superradiant instabilities in astrophysical systems · superradiant instabilities in astrophysical...

Superradiant instabilities in astrophysical systems

Helvi Witek,V. Cardoso, A. Ishibashi, U. Sperhake

CENTRA / IST,Lisbon, Portugal

work in progress

NRHEP network meeting, Aveiro, 9 July, 2012

H. Witek (CENTRA / IST) 1 / 29

Outline

1 Motivation

2 Massive scalar fields

3 Massive vector fields

4 Conclusions


Motivation


Superradiance effect

Penrose process(Penrose ’69, Christodoulou ’70)

scattering of particles off Kerr BH

⇒ reduction of BH mass

superradiant scattering(Misner ’72, Zeldovitch ’71)

scattering of wave packet off Kerr BHsuperradiance condition

ω < mΩH = ma

2Mr+

⇒ extraction of energy and angular momentum off BH⇒ amplification of energy and angular momentum of wave packet


Superradiance instability

“black hole bomb” (Press & Teukolsky ’72, Zeldovich ’71, Cardoso et al ’04)

consider Kerr BH surrounded bymirror

consider field with ω < mΩH

⇒ superradiant scattering

subsequequent amplification ofsuperradiant modes

⇒ exponential growth of modes⇒ instability due to superradiant scattering


Superradiance instability in physical systems

natural mirror provided by

anti-de Sitter spacetimes

massive fields with mass coupling Mµ(Damour et al. ’76, Detweiler ’80, Zouros & Eardley ’79)

Arvanitaki & Dubovsky ’11



growth rate of massive scalar fields

Detweiler ’80: Mµ << 1

τ ∼ 24( a

M

)−1

(Mµ)−9

(GM

c3

)Zouros & Eardley ’79 Mµ >> 1

τ ∼ 107 exp(1.84Mµ)

(GM

c3

)

for astrophysical BHs and known particles: Mµ ∼ 1018

⇒ insignificant for astrophysical systems?



most promising mass range: Mµ ∼ 1

ultralight bosons proposed by string theory compactifications: axions(Arvanitaki & Dubovsky ’10)

formation of bosonic bound statesaround astrophysical BHs

gravitational wave emission

“bosenova”-like particle bursts(Yoshino & Kodama ’12)

(Arvanitaki & Dubovsky ’11)



(Kodama & Yoshino ’11)

landscape of ultralight axions ⇒ “string axiverse”

bosonic fields with Mµ ∼ 10−22 as dark matter candidates

small, primordial BHs with M ∼ 10−18M

bosonic cloud around SMBHs (M ∼ 109M) if 10−21 ≤ Mµ ≤ 10−16

⇒ probe of photon mass (upper bound µγ ∼ 10−18 (Nakamura et al.’10))


Massive scalar field


Massive scalar fields - recent results

Klein-Gordon equation

(− µ2)ψ = 0 , with ψ = exp(imφ− iωt)Slm(θ)Rlm(r)

QNMs (Dolan ’07, Cardoso & Yoshida ’05, Berti et al ’09)

Mµ a/M Mω11 (n = 0) Mω11 (n = 1)0.00 0.00 0.2929− i0.09766 0.2645− i0.30630.00 0.99 0.4934− i0.03671 0.4837− i0.098800.42 0.00 0.4075− i0.001026 0.4147− i0.00040530.42 0.99 0.4088 + i1.504 · 10−7 0.4151 + i5.364 · 10−8

Tailsmassless field (Price ’71, Leaver ’86, Ching et al ’95)

ψ ∼t−2l−3

massive fields (Koyama & Tomimatsu ’01,’02, Burko & Khanna ’04)

ψ ∼t−l−3/2 sin(µt) , at intermediate times

ψ ∼t−5/6 sin(µt) , at very late times


Massive scalar fields - recent results

bound states: maximum instability growth rate for(Dolan ’07, Cardoso & Yoshida ’05)

l = m = 1 , a/M = 0.99 , Mµ = 0.42 :1

τ= ωI ∼ 1.5 · 10−7

(GM

c3

)−1

numerical results:Strafuss & Khanna ’05: MωI ∼ 2 · 10−5 (Gaussian initial data)Kodama & Yoshino ’12: MωI ∼ 3.2 · 10−7 (bound state initial data)

Dolan ’07


Massive scalar fields - Code setup

goal: study time evolution of massive scalar field in Kerr background

Kerr background in Kerr-Schild coordinates → excision of BH region

Klein-Gordon equation (− µ2)ψ = 0 as 3 + 1 time evolution problem

dtψ = −αΠ

dtΠ = −α(D iDiψ − µ2ψ − KΠ)− D iαDiψ

initial data:

Gaussian wave packet with r0 = 12M, w = 2Mquasi-bound state

4th finite differences in space, 4th order Runge-Kutta time-integrator

extraction of scalar field at fixed rex , mode decomposition

ψlm(t) =

∫dΩψ(t, θ, φ)Y ∗

lm(θ, φ)


Code tests I - space dependent mass coupling

consider spherically symmetric scalar field with unphysical mass

(Mµ)2 =− 10

r4

theoretical prediction: ψ ∼ exp(0.071565 t)

numerical result: ψ ∼ exp(0.07161 t) ⇒ agreement within 0.06%

0 100 200 300 400t/M

0

10

20

30

ln |

ψ00|

ln |ψ00

|

e0.07161 t

e0.071565 t


Code tests II - massless scalar fields

a/M = 0 a/M = 0.99

0 50 100 150t / M

-14

-12

-10

-8

-6

-4

-2

0

2

ln|

ψlm

|

l = m = 0l = m = 1

0 50 100 150 200 250t / M

-14

-12

-10

-8

-6

-4

-2

0

2

ln|

ψlm

|

l = m = 0l = m = 1

QNM frequencies

a/M = 0.0 Mω11 = 0.294− ı0.096 (0.2929− ı0.09766)a/M = 0.99 Mω11 = 0.493− ı0.0368 (0.4934− ı0.03671)

tail: ψ00 ∼ t−3.04 (ψ00 ∼ t−3)

numerical error: ∆ψ00/ψ00 ≤ 3%, . ∆ψ11/ψ11 ≤ 8%


Massive scalar fields in Schwarzschild

consider mass coupling Mµ = 0.1, 0.42, 1

0 50 100 150 200 250 t/M

-6

-5

-4

-3

-2

-1

0

1

2

log

|ψ

11|

µM = 0.1µM = 0.42µM = 1

tails in agreement with (Koyama & Tomimatsu ’02, Burko & Khanna ’04):

Mµ = 0.1 Mω11 = 0.293− ı0.036 ψ11 ∼ t−2.543 sin(0.1t)Mµ = 1.0 Mω11 = 0.965− ı0.0046 ψ11 ∼ t−0.873


Massive scalar field with Mµ = 0.42 in Schwarzschild

0 1000 2000 3000t / M

-0.2

-0.1

0

0.1

0.2

Re(

ψ1

1 )

rex

/ M = 22.5

0 1000 2000 3000t / M

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Re(

ψ1

1 )

rex

/ M = 40

dependent on location of measurement

beating of modes:

similar frequency MωR of fundamental and overtone modeωR,n = ωR,0 + δ, δ << 1

ψ ∼A0 exp(−ıω0t) + An exp(−ıωnt)

∼ exp(−ıωR,0t)(B0 + Bn cos(δt)− ıBn sin(δt)

)H. Witek (CENTRA / IST) 17 / 29

Massive scalar field in Kerr - bound states

Evolution of bound-state scalar field with Mµ = 0.42, a/M = 0.99

localized in the vicinity of the BH

node of overtone at x = 26.5 M

by construction Ψ11Ψ∗11 ∼ exp(−ıωt) exp(ıωt) = const.

Initial data Evolution

-100 -75 -50 -25 0 25 50 75 100x / M

0

0.5

1

1.5

2

2.5

| ψ

11 |

2

fundamental modeovertone mode

0 50 100 150 200 250t / M

-0.03

-0.02

-0.01

0

0.01

0.02

| ψ

11 | / | ψ

t=0 | -

1

h/M = 1/60h/M = 1/72


Massive scalar field in Kerr - Gaussian

Evolution of a scalar field with Mµ = 0.42 in Kerr a/M = 0.99 with Gaussianinitial data

0 2000 4000 6000 8000t / M

-0.4

-0.2

0

0.2

0.4

Re(

ψ11 )

rex

/ M = 20

0 2000 4000 6000 8000t / M

-0.2

-0.1

0

0.1

0.2

Re(

ψ11 )

rex

/ M = 26.5

0 2000 4000 6000 8000t / M

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Re(

ψ11 )

rex

/ M = 50

beating of modes if ωR,n = ωR,0 + δ, δ << 1

ψ ∼A0 exp(−ıω0t) + An exp(−ıωnt)

∼ exp(−ıωR,0t)(B0 + Bn cos(δt)− ıBn sin(δt)

)amplitude of modes dependent on location of measurement

possible explanation for result by Strafuss & Khanna ’05


Massive vector fields


Massive vector fields

massive hidden U(1) vector fields from string theory compactification(e.g., Jaeckel & Ringwald ’10)

expected: superradiance effect stronger than in scalar field case

rich phenomenology

studied by Galt’sov et al ’84, Konoplya ’06, Konoplya et al ’07,Herdeiro et al ’11, Rosa & Dolan ’11

vector field eqs. in Kerr non-separable ⇒ challenging problem


Massive vector fields in Schwarzschild background

Rosa & Dolan ’11:

Proca field equations

∇νFµν + µ2AA

µ = 0 Fµ = ∇µAν −∇νAµ

Lorenz condition has to be satisfied ∇µAµ = 0⇒ scalar mode gains physical meaning

decomposition of Aµ in vector spherical harmonics Z(i)lmµ

continued fraction method and forward integration


Massive vector fields in Schwarzschild background

Rosa & Dolan ’11: QNM spectrum

for given l , n:2 even parity modes(scalar and vector field modes),1 odd parity mode (vector field mode)

in electromagnetic limit (MµA → 0):

scalar mode = gauge modeeven and odd vector mode degenerate

field mass - breaking of degeneracy

distinct frequencies of even parity modes


Massive vector fields in Kerr - Code setup

goal: study time evolution of Proca field in Kerr background(work in progress)

Kerr background in Kerr-Schild coordinates → excision of BH region

Proca equation ∇νFµν + µ2A = 0

Lorenz condition ∇µAµ = 0

define Aµ = Aµ + nµϕ, Eµ = Fµνnν

⇒ formulation as 3 + 1 time evolution problem

initial data: gaussian wave packet

4th finite differences in space, 4th order Runge-Kutta time-integrator

extraction of Newman-Penrose scalar Φ2 at fixed rex , mode decomposition

Φ2,lm(t) =

∫dΩΦ2(t, θ, φ) −1Y

∗lm(θ, φ)


Massless vector field in Kerr

vector field with µA = 0 in Kerr with a/M = 0.99

0 50 100 150 200 t/M

-16

-14

-12

-10

-8

-6

-4

-2

log

|Φ

2,1

1|

h / M = 1 / 60h / M = 1 / 72

QNM frequencies: Mω11 = 0.461− i0.041 (0.463− i0.031)

in agreement with theoretical prediction (Berti et al, ’09)

improving with increasing resolution


Massive vector fields in Kerr

vector field with µA = 0.01, 0.42, 1 in Kerr with a/M = 0.99

0 100 200 300 t/M

-7

-6

-5

-4

-3

-2

-1

0

log

|Φ

2,1

1|

Mµ = 0.01Mµ = 0.40Mµ = 1

QNM frequencies

MµA = 0.01 ω11M = 0.462− i0.041MµA = 0.40 ω11M = 0.415MµA = 1.0 ω11M = 0.959− i0.004


Proca field in Kerr

Evolution of a Proca field with Mµ = 0.40 in Kerr a/M = 0.99beating of modesgrowth of the mode - signature of instability?

0 500 1000 1500 2000 2500t / M

-0.2

-0.1

0

0.1

0.2

Re

( Φ

2,1

1 )

rex

/ M = 10

0 500 1000 1500 2000 2500t / M

-0.05

0

0.05

Re

( Φ

2,1

1 )

rex

/ M = 15

0 500 1000 1500 2000 2500t / M

-0.04

-0.02

0

0.02

0.04

Re

( Φ

2,1

1 )

rex

/ M = 25

0 500 1000 1500 2000 2500t / M

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Re

( Φ

2,1

1 )

rex

/ M = 40


Conclusions

massive fields in Kerr spacetimes exhibit extremely rich spectra

evolution of scalar field wave packets

extensive code testingbeating of fundamental and overtone modes

first evolutions of massive vector fields in Kerr background

low mass fields MµA = 0.01 dampedbeating effect for MµA = 0.42growing of l = m = 1 mode → possible signature of instabilitystill in its infantry ⇒ more results to come soon


Thank you!

http://blackholes.ist.utl.pt


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