Solving the binary black hole problem. (again and again and again....) Mark
Hannam. Cardiff University. ACCGR Workshop. Brown University, May 21 2011.
Solving the binary black hole problem (again and again and again....) Mark Hannam Cardiff University
ACCGR Workshop Brown University, May 21 2011
Mark Hannam (Cardiff)
ACCGR Workshop, Brown University
1 / 17
Work in collaboration with...
Parameswaran Ajith, Caltech C´ eline Catto¨ en, Alberta Sascha Husa, UIB Frank Ohme, AEI Potsdam Michael P¨ urrer, Vienna Patricia Schmidt, Cardiff
Mark Hannam (Cardiff)
ACCGR Workshop, Brown University
2 / 17
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
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 [Br¨ ugmann, et. al., PRD 77 024027 (2008)]
Improvement: integrate full PN equations for inspiral. e ∼ 10−3 Husa, MDH, et. al., PRD 77, 044037 (2008)]
Mark Hannam (Cardiff)
ACCGR Workshop, Brown University
4 / 17
Reducing eccentricity further
It is difficult to improve on e ∼ 0.005 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
Reducing eccentricity further Comparison of ΩGW with EOB
0.0003 0.0002
MDΩ @radD
0.0001
ΩGW raw iteration 0
0.0000
iteration 1 iteration 2
-0.0001 -0.0002 -0.0003
200
400
600 t@MD
800
1000
See poster by Michael P¨ urrer Mark Hannam (Cardiff)
ACCGR Workshop, Brown University
5 / 17
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 0.030
0.12 TD hybrids
0.10
TD vs FD
0.020
Mismatch H%L
Mismatch H%L
0.025
0.015 0.010 0.005
0.08 0.06 0.04 0.02
10
15
20 30 50 Binary Mass HML
70
100
0.00
10
15
20
30 50 Binary Mass HML
70
100
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) 124052] Mark Hannam (Cardiff)
ACCGR Workshop, Brown University
6 / 17
q = 1 nonspinning 3.5 MΩm =0.045 MΩm =0.05 MΩm =0.06 MΩm =0.07 MΩm =0.09
Mismatch H%L
3.0 2.5 2.0 1.5 1.0 0.5 0.0 10
15
20
30 50 Binary Mass HML
Mark Hannam (Cardiff)
70
100
ACCGR Workshop, Brown University
7 / 17
q = 1 nonspinning
q = 4 nonspinning
3.5
3.5 MΩm =0.045 MΩm =0.05 MΩm =0.06 MΩm =0.07 MΩm =0.09
2.5 2.0 1.5 1.0 0.5 0.0 10
2.5 2.0 1.5 1.0 0.5
15
20
30 50 Binary Mass HML
70
0.0 10
100
q = 1, χ1 = χ2 = −0.5
20
30 50 Binary Mass HML
70
100
3.5 MΩm =0.04
3.0
MΩm =0.06
2.0
MΩm =0.07
MΩm =0.035
3.0
MΩm =0.05
2.5
Mismatch H%L
Mismatch H%L
15
q = 1, χ1 = χ2 = +0.5
3.5
1.5 1.0 0.5 0.0 10
MΩm =0.04 MΩm =0.045 MΩm =0.05 MΩm =0.06 MΩm =0.07
3.0 Mismatch H%L
Mismatch H%L
3.0
MΩm =0.04
2.5
MΩm =0.05
2.0
MΩm =0.06
1.5 1.0 0.5
15
20
30 50 Binary Mass HML
Mark Hannam (Cardiff)
70
100
0.0 10
15
20
30 50 Binary Mass HML
70
100
ACCGR Workshop, Brown University
7 / 17
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
Length requirements q = 4, nonspinning
opt. mismatch @%D
30 25 20 15 10 5 0 10
20
30 M M
40
50
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
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
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
Parameter bias with hybrids (q = 4 nonspinning)
bias @%D
opt. mismatch @%D
30 25 20 15
1.5 DΗ Η 1.0 DM M 0.5 0.0 0.1DΧ Χ -0.5 -1.0 10 20 30 40
50
10 5 0 10
Mark Hannam (Cardiff)
20
30 M M
40
50
ACCGR Workshop, Brown University
10 / 17
Waveform models The spinning-binary parameter space is vast: Binaries are characterized by the mass ratio η = m1 m2 /(m1 + m2 )2 . (1 parameter) the spins S1 and S2 of each BH. (6 parameters)
Assume we find a model with a cubic dependence on the parameters Then we need at least 47 ≈ 16, 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
Waveform models The spinning-binary parameter space is vast: Binaries are characterized by the mass ratio η = m1 m2 /(m1 + m2 )2 . (1 parameter) the spins S1 and S2 of each BH. (6 parameters)
Assume we find a model with a cubic dependence on the parameters Then we need at least 47 ≈ 16, 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, S1 , S2 . 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
Features of the new spinning-binary model What would we lose if we used only nonspinning waveforms in searches? " =−0.85 " =−0.75 " = −0.5 " =−0.25 "= 0 " = 0.25 " = 0.5 " = 0.75 " = 0.85
FF
1 0.9 0.8 400
0.16
300
0.18 0.2 0.22 !
200 100
0.24
M/M
sun
Match
1 0.5 400 0.16
300 M/M
sun
0.18
200
0.2 100
0.22
!
0.24
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
Features of the new spinning-binary model Can we see any further with these new waveforms?
Horizon distance (Mpc)
1200
! =−0.85 ! = −0.5 != 0 ! = 0.5 ! = 0.75 ! = 0.85
1000 800 600 400 200 0
100
200
300 M/Msun
400
500
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
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 0.965 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:0909.2867, PRL, in press] Mark Hannam (Cardiff)
ACCGR Workshop, Brown University
13 / 17
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) 10
The orbital plane will also precess
y 0
The resulting dynamics are complex -10
Consider q=3 χ1 = 0, χ2 = 0.75
5 z
0 -5
S·L=0 A maximally precessing case
-10 x
0 10
Mark Hannam (Cardiff)
ACCGR Workshop, Brown University
14 / 17
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 Ψ4,22 . 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
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 0.0025 Y'4,22
0.0020
Y4,22 Y4,21 Y4,33 Y4,44
0.001
0.0015
⇒
Y4,lm ¤
Y4,2 m ¤
0.01
Y'4,21
0.0010
10-4
0.0005
10-5
0.0000 200
400
600 t @MD
Mark Hannam (Cardiff)
800
1000
10-6 200
400
600
800
1000
1200
t @MD
ACCGR Workshop, Brown University
15 / 17
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 0.01
q = 3 nonspinning
Y4,22 Y4,21 Y4,33 Y4,44
0.01 0.001 ÈY4,lmÈ
Y4,lm ¤
0.001 10-4
10-5
10-5 10-6 200
10-4
400
600
800
1000
1200
10-6 200
400
600
t @MD
800 t @MD
1000
1200
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:1012.2879] Mark Hannam (Cardiff)
ACCGR Workshop, Brown University
15 / 17
Spectral element methods (SEM) (With C´eline Catto¨en) First test case: stationary 1+log trumpet solution of Schwarzschild Ideal test case: can treat each variable separately Preliminary results: Look at Axx , which is badly discontinuous at the puncture. Near the puncture, Axx = A
−2x 2 + y 2 + z 2 , r2
so Axx |y =z=0
=
−2A
(1) (2)
Axx |x=y =0 Mark Hannam (Cardiff)
=
A.
(3) ACCGR Workshop, Brown University
16 / 17
Spectral element methods (SEM) (With C´eline Catto¨en) First test case: stationary 1+log trumpet solution of Schwarzschild Ideal test case: can treat each variable separately Axx near the puncture, with and without filtering: A
−A
xy num
xy exact
at y=0.01495 z=0.01495 at t~2.4049
Axy num−Axy exact at y=0.01495 z=0.01495 at t~2.3934
1.5
0.05
0.04
acc P7 NELX9 NELXIn7 evenBoxIn
1 0.03
0.02
Axy num−Axy exact
Axy num−Axy exact
0.5
0
0.01
0
−0.01
−0.5 −0.02
−0.03
−1 −0.04
acc P7 NELX9 NELXIn7 evenBoxIn −1.5 −0.5
−0.4
−0.3
Mark Hannam (Cardiff)
−0.2
−0.1
0 x
0.1
0.2
0.3
0.4
0.5
−0.05 −0.5
−0.4
−0.3
−0.2
−0.1
0 x
0.1
0.2
0.3
0.4
0.5
ACCGR Workshop, Brown University
16 / 17
Spectral element methods (SEM) (With C´eline Catto¨en) 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 L2 norm centre excised of error on χ at t~0.10804, L=64 ALGEBRAIC −7
10
−8
L2 norm
10
NE=9 NE=11 NE=13 NE=15 17 N=5 N=7 N=9 N=11 N=13 N=15
−9
10
EXPONENTIAL
−10
10
5
10
Mark Hannam (Cardiff)
6
10 Degree of freedom Ng
7
10
ACCGR Workshop, Brown University
16 / 17
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
