Axion Bosenova Hirotaka Yoshino Hideo Kodama (KEK) Prog. Thoer. - PowerPoint PPT Presentation

Axion Bosenova Hirotaka Yoshino Hideo Kodama (KEK) Prog. Thoer. Phys. 128, 153-190 (2012), arXiv:1203.5070[gr-qc] JGRG22 @ RESCEU University of Tokyo ( November 14, 2012 )

Contents Introduction Simulation Discussion Summary

Introduction

Axiverse QCD axion 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, String axions PRD81 ( 2010 ) , 123530. String theory predicts the existence of 10 - 100 axion - like massive scalar fields. There are various expected phenomena of string axions. Anthropically Constrained CMB Matter Polarization Power Spectrum Inflated Black Hole Super-radiance Decays Away 10 -33 4 � 10 -28 3 � 10 -18 10 8 2 � 10 -20 3 � 10 -10 QCD axion Axion Mass in eV

Axion field around a rotating black hole Axion field forms a cloud around a rotating BH and extract energy of the BH by “superradiant instability”. Arvanitaki, Dimopoulos, Dubvosky, Kaloper, March - Russel, PRD81 ( 2010 ) , 123530. Arvanitaki and Dubovsky, PRD83 ( 2011 ) , 044026.

Superradiance Ze ľ dovich ( 1971 ) ∇ 2 Φ =0 Massless Klein - Gordon field Φ = Re[ e − i ω t R ( r ) S ( θ ) e im φ ] u d 2 u R = ω 2 − V ( ω ) � � + u = 0 √ r 2 + a 2 dr 2 ∗ A out A 1 in horizon u ∼ A out e i ω r ∗ + A in e − i ω r ∗ u ∼ e − i ( ω − m Ω H ) r ∗ � � 1 − m Ω H | T | 2 = 1 − | R | 2 Superradiant condition: ω < Ω H m ω

Bound state Zouros and Eardley, Ann. Phys. 118 ( 1979 ) , 139. Detweiler, PRD22 ( 1980 ) , 2323. ∇ 2 Φ − µ 2 Φ =0 Massive Klein - Gordon field 0.22 0.21 I II III IV 0.2 near horizon 0.19 u ∼ e − i ( ω − m Ω H ) r ∗ distant region 0.18 V u ∼ e − √ µ 2 − ω 2 r ∗ 0.17 Superradiant condition: 0.16 V ω < Ω H m ! 2 0.15 0.14 -100 -50 0 50 100 r * / M � 50 s ( M = 10 M ⊙ ) Typical time scale ∼ 10 7 M ∼ ( M = 10 6 M ⊙ ) 1 . 6 year

BH - axion system Super-Radiant Modes Decaying Modes Gravitons Rotating Black Hole Accretion Arvanitaki and Dubovsky, PRD83 ( 2011 ) , 044026. Superradiant instability Emission of gravitational waves ( Level transition, Pair annihilation of axions ) E ff ects of nonlinear self - interaction Bosenova Mode mixing

⇒ ⇒ Nonlinear e ff ect Typically, the potential of axion field becomes periodic V = f 2 a µ 2 [1 − cos( Φ /f a )] ϕ ≡ Φ ∇ 2 ϕ − µ 2 sin ϕ = 0 f a c.f., QCD axion QCD phase transition PQ phase transition Potential becomes like a wine U ( 1 ) PQ symmetry Z ( N ) symmetry bottle

BH - axion system Super-Radiant Modes Decaying Modes Gravitons Rotating Black Hole Accretion Arvanitaki and Dubovsky, PRD83 ( 2011 ) , 044026. Superradiant instability Emission of gravitational waves ( Level transition, Pair annihilation of axions ) E ff ects of nonlinear self - interaction Bosenova Mode mixing

Bosenova in condensed matter physics http://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 W e 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. W e adopt the background spacetime as the Kerr spacetime, and solve the axion field as a test field.

Simulations Typical two simulations Does the bosenova really happen?

Numerical simulation Sine - Gordon equation ∇ 2 ϕ − µ 2 sin ϕ = 0 a � M � 0.99, M Μ� 0.4 �� 0 40 1 a/M = 0 . 99 , Mµ = 0 . 4 Setup 2 20 3 4 � 1 r Sin � Φ � � 2 0 BH � 3 As the initial condition, we choose the bound state of � 4 the Klein - Gordon field of the mode. l = m = 1 � 20 � 40 � 40 � 20 0 20 40 r Cos � Φ � E/ [( f a /M p ) 2 M ] Initial peak value ( A ) 0.6 1370 ( B ) 0.7 1862

ϕ peak (0) = 0 . 6 Simulation ( A ) ( θ = π / 2) Axion field on the equatorial plane ( φ = 0) ( Equatorial plane ) r sin φ Φ r cos φ − 200 ≤ r ∗ /M ≤ 200

Simulation ( A ) Peak value and peak location Energy and angular momentum distribution (a) 2 1.5 ! peak 70 1 60 0.5 1 t = 1000 M 50 0 0.5 t = 1000 M 40 0 200 400 600 800 1000 dE / dr * 0 t = 0 30 t / M -0.5 (b) 20 50 100 150 200 20 15 10 t = 0 (peak) 0 10 r * -10 5 -200 -100 0 100 200 300 0 r * / M 0 200 400 600 800 1000 180 t / M 160 2 t = 1000 M 140 Fluxes toward the horizon t = 1000 M 1 120 100 0 dJ / dr * 0.002 t = 0 80 F E F E and F J 0 -1 50 100 150 200 60 -0.002 F J 40 -0.004 t = 0 -0.006 20 -0.008 0 -0.01 -20 0 200 400 600 800 1000 -200 -100 0 100 200 300 r * / M t / M

ϕ peak (0) = 0 . 7 Simulation ( B ) ( θ = π / 2) Axion field on the equatorial plane ( φ = 0) ( Equatorial plane ) r sin φ Φ r cos φ − 200 ≤ r ∗ /M ≤ 200

Simulation ( B ) Peak value and peak location Energy and angular momentum distribution (a) 4 3 ! peak 100 2 2 0.5 1.5 0.4 t = 750 M 80 t = 1500 M 1 0.3 1 0.2 t = 750 M 0.5 60 0 0.1 0 0 dE / dr * 0 200 400 600 800 1000 t = 1500 M t = 0 -0.1 40 -0.5 -200 -150 -100 -50 0 100 200 300 400 500 600 t / M (b) 16 t = 0 20 t = 750 M 12 (peak) 0 8 t = 1500 M -20 r * 4 -200 -100 0 100 200 300 0 r * / M 0 200 400 600 800 1000 300 t / M 8 0.5 250 6 t = 1500 M Fluxes toward the horizon 0 t = 1500 M 4 200 -0.5 t = 750 M 2 0.8 -1 t = 750 M 0 dJ / dr * 150 F E t = 0 0.4 -1.5 F E and F J -200 -150 -100 -50 0 -2 100 200 300 400 500 600 0 100 F J -0.4 t = 0 50 -0.8 t = 750 M -1.2 0 -1.6 t = 1500 M -50 0 200 400 600 800 1000 -200 -100 0 100 200 300 r * / M t / M

Simulation ( B ) Energy distribution 100 2 0.5 1.5 t = 750 M 0.4 80 t = 1500 M 0.3 1 0.2 t = 750 M 0.5 60 0.1 0 0 dE / dr * t = 1500 M t = 0 -0.1 -0.5 40 -200 -150 -100 -50 0 100 200 300 400 500 600 t = 0 20 t = 750 M 0 t = 1500 M -20 -200 -100 0 100 200 300 r * / M

Simulations Typical two simulations Does the bosenova really happen?

Does bosenova really happen? Bosenova??? amplitude Saturation??? (A) time Additional simulation:  1 . 05 ϕ (0) = C ϕ (A) (1000 M )  C = 1 . 08 ϕ (A) (1000 M ) ϕ (0) = C ˙ ˙ 1 . 09 

ϕ (0) = C ϕ (A) (1000 M ) Supplementary simulation ϕ (A) (1000 M ) ϕ (0) = C ˙ ˙  1 . 05 � t  1 . 08 C = ∆ E := F E dt Energy absorbed by the black hole 1 . 09 0  80 60 40 � E 1.09 1.08 20 0 C � 1.05 � 20 0 2000 4000 6000 8000 10000 t � M The bosenova happens when E ≃ 1600 × ( f a /M p ) 2 M

Discussions E ff ective theory Gravitational waves

E ff ective theory ( 1 ) � ϕ 2 � d 4 x √− g � − 1 �� Action 2( ∇ ϕ ) 2 − µ 2 ˆ 2 + ˆ S = U NL ( ϕ ) , δ ν Non - relativistic approximation r p δ r 1 e − iµt ψ + e iµt ψ ∗ � � ϕ = √ 2 µ � i − 1 � 2 µ ∂ i ψ∂ i ψ ∗ + α g � ψ ∗ ˙ ψ ∗ � r ψ ∗ ψ − µ 2 ˜ ˆ ψ − ψ ˙ d 4 x U NL ( | ψ | 2 /µ ) S NR = 2 ∞ ( − 1 / 2) n ˜ � U NL ( x ) = − x n . ( n !) 2 n =2 Approximate the axion cloud as a Gaussian wavepacket ψ = A ( t, r, ν ) e iS ( t,r, ν )+ im φ ( ν = cos θ ) − ( r − r p ) 2 − ( ν − ν p ) 2 � � A ( t, r, ν ) ≈ A 0 exp , 4 δ r r 2 4 δ ν p S ( t, r, ν ) ≈ S 0 ( t ) + p ( t )( r − r p ) + P ( t )( r − r p ) 2 + π ν ( t )( ν − ν p ) 2 + · · · ,

E ff ective theory ( 2 ) E ff ective Lagrangian: L = T − V δ ν T = 1 r p + 1 p + 1 2 A ˙ r + B ˙ 2 D ˙ δ 2 r 2 δ 2 δ r ˙ 2 C ˙ ν , r p δ r 1 � 1 + δ ν + 1 1 � 1 V = + − µ α 2 2( α g µr p ) 2 (1 + δ r ) 4 δ r 4 δ ν ( α g µr p )(1 + δ r ) g � n − 1 ∞ � ( − 1 / 2) n N ∗ � − α − 2 . √ δ r δ ν ( α g µr p ) 3 (1 + δ r ) g ( n !) 2 n n =2 Α G � 0.1 Potential 5 N � � 0.02 Α G � 0.1 3.0 0 α g = 0 . 1 2.5 2.0 2 V � ΜΑ G Α G Μ r p 1.5 � 5 1.0 0.5 � 10 0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 N � � 0.08 N � 0.0 0.5 1.0 1.5 2.0 2.5 Α G Μ r p

Small oscillations Oscillation around a equilibrium point ∆ q i = ( ∆ δ r , ∆ δ ν , α g µ ∆ r p ) d 2 ( ∆ q i ) � = − ω ij ∆ q j dt 2 j Oscillation frequencies α g = 0 . 4 , N ∗ = 1 . 1 ω 2 = 1 . 141 , 0 . 249 , 0 . 0166 , ∆ t ≈ 761 M       0 . 110 0 . 075 − 0 . 378 δ q = − 0 . 027  , 0 . 724  , − 0 . 005  .    0 . 994 0 . 686 0 . 925 α g = 0 . 4 , N ∗ = 1 . 3 ω 2 = 14 . 06 , ∆ t ≈ 26 M 5 . 59 , 0 . 175 ,       0 . 218 0 . 070 − 0 . 640 ∆ q = − 0 . 030  , 0 . 927  , − 0 . 085  .    0 . 975 0 . 367 0 . 763

Discussions Axion cloud model Gravitational waves