Simulations of a BH-axion system Hirotaka Y oshino ( KEK ) Hideo Kodama Prog. Thoer. Phys. 128, 153-190 (2012), arXiv:1203.5070[gr-qc] 3rd ExDiP2012 conference @ Grandvrio Hotel, Obihiro, Hokkaido ( August 8, 2012 )
Contents Introduction Code Simulation Typical two simulations Does the bosenova really happen? Discussion Comparison with BEC system Gravitational waves Summary
Contents Introduction Code Simulation Typical two simulations Does the bosenova really happen? Discussion Comparison with BEC system Gravitational waves Summary
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.
Kerr BH Metric Ergo region � ∆ − a 2 sin 2 θ dt 2 − 2 a sin 2 θ ( r 2 + a 2 − ∆ ) � ds 2 = − dtd φ Σ Σ g tt > 0 � ( r 2 + a 2 ) 2 − ∆ a 2 sin 2 θ � sin 2 θ d φ 2 + Σ ∆ dr 2 + Σ d θ 2 + Σ Σ = r 2 + a 2 cos 2 θ , ∆ = r 2 + a 2 − 2 Mr. J = Ma BH
Energy extraction BH’s rotational energy M rot = M − M irr BH � A H M irr = 16 π Methods of energy extraction Penrose process Blandford - Znajek process Superradiance ( Next slide )
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
Black hole bomb Press and Teukolsky ( 1972 ) mirror BH
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 0.19 distant region near horizon 0.18 V Superradiant 0.17 u ∼ e − √ condition: u ∼ e − i ( ω − m Ω H ) r ∗ µ 2 − ω 2 r ∗ 0.16 V ω < Ω H m 0.15 ! 2 0.14 -100 -50 0 50 100 r * / M 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 ∗
Growth rate Growth rate calculated by Time evolution continued fraction method l = m = 1 Mµ = 0 . 4 Dolan, PRD76 ( 2007 ) , 084001. 1e-06 a = 0.999 l = 1, m = 1 a = 0.99 a = 0.95 a = 0.9 a = 0.8 a = 0.7 1e-07 l = 2, m = 2 Φ 1e-08 Im( ! / ! ) l = 3, m = 3 1e-09 1e-10 r ∗ 1e-11 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 M ! Φ ∼ e − i ω t e − i ˜ ω r ∗ ( Near the horizon ) ω = ω − m Ω H ˜
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 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 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.
Contents Introduction Code Simulation Typical two simulations Does the bosenova really happen? Discussion Comparison with BEC system Gravitational waves Summary
Our code 3D code of coordinates ( r ∗ , θ , φ ) Comparison with semianalytic solution of the Klein - Gordon case t/M = 0 ∼ 100 Φ r ∗ ω (CF) /µ = 3 . 31 × 10 − 7 ˙ 2 E ≃ E (100 M ) − E (0) E I ω I = 200 ME (0) ω (Numerical) /µ = 3 . 26 × 10 − 7 I
Contents Introduction Code Simulation Typical two simulations Does the bosenova really happen? Discussion Comparison with BEC system Gravitational waves Summary
Contents Introduction Code Simulation Typical two simulations Does the bosenova really happen? Discussion Comparison with BEC system Gravitational waves Summary
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
Simulation ( B ) Snapshots t = 500 M 4 6 1 3 t = 350 M 5 0.5 t = 700 M 2 4 0 t = 0 -0.5 1 ! ! 3 -1 0 2 -1 1 -2 0 -200 -150 -100 -50 0 50 100 150 200 -100 -50 0 50 100 r * / M r * / M Φ ∼ e − i ω t e − i ˜ ω r ∗ m= - 1 mode is generated! ( Near the horizon ) ω = ω − m Ω H ˜ M ω KG = 0 . 39 M ω NL = 0 . 35 M ˜ ω NL = 0 . 87 M ˜ ω KG = − 0 . 04
Summary of the simulations ( A ) and ( B ) ( A ) When the peak value is not very large, the nonlinear term enhances the rate of superradiant instability. The nonlinear e ff ect makes energy distribute in the neighborhood of the black hole. ( B ) When the peak value is su ffi ciently large, the bosenova collapse happens. Once the bosenova happens, positive energy falls into the black hole, while the angular momentum continues to be extracted.
Contents Introduction Code Simulation Typical two simulations Does the bosenova really happen? Discussion Comparison with BEC system Gravitational waves Summary
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
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