simulations of a bh-axion system - kekintroduction simulation code summary discussion typical two...

Simulations of a BH-axion system

Hirotaka Yoshino

3rd ExDiP2012 conference @ Grandvrio Hotel, Obihiro, Hokkaido

(August 8, 2012)

Hideo Kodama(KEK)

Prog. Thoer. Phys. 128, 153-190 (2012), arXiv:1203.5070[gr-qc]

ContentsIntroduction

Simulation

Code

Summary

Discussion

Typical two simulations

Does the bosenova really happen?

Comparison with BEC system

Gravitational waves

CMB Polarization

10-33 4 ! 10-28

Axion Mass in eV

108

Inflated Away

Decays

3 ! 10-10

QCD axion2 ! 10-20

3 ! 10-18

Anthropically ConstrainedMatter

Power SpectrumBlack Hole Super-radiance

Axiverse

QCD axion

String axions

QCD axion was introduced to solve the Strong CP problem.

It is one of the candidates of dark matter.

Arvanitaki, Dimopoulos, Dubvosky, Kaloper, March-Russel, PRD81 (2010), 123530.

String theory predicts the existence of 10-100 axion-like massive scalar fields.

There are various expected phenomena of string axions.

Axion field around a rotating black hole

Axion field forms a cloud around a rotating BH andextract energy of the BH by “superradiant instability”.

Arvanitaki, Dimopoulos, Dubvosky, Kaloper, March-Russel, PRD81 (2010), 123530.Arvanitaki and Dubovsky, PRD83 (2011), 044026.

Kerr BH

Ergo region

BH

Metric

ds2 = !!

!! a2 sin2 !

"

"dt2 ! 2a sin2 !(r2 + a2 !!)

"dtd"

+!(r2 + a2)2 !!a2 sin2 !

"

"sin2 !d"2 +

"!

dr2 + "d!2

! = r2 + a2 cos2 !,! = r2 + a2 ! 2Mr.

gtt > 0

J = Ma

Energy extraction

Penrose process Blandford-Znajek process Superradiance

BH’s rotational energy

Methods of energy extraction

(Next slide)

Mrot = M !MirrBH

Mirr =!

AH

16!

Superradiance

R =u!

r2 + a2

d2u

dr2!

+!!2 ! V (!)

"u = 0

! < !Hm

Superradiant condition:

! = Re[e!i!tR(r)S(!)eim"]

u ! e!i(!!m!H)r!

!2! =0Massless Klein-Gordon field

!1! m!H

!

"|T |2 = 1! |R|2

u ! Aoutei!r! + Aine!i!r!

A1horizon

in

Aout

Zeľdovich (1971)

Black hole bomb Press and Teukolsky (1972)

BH

mirror

Bound stateZouros and Eardley, Ann. Phys. 118 (1979), 139.

R =u!

r2 + a2

d2u

dr2!

+!!2 ! V (!)

"u = 0

! < !Hm

Superradiant condition:

0.14 0.15 0.16 0.17 0.18 0.19 0.2

0.21 0.22

-100 -50 0 50 100

V

r*/M

!2

V

I II III IV

! = Re[e!i!tR(r)S(!)eim"]

near horizon

u ! e!i(!!m!H)r!

distant region

u ! e!"

µ2!!2r!

Massive Klein-Gordon field !2!" µ2! =0

Detweiler, PRD22 (1980), 2323.

Growth rate

Dolan, PRD76 (2007), 084001.

1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Im(!

/ !

)

M !

l = 1, m = 1

l = 2, m = 2

l = 3, m = 3

a = 0.999a = 0.99a = 0.95

a = 0.9a = 0.8a = 0.7

l = m = 1 Mµ = 0.4

Growth rate calculated by continued fraction method

Time evolution

r!

!

(Near the horizon)! ! e!i!te!i!r!

! = ! !m!H

Accretion

Rotating Black Hole

Super-Radiant Modes

Decaying Modes

Gravitons

BH-axion system

Superradiant instabilityEmission of gravitational wavesPair annihilation of axions

Effects of nonlinear self-interactionBosenovaMode mixing

Arvanitaki and Dubovsky, PRD83 (2011), 044026.

Nonlinear effect

c.f., QCD axion

Typically, the potential of axion field becomes periodic

U(1)PQ symmetry

PQ phase transition

V = f2aµ2[1! cos(!/fa)]

! ! !fa

!2!" µ2 sin! = 0

QCD phase transition

Z(N) symmetrypotential becomes like a wine bottle⇒ ⇒

Accretion

Rotating Black Hole

Super-Radiant Modes

Decaying Modes

Gravitons

BH-axion system

Superradiant instabilityEmission of gravitational wavesPair annihilation of axions

Effects of nonlinear self-interactionBosenovaMode mixing

Arvanitaki and Dubovsky, PRD83 (2011), 044026.

Bosenova in condensed matter physicshttp://spot.colorado.edu/~cwieman/Bosenova.html

BEC state of Rb85（interaction can be controlled）Switch from repulsive interaction to attractive interaction

Wieman et al., Nature 412 (2001), 295

What we would like to do

We would like to study the phenomena caused by axion cloud generated by the superradiant instability around a rotating black hole.

In particular, we study numerically whether “Bosenova” happens when the nonlinear interaction becomes important.

We adopt the background spacetime as the Kerr spacetime, and solve the axion field as a test field.


Simulation

Code

Summary

Discussion




Gravitational waves

Our code

3D code of coordinates (r!, !,")

r!

!

t/M = 0 ! 100

Comparison with semianalytic solution of the Klein-Gordon case

!(CF)I /µ = 3.31! 10!7

!(Numerical)I /µ = 3.26! 10!7

!I =E

2E! E(100M)" E(0)

200ME(0)


Simulation

Code

Summary

Discussion




Gravitational waves

Setup

Numerical simulation

Sine-Gordon equation

!2!" µ2 sin! = 0

a/M = 0.99, Mµ = 0.4

As the initial condition, we choose the bound state of the Klein-Gordon field of the mode.l = m = 1

!"01

2

3

4#1

#2#3 #4

BH

#40 #20 0 20 40

#40

#20

0

20

40

r Cos!$"

rSin!$"

a#M"0.99, M%"0.4

Initial peak value(A) 0.6 1370(B) 0.7 1862

E/[(fa/Mp)2M ]

Axion field on the equatorial plane

Simulation (A)

!200 " r!/M " 200

(! = 0)

!

(! = "/2)

!peak(0) = 0.6

(Equatorial plane)

r cos !

r sin!

Simulation (A) Peak value and peak location

0

5

10

15

20

0 200 400 600 800 1000

r*

(pea

k)

t/M

(b)

0

0.5

1

1.5

2

0 200 400 600 800 1000

!pea

k

t/M

(a)

Fluxes toward the horizon

Energy and angular momentum distribution

-0.01-0.008-0.006-0.004-0.002

0 0.002

0 200 400 600 800 1000

FE a

nd

FJ

t/M

FE

FJ

-10

0

10

20

30

40

50

60

70

-200 -100 0 100 200 300

dE

/dr*

r*/M

t = 0

t = 1000M

-0.5

0

0.5

1

50 100 150 200

t = 0

t = 1000M

-20

0

20

40

60

80

100

120

140

160

180

-200 -100 0 100 200 300

dJ/dr*

r*/M

t = 0

t = 1000M

-1

0

1

2

50 100 150 200

t = 0

t = 1000M

Axion field on the equatorial plane

Simulation (B)

!200 " r!/M " 200

(! = 0)

!

(! = "/2)

!peak(0) = 0.7

(Equatorial plane)

r cos !

r sin!

0

1

2

3

4

0 200 400 600 800 1000

!pea

k

t/M

(a)

0

4

8

12

16

0 200 400 600 800 1000

r*(p

eak)

t/M

(b)

-1.6-1.2-0.8-0.4

0 0.4 0.8

0 200 400 600 800 1000

FE a

nd FJ

t/M

FE

FJ

-20

0

20

40

60

80

100

-200 -100 0 100 200 300

dE

/dr*

r*/M

t = 0

t = 750M

t = 1500M

-0.1

0

0.1

0.2

0.3

0.4

0.5

-200 -150 -100 -50 0

t = 750M

t = 1500M

-0.5

0

0.5

1

1.5

2

100 200 300 400 500 600

t = 0

t = 750M

t = 1500M

-50

0

50

100

150

200

250

300

-200 -100 0 100 200 300

dJ/dr*

r*/M

t = 0

t = 750M

t = 1500M

-1.5

-1

-0.5

0

0.5

-200 -150 -100 -50 0

t = 750M

t = 1500M

-2

0

2

4

6

8

100 200 300 400 500 600

t = 0

t = 750M

t = 1500M

Peak value and peak location

Fluxes toward the horizon

Energy and angular momentum distribution

Simulation (B)

-20

0

20

40

60

80

100

-200 -100 0 100 200 300

dE

/dr*

r*/M

t = 0

t = 750M

t = 1500M

-0.1

0

0.1

0.2

0.3

0.4

0.5

-200 -150 -100 -50 0

t = 750M

t = 1500M

-0.5

0

0.5

1

1.5

2

100 200 300 400 500 600

t = 0

t = 750M

t = 1500M

Energy distribution

Simulation (B)

m=-1 mode is generated!

Simulation (B)

-1

-0.5

0

0.5

1

r*/M

!

t = 500M

-100 -50 0 50 100

0

1

2

3

4

5

6

-2

-1

0

1

2

3

4

-200 -150 -100 -50 0 50 100 150 200

!

r*/M

t = 0

t = 350M

t = 700M

Snapshots

M!KG = 0.39M !KG = !0.04 M !NL = 0.87

M!NL = 0.35

(Near the horizon)! ! e!i!te!i!r!

! = ! !m!H

Summary of the simulations (A) and (B)

When the peak value is not very large, the nonlinear term enhances the rate of superradiant instability.

When the peak value is sufficiently large, the bosenova collapse happens.

The nonlinear effect makes energy distribute in the neighborhood of the black hole.

Once the bosenova happens, positive energy falls into the black hole, while the angular momentum continues to be extracted.

(A)

(B)


Simulation

Code

Summary

Discussion




Gravitational waves

Does bosenova really happen?

time

amplitude

Bosenova???

(A)

Saturation???

Additional simulation:

!(0) = C!(A)(1000M)

!(0) = C!(A)(1000M)C =

!"

#

1.051.081.09

Supplementary simulation !(0) = C!(A)(1000M)

!(0) = C!(A)(1000M)

Energy absorbed by the black hole !E :=! t

0FEdt

C!1.05

1.08

1.09

0 2000 4000 6000 8000 10000

"20

0

20

40

60

80

t!M

#E

C =

!"

#

1.051.081.09

The bosenova happens when E ! 1600" (fa/Mp)2M


Simulation

Code

Summary

Discussion




Gravitational waves

Action

BEC BH-axion

S = Nh

!d3xdt

"i!!! +

12!!!2! " r2

2!!! " g

2(!!!)2

#

i! = !12"2! +

r2

2! + g|!|2!

Gross-Pitaevskii equation

Action

S =!

d4x!"g

""1

2(#!)2 " µ2

#!2

2+ UNL(!)

$%,

Non-relativistic approximation

UNL(x) = !!!

n=2

(!1/2)n

(n!)2xn.

! =1!2µ

!e!iµt" + eiµt""

"

+!g

r"!" ! µ2UNL(|"|2/µ)

!

SNR =!

d4x

"i

2

#!!! ! !!!

$! 1

2µ"i!"i!

!

Saito and Ueda, PRA63 (2001), 043601

Action

Effective theory

BEC BH-axion! = A(x, y, z, t)ei!(x,y,z,t)

A =exp

!!( x2

2d2x(t) + y2

2d2y(t) + z2

2d2z(t) )

"

#!3/2dx(t)dy(t)dz(t)

! =dx(t)2dx(t)

x2 +dy(t)2dy(t)

y2 +dz(t)2dz(t)

z2

dx = dy = dz = r(t)Spherical case

S =Nh

4

!dt

"3r2 + 3r ! f(r)

#

0

2

4

6

8

0 0.5 1 1.5

f!r"

r

rc

!=#.$!c!=!c

!=%.%!c

! = A(t, r, ")eiS(t,r,!)+im"

A(t, r, !) ! A0 exp!" (r " rp)2

4"rr2p

" (! " !p)2

4"!

",

S(t, r, !) ! S0(t) + p(t)(r " rp) + P (t)(r " rp)2 + "!(t)(! " !p)2 + · · · ,

N!"0.02

N!"0.08

0.0 0.5 1.0 1.5 2.0 2.5

#10

#5

0

5

$G%rp

V!%$G2

$G"0.1

Saito and Ueda, PRA63 (2001), 043601 (! = cos ")

Simulation results

BEC BH-axion

i! = !12"2! +

r2

2! + g|!|2!

0

100

200

300

0 1 2 3

|!!r="#|

t

2.5 2.6 0

1

2

3

4

0 200 400 600 800 1000!

pea

kt/M

(a)

Saito and Ueda, PRA63 (2001), 043601

Our simulation results

! i

2

!L2

2|!|2 +

L3

6|!|4

"!


Simulation

Code

Summary

Discussion




Gravitational waves

GWs emitted in the bosenova (rough estimate)Quadrupole moment

C!1.05

1.08

1.09

0 2000 4000 6000 8000 10000

"20

0

20

40

60

80

t!M

#EQij ! r2pE

rp ! 10M

About 5% of energy falls into the BH

E0 ! 10!3M

E = E0 + (!E/2) [cos(!t/!t)! 1]

!E ! 0.05E0 !t ! 500M

Amplitude of generated GWs

h ! Qij

robs! 10!7 M

robs

Detectability

Supermassive BH of our galaxy（Sagittarius A*）

Solar-mass BH (e.g., Cygnus X-1)

h ! Qij

robs! 10!7 M

robs

Detectable by the eLISA

hrss ! 10!24(Hz)!1/2

below the sensitivity of the Advanced LIGO, KAGRA (LCGT), etc.

(10!4 Hz)

(102 Hz)

Angular frequency of GW

hrss :=!"

|h|2dt

#1/2

! 10!16(Hz)!1/2


Simulation

Code

Summary

Discussion




Gravitational waves

Summary

We developed a reliable code and numerically studied the behaviour of axion field around a rotating black hole.

The nonlinear effect enhances the rate of superradiant instability when the amplitude is not very large.

Calculation of the gravitational waves emitted in bosenova.

Issues for future

The case where axions couple to magnetic fields.

The bosenova collapse would happen as a result of superradiant instability.

simulations of a bh-axion system - kekintroduction simulation code summary discussion typical two...

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