Model Waveform Accuracy Standards for Gravitational Wave Data Analysis

Transcription

1 Model Waveform Accuracy Standards for Gravitational Wave Data Analysis Lee Lindblom Theoretical Astrophysics, Caltech Numerical Relativity and Data Analysis Meeting Syracuse University 1 August 008 Collaborators: Duncan Brown (Syracuse) Benjamin Owen (Penn State) Lee Lindblom (Caltech) Waveform Standards NRDA008 1 / 18

2 Outline Derive model waveform accuracy requirements for ideal detectors: Standards for measurement. Standards for detection. Lee Lindblom (Caltech) Waveform Standards NRDA008 / 18

3 Outline Derive model waveform accuracy requirements for ideal detectors: Standards for measurement. Standards for detection. Transform requirements into more user-friendly forms. Lee Lindblom (Caltech) Waveform Standards NRDA008 / 18

4 Outline Derive model waveform accuracy requirements for ideal detectors: Standards for measurement. Standards for detection. Transform requirements into more user-friendly forms. Determine effect of calibration errors on accuracy requirements. Lee Lindblom (Caltech) Waveform Standards NRDA008 / 18

5 Outline Derive model waveform accuracy requirements for ideal detectors: Standards for measurement. Standards for detection. Transform requirements into more user-friendly forms. Determine effect of calibration errors on accuracy requirements. Evaluate standards for the Advanced LIGO case. Lee Lindblom (Caltech) Waveform Standards NRDA008 / 18

6 Outline Derive model waveform accuracy requirements for ideal detectors: Standards for measurement. Standards for detection. Transform requirements into more user-friendly forms. Determine effect of calibration errors on accuracy requirements. Evaluate standards for the Advanced LIGO case. What this talk will not cover How to measure NR waveform errors. How well do current NR waveforms satisfy these standards.... Lee Lindblom (Caltech) Waveform Standards NRDA008 / 18

7 Accuracy Standards for Measurement How close must two waveforms, h e (f ) and h m (f ), be to each other so that observations are unable to distinguish them? Lee Lindblom (Caltech) Waveform Standards NRDA008 3 / 18

8 Accuracy Standards for Measurement How close must two waveforms, h e (f ) and h m (f ), be to each other so that observations are unable to distinguish them? Consider the one parameter family of waveforms: h(λ, f ) = h e (f ) + λ[h m (f ) h e (f )] = h e (f ) + λδh(f ) Lee Lindblom (Caltech) Waveform Standards NRDA008 3 / 18

9 Accuracy Standards for Measurement How close must two waveforms, h e (f ) and h m (f ), be to each other so that observations are unable to distinguish them? Consider the one parameter family of waveforms: h(λ, f ) = h e (f ) + λ[h m (f ) h e (f )] = h e (f ) + λδh(f ) The variance for measuring the parameter λ is given by 1 h = h λ = δh δh, λ σ λ where the noise weighted inner product is defined by h e h m = 0 he(f )h m (f ) + h e (f )hm(f ) df. S n (f ) Lee Lindblom (Caltech) Waveform Standards NRDA008 3 / 18

10 Accuracy Standards for Measurement How close must two waveforms, h e (f ) and h m (f ), be to each other so that observations are unable to distinguish them? Consider the one parameter family of waveforms: h(λ, f ) = h e (f ) + λ[h m (f ) h e (f )] = h e (f ) + λδh(f ) The variance for measuring the parameter λ is given by 1 h = h λ = δh δh, λ σ λ where the noise weighted inner product is defined by h e h m = 0 he(f )h m (f ) + h e (f )hm(f ) df. S n (f ) Two waveforms are indistinguishable iff the variance σλ is larger than the parameter distance between the waveforms: ( λ) = 1 < σλ = 1/ δh δh, that is iff 1 > δh δh. Lee Lindblom (Caltech) Waveform Standards NRDA008 3 / 18

11 Accuracy Standards for Detection The signal-to-noise ratio ρ m for detecting a signal h e using a filter constructed from a model waveform h m is ρ m = h e ĥm = h e h m h m h m 1/. Lee Lindblom (Caltech) Waveform Standards NRDA008 4 / 18

12 Accuracy Standards for Detection The signal-to-noise ratio ρ m for detecting a signal h e using a filter constructed from a model waveform h m is ρ m = h e ĥm = h e h m h m h m 1/. Errors in model waveform, h m = h e + δh, result in reduction of ρ m compared to the optimal signal-to-noise ratio: ρ m = ρ (1 ɛ) = h e h e 1/ (1 ɛ). Lee Lindblom (Caltech) Waveform Standards NRDA008 4 / 18

13 Accuracy Standards for Detection The signal-to-noise ratio ρ m for detecting a signal h e using a filter constructed from a model waveform h m is ρ m = h e ĥm = h e h m h m h m 1/. Errors in model waveform, h m = h e + δh, result in reduction of ρ m compared to the optimal signal-to-noise ratio: ρ m = ρ (1 ɛ) = h e h e 1/ (1 ɛ). Evaluate mismatch ɛ in terms of the waveform error: ɛ = δh δh h e h e, where δh = δh h e h e δh /ρ. Lee Lindblom (Caltech) Waveform Standards NRDA008 4 / 18

14 Accuracy Standards for Detection The signal-to-noise ratio ρ m for detecting a signal h e using a filter constructed from a model waveform h m is ρ m = h e ĥm = h e h m h m h m 1/. Errors in model waveform, h m = h e + δh, result in reduction of ρ m compared to the optimal signal-to-noise ratio: ρ m = ρ (1 ɛ) = h e h e 1/ (1 ɛ). Evaluate mismatch ɛ in terms of the waveform error: ɛ = δh δh h e h e, where δh = δh h e h e δh /ρ. For detection, model waveform accuracy must satisfy the requirement δh δh < ɛ max ρ. Lee Lindblom (Caltech) Waveform Standards NRDA008 4 / 18

15 Sufficient Conditions for Waveform Accuracy The optimal accuracy standards δh δh < 1 and δh δh < ɛ max ρ depend in the details of the waveform model (e.g. the total mass and location of the source) as well as the details of the detector noise spectrum S n (f ). Construct slightly stronger sufficient conditions that are easier for the NR community to use. Lee Lindblom (Caltech) Waveform Standards NRDA008 5 / 18

16 Sufficient Conditions for Waveform Accuracy The optimal accuracy standards δh δh < 1 and δh δh < ɛ max ρ depend in the details of the waveform model (e.g. the total mass and location of the source) as well as the details of the detector noise spectrum S n (f ). Construct slightly stronger sufficient conditions that are easier for the NR community to use. One simplification can be made by noting that δh δh δh δh, So a sufficient condition for detection is: δh δh < ɛ max ρ. Lee Lindblom (Caltech) Waveform Standards NRDA008 5 / 18

17 Sufficient Conditions for Waveform Accuracy II Define the model waveform (logarithmic) amplitude δχ and phase δφ errors: δh = h e (δχ + iδφ). Lee Lindblom (Caltech) Waveform Standards NRDA008 6 / 18

18 Sufficient Conditions for Waveform Accuracy II Define the model waveform (logarithmic) amplitude δχ and phase δφ errors: δh = h e (δχ + iδφ). The basic accuracy requirements can be written as { δh δh = δχ + δφ 1/ρ measurement, < ρ ɛ max detection, where the signal-weighted average errors are defined as δχ = 0 δχ 4 h e ρ S n df, and δφ = 0 δφ 4 h e ρ S n df. Lee Lindblom (Caltech) Waveform Standards NRDA008 6 / 18

19 Sufficient Conditions for Waveform Accuracy II Define the model waveform (logarithmic) amplitude δχ and phase δφ errors: δh = h e (δχ + iδφ). The basic accuracy requirements can be written as { δh δh = δχ + δφ 1/ρ measurement, < ρ ɛ max detection, where the signal-weighted average errors are defined as δχ = 0 δχ 4 h e ρ S n df, and δφ = 0 δφ 4 h e ρ S n df. These averages satisfy δφ max δφ, etc., so a set of sufficient accuracy requirements are { ( ) ( ) 1/ρ measurement, max δχ + max δφ < ɛ max detection. Lee Lindblom (Caltech) Waveform Standards NRDA008 6 / 18

20 Sufficient Conditions for Waveform Accuracy III We can derive another sufficient waveform accuracy requirement by noting that: δh δh(f ) δh δh = 4 df S n (f ) min S n (f ), 0 where δh(f ) = 0 δh df is the L norm of δh(f ). Lee Lindblom (Caltech) Waveform Standards NRDA008 7 / 18

21 Sufficient Conditions for Waveform Accuracy III We can derive another sufficient waveform accuracy requirement by noting that: δh δh(f ) δh δh = 4 df S n (f ) min S n (f ), 0 where δh(f ) = 0 δh df is the L norm of δh(f ). We can therefore convert the basic accuracy requirements into the following sufficient conditions: { δh(f ) C h e (f ) < /ρ measurement, ɛ max C detection, where C is defined as C ρ = h e (f ) / min S n (f ) 1. Lee Lindblom (Caltech) Waveform Standards NRDA008 7 / 18

22 Sufficient Conditions for Waveform Accuracy III We can derive another sufficient waveform accuracy requirement by noting that: δh δh(f ) δh δh = 4 df S n (f ) min S n (f ), 0 where δh(f ) = 0 δh df is the L norm of δh(f ). We can therefore convert the basic accuracy requirements into the following sufficient conditions: { δh(f ) C h e (f ) < /ρ measurement, ɛ max C detection, where C is defined as C ρ = h e (f ) / min S n (f ) 1. The quantity C can be evaluated once-and-for-all for a given type of waveform and a given detector noise spectrum. Lee Lindblom (Caltech) Waveform Standards NRDA008 7 / 18

23 Sufficient Conditions for Waveform Accuracy III We can derive another sufficient waveform accuracy requirement by noting that: δh δh(f ) δh δh = 4 df S n (f ) min S n (f ), 0 where δh(f ) = 0 δh df is the L norm of δh(f ). We can therefore convert the basic accuracy requirements into the following sufficient conditions: { δh(t) δh(f ) C = h e (t) h e (f ) < /ρ measurement, ɛ max C detection, where C is defined as C ρ = h e (f ) / min S n (f ) 1. The quantity C can be evaluated once-and-for-all for a given type of waveform and a given detector noise spectrum. Lee Lindblom (Caltech) Waveform Standards NRDA008 8 / 18

24 Effects of Calibration Error Let v(f ) denote the raw detector output, and R(f ) denote the response function used to convert v(f ) to the observed gravitational wave signal: h(f ) = R(f )v(f ). Lee Lindblom (Caltech) Waveform Standards NRDA008 9 / 18

25 Effects of Calibration Error Let v(f ) denote the raw detector output, and R(f ) denote the response function used to convert v(f ) to the observed gravitational wave signal: h(f ) = R(f )v(f ). Assume the measured response function R(f ) has errors that cause errors in the inferred gravitational waveform: h(f ) = R(f )v(f ) = [R e (f ) + δr(f )]v(f ) = h e (f ) + δh R (f ). Lee Lindblom (Caltech) Waveform Standards NRDA008 9 / 18

26 Effects of Calibration Error Let v(f ) denote the raw detector output, and R(f ) denote the response function used to convert v(f ) to the observed gravitational wave signal: h(f ) = R(f )v(f ). Assume the measured response function R(f ) has errors that cause errors in the inferred gravitational waveform: h(f ) = R(f )v(f ) = [R e (f ) + δr(f )]v(f ) = h e (f ) + δh R (f ). Evaluate the signal-to-noise ratio for an observed signal h with a filter based on the model waveform h m = h e + δh m. Keep terms through quadratic order in δr and δh m : ρ m = h h m h m h m 1/ = ρ 1 ρ (δh m δh R ) (δh m δh R ), where (δh m δh R ) = δh m δh R h e h e (δh m δh R ) /ρ. Lee Lindblom (Caltech) Waveform Standards NRDA008 9 / 18

27 Effects of Calibration Error II Determine the maximum effect of response function error, δh R, and modeling error, δh m, on the signal-to-noise ratio ρ: ρ m = ρ (δh m δh R ) (δh m δh R ) /ρ, ρ [ δh m δh m 1/ + δh R δh R 1/] /ρ. Lee Lindblom (Caltech) Waveform Standards NRDA / 18

28 Effects of Calibration Error II Determine the maximum effect of response function error, δh R, and modeling error, δh m, on the signal-to-noise ratio ρ: ρ m = ρ (δh m δh R ) (δh m δh R ) /ρ, ρ [ δh m δh m 1/ + δh R δh R 1/] /ρ. Define η, the ratio of model waveform error to response function error: δh m δh m = η δh R δh R. Re-express ρ m as, ρ m ρ (1 + η) δh R δh R /ρ. Lee Lindblom (Caltech) Waveform Standards NRDA / 18

29 Effects of Calibration Error II Determine the maximum effect of response function error, δh R, and modeling error, δh m, on the signal-to-noise ratio ρ: ρ m = ρ (δh m δh R ) (δh m δh R ) /ρ, ρ [ δh m δh m 1/ + δh R δh R 1/] /ρ. Define η, the ratio of model waveform error to response function error: δh m δh m = η δh R δh R. Re-express ρ m as, ρ m ρ (1 + η) δh R δh R /ρ. Waveform model errors less than maximum fraction of the calibration error, η η max, are swamped by calibration error. Lee Lindblom (Caltech) Waveform Standards NRDA / 18

30 Effects of Calibration Error II Determine the maximum effect of response function error, δh R, and modeling error, δh m, on the signal-to-noise ratio ρ: ρ m = ρ (δh m δh R ) (δh m δh R ) /ρ, ρ [ δh m δh m 1/ + δh R δh R 1/] /ρ. Define η, the ratio of model waveform error to response function error: δh m δh m = η δh R δh R. Re-express ρ m as, ρ m ρ (1 + η) δh R δh R /ρ. Waveform model errors less than maximum fraction of the calibration error, η η max, are swamped by calibration error. Natural choices for η max are η max = 1, or η max = Lee Lindblom (Caltech) Waveform Standards NRDA / 18

31 Summary of Model Waveform Accuracy Standards The basic model waveform accuracy standards are: { δh δh 1/ρ measurement, = δχ ρ m + δφm < ɛ max detection. Simpler conditions that guarantee the basic standards are: { δh(t) δh(f ) C = h e (t) h e (f ) < /ρ measurement, ɛ max C detection. Lee Lindblom (Caltech) Waveform Standards NRDA / 18

32 Summary of Model Waveform Accuracy Standards The basic model waveform accuracy standards are: { δh δh 1/ρ measurement, = δχ ρ m + δφm < ɛ max detection. Simpler conditions that guarantee the basic standards are: { δh(t) δh(f ) C = h e (t) h e (f ) < /ρ measurement, ɛ max C detection. The basic waveform accuracy standards need not be enforced when they are stricter than the response-function error condition: δh(t) [ ] h e (t) η maxc δχ R + δφr. Lee Lindblom (Caltech) Waveform Standards NRDA / 18

33 Summary of Model Waveform Accuracy Standards The basic model waveform accuracy standards are: { δh δh 1/ρ measurement, = δχ ρ m + δφm < ɛ max detection. Simpler conditions that guarantee the basic standards are: { δh(t) δh(f ) C = h e (t) h e (f ) < /ρ measurement, ɛ max C detection. The basic waveform accuracy standards need not be enforced when they are stricter than the response-function error condition: δh(t) [ ] h e (t) η maxc δχ R + δφr. To use these standards, we must determine the ranges for the quantities ρ, ɛ max, C, η max, δχ R, and δφ R appropriate for LIGO. Lee Lindblom (Caltech) Waveform Standards NRDA / 18

34 Measurement Standards for LIGO The most restrictive waveform standards are needed for the strongest gravitational wave signals. For Advanced LIGO the maximum signal-to-noise ratio unlikely larger than ρ max 80. Lee Lindblom (Caltech) Waveform Standards NRDA008 1 / 18

35 Measurement Standards for LIGO The most restrictive waveform standards are needed for the strongest gravitational wave signals. For Advanced LIGO the maximum signal-to-noise ratio unlikely larger than ρ max 80. The signal-to-noise quantity C = ρ min S n / h e 1 has been evaluated for equal-mass non-spinning black hole binaries using LIGO noise curves C Initial LIGO Advanced LIGO M / M Lee Lindblom (Caltech) Waveform Standards NRDA008 1 / 18

36 Measurement Standards for LIGO The most restrictive waveform standards are needed for the strongest gravitational wave signals. For Advanced LIGO the maximum signal-to-noise ratio unlikely larger than ρ max 80. The signal-to-noise quantity C = ρ min S n / h e 1 has been evaluated for equal-mass non-spinning black hole binaries using LIGO noise curves. The accuracy requirement for BBH waveforms for Advanced LIGO measurements is therefore: δh m (t) h e (t) < C ρ C Initial LIGO Advanced LIGO M / M Lee Lindblom (Caltech) Waveform Standards NRDA008 1 / 18

37 Detection Standards for LIGO Accuracy requirement for detection depends on the parameter ɛ max, the maximum allowed mismatch between an exact waveform and its model counterpart. The maximum mismatch is chosen to assure searches miss only a small fraction of real signals. The common choice ɛ max = limits the loss rate to about 10%. Lee Lindblom (Caltech) Waveform Standards NRDA / 18

38 Detection Standards for LIGO Accuracy requirement for detection depends on the parameter ɛ max, the maximum allowed mismatch between an exact waveform and its model counterpart. The maximum mismatch is chosen to assure searches miss only a small fraction of real signals. The common choice ɛ max = limits the loss rate to about 10%. h e Real searches are more complicated: comparing signals with a discrete template bank of model waveforms. h b1 ε bank ε max h m h b In this case ɛ max must be chosen so that ɛ max = ɛ match ɛ bank. ε match Lee Lindblom (Caltech) Waveform Standards NRDA / 18

39 Detection Standards for LIGO Accuracy requirement for detection depends on the parameter ɛ max, the maximum allowed mismatch between an exact waveform and its model counterpart. The maximum mismatch is chosen to assure searches miss only a small fraction of real signals. The common choice ɛ max = limits the loss rate to about 10%. h e Real searches are more complicated: ε max comparing signals with a discrete template bank of model waveforms. h ε bank h h b1 m b In this case ɛ max must be chosen so that ɛ max = ɛ match ɛ bank. For Initial LIGO, template banks are constructed with ɛ bank = 0.03, so ɛ max = = is the appropriate choice. Accuracy requirement for BBH waveforms for detection in LIGO: δh m (t) h e (t) ε match < ɛ max C = Lee Lindblom (Caltech) Waveform Standards NRDA / 18

40 Calibration Error Effects in LIGO Model waveform errors are insignificant when they are smaller than some fraction (η max 0.4) of the response function error, δχ m + δφm η max [δχ R + δφr ]. Lee Lindblom (Caltech) Waveform Standards NRDA / 18

41 Calibration Error Effects in LIGO Model waveform errors are insignificant when they are smaller than some fraction (η max 0.4) of the response function error, δχ m + δφm η max [δχ R + δφr ]. For the Initial LIGO S4 data, the calibration errors are: 0.03 δχ R + δφ R 0.06 δχ R + δφ R 0.09 L1 0.1 H1 Lee Lindblom (Caltech) Waveform Standards NRDA / 18

42 Calibration Error Effects in LIGO Model waveform errors are insignificant when they are smaller than some fraction (η max 0.4) of the response function error, δχ m + δφm η max [δχ R + δφr ]. For the Initial LIGO S4 data, the calibration errors are: 0.03 δχ R + δφ R 0.09 L δχ R + δφ R 0.1 H1 A sufficient condition for model waveform error to be insignificant compared to calibration error is therefore: δh m (t) η max C min δχ R h e (t) + min δφ R = Lee Lindblom (Caltech) Waveform Standards NRDA / 18

43 Calibration Error Effects in LIGO Model waveform errors are insignificant when they are smaller than some fraction (η max 0.4) of the response function error, δχ m + δφm η max [δχ R + δφr ]. For the Initial LIGO S4 data, the calibration errors are: 0.03 δχ R + δφ R 0.09 L δχ R + δφ R 0.1 H1 A sufficient condition for model waveform error to be insignificant compared to calibration error is therefore: δh m (t) η max C min δχ R h e (t) + min δφ R = The ideal-detector measurement standard requires waveform errors smaller than calibration errors for ρ 80, so calibration errors prevent optimal accuracy measurements for these sources. Lee Lindblom (Caltech) Waveform Standards NRDA / 18

44 The End Lee Lindblom (Caltech) Waveform Standards NRDA / 18

45 Extra Slides for Discussion Lee Lindblom (Caltech) Waveform Standards NRDA / 18

46 Frequency Domain BBH Waveforms (Equal Mass Non-Spinning) r A h / M 10 Φ h M f FFT of BBH waveform from Scheel, et al. (008) M f Lee Lindblom (Caltech) Waveform Standards NRDA / 18

47 Summary LIGO LISA Waveform Error Measurement Detection Diagnostic Requirement Requirement δφ 1/ ρ ɛmax max δφ 1/ ρ ɛmax δh(t) / h e (t) C/ρ ɛmax C Waveform Error Measurement Detection Diagnostic Requirement Requirement δφ max δφ δh(t) / h e (t) Waveform Error Measurement Detection Diagnostic Requirement Requirement δφ max δφ δh(t) / h e (t) Lee Lindblom (Caltech) Waveform Standards NRDA / 18