Audio
​Reference
​Radial infall
gamma=1.1

R_BH Exp[-2 i w x0]
~13.2
echo.dat / echo.wav
[1]
Inspiral+plunge
1 (Neumann)
100
echo.dat
[7]
Inspiral+plunge
-1 (Dirichlet)
100
echo.dat
[7]

​Other sources 
(nonspinning remnant): ​
Prompt signal
Reflection coefficient (R)
Width of the cavity
​(d/M)
Audio
​Reference
Gaussian
​(r0=9M, sigma=6M)
1
~48.66
​echo.dat / echo.wav
[3]

​Wormhole:
(under development, static case)
(The reflectivity in this case is R=R_BH * Exp[-2i w x0], where R_BH is the reflection coefficient for left-moving waves scattered off a black hole, w is the frequency, x0 is the location of the throat [6])

​C-stars (strongly-anisotropic stars,
static case): (under development)
(The reflectivity in this case is R=R_BH * Exp[-2i w x0], where R_BH is the reflection coefficient for left-moving waves scattered off a black hole, w is the frequency, x0 is the location of the throat [6])
Prompt signal
Reflection coefficient (R)
Width of the cavity
​(d/M)
Audio
​Reference
Scalar perturbation; Gaussian
​(r0=5M, sigma=4M)
regularity at the center (|R|^2=1)
corr. M/R=0.4872
​echo.dat / echo.wav
[8]
​Scalar perturbation; Gaussian
​(r0=5M, sigma=4M)
​regularity at the center (|R|^2=1)
corr. M/R=0.4906
​echo.dat / echo.wav
[8]

Playing echo waveforms

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