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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