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1 Black Holes in Modified Gravity Takahiro Tanaka (YITP)
2 Inspiraling-coalescing binaries 連星系からの重力波からは様々な情報を引き出せる Inspiral phase (large separation) クリーンな系 質点近似がよい星の内部構造はほとんど無視できる 正確な波形の予測が可能 for detection for parameter extraction for precision test of general relativity Merging phase 近年の数値相対論の目覚ましい発展 (Cutler et al, PRL (1993)) EOS of nuclear matter Electromagnetic counterpart Ringing tail - quasi-normal oscillation of BH
3 強い重力場で一般相対論は本当に正しいのか? Inspiraling binary は合体までに何周期もの重力波を出す およそ 1 周期程度のphaseのずれがあると区別できる 高精度な軌道パラメータの決定 ブラックホール時空の強重力場領域をマップ 3
4 Modified Gravity theory ダークマター / ダークエネルギーに対する alternative 重力理論の修正 どのような重力理論の修正が重力波で検出可能か? 重力波はどのような重力理論の修正に新た 重力波はどのような重力理論の修正に新たな制限を加えることができるのか?
5 Contents Randall Sundrum braneworld これはUV 側での modification という意味で他のものと毛色が違う Einstein dilaton Gauss Bonnet, Dynamical Chern Simons gravity Massive gravity Black holes in modified gravity Propagation of gravitational waves
6 Infinite extra dimension: Randall Sundrum II model Volume of the bulk is finite due to warped geometry although its extension is infinite. = l ( μ ν dz η ) μν dx dx l ds + z μ x?? Λ l 3 z :AdS curvature radius = 6 Λ Negative cosmological constant σ = Brane tension 4πG 5 l z Brane = l AdS Bulk Et Extension is infinite, it but t4 D GR seems to be recovered! 6
7 Gravity on the brane looks like 4D GR approximately, BUT for many years Schwarzschild like like BH solution had been unknown. 7
8 Black string solution ( Chamblin, Hawking, Reall ( 00) ) ds l = z ( ( ) ) Sch μ ν dz + g dx dx μν g μν ( x) Metric induced on the brane is exactly Schwarzschild solution. z However, this solution is singular. C μνρσ C μνρσ z 4 behavior of zero mode Kintaro candy solution Moreover, this solution is unstable. Gregory Laflamme instability 8
9 AdS/CFT correspondence Z[q]= d[φ] exp( S CFT [φ,q]) ( Maldacena ( 98) ) ( Gubser ( 01) ) ( Hawking, Hertog, Reall ( 00) ) Boundary = d[g] exp( S HE S GH +S 1 +S +S 3 ) exp(-w W CFT [q]) metric Counter terms 1 5 ( 5) 1 S 3 4 EH = d x g R + S κ 5 l 1 = d x q κ5 l 1 S GH = 4 d x q K l 4 κ 5 S = d x q 4 κ5 = L z 0 0 limit is well defined with the counter terms Brane position brane tension S 3 ( 4 ) R d[g] exp( S RS ) = d[g] exp( (S EH + S GH )+S 1 -S matter ) = exp( S S matter (W CFT + S 3 )) z 0 cutoff scale parameter 4D Einstein Hilbert action 9
10 Classical black hole evaporation conjecture 4D Einstein+CFT with the lowest order quantum correction number of field of CFT l κ 4 4D BH with CFT Hawking radiation in 4D Einstein+CFT picture Time scale of BH evaporation τ = M M M Solar 3 1mm l 10year (T.T. T ( 0) 0), Emparan et al ( 0)) AdS/CFT correspondence equivalent equivalent equivalent M& M ( Number ) species of Classical 5D dynamics in RS II model 5D BH on brane Classical evaporation of 5D BH G N 1 M 3 l ( G M ) 3 DECIGO/BBOで l <1.5μmまで制限がつけられる(Yagi, Tanahashi, Tanaka(011)) 10 N
11 However, static tti brane localized li dblack hole was finally obtained numerically Pau Figueras, Toby Wiseman (011) Abdolrahimi, Cattoen, Page, Yaghoobpour Tari (01) 得られた解はブレーン上でみると Schwarzschild 解に非常に近い この解が安定な物理的な解であるのか? 物理的な解であるなら CFT 的解釈はどうなるのか? なぜ ホーキング輻射が抑制されるのか? 11
12 Scalar tensor 理論 BH no hi hair: BH はスカラー hi hair を持たない ツルン 星はスカラー hair を持てる Einstein dilaton Gauss Bonnet, Dynamical Chern Simons gravity S G g θ R GB R R α d x d x g ( θ ) + V ( θ ) * G N N μν R GB = R 4R μν R + R αβ μν R μν αβ * R R = ε αβ σχ R [ ] σχ μν R μν αβ 相互作用が θ 曲率の高次項 θ が定数だと topological invariant で寄与しない 曲率の高次なので弱重力ではなかなか検出できない
13 Effective theory としての限界 古典的には微小パラメータはα r curvature 量子論的には x x h h θ h h x x α r curvature <<1 << 1 が要求される のような条件が必要 n 点の相互作用頂点をv 個挿入する h スカラーの propagator の数の増加 :P θ = v/ n 1 本 Graviton propagatorの数の増加 :P h = (n-1)v/ θ Strong coupling energy scale: 一番厳しい制限は n=3 3 のとき Λ c > ( 10μm) 1 重力の精密測定 Suppression: ( v ( Pθ + Ph ) v ( Pθ + Ph ) α M ) pl M pl Λ Λ = α 1 / n 1/ n c M pl α <10 13 cm
14 EDGB Cassini 衛星 BH の存在 (4Msol) 現時点での制限 1 α 1/ < cm (Amendola, Charmousis, Davis (007)) EDGB δ Φ 1 r 7 Newton θ " R " 1 r 1 r 1/ α EDGB < cm δ Φ Newton 4 θ 1 r " θ R" しかし EDGBにおいて 小質量 BH 解が存在しないという statement はeffective th. の適用外 CS Gravity Probe B, LAGEOS (Ali Haimound, Chen (011)) α 1 / CS < <10 13 cm
15 BH には毛があっても NS にはない EDGB, CS(slow rotation) のいずれもBH 解が知られており それぞれ scalar monopole, dipole をもつ EDGB, CSのいずれの場合もNSはmonopole chargeを持てない θ " R " 1 3 Q = 4 d x " R " = d x " R " T Topological invariantなので自明なtopology をもつ時空では d 1 θ zone x + t n x + t n x + L r EDGB: monopole source δ 3 (x z(t)) dipole radiation ( 1PN order) ( ) ( ) (source) far CS:dipole charge δ 3 (x z(t)) PN order (Yagi, Stein, Yunes, Tanaka (01))
16 Dipole radiation による EDGBへの制限 Low mass X ray binary, A / 5 < cm α (Yagi (01)) EDGB 将来の地上重力波観測 SNR=0, 6Msol+1Msol 1/ α EDGB < cm PN correction による CS への制限 dipole dipole が作る T μν 将来の地上重力波観測 : 距離 =100Mpc, a~0.4m α 1/ CS < cm metric の quadrupole 成分運動方程式への PN correction 低い PN 次数の補正に対しては合体直前でない連星でも制限がつけられる
17 重力波の伝播 重力波の伝播が普通ではないモデルで 観測的に矛盾のないものはあるだろうか? Chern Simons Modified Gravity 背景のθ の宇宙論的な時間変化があるとしてもなかなか見えない & 66 αθ < 3 10 cm : J (double 3039(d pulsar) (Yunes & Spergel, arxiv: ) L, R 1 右巻きと左巻きの重力波で振動数に依存して振幅が異なる h = h + + ih ( ) ( ) ( ) ( ) = ± z L, R L, R 5/ 7 dθ d θ h hgr exp 64π α f H 0 dz 1+ z + 1+ z 0 dz dz L, R h exp f O 10 3 α & GR ± θ Massive bi gravity ( ) ( ( )) massive と massless の gravitonが両方存在 ν 振動のようなことが期待できる 問題のない宇宙モデルがつくれるのか? ( ) ( ) ( ) ( )
18 L = Ghost free massive bi gravity ~ ~ g R g R + + g 16πG 16πG κ n 4 V n n N N = 0 V =1 0 V1 = τ1 V =τ1 τ n [ ] i ik n Tr γ j kj τ γ c + g g ~ L matter L ~ g を固定した場合がde Rham Gabadadze Tolley massive gravity Metric 10 成分 -constarints 4 成分 =6 成分 massive spin は5 成分なので 1 成分余っている scalar singlet でghost( 運動項が逆符号 ) になる constraintがlagrange multiplierであるlapseやshiftを完全に決めない場合 constraint が閉じず 余分の consistency 条件が現れ 余分の1 成分が消えるこのmodelで g ~ もdynamicalにしてもやはりghost freeであることが示された (Hassan, Rosen (01))
19 ()( t dt dx ) ( t) c ( t) dt ds = a + ~ d s = b + dx ( mass ) FLRW background ( ) ( ) δ S T mass μν = δ g (Comelli, Crisostomi, Nesti, Pilo (01)) ( mass) T μν = 0 6c 3ξ + 4c ξ + c 1 cba ab = μ ( )( ) 0 μν branch 1 branch ξ b branch 1: 線形摂動を考えると期待されるスカラーやベクトルタイプの重力のモードが現れない Strong coupling? branch : 線形摂動で期待される全てのモードが現れるbranch a
20 まとめ 重力の修正を重力波で捉えるということを考えると 奥が深い Randall Sundrum braneworld 静的解の発見で BH の古典的蒸発仮説は間違っているように思われるが どのように解釈すべきかは依然謎である Einstein dilaton Gauss Bonnet, Dynamical Chern Simons gravity 4 次元の簡単なモデルであるが 最近まで調べられておらず モデルに対する現状の制限が弱く 重力波で飛躍的に強い制限が得られる例を与えている Massive gravity 観測可能な graviton 振動を与える可能性のあるモデル Black holes in modified gravity Propagation of gravitational ti waves
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