Mining information from unequal-mass binaries
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1 Mining information from unequal-mass binaries U. Sperhake Theoretisch-Physikalisches Institut Friedrich-Schiller Universität Jena SFB/Transregio 7 02 th July 2007 B. Brügmann, J. A. González, M. D. Hannam, S. Husa E. Berti, V. Cardoso U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
2 Overview Introduction and motivation Numerical simulations Global properties Post-Newtonian comparison The black-hole ringdown Conclusion U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
3 Introduction BBH-simulations successful: 2 families of codes generalized harmonic formulation Pretorius 05 moving punctures, UTB 06, Goddard 06 Results 3 categories Astrophysics: kicks, spin-flips, BH-properties,... GW-detection: PN-comparison, template banks, accuracy,... Mathematical : Consistency, num.techniques, non-linearities,... Comparison with analytic results: Buonanno et al. 06 Extend to unequal masses Key purposes improve physical understanding check accuracy of results validity of approximation methods U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
4 Numerical framework BAM code Moving puncture technique χ-version of BSSN equations Fourth-order discretization (Runge-Kutta) Mesh-refinement: Nested boxes, Berger-Olliger-MR Innermost boxes follow BH-motion Outer boundary: Sommerfeld Wave extraction: Newman-Penrose scalar Ψ 4 Equatorial symmetry for all simulations U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
5 Unequal mass sequence Total recoil sequence González et al. 07 Mass ratio: q := M 1 M 2 = η := q (1+q) 2 = Resolutions at punctures h = 1 45, 1 51, 1 58 low, medium, high Larger separation runs for q = 1 1, 1 2, 1 3 U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
6 Typical waveforms U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
7 Global properties: Total radiated energy E rad M = [ 4q (1+q) 2 ] 2 U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
8 Global properties: Enery radiated in l = 2, 3, 4, 5 Higher modes become more important for unequal masses Supression of odd l in equal mass limit U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
9 Final angular momentum of black hole Measured by: 1. conservation of J, Fit: j fin = α + β (1+q) 2 q q2 (1+q) 4 2. QNM fitting U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
10 The Post-Newtonian inspiral Numerical frequency estimates Write ψ lm = A lm e iφ lm dφ lm dt ω Dm = 1 m Assume, one frequency dominates [ ] ω Dm = 1 m Im ψ lm ψ lm Virtually identical Orbital frequency of the puncture motion Convert (x, y) (r, φ) ω c = dφ dt U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
11 Comparison of frequencies Buonanno et al. 06: Take quadrupole formula Orbital frequency Wave amplitude Approximates inspiral waveform even close to merger This Newtonian Quasi-Circular (NQC) approximation works both ways. Extend this comparison to Unequal masses Higher l, m 2.5 PNQC 2.5 PN quasi-circular waveforms by Arun et al. 04 U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
12 PN modes: l = 2, l = 3 U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
13 PNQC comparison: q=3 U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
14 PNQC comparison: q=3 (Zoom) U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
15 PNQC comparison: q=3 (Differences) PN convergence non-monotonic (1 PN poor) Same result for for q = 2 U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
16 Black hole ringdown An excited black hole will ring down in the form of quasinormal modes Modes characterized by angular dependence, damping: l, m, n U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
17 Black hole ringdown An excited black hole will ring down in the form of quasinormal modes Modes characterized by angular dependence, damping: l, m, n No-hair theorem frequencies depend only on (j, M) U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
18 Black hole ringdown An excited black hole will ring down in the form of quasinormal modes Modes characterized by angular dependence, damping: l, m, n No-hair theorem frequencies depend only on (j, M) Smoking gun of black hole U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
19 Black hole ringdown An excited black hole will ring down in the form of quasinormal modes Modes characterized by angular dependence, damping: l, m, n No-hair theorem frequencies depend only on (j, M) Smoking gun of black hole LISA high SNR test no-hair and GR U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
20 Black hole ringdown An excited black hole will ring down in the form of quasinormal modes Modes characterized by angular dependence, damping: l, m, n No-hair theorem frequencies depend only on (j, M) Smoking gun of black hole LISA high SNR test no-hair and GR Frequencies calculated for Schwarzschild: e.g. Chandrasekhar & Detweiler 75 Kerr: e.g. Leaver 86, Berti & Cardoso 06 U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
21 Strategy for fitting QNMs Slightly different from Buonanno et al. 06 Focus on fundamental modes: no overtones Slide window through signal: t ini variable, time dependency of frequencies Use Prony-methods Berti et al. 07 Robust against initial guess t fin = t 3 Straightforwardly accomodates complex signals Generalizes to include overtones Time dependence of frequencies may be caused by Neglecting overtones Time evolution of j, M U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
22 QNM fit: Quality factor Quality factor: Q := πf τ depends only on j Pure ringdown constant at large t ini. U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
23 Estimate of the Kerr parameter Deviations from constant significant? Modes l = 2, 3, 4 agree at same time U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
24 Estimate of the Kerr parameter Deviations decrease at higher resolutions Modes l = 2, 3, 4 still agree at same time U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
25 Estimate of the Kerr parameter Richardson extrapolation inconclusive due to numerical noise Our results are compatible with constant j U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
26 Estimate of final BH mass Same behaviour as for j Higher numerical accuracy needed U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
27 Determining the start time of the ringdown Helps comparison numerical vs. approximative techniques Breaking effect due to ringdown in recoil? Fundamental difficulties: QNMs do not form a complete basis There is no pure ringdown phase Ambiguities in ring-down time U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
28 Determining the start time of the ringdown Helps comparison numerical vs. approximative techniques Breaking effect due to ringdown in recoil? Fundamental difficulties: QNMs do not form a complete basis There is no pure ringdown phase Ambiguities in ring-down time Use deviations of waveform from QNM fit Dorband et al. 06 Minimize norm of deviation over t 0 Problems: norm oscillates, no promounced global minima. U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
29 Determining the start time of the ringdown Helps comparison numerical vs. approximative techniques Breaking effect due to ringdown in recoil? Fundamental difficulties: QNMs do not form a complete basis There is no pure ringdown phase Ambiguities in ring-down time Use deviations of waveform from QNM fit Dorband et al. 06 Minimize norm of deviation over t 0 Problems: norm oscillates, no promounced global minima. Alternatives: Energy maximized orthogonal projection (EMOP) Nollert 00 Detection based approach U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
30 Energy maximized orthogonal projection (EMOP) Decompose waveform into QNM part and orthogonal part U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
31 Energy maximized orthogonal projection (EMOP) Decompose waveform into QNM part and orthogonal part With U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
32 Energy maximized orthogonal projection (EMOP) Decompose waveform into QNM part and orthogonal part With t 0 : maximize fraction of the energy contained in ring-down. U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
33 EMOP times Same behaviour for h t EMOP almost constant for different mass ratios! U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
34 Matched filtering based approach Take numerical waveform h and template bank T λ1,λ 2,... U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
35 Matched filtering based approach Take numerical waveform h and template bank T λ1,λ 2,... Maximize SNR over t 0 U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
36 Matched filtering based approach Take numerical waveform h and template bank T λ1,λ 2,... Maximize SNR over t 0 Define ring-down time t MF as the time which satisfies Note: t 0 t MF U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
37 Ring-down time based on matched filtering t MF are systematically earlier than t EMOP Ring-down energy agrees well with Flanagan & Hughes 98 Fit for ring-down energy as function of mass ratio U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
38 Conclusions Unequal-mass sequence of non-spinning BBH inspiral Mass-ratio q = Total E rad, j fin decrease at larger mass ratios E rad in higher modes: up to 10 % in l = 3, m = 3 for q = 4 Numerical memory effect decreases at larger radii 2.5 PN inspiral frequencies show excellent agreement for higher mass ratios and higher l, m Small extraction radii problematic for large wavelengths Ring-down predictions for j fin agree well with radiation estimates Quality factor nearly constant. Small oscillations non-linearities? Estimate ring-down time from EMOP, Matched filtering U. Sperhake (FSU Jena) Mining information from unequal-mass binaries 02/07/ / 28
Mining information from unequal-mass binaries
Mining information from unequal-mass binaries U. Sperhake Theoretisch-Physikalisches Institut Friedrich-Schiller Universität Jena SFB/Transregio 7 19 th February 2007 B. Brügmann, J. A. González, M. D.
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