Download Surrogate model of BNS waveforms for measuring the EOS

Surrogate model of BNS
waveforms for measuring the EOS
Ben Lackey, Michael Pürrer, Andrea Taracchini
Albert Einstein Institute-Potsdam, Germany
Trento, Italy, 6 June 2017
Measuring the EOS with BNS inspiral
•
•
Tidal field Eij from companion star induces a quadrupole moment Qij
in the NS
Amount of deformation depends on stiffness of EOS via the tidal
deformability ⇤:
Qij =
NS
⇤(EOS, m)m5 Eij
NS
Measuring the EOS with BNS inspiral
•
•
Tidal effect enters at 5th post-Newtonian order
Results in phase shift of ~10 radians
(v/c)2
(v/c)7
(v/c)10
(f ) = 0PN(f ; M) [1 + 1PN(f ; ⌘) + 1.5PN(f ; ⌘, S1 , S2 ) + · · · + 3.5PN(f ; ⌘, S1 , S2 ) + 5PN(f ; ⌘, ⇤1 , ⇤2 )]
400Hz up to merger
Measuring the EOS with BNS inspiral
˜
Can measure a linear combination of the tidal parameters ⇤
h
8
˜=
⇤
(1 + 7⌘
13
31⌘ 2 )(⇤1 + ⇤2 ) +
p
1
4⌘(1 + 9⌘
11⌘ 2 )(⇤1
Network
m1 = 1.35
M , SNR=30
m2 = 1.35 M
400
28
24
200
˜
⇤
•
20
0
16
12
200
400
0
8
4
1000
˜
⇤
2000
3000
L. Wade et al. PRD 89, 103012 (2014)
0
⇤2 )
i
Measuring the EOS with BNS inspiral
•
•
˜
Can measure a linear combination of the tidal parameters ⇤
Stacking observations allows us to directly measure a parameterized EOS
40 detected events at design sensitivity
B. Lackey, L. Wade. PRD 91, 043002 (2015)
The problem
•
Waveforms fast enough for Bayesian parameter estimation are too
7
inaccurate ( 10 samples at 0.1s each)
4 different
PN models
The problem
Waveforms fast enough for Bayesian parameter estimation are too
7
inaccurate ( 10 samples at 0.1s each)
0.012
4 different
PN models
Probability density
•
m1 = 1.35 M , m2 = 1.35 M
F2 Injection
T1 Injection
T2 Injection
T3 Injection
T4 Injection
0.010
0.008
0.006
0.004
0.002
0.0000
200
400
600
˜
⇤
800 1000
L. Wade et al. PRD 89, 103012 (2014)
The problem
•
Waveforms fast enough for Bayesian parameter estimation are too
7
inaccurate ( 10 samples at 0.1s each)
Accurate waveforms are too slow (10 minutes each)
0.012
4 different
PN models
Probability density
•
m1 = 1.35 M , m2 = 1.35 M
F2 Injection
T1 Injection
T2 Injection
T3 Injection
T4 Injection
0.010
0.008
0.006
6 different
0.004
EOB models
0.002
0.0000
200
400
600
˜
⇤
800 1000
L. Wade et al. PRD 89, 103012 (2014)
Spin-tidal-EOB waveform model
•
Effective-one-body model with aligned spin and dynamic tides
T. Hinderer et al. PRL 116, 181101 (2016)
• ` = 2, 3 tidal parameters
• ` = 2, 3 f-mode frequencies
• End is tapered to agree with numerical BNS simulations (In progress)
• 12-dimensional parameter space (1 mass, 1 spin, 4 matter parameters per
NS)
Spin-tidal-EOB waveform model
•
•
Effective-one-body model with aligned spin and dynamic tides
T. Hinderer et al. PRL 116, 181101 (2016)
• ` = 2, 3 tidal parameters
• ` = 2, 3 f-mode frequencies
• End is tapered to agree with numerical BNS simulations (In progress)
• 12-dimensional parameter space (1 mass, 1 spin, 4 matter parameters per
NS)
Can reduce number of parameters to 5: x = {q, S1 , S2 , ⇤1 , ⇤2 }
• Rescale waveform with total mass M, and just use mass ratio q
• Universal relations relate ` = 3 tidal parameter and ` = 2, 3 f-mode
frequencies to ` = 2 tidal parameter (Kent Yagi. PRD 89, 043011 (2014))
Spin-tidal-EOB waveform model
•
•
•
Effective-one-body model with aligned spin and dynamic tides
T. Hinderer et al. PRL 116, 181101 (2016)
• ` = 2, 3 tidal parameters
• ` = 2, 3 f-mode frequencies
• End is tapered to agree with numerical BNS simulations (In progress)
• 12-dimensional parameter space (1 mass, 1 spin, 4 matter parameters per
NS)
Can reduce number of parameters to 5: x = {q, S1 , S2 , ⇤1 , ⇤2 }
• Rescale waveform with total mass M, and just use mass ratio q
• Universal relations relate ` = 3 tidal parameter and ` = 2, 3 f-mode
frequencies to ` = 2 tidal parameter (Kent Yagi. PRD 89, 043011 (2014))
Will use aligned-spin TaylorT4 model for rest of talk since EOB model is still in
progress
Surrogate model overview
Interpolate amplitude and phase of the accurate waveform as a
function of frequency f and waveform parameters x :
•
h̃(f ; x) = A(f ; x)ei
•
(f ;x)
Interpolate between frequency nodes Fj with reduced bases
Precomputed
Calculated online
X
A
Bj (f )A(Fj ; x)
A(f ; x) =
j
(f ; x) =
X
Bj (f ) (Fj ; x)
j
•
Interpolate amplitude and phase at nodes Fj as functions of x
• Rectangular grid is prohibitive in 5-dimensions (~105 points)
• Gaussian Process Regression (GPR) does not require
rectangular grid
Initial training set
•
•
Draw 128 points from space-filling latin hypercube design (LHD)
Add the 32 corners of parameter space
q 2 [1/3, 1]
S1 2 [ 0.7, 0.7], S2 2 [ 0.7, 0.7]
4
4
⇤1 2 [0, 10 ], ⇤2 2 [0, 10 ]
Initial training set
•
•
Phase of waveform spans ~10,000 radians over parameter space
5
10
Requires interpolation accuracy of
for 0.1rad phase error
} 10,000 rad
Initial training set residual
•
•
Phase of residual relative to TaylorF2 spans ~100 radians
3
Requires interpolation accuracy of 10 for 0.1rad phase error
ln A+i
h̃ = h̃F2 e
}
100 rad
ln A(f ; x) =
X
A
Bj (f )
ln A(Fj ; x)
j
(f ; x) =
X
j
Bj (f )
(Fj ; x)
Interpolating frequency dependence
Amplitude basis
•
•
ln A(f ; x) =
X
A
Bj (f )
ln A(Fj ; x)
j
Phase basis
(f ; x) =
X
j
B0 (f )
F0
Bj (f )
(Fj ; x)
Gaussian Process Regression (GPR)
•
GPR assumes the values of a function yi at the points xi are random
variables that follow a multivariate Gaussian: p(yi |xi ) = N (0, k(xi , xj ))
• Covariance between points k(xi , xj )
0
0
• Can estimate the functional value y at x given sampled data (xi , yi )
• Mean:
E[p(y 0 |xi , x0 , yi )]
• Uncertainty: Var[p(y 0 |xi , x0 , yi )]
x3
x0
x2
x1
x0
x4
Gaussian Process Regression (GPR)
•
•
GPR assumes the values of a function yi at the points xi are random
variables that follow a multivariate Gaussian: p(yi |xi ) = N (0, k(xi , xj ))
• Covariance between points k(xi , xj )
0
0
• Can estimate the functional value y at x given sampled data (xi , yi )
• Mean:
E[p(y 0 |xi , x0 , yi )]
• Uncertainty: Var[p(y 0 |xi , x0 , yi )]
Covariance (kernel) here assumes function is twice differentiable:
2 ⌫=5/2
f kMatern (r)
k(xi , xj ) =
+ n2 ij
✓
◆
⇣
⌘
2
p
p
5r
⌫=5/2
Correlation decreases
kMatern (r) = 1 + 5r +
exp
5r
with distance
3
r2 = (x
x0 )T M (x
x0 )
M = diag(`q 2 , `S12 , `S22 , `⇤12 , `⇤22 )
Tunable hyperparameters
Surrogate of initial training set
•
•
Compared surrogate to 1000 test-set waveforms
Maximum phase error of 2.9 rad
Surrogate error
Updating surrogate with active learning
•
GPR provides inexpensive estimate of surrogate model error
Updating surrogate with active learning
•
GPR provides inexpensive estimate of surrogate model error
1. Iteratively choose 200 new points that maximize the GPR error
Updating surrogate with active learning
•
GPR provides inexpensive estimate of surrogate model error
1. Iteratively choose 200 new points that maximize the GPR error
2. Generate those 200 new waveforms
3. Reconstruct the surrogate model
Updating surrogate with active learning
•
•
Compared surrogate to 1000 test-set waveforms
Maximum phase error of 0.6 rad (5x improvement)
Surrogate error
Updating with random training set
•
•
•
Compare updated surrogate to 1000 test-set waveforms
Errors are concentrated at unequal mass ratio and large spin magnitudes
Add the 71 waveforms with RMS phase error >0.1rad
Test-set waveforms with
RMS phase error >0.1 rad
Updating with random training set
•
•
•
•
Compare updated surrogate to 1000 test-set waveforms
Errors are concentrated at unequal mass ratio and nonzero spins
Add the 71 waveforms with RMS phase error > 0.1rad
Maximum phase error of 0.4rad compared to a 2nd test set
Surrogate error
Accuracy of surrogate
•
Estimate final error with 2nd test set of 1000 waveforms
Accuracy of surrogate
•
Estimate final error with 2nd test set of 1000 waveforms
Accuracy of surrogate
•
Estimate final error with 2nd test set of 1000 waveforms
• Maximum phase error: 0.36 rad
• Maximum fractional amplitude error: 2.6%
Accuracy of surrogate
•
•
Estimate final error with 2nd test set of 1000 waveforms
Mismatch is the loss in SNR resulting from surrogate error
Accuracy of surrogate
•
•
Estimate final error with 2nd test set of 1000 waveforms
Mismatch is the loss in SNR resulting from surrogate error
Accuracy of surrogate
•
•
Estimate final error with 2nd test set of 1000 waveforms
Mismatch is the loss in SNR resulting from surrogate error
• Maximum mismatch: 2.6 ⇥ 10 4
Accuracy of surrogate
•
•
Estimate final error with 2nd test set of 1000 waveforms
Maximum phase error is comparable to tidal effect for ⇤ = 50
Speed of surrogate
•
•
Spin-tidal-EOB code takes ~10 minutes
GPR Surrogate (written in Python):
• Fixed cost of ~50 ms for computing surrogate (not optimized)
• Additional cost for resampling to desired frequencies (interpolation)
• Faster than all waveforms in LALSuite (written in C) below ~18 Hz
Surrogate
Future work
•
•
Regenerate surrogate for Spin-Tidal-EOB model instead of TaylorT4
NR simulations will provide a small improvement to EOB model (1-2 rad)
• Can generate surrogate of the difference (as was done with TaylorF2)
• Could cover more limited parameter space with ~50 NR simulations
Conclusions
•
•
•
BNS inspiral observations can be used to measure the EOS
But, they require accurate and inexpensive waveform models
Surrogate of spin-tidal-EOB model can make this possible, and preliminary model should be available within a month
Thank you
Extra slides
Latin hypercube design
•
Begin with space-filling latin hypercube design (LHD)
• Samples each parameter uniformly
• Noncollapsing (subspace also a LHD)
• Doesn’t waste samples if waveform depends weekly on a parameter