GW Echoes

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]

References:

Adriano Testa & Paolo Pani, An analytical template for echoes, arXiv:1806.04253, Phys. Rev. D 98, 044018 (2018)
Adriano Testa, Master thesis, Sapienza University of Rome (2018)
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
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)
Vitor Cardoso, Edgardo Franzin, Paolo Pani, Is the gravitational-wave ringdown a probe of the event horizon? - Phys. Rev. Lett. 116, 171101 (2016)
Zachary Mark, Aaron Zimmerman, Song Ming Du, and Yanbei Chen, A recipe for echoes from exotic compact objects, Phys. Rev. D 96, 084002 (2017)
Gaurav Khanna, http://gravity.umassd.wikispaces.net/Echoes
Raposo, Pani, Bezares, Palenzuela, Cardoso, "Anisotropic stars as ultracompact objects in General Relativity" Phys. Rev. D 99, 104072 (2019)
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]

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

Datafiles

Playing echo waveforms

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