Gravitational waves from compact binary mergers:a phenomenological approach
Sourabh Nampalliwar, Richard Price, Gaurav Khanna
1U. Texas Brownsville
2Fudan University
2U. Mass. Dartmouth
April 13, 2016
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Gravitational waves
Travel at the speed of light.
Carry energy.
via learner.org
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Gravitational wave detectors
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Gravitational wave sources
via LIGO
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Binary black hole coalescence - the standard picture
via T. W. Baumgarte, S. L. Shapiro, Numerical Relativity, Cambridge U. Press, New York (2010)
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Quasinormal ringing
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Quasinormal ringing during the ringdown
via Berti et al., CQG 26 (2009) 163001
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Binary black hole coalescence
Goal: Understand plunge, in particular the excitation of QNR.
Approach: Get insights, test insights.
Principle: Simplify, simplify, simplify.
8 / 32 Sourabh Nampalliwar Gravitational waves: phenomenological understanding of merger
Modelling
Complex frequencies → FDGF → Linearized Einstein equations
− ∂2Ψ`m
∂t2+∂2Ψ`m
∂x2− V`(x)Ψ`m = S`m(x, t) (1)
r → r∗ → x (2)
Φ =1
r
∑l,m
Ψlm(r, t)Ylm(θ, φ) (3)
Ψ(t, x) = Integral over Green function (4)
9 / 32 Sourabh Nampalliwar Gravitational waves: phenomenological understanding of merger
Modelling
Complex frequencies → FDGF → Linearized Einstein equations
− ∂2Ψ`m
∂t2+∂2Ψ`m
∂x2− V`(x)Ψ`m = S`m(x, t) (1)
r → r∗ → x (2)
Φ =1
r
∑l,m
Ψlm(r, t)Ylm(θ, φ) (3)
Ψ(t, x) = Integral over Green function (4)
9 / 32 Sourabh Nampalliwar Gravitational waves: phenomenological understanding of merger
Modelling
Complex frequencies → FDGF → Linearized Einstein equations
− ∂2Ψ`m
∂t2+∂2Ψ`m
∂x2− V`(x)Ψ`m = S`m(x, t) (1)
r → r∗ → x (2)
Φ =1
r
∑l,m
Ψlm(r, t)Ylm(θ, φ) (3)
Ψ(t, x) = Integral over Green function (4)
9 / 32 Sourabh Nampalliwar Gravitational waves: phenomenological understanding of merger
Modelling
Complex frequencies → FDGF → Linearized Einstein equations
− ∂2Ψ`m
∂t2+∂2Ψ`m
∂x2− V`(x)Ψ`m = S`m(x, t) (1)
r → r∗ → x (2)
Φ =1
r
∑l,m
Ψlm(r, t)Ylm(θ, φ) (3)
Ψ(t, x) = Integral over Green function (4)
9 / 32 Sourabh Nampalliwar Gravitational waves: phenomenological understanding of merger
Modelling
Complex frequencies → FDGF → Linearized Einstein equations
− ∂2Ψ`m
∂t2+∂2Ψ`m
∂x2− V`(x)Ψ`m = S`m(x, t) (1)
r → r∗ → x (2)
Φ =1
r
∑l,m
Ψlm(r, t)Ylm(θ, φ) (3)
Ψ(t, x) = Integral over Green function (4)
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Comparison - TDP vs Schwarzschild
0
0.2
0.4
0.6
0.8
x0
V(x)
Tortoise radial coordinate x
−∞← →∞
l = 1
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Analysis - QNR from radial infall in TDP
Ψ(t− x) =x02
∫ T6
Tcross
e−ωdθ
[1
x0(sinωoθ − cosωoθ) −
2
xsinωoθ
]dT
− 1
2
∫ Tcross
−∞e−ωdξ
[( 2x0F (T )
− 1)
cosωoξ − sinωoξ
− 2x20xF (T )
(cosωoξ − sinωoξ)
]dT . (5)
where
θ = u− T + F (T ), ξ = u− T − F (T ) + 2x0.
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Analysis - QNR from radial infall in TDP
T,F(T)
T6
xx0
t,x
Tcross
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Analysis - QNR from radial infall in TDP
Ψ(u) =x02
∫ T6
Tcross
e−ωdθ
[1
x0(sinωoθ − cosωoθ) −
2
xsinωoθ
]dT
− 1
2
∫ Tcross
−∞e−ωdξ
[( 2x0F (T )
− 1)
cosωoξ − sinωoξ
− 2x20xF (T )
(cosωoξ − sinωoξ)
]dT . (6)
where
θ = u− T + F (T ), ξ = u− T − F (T ) + 2x0
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Analysis - QNR from radial infall in TDP
Ψ(u) =x02
∫ T6
Tcross
e−ωdθ
[1
x0(sinωoθ − cosωoθ) −
2
xsinωoθ
]dT
− 1
2
∫ Tcross
−∞e−ωdξ
[( 2x0F (T )
− 1)
cosωoξ − sinωoξ
− 2x20xF (T )
(cosωoξ − sinωoξ)
]dT . (7)
where
θ = u− T + F (T ), ξ = u− T − F (T ) + 2x0
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Analysis - QNR from radial infall in TDP
-0.04
-0.02
0
0.02
10 20 30 40 50 60
Ψ
Shifted u
vcross
across = 0.0005
12QN
0.30.310.320.330.34
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Analysis - QNR from radial infall in TDP
-1.5
-1.2
-0.9
-0.6
-0.3
-20 -10 0 10 20 30 40 50
Ψ
Shifted retarded time u
v = 0.31, TLR = 300
aLR0.0010.0050.010.050.1
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Analysis - QNR from radial infall in Schwarzschild
−∂2Ψ`m
∂t2+∂2Ψ`m
∂x2− V`(x)Ψ`m = S`m(x, t) (8)
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Analysis - QNR from radial infall in Schwarzschild
100 200 300Retarded time u (shifted)
-6e-05
-4e-05
-2e-05
0
2e-05
4e-05
Ψ
τ=250
τ=450
Data courtesy Gaurav Khanna
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Analysis - QNR from orbital infall in Schwarzschild
500 550 600 650
t/M
-0.0002
-0.0001
0
0.0001
ψ
0.04
0.02
0.005
0.01
Data courtesy Gaurav Khanna
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Analysis - QNR from orbital infall in Schwarzschild
500 550 600
t/M
-0.004
-0.002
0
0.002
0.004
ψ
-0.30
2
Data courtesy Gaurav Khanna
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Conclusion
During a binary black hole coalescence, the transition from theinspiral phase to the ringdown phase is completely determined bythe local properties at the photon orbit.
Thank you!
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BONUS!
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Analysis - QNR from orbital infall in TDP
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0 10 20 30 40 50 60 70
Ψ
Shifted retarded time u [/M]
ω = 4, v = 0.3
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Comparison - Evolution vs FDGF
0 10 20 30 40Retarded time u
-1
-0.5
0
Ψ
EvolutionFDGF
a0=4 τ =1 x
0=2 x
obs=6
Evolution data courtesy Gaurav Khanna
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Assumption - Trajectories: Orbital
Redshift in angular velocity.
dω/d t = 0 at the light ring.
Speed of light should not be breached.
ω(T ) = ωLR27M2
(1 + σ)r(T )2
(1− 2M
r(T )
)(1 +
3σM
r(T )
)(9)
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Analysis - QNR from orbital motion in Schwarzschild
500 550 600 650
t/M
-0.0002
-0.0001
0
0.0001
ψ
0.04
0.02
0.005
0.01
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Interesting features of orbital trajectories
Speed of light constraint:
ω2LRM
2 < (1− v2LR)/27. (10)
Here this impliesωLR < 0.1836/M. (11)
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Direct radiation
540 560 580 600
t/M
-0.005
0
0.005
ψ
0.5
0.1 0.05
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Green’s function
∂2G
∂x2− ∂2G
∂t2− V (x)G = δ(x− a)δ(t− T ) (12)
Ψ(t, x) =
∫ ∞−∞
∫ ∞−∞
G(x, a; t− T )f(T )δ(a− F [T ]) da dT
(13)
G(x, a; t− T ) =1
2π
∫ ∞−∞
e−iω(t−T )G(x, a;ω)dω , (14)
Ψ(t, x) =1
2π
∫ ∫ ∫ ∞−∞
e−iω(t−T )G(x, a;ω)δ(a−F [T ]) dω da dT .
(15)
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Assumption - Trajectories: Radial
dx/dt→ −1 near the horizon.
At least two parameters.
Static initial data.
F (T ) =
{a0 + τ −
(T 3 + τ3
)1/3T ≥ 0
a0 T < 0 .(16)
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Approximation - Truncated Multipole Potential
Features of curvature potential*
V` =
`(`+1)x2 for r � 2M
(1− 2Mr ) for r ∼ 2M
(17)
Truncated Dipole Potential (TDP)
V =
{2/x2 for x ≥ x0
0 for x < x0(18)
where x0 is the light ring aka photon orbit aka UCO.
*Cunningham, Price & Moncrief, ApJ, 224, 643-667 (1978)
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Approximation - Truncated Multipole Potential
Features of curvature potential*
V` =
`(`+1)x2 for r � 2M
(1− 2Mr ) for r ∼ 2M
(17)
Truncated Dipole Potential (TDP)
V =
{2/x2 for x ≥ x0
0 for x < x0(18)
where x0 is the light ring aka photon orbit aka UCO.
*Cunningham, Price & Moncrief, ApJ, 224, 643-667 (1978)
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