binary black hole coalescence: analysis of the plunge · 8/30/2016 · binary black hole...

Binary black hole coalescence: Analysis of theplunge

Sourabh Nampalliwar1,2

with Richard Price2,3 and Gaurav Khanna4

1Fudan University, Shanghai

2U. Texas, Brownsville

3Massachusetts Institute of Technology

4U. Mass., Dartmouth

August 30, 2016

1 / 31 Sourabh Nampalliwar BBH coalescence: Analysis of the plunge

Gravitational wave sources

via LIGO


Binary black hole coalescence - the standard picture

via T. W. Baumgarte, S. L. Shapiro, Numerical Relativity, Cambridge U. Press, New York (2010)


Quasinormal ringing during the ringdown

via Berti et al., CQG 26 (2009) 163001


Binary black hole coalescence

Goal: Understand plunge, in particular the excitation of QNR.

Approach: Get insights, test insights.

Principle: Simplify.


Modelling

Complex frequencies → FDGF → Linearized Einstein equations

Φ =1

r

∑l,m

Ψlm(r, t)Ylm(θ, φ) (1)

− ∂2Ψ`m

∂t2+∂2Ψ`m

∂x2− V`(x)Ψ`m = S`m(x, t) (2)

r → r∗ → x (3)

Ψ(t, x) = Integral over Green function (4)


Comparison - TDP vs Schwarzschild

0

0.2

0.4

0.6

0.8

x0

V(x)

Tortoise radial coordinate x

−∞← →∞

l = 1


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.



T,F(T)

T6

xx0

t,x

Tcross



Ψ(u) =x02

∫ T6

Tcross

e−ωdθ

[1


2

xsinωoθ

]dT

− 1

2

∫ Tcross

−∞e−ωdξ

[( 2x0F (T )

− 1)


− 2x20xF (T )


]dT . (6)

where

θ = u− T + F (T ), ξ = u− T − F (T ) + 2x0



Ψ(u) =x02

∫ T6

Tcross

e−ωdθ

[1


2

xsinωoθ

]dT

− 1

2

∫ Tcross

−∞e−ωdξ

[( 2x0F (T )

− 1)


− 2x20xF (T )


]dT . (7)

where

θ = u− T + F (T ), ξ = u− T − F (T ) + 2x0



-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



-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


Analysis - QNR from radial infall in Schwarzschild

−∂2Ψ`m

∂t2+∂2Ψ`m

∂x2− V`(x)Ψ`m = S`m(x, t) (8)


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


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



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



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!


BONUS!


Gravitational waves

Travel at the speed of light.

Carry energy.

via learner.org


Quasinormal ringing


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


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


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)


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


Interesting features of orbital trajectories

Speed of light constraint:

ω2LRM

2 < (1− v2LR)/27. (10)

Here this impliesωLR < 0.1836/M. (11)


Direct radiation

540 560 580 600

t/M

-0.005

0

0.005

ψ

0.5

0.1 0.05


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)


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)


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)


binary black hole coalescence: analysis of the plunge · 8/30/2016 · binary black hole...

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