Basic idea Inspiral results Systematic errors IMR models Conclusion Navigating in waveform space Frank Ohme Cardiff University 20/09/2013 @ NRDA / Mallorca In collaboration with A. Nielsen, A. Lundgren, D. Keppel, M. Pürrer, M. Hannam and S. Fairhurst Phys.Rev. D88 , 042002 (2013), ArXiv:1304.7017 Frank Ohme Navigating in waveform space 1 / 11
Basic idea Inspiral results Systematic errors IMR models Conclusion Waveform distances Frank Ohme Navigating in waveform space 2 / 11
Basic idea Inspiral results Systematic errors IMR models Conclusion Waveform distances Waveform neighbourhood � h 1 − h 2 � 2 = � h 1 − h 2 , h 1 − h 2 � � f 2 h 1 ( f ) ˜ ˜ h ∗ 2 ( f ) � h 1 , h 2 � = 4 Re df S n ( f ) f 1 S n : aLIGO (zero detuned, high power) f 1 = 15 Hz , f 2 = f ISCO Frank Ohme Navigating in waveform space 2 / 11
Basic idea Inspiral results Systematic errors IMR models Conclusion Waveform distances Waveform neighbourhood � h 1 − h 2 � 2 = � h 1 − h 2 , h 1 − h 2 � � f 2 h 1 ( f ) ˜ ˜ h ∗ 2 ( f ) � h 1 , h 2 � = 4 Re df S n ( f ) f 1 Why bother? Template bank spacing Close signals may be confused for each other. ⇒ Implications for parameter estimation. S n : aLIGO (zero detuned, high power) f 1 = 15 Hz , f 2 = f ISCO Frank Ohme Navigating in waveform space 2 / 11
Basic idea Inspiral results Systematic errors IMR models Conclusion Fisher matrix and coordinate choice Metric/Fisher matrix � 2 ≈ � Γ ij ∆ θ i ∆ θ j � � � h ( θ ) − h ( θ + ∆ θ ) ( Γ ij Fisher matrix/metric) i , j Coordinate and waveform choice Aligned spins, θ ph = { m 1 , m 2 , χ 1 , χ 2 , t 0 , φ 0 } � f � f � 7 / 6 � 7 � ( k − 5 ) / 3 � �� � ψ k ( θ ph ) + ψ log TaylorF2: h = A exp i k ( θ ph ) f 0 f 0 k Frank Ohme Navigating in waveform space 3 / 11
Basic idea Inspiral results Systematic errors IMR models Conclusion Fisher matrix and coordinate choice Metric/Fisher matrix � 2 ≈ � Γ ij ∆ θ i ∆ θ j � � � h ( θ ) − h ( θ + ∆ θ ) ( Γ ij Fisher matrix/metric) i , j Coordinate and waveform choice Aligned spins, θ ph = { m 1 , m 2 , χ 1 , χ 2 , t 0 , φ 0 } � f � f � 7 / 6 � 7 � ( k − 5 ) / 3 � �� � ψ k ( θ ph ) + ψ log TaylorF2: h = A exp i k ( θ ph ) f 0 f 0 k Adapted coordinate choice Use PN coefficients ψ ( log ) as variables k ⇒ Increased dimensionality, unphysical waveforms included ⇒ Γ ij (almost) coordinate-independent ( → flat manifold) [ Tanaka & Tagoshi 2000, Sathyaprakash & Schutz 2003, Pai & Arun 2013, Brown et al 2012 ] Frank Ohme Navigating in waveform space 3 / 11
Basic idea Inspiral results Systematic errors IMR models Conclusion PCA Principal component analysis (PCA) Diagonalise Γ ij : Eigenvectors µ i represent principal directions ranked by their eigenvalues λ i � 2 = � � � λ i (∆ µ i ) 2 � ∆ h i Frank Ohme Navigating in waveform space 4 / 11
Basic idea Inspiral results Systematic errors IMR models Conclusion PCA Principal component analysis (PCA) Diagonalise Γ ij : Eigenvectors µ i represent principal directions ranked by their eigenvalues λ i � 2 = � � � λ i (∆ µ i ) 2 � ∆ h i Accurate match predictions Geometric template placement � � � � � � � � � � 0.5 � � � � � � Ξ 2 � Λ 21 � 2 Μ 2 � 97 � match � � � � � � � � � � � � � � 0.0 � � � � � � � � � � � � � � � � � � � � � 0.5 � � � � � � � � � � � � 1.0 � 0.5 0.0 0.5 1.0 Ξ 1 � Λ 11 � 2 Μ 1 Application: [ Brown et al 2012, Harry et al 2013 ] Frank Ohme Navigating in waveform space 4 / 11
Basic idea Inspiral results Systematic errors IMR models Conclusion Physical meaning of principal components Principal directions in physical coordinate space First principal component 0.25 const. M c symmetric mass ratio ( m 1 m 2 ) 3 / 5 0.20 µ 1 ∼ M c = ( m 1 + m 2 ) 1 / 5 0.15 (+ higher-order corrections) 0.10 µ 1 extremely well measurable 0.05 through GWs (order of 6 8 10 12 14 16 18 20 magnitude better than M c ) total mass � M � � Frank Ohme Navigating in waveform space 5 / 11
Basic idea Inspiral results Systematic errors IMR models Conclusion Physical meaning of principal components Principal directions in physical coordinate space First principal component 0.25 const. M c symmetric mass ratio ( m 1 m 2 ) 3 / 5 0.20 µ 1 ∼ M c = ( m 1 + m 2 ) 1 / 5 0.15 (+ higher-order corrections) 0.10 µ 1 extremely well measurable 0.05 through GWs (order of 6 8 10 12 14 16 18 20 magnitude better than M c ) total mass � M � � Second principal component 1.0 0.5 Combination of 1PN, 1.5PN and BH spin 2.5PN terms 0.0 ⇒ Dominated by spin-orbit effects � 0.5 GW measurements restrict � 1.0 0.05 0.10 0.15 0.20 0.25 parameters to narrow band symmetric masss ratio Frank Ohme Navigating in waveform space 5 / 11
Basic idea Inspiral results Systematic errors IMR models Conclusion Mass ratio/spin degeneracy 90% confidence interval at SNR 10 Spin(s) and masses weakly constrained → “mass ratio/spin degeneracy” [ Ohme et al 2013, Hannam et al 2013, Cutler & Flanagan 1994 ] ( 1 . 35 + 5 ) M ⊙ , spin 0 . 3 0 50 100 150 t � ms � Frank Ohme Navigating in waveform space 6 / 11
Basic idea Inspiral results Systematic errors IMR models Conclusion Mass ratio/spin degeneracy 90% confidence interval at SNR 10 Spin(s) and masses weakly constrained → “mass ratio/spin degeneracy” [ Ohme et al 2013, Hannam et al 2013, Cutler & Flanagan 1994 ] ( 2 . 4 + 2 . 6 ) M ⊙ , spin 0 . 08 0 50 100 150 t � ms � Frank Ohme Navigating in waveform space 6 / 11
Basic idea Inspiral results Systematic errors IMR models Conclusion Aside: Systematic errors Previous approach ∆ θ ph → ∆ ψ ( log ) → ∆ µ i → ∆ h k Frank Ohme Navigating in waveform space 7 / 11
Basic idea Inspiral results Systematic errors IMR models Conclusion Aside: Systematic errors Previous approach ∆ θ ph → ∆ ψ ( log ) → ∆ µ i → ∆ h k Different waveform models ∆ h → ∆ ψ ( log ) min → ∆ µ i → ∆ θ ph k Frank Ohme Navigating in waveform space 7 / 11
Basic idea Inspiral results Systematic errors IMR models Conclusion Aside: Systematic errors Systematic bias: nonspinning search Previous approach 150 1.5 symmetric ∆ θ ph → ∆ ψ ( log ) → ∆ µ i → ∆ h mass ratio 100 1. k M c bias � � � bias � � � total mass 50 0.5 Different waveform models M c 0 0. ∆ h → ∆ ψ ( log ) min → ∆ µ i → ∆ θ ph � 50 � 0.5 k � 0.5 0.0 0.5 BH spin Simple and very efficient algorithm to study systematic errors Accurate as long as resulting SNR loss � ∆ h � is small Initial results: Spin-orbit coupling at leading and next-to-leading order are crucial (“more important” than 3PN and 3.5PN non-sp. terms) Optimal detection in many cases outside physical spin range, i.e., | χ recov | > 1 Frank Ohme Navigating in waveform space 7 / 11
Basic idea Inspiral results Systematic errors IMR models Conclusion Extrapolating inspiral results Thoughts about late inspiral (merger, ringdown) Construction of inspiral-merger-ringdown models NR waveforms: Total mass is trivial scaling factor ⇒ One simulation represents family of waveforms ⇒ Coverage illustrated by mass-optimized matches Frank Ohme Navigating in waveform space 8 / 11
Basic idea Inspiral results Systematic errors IMR models Conclusion Extrapolating inspiral results Thoughts about late inspiral (merger, ringdown) Construction of inspiral-merger-ringdown models NR waveforms: Total mass is trivial scaling factor ⇒ One simulation represents family of waveforms ⇒ Coverage illustrated by mass-optimized matches Frank Ohme Navigating in waveform space 8 / 11
Basic idea Inspiral results Systematic errors IMR models Conclusion Extrapolating inspiral results Thoughts about late inspiral (merger, ringdown) Construction of inspiral-merger-ringdown models NR waveforms: Total mass is trivial scaling factor ⇒ One simulation represents family of waveforms ⇒ Coverage illustrated by mass-optimized matches New strategy? Base parameter-space coverage on known degeneracies Rather than individually fitting highly correlated coefficients, PCA could help to make phenomenological fits more accurate Frank Ohme Navigating in waveform space 8 / 11
Basic idea Inspiral results Systematic errors IMR models Conclusion Degeneracies in the last stages of the signal Principal directions of PhenomC Inspiral-merger-ringdown model PhenomC [ Santamaría, FO et al 2010 ] Suitable for calculating principal directions locally Total mass variations are projected out Frank Ohme Navigating in waveform space 9 / 11
Basic idea Inspiral results Systematic errors IMR models Conclusion Degeneracies in the last stages of the signal Principal directions of PhenomC Inspiral-merger-ringdown model PhenomC [ Santamaría, FO et al 2010 ] Suitable for calculating principal directions locally Total mass variations are projected out Frank Ohme Navigating in waveform space 9 / 11
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