Solving the binary black hole problem (again and again and again...)

Transcription

1 Solving the binary black hole problem (again and again and again...) Mark Hannam Cardiff University ACCGR Workshop Brown University, May Mark Hannam (Cardiff) ACCGR Workshop, Brown University 1 / 17

2 Work in collaboration with... Parameswaran Ajith, Caltech Céline Cattoën, Alberta Sascha Husa, UIB Frank Ohme, AEI Potsdam Michael Pürrer, Vienna Patricia Schmidt, Cardiff Mark Hannam (Cardiff) ACCGR Workshop, Brown University 2 / 17

3 Science from GW observations of BH mergers requires accurate theoretical waveforms across the entire parameter space The parameter space is large Modeling generic systems is poorly understood NR simulations are computationally expensive Length and accuracy requirements are unclear Even setting up the simulations is difficult... Mark Hannam (Cardiff) ACCGR Workshop, Brown University 3 / 17

4 Problems from the outset: initial parameters We need to determine initial parameters for low-eccentricity inspiral Iterative procedures have been proposed works with quasi-equilibrium conformal-thin-sandwich excision data (SpEC) [Pfeiffer, et. al., 2007; Mroue, et. al., 2010; Buonanno, et. al., 2010] For puncture data, we have used PN-based methods First guess: use PN prediction for circular orbits (no inspiral). e 10 2 [Brügmann, et. al., PRD (2008)] Improvement: integrate full PN equations for inspiral. e 10 3 Husa, MDH, et. al., PRD 77, (2008)] Mark Hannam (Cardiff) ACCGR Workshop, Brown University 4 / 17

5 Reducing eccentricity further It is difficult to improve on e with a naive iterative procedure Numerical noise in the waveforms Gauge adjustments over the first orbit Long-term gauge effects in the puncture motion A cleaner eccentricity measure can be made from the (filtered) GW phase An iterative procedure can be based on full PN information Mark Hannam (Cardiff) ACCGR Workshop, Brown University 5 / 17

6 Reducing eccentricity further ΩGW raw iteration 0 iteration 1 iteration See poster by Michael Pürrer Mark Hannam (Cardiff) ACCGR Workshop, Brown University 5 / 17

7 How long do numerical waveforms need to be? We can produce complete IMR waveforms by making PN + NR hybrids PN errors dominate the errors in these hybrids The longer the NR waveform, the better the hybrid TD hybrids TD vs FD Mismatch Without PN error estimates, how do we decide NR waveform lengths? First attempt: make hybrids with several approximants and compare MDH, Husa, Ohme, Ajith, PRD 82 (2010) ] Mark Hannam (Cardiff) ACCGR Workshop, Brown University 6 / 17

8 q = 1 nonspinning MΩm MΩm 0.05 MΩm 0.06 MΩm 0.07 MΩm Mark Hannam (Cardiff) ACCGR Workshop, Brown University 7 / 17

9 q = 1 nonspinning MΩm MΩm 0.05 MΩm 0.06 MΩm 0.07 MΩm q = 4 nonspinning MΩm 0.04 MΩm MΩm 0.05 MΩm 0.06 MΩm q = 1, χ 1 = χ 2 = 0.5 q = 1, χ 1 = χ 2 = MΩm MΩm MΩm 0.05 MΩm MΩm MΩm MΩm MΩm Mark Hannam (Cardiff) ACCGR Workshop, Brown University 7 / 17

10 Length requirements In a search, optimize match calculation over all parameters (with only one NR waveform, can only optimize over mass) But, we can compute matches without NR waveforms. The fully optimized match depends on only the match between the two PN approximants the phase difference of the approximants at the matching frequency the ratio of the total power in the PN and NR parts the accumulated phase difference of the PN parts (when considering different parameters) All of these are integrated quantities, that can be accurately modeled already by phenomenological waveforms! Mark Hannam (Cardiff) ACCGR Workshop, Brown University 8 / 17

11 Length requirements q = 4, nonspinning 30 opt. mismatch M M Nonspinning cases: < 10 orbits are acceptable up to q = 10 Spinning cases: not so good but higher order spin terms may improve the PN accuracy Mark Hannam (Cardiff) ACCGR Workshop, Brown University 8 / 17

12 Parameter estimation Parameter estimation errors depend on signal strength and waveform errors The waveform errors are irrelevant if the waveforms are indistinguishable Several studies have considered length requirements for indistinguishability Bad news: we need 100s or 1000s of NR orbits [Damour, et. al. (2010); MacDonald, et. al. (2010); Boyle (2011)] Obvious conclusion: we need longer waveforms Mark Hannam (Cardiff) ACCGR Workshop, Brown University 9 / 17

13 Parameter estimation Parameter estimation errors depend on signal strength and waveform errors The waveform errors are irrelevant if the waveforms are indistinguishable Several studies have considered length requirements for indistinguishability Bad news: we need 100s or 1000s of NR orbits [Damour, et. al. (2010); MacDonald, et. al. (2010); Boyle (2011)] Obvious conclusion: we need longer waveforms Too bad: we won t have those waveforms by 2015! The real question is: what will be the errors from waveforms with 10 cycles? Mark Hannam (Cardiff) ACCGR Workshop, Brown University 9 / 17

14 Parameter bias with hybrids (q = 4 nonspinning) opt. mismatch bias M M Η Η 0.1 Χ Χ M M Mark Hannam (Cardiff) ACCGR Workshop, Brown University 10 / 17

15 Waveform models The spinning-binary parameter space is vast: Binaries are characterized by the mass ratio η = m 1m 2/(m 1 + m 2) 2. (1 parameter) the spins S 1 and S 2 of each BH. (6 parameters) Assume we find a model with a cubic dependence on the parameters Then we need at least , 000 simulations! And the physics now includes precession of the black-hole spins, and of the binary s orbital plane... Mark Hannam (Cardiff) ACCGR Workshop, Brown University 11 / 17

16 Waveform models The spinning-binary parameter space is vast: Binaries are characterized by the mass ratio η = m 1m 2/(m 1 + m 2) 2. (1 parameter) the spins S 1 and S 2 of each BH. (6 parameters) Assume we find a model with a cubic dependence on the parameters Then we need at least , 000 simulations! And the physics now includes precession of the black-hole spins, and of the binary s orbital plane... However... As a first approximation, describing non-precessing binaries may be enough Now we have to deal with only three parameters, η, S 1, S 2. Plus: the waveforms vary most strongly with the total spin, χ. Now we have only two parameters, η, χ. Mark Hannam (Cardiff) ACCGR Workshop, Brown University 11 / 17

17 Features of the new spinning-binary model What would we lose if we used only nonspinning waveforms in searches? FF M/M sun ! " = 0.85 " = 0.75 " = 0.5 " = 0.25 " = 0 " = 0.25 " = 0.5 " = 0.75 " = 0.85 Match M/M sun ! FF < 0.8 suggests that over 50% of high-spin binaries may not be detected! Match 0.3 means that parameters will be significantly biased. Mark Hannam (Cardiff) ACCGR Workshop, Brown University 12 / 17

18 Features of the new spinning-binary model Can we see any further with these new waveforms? Horizon distance (Mpc) ! = 0.85! = 0.5! = 0! = 0.5! = 0.75! = M/M sun We may see high-spin binaries twice as far away as nonspinning binaries This is excellent news: most astrophysical BHs may be highly spinning! Mark Hannam (Cardiff) ACCGR Workshop, Brown University 12 / 17

19 Can this simple model also detect generic binaries? A preliminary study: We do not have generic merger waveforms We can produce generic PN waveforms Perform a monte-carlo comparison of generic against non-precessing waveforms With matches above we have: 85% at q = 1 62% at q = 4 37% at q = 9. We find that For comparable mass binaries, the non-precessing model finds most signals For larger mass ratios, a full generic model is needed [Ajith, MDH, Husa, et al, arxiv: , PRL, in press] Mark Hannam (Cardiff) ACCGR Workshop, Brown University 13 / 17

20 Eventually we have to face up to generic spins If the spins are not (anti-)aligned with the orbital angular momentum The spin directions will evolve (precess) The orbital plane will also precess The resulting dynamics are complex 10 y 0 10 Consider q = 3 χ 1 = 0, χ 2 = 0.75 S L = 0 A maximally precessing case 5 z x 0 10 Mark Hannam (Cardiff) ACCGR Workshop, Brown University 14 / 17

21 How can we possibly model such systems? Precession of the orbital plane mixes up the mode structure The quadrupole mode is no longer clearly dominant Energy is distributed across all modes. A simple phenomenological ansatz looks to be out of the question... BUT... Rotate frame of reference to maximize Ψ 4,22. Maximization is accurate to within fractions of a radian Locates direction of the orbital angular momentum Defines a unique quadrupole aligned waveform Mark Hannam (Cardiff) ACCGR Workshop, Brown University 15 / 17

22 How can we possibly model such systems? Precession of the orbital plane mixes up the mode structure The quadrupole mode is no longer clearly dominant Energy is distributed across all modes. A simple phenomenological ansatz looks to be out of the question... BUT... It is possible to disentangle the complex motion And the mode structure can be made simpler '4,22 '4, ,21 4, Mark Hannam (Cardiff) ACCGR Workshop, Brown University 15 / 17

23 How can we possibly model such systems? Non-precessing mode structure is retained q = 3 with S L = 0 comparable to q = 3 nonspinning Waveform modeling may now be simplified q = 3, S L = 0 q = 3 nonspinning ,21 4, ,lm How well does this extend to more general cases? How well do QA and non-precessing waveforms agree? See the poster by Patricia Schmidt [Schmidt, et. al., arxiv: ] Mark Hannam (Cardiff) ACCGR Workshop, Brown University 15 / 17

24 Spectral element methods (SEM) (With Céline Cattoën) First test case: stationary 1+log trumpet solution of Schwarzschild Ideal test case: can treat each variable separately Preliminary results: Look at A xx, which is badly discontinuous at the puncture. Near the puncture, so A xx = A 2x 2 + y 2 + z 2 r 2, A xx y=z=0 = 2A (1) A xx x=y=0 = A. (3) (2) Mark Hannam (Cardiff) ACCGR Workshop, Brown University 16 / 17

25 Spectral element methods (SEM) (With Céline Cattoën) First test case: stationary 1+log trumpet solution of Schwarzschild Ideal test case: can treat each variable separately A xx near the puncture, with and without filtering: 1.5 A xy num A xy exact at y= z= at t~ A xy num A xy exact at y= z= at t~ acc P7 NELX9 NELXIn7 evenboxin A xy num A xy exact 0 A xy num A xy exact acc P7 NELX9 NELXIn7 evenboxin x x Mark Hannam (Cardiff) ACCGR Workshop, Brown University 16 / 17

26 Spectral element methods (SEM) (With Céline Cattoën) First test case: stationary 1+log trumpet solution of Schwarzschild Ideal test case: can treat each variable separately Exponential convergence is maintained away from the puncture hp convergence of the L 2 norm centre excised of error on χ at t~ , L=64 L 2 norm EXPONENTIAL ALGEBRAIC NE=9 NE=11 NE=13 NE=15 17 N=5 N=7 N=9 N=11 N=13 N= Degree of freedom N g Mark Hannam (Cardiff) ACCGR Workshop, Brown University 16 / 17

27 Conclusions By the time Advanced LIGO is operational we won t have perfect waveform models but acceptable models are feasible How to produce those models remains an open question Mark Hannam (Cardiff) ACCGR Workshop, Brown University 17 / 17