Navigating in waveform space Frank Ohme Cardiff University - - PowerPoint PPT Presentation

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

h1 − h22 = h1 − h2, h1 − h2 h1, h2 = 4 Re f2

f1

˜ h1(f) ˜ h∗

2(f)

Sn(f) df

Sn: aLIGO (zero detuned, high power) f1 = 15 Hz, f2 = fISCO

Frank Ohme Navigating in waveform space 2 / 11

Basic idea Inspiral results Systematic errors IMR models Conclusion Waveform distances

Waveform neighbourhood

h1 − h22 = h1 − h2, h1 − h2 h1, h2 = 4 Re f2

f1

˜ h1(f) ˜ h∗

2(f)

Sn(f) df

Why bother?

Template bank spacing Close signals may be confused for each other. ⇒ Implications for parameter estimation.

Sn: aLIGO (zero detuned, high power) f1 = 15 Hz, f2 = fISCO

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Basic idea Inspiral results Systematic errors IMR models Conclusion Fisher matrix and coordinate choice

Metric/Fisher matrix

• h(θ) − h(θ + ∆θ)
• 2 ≈
• i,j

Γij ∆θi∆θj

(Γij Fisher matrix/metric)

Coordinate and waveform choice Aligned spins, θph = {m1, m2, χ1, χ2, t0, φ0} TaylorF2: h = A

f f0 7/6 exp

• i

7

• k

f f0 (k−5)/3 ψk(θph) + ψlog

k (θph)

• 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

• h(θ) − h(θ + ∆θ)
• 2 ≈
• i,j

Γij ∆θi∆θj

(Γij Fisher matrix/metric)

Coordinate and waveform choice Aligned spins, θph = {m1, m2, χ1, χ2, t0, φ0} TaylorF2: h = A

f f0 7/6 exp

• i

7

• k

f f0 (k−5)/3 ψk(θph) + ψlog

k (θph)

• Adapted coordinate choice

Use PN coefficients ψ(log)

k

as variables ⇒ Increased dimensionality, unphysical waveforms included ⇒ Γij (almost) coordinate-independent (→ flat manifold)

[Tanaka & Tagoshi 2000, Sathyaprakash & Schutz 2003, Pai & Arun 2013, Brown et al 2012]

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Basic idea Inspiral results Systematic errors IMR models Conclusion PCA

Principal component analysis (PCA) Diagonalise Γij : Eigenvectors µi represent principal directions ranked by their eigenvalues λi

• ∆h
• 2 =
• i

λi(∆µi)2

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Basic idea Inspiral results Systematic errors IMR models Conclusion PCA

Principal component analysis (PCA) Diagonalise Γij : Eigenvectors µi represent principal directions ranked by their eigenvalues λi

• ∆h
• 2 =
• i

λi(∆µi)2 Accurate match predictions Geometric template placement

• 97 match

1.0 0.5 0.0 0.5 1.0 0.5 0.0 0.5 Ξ1 Λ112 Μ1 Ξ2 Λ212 Μ2 Application:[Brown et al 2012, Harry et al 2013]

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Basic idea Inspiral results Systematic errors IMR models Conclusion Physical meaning of principal components

Principal directions in physical coordinate space

First principal component µ1 ∼ Mc = (m1 m2)3/5 (m1 + m2)1/5 (+ higher-order corrections) µ1 extremely well measurable through GWs (order of magnitude better than Mc)

• const. Mc

6 8 10 12 14 16 18 20 0.05 0.10 0.15 0.20 0.25 total mass M symmetric mass ratio

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Basic idea Inspiral results Systematic errors IMR models Conclusion Physical meaning of principal components

Principal directions in physical coordinate space

First principal component µ1 ∼ Mc = (m1 m2)3/5 (m1 + m2)1/5 (+ higher-order corrections) µ1 extremely well measurable through GWs (order of magnitude better than Mc)

• const. Mc

6 8 10 12 14 16 18 20 0.05 0.10 0.15 0.20 0.25 total mass M symmetric mass ratio

Second principal component Combination of 1PN, 1.5PN and 2.5PN terms ⇒ Dominated by spin-orbit effects GW measurements restrict parameters to narrow band

0.05 0.10 0.15 0.20 0.25 1.0 0.5 0.0 0.5 1.0 symmetric masss ratio BH spin

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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

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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

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Basic idea Inspiral results Systematic errors IMR models Conclusion

Aside: Systematic errors

Previous approach ∆θph → ∆ψ(log)

k

→ ∆µi → ∆h

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Basic idea Inspiral results Systematic errors IMR models Conclusion

Aside: Systematic errors

Previous approach ∆θph → ∆ψ(log)

k

→ ∆µi → ∆h Different waveform models ∆h → ∆ψ(log)

k

→ ∆µi

min

→ ∆θph

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Basic idea Inspiral results Systematic errors IMR models Conclusion

Aside: Systematic errors

Previous approach ∆θph → ∆ψ(log)

k

→ ∆µi → ∆h Different waveform models ∆h → ∆ψ(log)

k

→ ∆µi

min

→ ∆θph

0.5 0.0 0.5 50 50 100 150 0.5 0. 0.5 1. 1.5 BH spin bias Mc bias

Systematic bias: nonspinning search

total mass symmetric mass ratio Mc

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., |χrecov| > 1

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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

• f waveforms

⇒ Coverage illustrated by mass-optimized matches

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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

• f 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

• f 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

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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

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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

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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 SEOBNR match contours

[courtesy M. Pürrer]

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Basic idea Inspiral results Systematic errors IMR models Conclusion Degeneracies in the last stages of the signal

Different degeneracies

Ringdown frequency

0.05 0.10 0.15 0.20 0.25 1.0 0.5 0.0 0.5 1.0 symmetric mass ratio spin parameter constant Mc [Rezzolla et al 2008, Berti et al 2006]

Dominant mass parameter

chirp mass

• sym. mass ratio

total mass 50 100 150 200 total mass M relative parameter error

(Equal mass PhenomC Fisher matrix)

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Basic idea Inspiral results Systematic errors IMR models Conclusion

Conclusions

Understanding the waveform structure in terms of principal directions in parameter space is useful for.. . GW detection (template spacing) parameter estimation (degeneracies) astrophysical interpretation (measurability) efficient match calculations (including systematic errors) waveform modelling promoting a physical intuition of degeneracy breaking by merger & ringdown

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