Gravitational waves from binary black holes

Transcription

1 Gravitational waves from binary black holes Hiroyuki Nakano YITP, Kyoto University DECIGO workshop, October 27, 2013 Hiroyuki Nakano Gravitational waves from binary black holes

2 Binary black holes (BBHs) 1M 1.5 km s

3 Gravitational waves from BBHs Gravitational waveform Inspiral Merger Ringdown t/m Inspiral: Post-Newtonian Merger: Numerical Relativity Ringdown: Black hole perturbation

4 Gravitational wave observations of BBHs ( ) Various gravitational wave detectors and f merge M M Hz (from M 10 6 M 10 3 M

5 Misc. (from By Christopher Moore, Robert Cole and Christopher Berry from the Gravitational Wave Group at the Institute of Astronomy, University of Cambridge

6 Numerical INJection Analysis (NINJA): 1 NINJA-1 : Numerical relativity (NR) and Data analysis (DA) communities 23 numerical waveforms (10 NR groups) Injected into Gaussian noise colored with the frequency sensitivity of first generation detectors Search and parameter-estimation (9 DA groups) Aylott et al., Class. Quantum Grav. 26 (2009) Aylott et al., Class. Quantum Grav. 26 (2009) Major limitations No length or accuracy requirements for the NR waveforms No non-gaussian noise transients

7 Numerical INJection Analysis (NINJA): 2 NINJA-2 : for real science Use real noise through LSC/Virgo MOU More NR waveforms (13 NR groups) PN-NR stitching to cover mass down to 10M. Accuracy requirements 5 usable orbits before merger NR (2, 2) amplitude accuracy below 5% NR (2, 2) accumulated phase error 0.05 radian GW stitching at Mω 2, Hybrid GWs start at Mω 2, ( 10M at 20Hz) Highest PN order available for phase and amplitude Only l = m = 2 required, but all harmonic modes welcome.

8 Numerical INJection Analysis (NINJA): 3 Aligned (anti-aligned) spinning (non-precessing) cases 10 q Χ2 0.0 Χ q: mass ratio, χ i : dimensionless spins

9 Numerical INJection Analysis (NINJA): 4 Equal-mass, equal-spin waveforms (χ = (χ 1 + χ 2 )/2) 10M, 100Mpc Ajith et al., Class. Quantum Grav. 29 (2012) Ajith et al., Class. Quantum Grav. 30 (2013)

10 Numerical INJection Analysis (NINJA): 5 In progress Real data from GW detectors Recolored data for the sensitivities expected from aligo/avirgo in blind injections

11 Recent progress: 1 A catalog of 171 high-quality binary black-hole simulations for gravitational-wave astronomy, Mroue et al., [arxiv: ] π π e χ cos θ φ q Norbits Black hole A Black hole B χ: dimensionless spin, (θ, φ): initial spin direction, q = m 1 /m 2 : mass ratio, N orbits : number of orbits before merger, e: initial eccentricity

12 Recent progress: 2 A catalog of 171 high-quality binary black-hole simulations for gravitational-wave astronomy, Mroue et al., [arxiv: ] (See also Pekowsky et al., [arxiv: ], over 220 waveforms)

13 Numerical Relativity/Analytical Relativity (NRAR): 1 Collaboration between Numerical and Analytical Relativity Abbreviation JCP FAU GATech RIT Lean AEI PC SXS UIUC NR groups Group name Jena-Cardiff-Palma Florida Atlantic University Georgia Tech Rochester Institute of Technology Ulrich Sperhake Albert Einstein Institute Palma-Caltech Simulating extreme Spacetimes (Caltech, Cornell, CITA, CSU Fullerton) University of Illinois, Urbana-Champaign Hinder et al., [arxiv: ]

14 Numerical Relativity/Analytical Relativity (NRAR): 2 Plan: Targets: Stage 1: Basic coverage (q = 1, 2, 3, mild spins) Stage 2: Additional precessing configurations Stage 3: More challenging configurations Higher mass ratios (q 10) Large spins ( χ > 0.95) Long (> 40orbits) 20 usable GW cycles Eccentricity: Aim for e < as conservative target Phase error φ(t) < 0.25radians up to Mω GW = 0.2 (< 1orbit before merger) Relative amplitude error A/A < 0.01 [See Ian Hinder,

15 Numerical Relativity/Analytical Relativity (NRAR): 3 First stage of the NRAR collaboration (60 simulations, for spinning non-precessing configurations) Χ q Χ1 0.5

16 Numerical Relativity/Analytical Relativity (NRAR): 4 Analyzed configurations in Hinder et al., [arxiv: ] # Group Label q S 1 /m1 2 S 2 /m2 2 M f /M χ f 1 JCP J1p ( 0.128, 0.171, 0.494) (0.129, 0.149, 0.106) J FAU F1p (0.000, 0.520, 0.300) (0.520, 0.000, 0.300) F GATech G G G G RIT R Lean L4 * L AEI A A PC P P SXS S * S * S S S S3p (0.260, 0.005, 0.150) S1p (0.054, 0.514, 0.305) (0.054, 0.514, 0.305) S1p (0.000, 0.520, 0.300) (0.000, 0.520, 0.300) S UIUC U q = m 1 /m 2, S i /m 2 i : dimensionless spin, M = m 1 + m 2, M f and χ f : mass and dimensionless spin of the final BH

17 Numerical Relativity/Analytical Relativity (NRAR): 5 Error estimates for the NR simulations φ (radians) A/A usable GW cycles at ω 22, ref at ω 22, ref (aligned) early inspiral JCP FAU GATech Lean AEI PC SXS UIUC RIT Finite numerical resolution Waveform measurement at finite distance from the source Computation of h from ψ 4

18 Numerical Relativity/Analytical Relativity (NRAR): 6 Spinning non-precessing template models Comparison with time-domain SEOBNRv1 model F _ 30% 10% 3% 1% 0.3% 2: J : F : G : G : P : P : S : U % M (M sun ) M (M sun ) Spin-aligned EOBNR model. SEOBNRv1 1% 0.8% 0.6% 0.4% 0.2% 6: G : G : L4 * 12: A : A : S * (Calibrated) 17: S * (Calibrated) 18: S : S : S

19 Numerical Relativity/Analytical Relativity (NRAR): 7 Spinning non-precessing template models Comparison with Frequency-domain phenomenological IMRPhenomB F _ 10% 3% 4: F : G : A : P : P : S : S : S : S : S IMRPhenomB 3% 2% 2: J : G : G : G : L4 12: A : S : U % 1% 0.3% M (M sun ) M (M sun ) Frequency domain (non-precessing spins) inspiral-merger-ringdown templates of Ajith et al. (2011).

20 Numerical Relativity/Analytical Relativity (NRAR): 8 Spinning non-precessing template models Comparison with Frequency-domain phenomenological IMRPhenomC 10% 3% IMRPhenomC 2% 1.5% 2: J : F : G : G : G : G : L4 12: A : P : S : U F _ 1% 1% 13: A : P % 16: S : S : S % 20: S : S % M (M sun ) M (M sun ) Frequency domain (non-precessing spins) inspiral-merger-ringdown templates of with phenomenological coefficients defined in the Table II of Santamaria et al. (2010).

21 Numerical Relativity/Analytical Relativity (NRAR): 9 Non-spinning q = 10 waveform 30% 10% F _ 3% 1% 0.3% 0.1% EOBNRv2 SEOBNRv1 IHES EOB IMRPhenomB IMRPhenomC 0.03% 0.01% M (M sun ) Unfaithfulness of analytical non-spinning q = 10 waveforms generated by IMRPhenomB, IMRPhenomC and three versions of EOB models with the numerical non-spinning q = 10 waveform (R10).

22 Intermediate-mass-ratio BBHs Mass ratio: 1/10 q 1/100 Full numerical simulations and Analytic treatments NR: challenge in the exploration of BBH parameter space AR: Post-Newtonian approach (v 1) Effective one body approach Gravitational self-force (q 1) and so on. Simple as possible Regge-Wheeler-Zerilli formalism (BHP) + remnant BH s spin TaylorT4 orbital phase evolution (PN) + fitting parameters Lousto et al., Phys. Rev. Lett. 104, (2010). Lousto et al., Phys. Rev. D82, (2010). Nakano et al., Phys. Rev. D84, (2011).

23 Final remnant of BBH mergers Final remnant black hole s spin q = 1/10 q = 1/15 q = 1/100 Non-dim. spin ± ± ± We want to introduce the spin effect into the black hole perturbation approach. BH (M = m 1 + m 2 ) + particle (µ = m 1 m 2 /M) M µ

24 Analytic, perturbative approach Spin-Regge-Wheeler-Zerilli (SRWZ) formalism Extension of the RWZ for Schwarzschild perturbations. Include, perturbatively, a term linear in the remnant BH s spin. 2nd order perturbations. Ψ lm (t, r) = Ψ (1) lm Ψ (o) lm = Ψ(o,1) lm Ψ (1) lm, Ψ(2) lm : (t, r) + Ψ(2) lm (t, r), (t, r) + 2 Even parity Zerilli function dt Ψ (o,z,2) lm (t, r), Ψ (o,1) lm : Odd parity Regge-Wheeler function Ψ (o,z,2) lm : Odd parity Zerilli function

25 The SRWZ formalism Example (even parity wave equation) 2 t 2 Ψ lm (t, r) + 2 r 2 Ψ lm (t, r) V (even) l = S (even) lm (t, r; r p(t), φ p(t)), (r)ψ lm (t, r) + i m α ˆP (even) l Ψ lm (t, r) r = r + 2M ln[r/(2m) 1] α: Nondimensional spin parameter V (even/odd) l : Potentials ˆP (even) l : Differential operator S lm : Source term (2nd order) We need the particle s trajectory (r p (t), φ p (t)).

26 Trajectory: Orbital frequency Ω evolution Based on TaylorT4 evolution [Boyle et al. (2007)] dω = 96 [ dt 5 Ω11/3 M 5/3 η (1+B (Ω/Ω 0 ) β/3) ( ) 4 η (MΩ) 2/3 + 4 π MΩ ( η + 59 ) ( 18 η2 (MΩ) 4/ π 189 ) 8 η π (MΩ) 5/3 ( π γ ) ) η3 ( (MΩ) π η π η2 π ln (64 MΩ) η η π η2 ] (MΩ) 7/3 +A (Ω/Ω 0 ) α/3, Ω = dφ dt, M = m 1 + m 2, η = m 1m 2 M 2, A, α, B and β: Fitting parameters MΩ 0 = (1/3) 3/ at R Sch = 3M for circular orbit. α > 7 and β > 7 to be consistent with the 3.5PN formula.

27 Trajectory: Orbital radius Based on the ADM (Arnowitt, Deser and Misner)-TT PN (NR ADM-TT/ trumpet stationary 1 + log slice of Schwarzschild) M R = (MΩ) 2/3 + [ 1 + ( ) η (MΩ) 2/3 + ( η + 19 ) η2 (MΩ) 4/3 ( η η π2 3 2 η2 + 2 ) 81 η3 R and Ω: in the NR coordinates a 0, a 1, C: Fitting parameters MΩ 0 = (1/3) 3/ (MΩ) 2 ]/ ( 1+a 0 (Ω/Ω 0 ) a 1 ) +C, a 1 > 2 to be consistent with the 3PN calculation. C Inconsistent with the ADM-TT PN formula. But, we need!

28 Wave calculation in the SRWZ formalism Radial transformation to remove the offset C between the NR and the trumpet coordinates by assuming T NR = T Log, R NR R Log = R NR C. To the standard Schwarzschild coordinates, Final plunge trajectory : (T Log, R Log ) (T Sch, R Sch ), Plunging (Schwarzschild) orbit from a matching radius R M 3M to the horizon R = 2M: Geodesic without the radiation reaction. The SRWZ waveforms at a sufficiently distant location R Obs : R Obs M (h+ i h ) = lm (l 1)l(l + 1)(l + 2) ( 2M Ψ (even) lm ) i Ψ (odd) lm 2Y lm.

29 Short summary for wave calculation Extended TaylorT4 (Trajectory) + SRWZ formalism (Wave generation) h + (r obs /M) t/m

30 Wave extrapolation for NR waveforms Perturbative formula for NR waveforms We extrapolate waveforms to r, [ lim [r ψlm r 4 (r, t)] = r ψ4 lm + O(R 2 Obs ), (r, t) (l 1)(l + 2) 2 t 0 dt ψ lm 4 (r, t) ] r Obs : Approximate areal radius of the sphere R Obs =const. r=r Obs This formula gives reliable extrapolations for R Obs 100M. (Numerical study by [M. C. Babiuc et al. (2011)]) ψ 4 h: pygwanalysis code [Reisswig and Pollney (2011)] in EINSTEINTOOLKIT

31 Results: Gravitational wave phase (q = 1/10) NR vs. SRWZ waveforms q = 1/10, φ = 0 at t = 830M. (l = 2, m = 1) 0-10 NR SRWZ -20 (l = 2, m = 2) NR SRWZ t/m (l = 3, m = 3) 0 NR SRWZ t/m t/m

32 Results: Gravitational wave phase (q = 1/15) NR vs. SRWZ waveforms q = 1/15, φ = 0 at t = 600M. (l = 2, m = 1) 0-10 NR SRWZ (l = 2, m = 2) NR SRWZ t/m (l = 3, m = 3) 0 NR -20 SRWZ t/m t/m

33 Results: Gravitational wave phase (q = 1/100) NR vs. SRWZ waveforms q = 1/100, φ = 0 at t = 85M. (l = 2, m = 1) 0 NR SRWZ -5 (l = 2, m = 2) NR SRWZ t/m (l = 3, m = 3) NR SRWZ t/m t/m

34 Matching and next plan Match between the NR and SRWZ (l = 2, m = 2) GWs in aligo (Zero Det, High Power). (Integration from f low 10Hz.) q = 1/10 q = 1/15 q = 1/100 Range (MΩ 22 ) Total mass (M ) M Next plan: Longer NR simulations, Various q cases Fitting parameters in fitting functions for the trajectory (A, α, B, β, a 0, a 1, C) = c 0 η c 1.

35 Summary Gravitational waves from binary black holes Total mass Frequency Detectors Analytical Relativity Numerical Relativity Data Analysis Inspiral Merger Ringdown t/m

36 suppl.: Fitting for orbital frequency dω/dt(ω) for q = 1/10 Up to MΩ = 0.175: (NR orbital frequency around R Sch = 3M) End point of the NR curve: M 2 dω/dt e-05 1e-06 NR (q=1/10) Fitting PN R Sch = 2M 0.1 MΩ Black: NR, Red: Fitting, Blue: PN Mass-ratio A α B β q = 1/ (> 7) (> 7) q = 1/ (> 7) (> 7) q = 1/ (< 7) (> 7)

37 suppl.: Fitting for orbital radius 10 NR (q=1/10) Fitting PN R(Ω) for q = 1/10. Up to MΩ = (R Sch = 3M) End point of the NR curve: R Sch = 2M R/M 1 MΩ Black: NR, Red: Fitting, Blue: PN 0.1 Mass-ratio C a 0 a 1 q = 1/ ( 0) (> 2) q = 1/ ( 0) (> 2) q = 1/ ( 0) (> 2)