Superradiant Terahertz Sources and their Applications in ...
Superradiant instabilities in astrophysical systems · Superradiant instabilities in astrophysical...
Transcript of 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
H. Witek (CENTRA / IST) 2 / 29
Motivation
H. Witek (CENTRA / IST) 3 / 29
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
H. Witek (CENTRA / IST) 4 / 29
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
H. Witek (CENTRA / IST) 5 / 29
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
H. Witek (CENTRA / IST) 6 / 29
Superradiance instability in physical systems
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?
H. Witek (CENTRA / IST) 7 / 29
Superradiance instability in physical 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)
H. Witek (CENTRA / IST) 8 / 29
Superradiance instability in physical systems
(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))
H. Witek (CENTRA / IST) 9 / 29
Massive scalar field
H. Witek (CENTRA / IST) 10 / 29
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
H. Witek (CENTRA / IST) 11 / 29
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
H. Witek (CENTRA / IST) 12 / 29
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)
H. Witek (CENTRA / IST) 13 / 29
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
H. Witek (CENTRA / IST) 14 / 29
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%
H. Witek (CENTRA / IST) 15 / 29
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
H. Witek (CENTRA / IST) 16 / 29
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
H. Witek (CENTRA / IST) 18 / 29
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
H. Witek (CENTRA / IST) 19 / 29
Massive vector fields
H. Witek (CENTRA / IST) 20 / 29
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
H. Witek (CENTRA / IST) 21 / 29
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
H. Witek (CENTRA / IST) 22 / 29
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
H. Witek (CENTRA / IST) 23 / 29
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)
H. Witek (CENTRA / IST) 24 / 29
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
H. Witek (CENTRA / IST) 25 / 29
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
H. Witek (CENTRA / IST) 26 / 29
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
H. Witek (CENTRA / IST) 27 / 29
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
H. Witek (CENTRA / IST) 28 / 29
Thank you!
http://blackholes.ist.utl.pt
H. Witek (CENTRA / IST) 29 / 29