Model Waveform and Calibration Accuracy Standards for Gravitational Wave Data Analysis

Transcription

1 Model Waveform and Calibration Accuracy Standards for Gravitational Wave Data Analysis Lee Lindblom Theoretical Astrophysics, Caltech LIGO Seminar, Caltech 10 November 2009 Collaborators: Duncan Brown (Syracuse), Benjamin Owen (Penn State) How accurate must model waveforms and detector calibration be: to prevent a significant rate of missed detections? to prevent a significant accuracy loss for measurements? to avoid unnecessary costs of achieving excess accuracy? Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

2 A Theoretician s View of GW Data Analysis: Data analysis identifies and then measures the properties of signals in GW data by matching to model waveforms. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

3 A Theoretician s View of GW Data Analysis: Data analysis identifies and then measures the properties of signals in GW data by matching to model waveforms. Think of a waveform h(t) as a vector, h, whose components are the amplitudes of the waveform at each frequency: h(f ) = h(t)e 2πi f t dt A h (f )e iφ h(f ) Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

4 A Theoretician s View of GW Data Analysis: Data analysis identifies and then measures the properties of signals in GW data by matching to model waveforms. Think of a waveform h(t) as a vector, h, whose components are the amplitudes of the waveform at each frequency: h(f ) = h(t)e 2πi f t dt A h (f )e iφ h(f ) r A h / M M f Φ h 10 2 M f Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

5 A Theoretician s View of GW Data Analysis II: Let h e = h e (f ) denote the exact waveform for some source, and let h m = h m (f ) denote a model of this waveform. Define a waveform inner product that weights components (frequencies) in proportion to the detector s sensitivity: he h m = h e h m = he (f )h m (f ) + h e (f )hm(f ) df, S n (f ) where S n (f ) is the power spectral density of the detector noise. This inner product is normalized so that ρ = h e h e is the optimal signal-to-noise ratio for detecting the waveform h e. S n 1/2 (f) / Hz 1/ Initial LIGO Advanced LIGO f (Hz) Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

6 A Theoretician s View of GW Data Analysis III: Project the signal h e onto a model waveform, h m : ρ m h e ĥm = h e ĥm = h e h m hm h m. h m he normalized so that ĥm ĥm = 1. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

7 A Theoretician s View of GW Data Analysis III: Project the signal h e onto a model waveform, h m : ρ m h e ĥm = h e ĥm = h e h m hm h m. h m he normalized so that ĥm ĥm = 1. Search for signals by projecting data onto model waveforms: ρ m is the signal-to-noise ratio for h e projected onto h m. A detection is made when h e has a projected signal-to-noise ratio ρ m that exceeds a pre-determined threshold. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

8 A Theoretician s View of GW Data Analysis III: Project the signal h e onto a model waveform, h m : ρ m h e ĥm = h e ĥm = h e h m hm h m. h m he normalized so that ĥm ĥm = 1. Search for signals by projecting data onto model waveforms: ρ m is the signal-to-noise ratio for h e projected onto h m. A detection is made when h e has a projected signal-to-noise ratio ρ m that exceeds a pre-determined threshold. Measured signal-to-noise ratio, ρ m, is largest when the model waveform h m is proportional to the exact h e ; in this case ρ m equals the optimal signal-to-noise ratio ρ: ρ m = h e h e he h e = h e h e = ρ = 2 h e (f ) 2 df. S n (f ) Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

9 How accurate must model waveforms be for GW data analysis? Derive model waveform accuracy requirements for ideal detectors: Standards for detection. Standards for measurement. Determine effects of Detector Calibration Errors. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

10 How accurate must model waveforms be for GW data analysis? Derive model waveform accuracy requirements for ideal detectors: Standards for detection. Standards for measurement. Determine effects of Detector Calibration Errors. Evaluate standards for the LIGO case. Do current LIGO search templates meet the appropriate initial LIGO standards? Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

11 How accurate must model waveforms be for GW data analysis? Derive model waveform accuracy requirements for ideal detectors: Standards for detection. Standards for measurement. Determine effects of Detector Calibration Errors. Evaluate standards for the LIGO case. Do current LIGO search templates meet the appropriate initial LIGO standards? Possible misinterpretations and misapplications of the standards. Transform standards into more user-friendly forms. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

12 Accuracy Standards for Detection The measured signal-to-noise ratio ρ m for detecting the signal h e is the projection of h e onto ĥm: ρ m = h e ĥm = h e h m h m h m 1/2. Errors in model waveform, h m = h e + δh, result in reduction of ρ m compared to the optimal signal-to-noise ratio ρ: ρ m = ρ (1 ɛ) = h e h e 1/2 (1 ɛ). Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

13 Accuracy Standards for Detection The measured signal-to-noise ratio ρ m for detecting the signal h e is the projection of h e onto ĥm: ρ m = h e ĥm = h e h m h m h m 1/2. Errors in model waveform, h m = h e + δh, result in reduction of ρ m compared to the optimal signal-to-noise ratio ρ: ρ m = ρ (1 ɛ) = h e h e 1/2 (1 ɛ). Evaluate this mismatch ɛ in terms of the waveform error: ɛ = δh δh 2 h e h e, where δh = δh ĥe ĥe δh. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

14 Accuracy Standards for Detection II If the maximum range for detecting a signal using an exact model waveform is R, then the effective range for detections using an inexact model waveform will be R(1 ɛ). Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

15 Accuracy Standards for Detection II If the maximum range for detecting a signal using an exact model waveform is R, then the effective range for detections using an inexact model waveform will be R(1 ɛ). The rate of detections is proportional to the volume of space where sources can be seen, so when model waveform errors exist the effective rate of detections is reduced by the amount: R 3 R 3 (1 ɛ) 3 R 3 = 1 (1 ɛ) 3 3ɛ Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

16 Accuracy Standards for Detection II If the maximum range for detecting a signal using an exact model waveform is R, then the effective range for detections using an inexact model waveform will be R(1 ɛ). The rate of detections is proportional to the volume of space where sources can be seen, so when model waveform errors exist the effective rate of detections is reduced by the amount: R 3 R 3 (1 ɛ) 3 R 3 = 1 (1 ɛ) 3 3ɛ The loss of detections can be limited to an acceptable level, by limiting the mismatch ɛ to an acceptable range: ɛ < ɛ max. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

17 Accuracy Standards for Detection II If the maximum range for detecting a signal using an exact model waveform is R, then the effective range for detections using an inexact model waveform will be R(1 ɛ). The rate of detections is proportional to the volume of space where sources can be seen, so when model waveform errors exist the effective rate of detections is reduced by the amount: R 3 R 3 (1 ɛ) 3 R 3 = 1 (1 ɛ) 3 3ɛ The loss of detections can be limited to an acceptable level, by limiting the mismatch ɛ to an acceptable range: ɛ < ɛ max. Consequently model waveform accuracy must satisfy the requirement for detection: δh δh < 2ɛ max ρ 2. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

18 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 Accuracy Standards LIGO Seminar 11/10// / 23

19 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 Accuracy Standards LIGO Seminar 11/10// / 23

20 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 ρ m 1 h = h σ λ λ = δh δh, λ λ σ 2 λ where the noise weighted inner product is defined by h e h m = he (f )h m (f ) + h e (f )hm(f ) df. S n (f ) Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

21 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 ρ m 1 h = h σ λ λ = δh δh, λ λ σ 2 λ where the noise weighted inner product is defined by h e h m = he (f )h m (f ) + h e (f )hm(f ) df. S n (f ) Two waveforms are indistinguishable iff the variance σλ 2 is larger than the parameter distance between the waveforms: ( λ) 2 = 1 < σλ 2 = 1/ δh δh, that is iff 1 > δh δh. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

22 Effects of Detector Calibration Error The raw electronic output of the detector, v(f ), is converted to the measured gravitational wave signal, h(f ), using the response function: h(f ) = R(f )v(f ). Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

23 Effects of Detector Calibration Error The raw electronic output of the detector, v(f ), is converted to the measured gravitational wave signal, h(f ), using the response function: h(f ) = R(f )v(f ). Errors in the measured response function produce errors in the inferred waveform: or equivalently h = Rv e = (R e + δr) v e = h e + δh R, δh R = h e e δχ R+iδΦ R h e h e (δχ R + iδφ R ). Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

24 Effects of Detector Calibration Error The raw electronic output of the detector, v(f ), is converted to the measured gravitational wave signal, h(f ), using the response function: h(f ) = R(f )v(f ). Errors in the measured response function produce errors in the inferred waveform: or equivalently h = Rv e = (R e + δr) v e = h e + δh R, δh R = h e e δχ R+iδΦ R h e h e (δχ R + iδφ R ). Errors in the measured response function also affect the measured power spectral density of the detector noise, S n (f ) = e 2δχ R(f ) S e (f ), with resulting effects on the measured signal-to-noise ratio ρ m. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

25 Effects of Detector Calibration Error II Evaluate the measured signal-to-noise ratio: ρ m = h h m hm h m = h e + δh R h e + δh m he + δh m h e + δh m, ρ 1 2ρ (δh m δh R ) (δh m δh R ), where (δh m δh R ) = δh m δh R ĥe ĥe δh m δh R. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

26 Effects of Detector Calibration Error II Evaluate the measured signal-to-noise ratio: ρ m = h h m hm h m = h e + δh R h e + δh m he + δh m h e + δh m, ρ 1 2ρ (δh m δh R ) (δh m δh R ), where (δh m δh R ) = δh m δh R ĥe ĥe δh m δh R. Errors in the measured signal-to-noise ratio, δρ m, depend only on the difference between the waveform errors: δh m δh R. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

27 Effects of Detector Calibration Error II Evaluate the measured signal-to-noise ratio: ρ m = h h m hm h m = h e + δh R h e + δh m he + δh m h e + δh m, ρ 1 2ρ (δh m δh R ) (δh m δh R ), where (δh m δh R ) = δh m δh R ĥe ĥe δh m δh R. Errors in the measured signal-to-noise ratio, δρ m, depend only on the difference between the waveform errors: δh m δh R. Waveform accuracy standards are therefore just the ideal detector (δh R = 0) standards with δh m replaced by δh m δh R : δh m δh R δh m δh R < 1 for measurement, and δh m δh R δh m δh R < 2ɛ max ρ 2 for detection. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

28 Effects of Detector Calibration Error III The combined accuracy requirements { can be written as 1 measurement, δh m δh R δh m δh R < 2ɛ max ρ 2 detection. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

29 Effects of Detector Calibration Error III The combined accuracy requirements { can be written as 1 measurement, δh m δh R δh m δh R < 2ɛ max ρ 2 detection. Waveform modeling error, δh m, is uncorrelated with calibration error, δh R, so re-write the accuracy requirement using, δhm δh R δh m δh R < δh m δh m + δh R δh R, which leads to the new accuracy requirements: δhm δh m + { 1 measurement, δh R δh R < 2ɛmax ρ detection. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

30 Effects of Detector Calibration Error III The combined accuracy requirements { can be written as 1 measurement, δh m δh R δh m δh R < 2ɛ max ρ 2 detection. Waveform modeling error, δh m, is uncorrelated with calibration error, δh R, so re-write the accuracy requirement using, δhm δh R δh m δh R < δh m δh m + δh R δh R, which leads to the new accuracy requirements: δhm δh m + { 1 measurement, δh R δh R < 2ɛmax ρ detection. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

31 Effects of Detector Calibration Error III The combined accuracy requirements { can be written as 1 measurement, δh m δh R δh m δh R < 2ɛ max ρ 2 detection. Waveform modeling error, δh m, is uncorrelated with calibration error, δh R, so re-write the accuracy requirement using, δhm δh R δh m δh R < δh m δh m + δh R δh R, which leads to the new accuracy requirements: δhm δh m + { 1 measurement, δh R δh R < 2ɛmax ρ detection. Choose the relative size of the errors based on cost, or...? If comparable accuracy standards are adopted, then the calibration standard is δh R δh R < 1/2, and the waveform standards are: { 1/2 measurement, δhm δh m < 2ɛmax ρ 1/2 detection. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

32 Accuracy Standards for LIGO It is useful to define the model waveform (logarithmic) amplitude δχ m and phase δφ m errors: δh m = h e e δχm+iδφm h e h e (δχ m + iδφ m ). Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

33 Accuracy Standards for LIGO It is useful to define the model waveform (logarithmic) amplitude δχ m and phase δφ m errors: δh m = h e e δχm+iδφm h e h e (δχ m + iδφ m ). The basic accuracy requirements can be written as { δh δh 2 2 = δχ m + δφm < 1/(2ρ max ) measurement, ρ 2ɛmax 1/(2ρ max ) detection, where the signal-weighted average errors are defined as δχ m 2 = δχ 2 m 2 h e 2 ρ 2 S n df, and δφ m 2 = δφ 2 m 2 h e 2 ρ 2 S n df. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

34 Accuracy Standards for LIGO It is useful to define the model waveform (logarithmic) amplitude δχ m and phase δφ m errors: δh m = h e e δχm+iδφm h e h e (δχ m + iδφ m ). The basic accuracy requirements can be written as { δh δh 2 2 = δχ m + δφm < 1/(2ρ max ) measurement, ρ 2ɛmax 1/(2ρ max ) detection, where the signal-weighted average errors are defined as δχ m 2 = δφ 2 m δχ 2 2 h e h e 2 m df, and δφ ρ 2 m = df. S n ρ 2 S n The most restrictive measurement standards are needed for the strongest gravitational wave signals. For Advanced LIGO the maximum signal-to-noise ratio unlikely larger than ρ max δχ R + δφr δχ m + δφm < ρ max Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

35 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 Accuracy Standards LIGO Seminar 11/10// / 23

36 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 = h limits the loss rate to about 10%. e εmax Real searches are more complicated: ε FF comparing signals with a discrete template bank of model waveforms. h ε h b MM h m h m b For Initial LIGO, template banks are constructed with ɛ MM = 0.03, so ɛ FF = ɛ EFF ɛ MM = = ε EFF Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

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 = h limits the loss rate to about 10%. e εmax Real searches are more complicated: ε FF comparing signals with a discrete template bank of model waveforms. h ε h b MM h m h m b For Initial LIGO, template banks are constructed with ɛ MM = 0.03, so ɛ FF = ɛ EFF ɛ MM = = To ensure this condition, ɛ max must be chosen so that ɛ max Accuracy requirement for BBH waveforms for detection in LIGO: 2 2 δχ m + δφm < 2ɛmax 1/(2ρ max ) ε EFF Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

38 How good are current LIGO templates? Studies by Pan, et al. Phys.Rev. D77, (2008), and by Boyle, et al. CQG 26, (2009) suggest ɛ FF for current non-spinning LIGO templates may be as large as pn TaylorF2 Maximum overlap pn TaylorF Pseudo-4.0 pn TaylorF Total mass (M ) The effective range R BBH for BBH detections may therefore be reduced by up to (1 ɛ FF ɛ MM )R BBH 0.93R BBH, resulting in an event loss rate that may be as large as 1 (1 ɛ FF ɛ MM ) Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

39 Verifying Calibration Accuracy The standards place limits on the signal- and noise-weighted averages of the frequency-domain amplitude and phase errors of the response function R = R e e δχ R+iδΦ R : δχ R 2 + δφr 2 < 1/(4ρ 2 max ) These standards are difficult (impossible?) to enforce as written because they require the measured response function errors to be averaged with the (unknown) waveform h e. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

40 Verifying Calibration Accuracy The standards place limits on the signal- and noise-weighted averages of the frequency-domain amplitude and phase errors of the response function R = R e e δχ R+iδΦ R : δχ R 2 + δφr 2 < 1/(4ρ 2 max ) These standards are difficult (impossible?) to enforce as written because they require the measured response function errors to be averaged with the (unknown) waveform h e. This can be resolved by enforcing the somewhat stronger sufficient conditions: δχ R 2 + δφr 2 = 0 [ (δχr ) 2 + (δφ R ) 2] 4 h e 2 ρ 2 S n (f ) df, max [ (δχ R ) 2 + (δφ R ) 2] < 1/(4ρ 2 max). Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

41 Verifying NR Waveform Accuracy The standards also place limits on the signal- and noise-weighted averages of the waveform amplitude and phase errors: { δh m δh m 2 2 1/(2ρmax ) measurement, = δχ ρ 2 m + δφm < 2ɛmax detection. How can NR waveforms be checked against these standards? Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

42 Verifying NR Waveform Accuracy The standards also place limits on the signal- and noise-weighted averages of the waveform amplitude and phase errors: { δh m δh m 2 2 1/(2ρmax ) measurement, = δχ ρ 2 m + δφm < 2ɛmax detection. How can NR waveforms be checked against these standards? Express the time-domain waveform in terms of an amplitude A e (t) and phase Φ e (t) of the exact waveform, plus errors, h e (t) = A e (t) cos Φ e (t), h m (t) = A e (t) [1 + δµ χ g χ (t)] cos [Φ e (t) + δµ Φ g Φ (t)], where δµ χ and δµ Φ are the maximum amplitude and phase errors so that g χ (t) 1 and g Φ (t) 1. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

43 Verifying NR Waveform Accuracy II Some NR groups have estimated the maximum time-domain waveform errors δµ χ and δµ Φ, and compared them with the standards for δχ m and δφ m. Is this good enough? Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

44 Verifying NR Waveform Accuracy II Some NR groups have estimated the maximum time-domain waveform errors δµ χ and δµ Φ, and compared them with the standards for δχ m and δφ m. Is this good enough? Consider a model waveform: h m (t) with errors of the form: h m (t) = A e (t) [1 + δµ χ g χ (t)] cos [Φ e (t) + δµ Φ g Φ (t)], with g χ = g Φ = cos[λφ e (t)]. Compute ratio of frequency- to time-domain error measures, R = δχ m 2 +δφm 2 δµ 2 χ +δµ2 Φ using the PN+Caltech/Cornell M / M waveform for A e and Φ e R λ = 0 λ = 2 λ = 4 λ = 6 λ = 8 λ = 10 Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

45 Verifying NR Waveform Accuracy II Some NR groups have estimated the maximum time-domain waveform errors δµ χ and δµ Φ, and compared them with the standards for δχ m and δφ m. Is this good enough? Consider a model waveform: h m (t) with errors of the form: h m (t) = A e (t) [1 + δµ χ g χ (t)] cos [Φ e (t) + δµ Φ g Φ (t)], with g χ = g Φ = cos[λφ e (t)]. Compute ratio of frequency- to time-domain error measures, R = δχ m 2 +δφm 2 δµ 2 χ +δµ2 Φ using the PN+Caltech/Cornell M / M waveform for A e and Φ e R λ = 0 λ = 2 λ = 4 λ = 6 λ = 8 λ = 10 Bad News! Limiting δµ χ and δµ Φ to the standards is not sufficient. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

46 Verifying NR Waveform Accuracy III Additional knowledge of the full waveform errors, δµ χ g χ (t) and δµ Φ g Φ (t), is needed. Unfortunately the exact time dependencies, g χ (t) and g Φ (t), will never be known. Is a partial knowledge of g χ (t) and g Φ (t) sufficient? Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

47 Verifying NR Waveform Accuracy III Additional knowledge of the full waveform errors, δµ χ g χ (t) and δµ Φ g Φ (t), is needed. Unfortunately the exact time dependencies, g χ (t) and g Φ (t), will never be known. Is a partial knowledge of g χ (t) and g Φ (t) sufficient? Probably the most we will ever know will be local-in-time error envelope-functions G χ (t) and G Φ (t), that satisfy g χ (t) G χ (t) 1, and g Φ (t) G Φ (t) 1. Do time-domain bounds imply frequency-domain bounds, i.e., does g(t) G(t) imply g(f ) G(f )? Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

48 Verifying NR Waveform Accuracy III Additional knowledge of the full waveform errors, δµ χ g χ (t) and δµ Φ g Φ (t), is needed. Unfortunately the exact time dependencies, g χ (t) and g Φ (t), will never be known. Is a partial knowledge of g χ (t) and g Φ (t) sufficient? Probably the most we will ever know will be local-in-time error envelope-functions G χ (t) and G Φ (t), that satisfy g χ (t) G χ (t) 1, and g Φ (t) G Φ (t) 1. Do time-domain bounds imply frequency-domain bounds, i.e., does g(t) G(t) imply g(f ) G(f )? No! It is not possible to verify the accuracy of a waveform using a time-domain error-envelope function. 0 Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23 G(t) g(t) 0 0 G(f) g(f)

49 Alternate Waveform Accuracy Requirements This seems like a disaster: error envelope functions are probably the most we will ever know about waveform errors, yet they do not provide useful estimates of the relevant error norms. Is it possible to construct an alternate waveform accuracy requirement that relies only on a bound, g(t) G(t) 1, of the time-domain waveform error? Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

50 Alternate Waveform Accuracy Requirements This seems like a disaster: error envelope functions are probably the most we will ever know about waveform errors, yet they do not provide useful estimates of the relevant error norms. Is it possible to construct an alternate waveform accuracy requirement that relies only on a bound, g(t) G(t) 1, of the time-domain waveform error? A local-in-time error envelope G(t) does provide a bound on the L 2 norm of the frequency-domain waveform error: g(f ) 2 df = g(t) 2 dt G(t) 2 dt. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

51 Alternate Waveform Accuracy Requirements This seems like a disaster: error envelope functions are probably the most we will ever know about waveform errors, yet they do not provide useful estimates of the relevant error norms. Is it possible to construct an alternate waveform accuracy requirement that relies only on a bound, g(t) G(t) 1, of the time-domain waveform error? A local-in-time error envelope G(t) does provide a bound on the L 2 norm of the frequency-domain waveform error: g(f ) 2 df = g(t) 2 dt G(t) 2 dt. A waveform accuracy requirement based on L 2 norms, rather than the usual noise-weighted norm, could therefore be implemented using local-in-time error bounds Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

52 L 2 Norm Accuracy Standard We can derive an accuracy requirement based on L 2 norms: δh m δh m = 2 δh m 2 S n (f ) df 2 δh m(f ) 2 min S n (f ), where δh m (f ) 2 = δh m 2 df is the L 2 norm of δh m (f ). Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

53 L 2 Norm Accuracy Standard We can derive an accuracy requirement based on L 2 norms: δh m δh m = 2 δh m 2 S n (f ) df 2 δh m(f ) 2 min S n (f ), where δh m (f ) 2 = δh m 2 df is the L 2 norm of δh m (f ). We can therefore convert the basic accuracy requirements (on measurement in this case) into the following sufficient condition: δhm δh m 2 δhm (f ) min Sn (f ) < 1 2. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

54 L 2 Norm Accuracy Standard We can derive an accuracy requirement based on L 2 norms: δh m δh m = 2 δh m 2 S n (f ) df 2 δh m(f ) 2 min S n (f ), where δh m (f ) 2 = δh m 2 df is the L 2 norm of δh m (f ). We can therefore convert the basic accuracy requirements (on measurement in this case) into the following sufficient condition: δhm δh m 2 δhm (f ) min Sn (f ) < 1 2. This accuracy requirement demands the waveform h m and its error-envelope estimate δh m to have the proper scale. NR simulations only determine the scale invariant r h m /M and rδh m /M, so what value of the scale r should be used? Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

55 L 2 Norm Accuracy Standards II A scale invariant accuracy standard can be constructed by introducing the obvious L 2 norm waveform scale: δh(f ) h m (f ) = δh(t) min h m (t) < Sn 2 2 h m. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

56 L 2 Norm Accuracy Standards II A scale invariant accuracy standard can be constructed by introducing the obvious L 2 norm waveform scale: δh(f ) h m (f ) = δh(t) min h m (t) < Sn 2 2 h m. Unfortunately, the right side of this new condition depends on h m, which must still be scaled properly. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

57 L 2 Norm Accuracy Standards II A scale invariant accuracy standard can be constructed by introducing the obvious L 2 norm waveform scale: δh(f ) h m (f ) = δh(t) min h m (t) < Sn 2 2 h m. Unfortunately, the right side of this new condition depends on h m, which must still be scaled properly. Introduce the scale invariant quantity C, defined as C 2 ρ 2 = 2 h m (f ) 2 / min S n (f ) 1, and use it to re-write the accuracy standards, δh(f ) h m (f ) = δh(t) h m (t) < C 2ρ, in a way that depends on the waveform scale only through the standard signal-to-noise ratio ρ. Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

58 Sufficient Conditions for LIGO The signal-to-noise quantity C 2 = ρ 2 min S n /2 h m 2 1 has been evaluated for equal-mass non-spinning BBH waveforms using LIGO noise C Initial LIGO Advanced LIGO 0.01 M / M Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

59 Sufficient Conditions for LIGO The signal-to-noise quantity C 2 = ρ 2 min S n /2 h m 2 1 has been evaluated for equal-mass non-spinning BBH waveforms using LIGO noise. Sufficient accuracy requirements for BBH waveforms for Advanced LIGO are therefore: δh m (t) h m (t) measurement, 200 C 2ɛ max detection. { C/2ρ C Initial LIGO Advanced LIGO M / M Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

60 Sufficient Conditions for LIGO The signal-to-noise quantity C 2 = ρ 2 min S n /2 h m 2 1 has been evaluated for equal-mass non-spinning BBH waveforms using LIGO noise. Sufficient accuracy requirements for BBH waveforms for Advanced LIGO are therefore: δh m (t) h m (t) measurement, 200 C 2ɛ max detection. { C/2ρ C Initial LIGO Advanced LIGO M / M These requirements can be enforced as conditions on local-in-time bounds of the amplitude and phase errors: δh m (t) A ( m 2 δµ 2 χ G χ 2 + δµ Φ 2G 2 { Φ) dt C/2ρ measurement h m (t) A mdt 2 C 2ɛ max detection Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23

61 Summary and Questions A set of accuracy standards now exist for detector calibration, 2 2 δχ R + δφr < 1/(2ρmax ), and for model waveforms, { 2 2 1/(2ρ δχ m + δφm < max ) measurement, 2ɛmax 1/(2ρ max ) detection. These standards are difficult (impossible?) to enforce directly, so easier to enforce conditions have been derived, for calibration max[(δχr ) 2 + (δφ R ) 2 ] < 1/(2ρ max ), and for waveforms: δh m (t) h m (t) A 2 m ( δµ 2 χ G 2 χ + δµ 2 Φ G 2 Φ) dt A 2 mdt { C/(2ρmax ) measure, C 2ɛ max detection. Do the calibration and search template accuracies currently being used by LIGO satisfy these requirements? Do the waveforms produced by various NR groups satisfy the Advanced LIGO versions of these accuracy requirements? Lee Lindblom (Caltech) Waveform Accuracy Standards LIGO Seminar 11/10// / 23