Towards a Search for Stochastic Gravitational-Wave Backgrounds from Ultra-light Bosons

Transcription

1 Towards a Search for Stochastic Gravitational-Wave Backgrounds from Ultra-light Bosons Leo Tsukada RESCEU, Univ. of Tokyo The first annual symposium "Gravitational Wave Physics and Astronomy: Genesis" January 5, 2018 In collaboration with: Kipp Cannon Patrick Meyers (Univ. of Minnesota) 1/26

2 Outline Introduction Theory : Superradiant instability Formulation : Stochastic GW background Method : Bayesian Parameter Estimation Results : Injection tests, Scramble O1 data Conclusion Outline 2/26

3 Outline Introduction Theory : Superradiant instability Formulation : Stochastic GW background Method : Bayesian Parameter Estimation Results : Injection tests, Scramble O1 data Conclusion Outline 3/26

4 Gravitational waves Stochastic background (SGWB) : SNR below threshold and superposed unresolvable signals - large number of compact objects or bursts - inflation origin - axion-like bosons with low mass around BHs > explain a potential abundance of low-spin BHs 4/26

5 Basics Energy density spectrum of SGWB Ω GW ( f ) 1 ρ c dρ GW d ln f ρ GW = c2 32πG!h µν (t,! x)! h µν (t,! x) Naive statistics S SNR T /2 T /2 dts 1 (t)s 2 (t) S T df f 3 Ω GW ( f ) S 2 S 2 df P 1 ( f )P 2 ( f ) Both are random variables! s(t) = h(t) + n(t) 1/2 5/26

6 Outline Introduction Theory : Superradiant instability Formulation : Stochastic GW background Method : Bayesian Parameter Estimation Results : Injection tests, Scramble O1 data Conclusion Outline 6/26

7 Schematic picture Ω H Boson condensate M S continuous GWs ω GW 2m b! Brito. et al (2017) τ inst τ GW M 0.07χ 1 10M 0.1 M µ 9 yr M χ 1 10M 0.1 M µ 15 yr instability condition 0 < ω < mω H satisfied M / J extraction from a BH Boson condensate GW emission 7/26

8 Bosonic Cloud Formation Action S = Field equations d 4 x g Brito. et al (2017), Kodama. et al (2012) R 16π 1 2 gµν Ψ Ψ µ 2,µ,ν 2 Ψ2 µ = m b! i G µν = 8πT µν i µ µ Ψ = µ 2 Ψ scalar boson mass where T µν = Ψ,µ Ψ,ν 1 ( 2 gµν Ψ,α Ψ,α + µ 2 Ψ 2 ) (modified) Teukolsky equation (s = 0, mass term) 8/26

9 Potential Bosonic Cloud Formation Ψ = dωe iωt+imϕ 0 S lm (θ)r lm (r) Arvanitaki. et al (2009) Radial function r (Δ r )R lm + Δ 1 ω 2 (r 2 + a 2 ) 4aMrmω + a 2 m 2 Boundary conditions At the event horizon d 2 u(r * ) dr *2 ω < mω H d 2 u(r * ) dr *2 { } ( µ 2 r 2 + λ lm + a 2 ω 2 ) + ( ω 2 V(ω,r) )u(r * ) = 0 + ( ω mω ) 2 H u(r * ) = 0 u(r * ) e i(ω mω H )r * : Energy extraction! (Δ = r 2 2Mr + a 2 ) R lm = 0 (dr * = (r 2 + a 2 )dr / Δ, u = r 2 + a 2 R lm ) Ergo-region! Black Hole Horizon V(ω,r) Barrier region r * Potential Well Exponential growth region Mirror at r~1/μ 9/26

10 Bosonic Cloud Formation Spatial distribution Ψ :l = 1,m = 1 T µν :l = 2,m = 2 Low energy density Ignore its backreaction 10/26

11 Gravitational-wave emission Continuous waves Ψ = dω e i ω lmn t+imϕ 0 S lm (θ)r lm (r) T µν = Ψ,µ Ψ,ν 1 ( 2 gµν Ψ Ψ,α + µ 2 Ψ 2 ),α e i 2ω R t Constant frequency 2ω R 2m b! M BH O( 10M ) Unstable range M BH µ 0.5 µ M BH Dolan (2007) Brito. et al (2017) 11/26

12 Outline Introduction Theory : Superradiant instability Formulation : Stochastic GW background Method : Bayesian Parameter Estimation Results : Injection tests, Scramble O1 data Conclusion Outline 12/26

13 Overall formula Ω GW ( f ) = f dρ GW ρ c df = f ρ c SNR<8 contribution to SGWB dχ dm BH dz dt dz p(χ) d!n dm BH de s df s p(χ) :initial spin distribution d!n BH formation rate per unit M (Core collapse supernova model) dm BH considers M BH [3M, 50M ] Energy spectrum for a given M, z in the source frame de s ( ) df s = E GW δ f (1+ z) f s the detector frame 13/26

14 Radiated energy Boson condensate (t < τ inst )!M =!M S!J = m!m ω "M = ω R S R m " J M S max M i, J i τ inst 1 ω I M f, J f J f = m ω R (M f M i ) + J i At superradiance saturation ω = mω H J f = 4mM f 3 ω R m 2 + 4M f2 ω R 2 M max S M i M f = M i m3 m 6 16m 2 ω R (mm i ω R J i ) 2 8ω R 2 (mm i ω R J i ) GW τ GW Boson condensate M f, J f 14/26

15 Radiated energy GW emission!m S =!E GW = M S (t) = Δt E =!M (t)dt = GW S 0 = M max Δt S Δt + τ GW (τ inst < t < τ GW ) M S max 1+ t τ GW π (M M S µ)14 M BH 0 Δt M S max τ GW (t + τ GW ) 2 dt signal duration Δt = min(τ GW,1 H 0 ) M S max 0.1M i (a 1, M i 50M ) M τ GW χ 1 10M M µ 15 yr τ inst 1 ω I GW τ GW M i, J i Boson condensate M f, J f M S max 15/26

16 Energy density spectrum m b = [ev] m b = [ev] m b = [ev] 10 8 ΩGW p(χ) frequency(hz) 16/26

17 Outline Introduction Theory : Superradiant instability Formulation : Stochastic GW background Method : Bayesian Parameter Estimation Results : Injection tests, Scramble O1 data Conclusion Outline 17/26

18 Bayesian inference Parameter estimation ( ) = L ( d m b, χ; H )π ( m, χ H ) b Z ( d H ) P m b, χ d ; H posterior likelihood evidence posterior : probability of the values given data d prior m b, χ likelihood : probability of obtaining data given parameters m b, χ d 18/26

19 sgwb_model_selection Python package used for parameter estimation { log L( d m, χ; H )} b ( Y ˆ ) ( f ) γ ( f )Ω( f,m, χ) 2 b 2σ 2 ( f ) d f Y ˆ( f ) 20π 2 f 3!s 2 1 ( f )!s 2 ( f ) Y ˆ = γ ( f )Ω( f ) 3H 0 σ 2 ( f ) = 1 2T Δf 10π 2 2 3H 0 2 in the presence of a signal f 6 P 1 ( f )P 2 ( f ) π ( m, χ H ) uniform (10 13 ev m ev) b b uniform (0 χ 1) 19/26

20 Outline Introduction Theory : Superradiant instability Formulation : Stochastic GW background Method : Bayesian Parameter Estimation Results : Injection tests, Scramble O1 data Conclusion Outline 20/26

21 Initial spin distribution Optimistic case Uniform distribution -> Varying lower limit χ ll ( χ ul =1) 21/26

22 χ ll case m b = ev χ ll = m b = [ev] χ ll =0.8 fake noise 10 6 ΩGW frequency(hz) 22/26

23 Scramble O1 noise test Shift the timestamp of two data streams from HL. 1 second timeshift no astrophysical signal Y ˆ( f ) 20π 2 f 3!s ( f )!s ( f ) 3H 2 H L 0 σ 2 ( f ) = 1 2T Δf 10π 2 3H f 6 P H ( f )P L ( f ) 23/26

24 Result ( χ ll case) 24/26

25 Outline Introduction Theory : Superradiant instability Formulation : Stochastic GW background Method : Bayesian Parameter Estimation Results : Injection tests, Scramble O1 data Conclusion Outline 25/26

26 Conclusion Injection tests Consistent results are obtained -> More trials with different injections are necessary for statistics Scramble O1 data tests Reasonable constraints are placed -> Use the real O1 data and O2 data. 26/26

27 THANK YOU FOR LISTENING!

28 EXTRA 28/26

29 Why!E (!M ) M 2? GW S S Teukolsky formalism (s=-2) Newman-Penrose scalar (Schwarzshild approx.) l!m Ψ 4 (t,r,ω) = 1 dω e iωt Y ( r 4 l!m Ω)R l!m (r) G µν = 8πT µν r 2 Teukolsky equation -> radial equation + source term 1 2M r 2 r R 2 ( r M ) R + 1 2M r r 1 {!ω 2 r 2 4i!ω(r 3M )} (l 1)(l + 2) R = T l!m!ω R l!m 1 r T l!m!ω 1 r T µν 29/26

30 Why!E (!M ) M 2? GW S S Ψ 4(r ) 1 2 ( h!! i h!! ) + de GW dt ( ) = r 2 = r 2 16π dω h+! 2 + h! T Schematic picture l!m!ω M S 4π M S = dω ( Ψ 4 dt) Ψ 4 dt T t t gdrdθdϕ ( ) * total energy of the scalar field In the Schwarzschild limit and M µ 1 de GW dt π (M M S µ)14 M BH 2 30/26

31 Schwarzschild approx. T l!m!ω 2π = 2 ( l 1 )l( l +1) ( l + 2) 1/2 r 4 0T +2 ( l 1)l ( l +1) ( l + 2) 1/2 r 2 f Ĵ r 3 f 1 1T S T 1 2π +rf Ĵ r 4 f 1 dωdt!t S Y l!m e i!ωt Ĵ [ r T ] 2 where!t S = T nn,t nm,t mm for S = 0, 1, 2 f 1 2M r Ĵ f r + iω T nn T µν n µ n ν,t nm T µν n µ m ν,t mm T µν m µ m ν n µ = 1 ( 1, f,0,0), m µ = 1 2 2r 0,0,1, i sinθ 31/26

32 Solution to a Sturm-Liouville Eq. r ( Δ 1 )R + Δ 2 Δ 1 K 2 + 4i(r M )K r { } 8i!ωr λ R = T Δ 2 l!m!ω two independent homogeneous solutions R H R Δ2 e ikr * (r * ) r 3 B out e i!ωr * + r 1 B in e i!ωr * (r * ) A * out eikr + Δ 2 A in e ikr* (r * ) r 3 e i!ωr * (r * ) R lm (r * G(r *, x * )T ) = dx * l!m!ω Δ 2 = 1 R r * R H T dx * l!m!ω + R H R T dx * l!m!ω W l Δ 2 r * Δ 2 32/26

33 Gravitational-wave emission At infinity (Schwarzschild background) r R 1 dx RH T (r ) l!m!ω W Δ 2 r 3 e i!ωr = Z l!m!ω r 3 e i!ωr l Z l!m!ω r dxr H T l!m!ω / Δ2 W l δ 2l { δ δ (!ω 2ω )Z + δ δ (!ω } + 2ω )Z 2!m R 2,2,2ω R 2!m R 2, 2, 2ω R Wronskian W l 1 Δ R dr H dr dr RH dr 33/26

34 Potential Bosonic Cloud Formation Boundary conditions Complex eigenfrequency ω lmn = ω R + iω I ( M µ 1 ) Richard. et al (2017), Kodama. et al (2012) µ µ 2 M µ l + n +1 ω I > 0 ω R < mω H energy extraction + bound states 2 + i2µγ r ( mω µ )( M µ ) 4l+4 l + H Ergo-region V(ω,r) Barrier region Potential Well Exponential growth region Mirror at r~1/μ Superradiant instability! Black Hole Horizon r * Arvanitaki. et al (2009) 34/26

35 Bosonic Cloud Formation radial profile ( Ψ :l = 1,m = 1 mode) M µ 1 ( ) m b = ev M BH = 50M Low energy density Ignore its backreaction 35/26

36 Bosonic Cloud Formation Angular profile Ψ :l = 1,m = 1 T µν :l = 2,m = 2 m b = ev M BH = 50M Low energy density Ignore its backreaction 36/26

37 Connection between T Ψ. µν Gravitational atom in the limit M µ 1 { } ( µ 2 r 2 + λ lm + a 2 ω 2 ) r (Δ r )R lm + Δ 1 ω 2 (r 2 + a 2 ) 4aMrmω + a 2 m 2 d 2 dr 2 ( ) + ω 2 µ 2 + 2M µ 2 rr lm r l ( l +1 ) r 2 R lm A 0!r l e!r/2 L n 2l+1 (!r) 0 S lm P l m (cosθ) Ψ ( l=1,m=1) A 0!re!r/2 cos( ϕ ω R t)sinθ rr lm ( ) = 0 R lm = 0 T µν = Ψ,µ Ψ,ν 1 ( 2 gµν Ψ,α Ψ,α + µ 2 Ψ 2 ) Z l!m!ω r dxr H T l!m!ω / Δ 2 W l 37/26

38 Gravitational-wave emission Ψ 4(r ) 1 ( r Z 2,2,2ω R Y 2,2 (Ω)e i2ω R (r t ) + Z 2, 2, 2ω R Y 2, 2 (Ω)e i2ω R (r t )) = 1 2 h!! + i!! ( h ) de GW = r 2 Schematic dt 16π dω ( h+! 2 + h! 2 ) = r 2 picture 4π dω Ψ 4 dt 1 = 4π ( 2ω ) Z 2 2 2,m,mω R M 2 S m=±2 R M BH ( )( Ψ 4 dt) * M S = T t t gdrdθdϕ total energy of the scalar field de GW dt π (M M S µ)14 M BH 2 38/26

39 BH formation rate star formation rate initial mass fraction function a relation between stellar mass and BH mass 39/26

40 Initial spin distribution Three types Uniform distribution χ ul -> Varying upper limit (lower limit =0) χ ll -> Varying lower limit (upper limit =1) Delta-function distribution -> Varying peak value χ delta 40/26

41 χ ul case m b = ev χ ul = m b = [ev] χ ul =0.8 fake noise 10 6 ΩGW frequency(hz) 41/26

42 Nested sampling Compute evidence efficiently Evidence P( d H ) = P( d θ, H )P( θ H )d D θ Z = M i=1 1 = L( X)dX L i(x i 1 X i ) θ 2 X(λ) = L(θ )>λ P( θ H )d D θ L Posterior 4 L p j = 1 3 L 2Z L (X X ) 2 L 1 j j 1 j L 0 L 0 X 0 = 1 θ 1 L 4 L 3!L L 2 L 1 evidence Z X 4 X 3! X X 2 X 1 42/26