6.4 LIGO accuracy standards

The strong emission of gravitational waves from the inspiral and merger of binary black holes has been a dominant motivation for the construction of the LIGO and Virgo gravitational wave observatories. The precise detail of the waveform obtained from numerical simulation is a key tool to enhance detection and allow useful scientific interpretation of the gravitational signal. The first derivation [107] of the accuracy required for numerically-generated black-hole waveforms to be useful as templates for gravitational-wave data analysis was carried out in the frequency domain. Proper accuracy standards must take into account the power spectral density of the detector noise Sn (f), which is calibrated with respect to the frequency domain strain ˆh(f). Consequently, the primary accuracy standards must be formulated in the frequency domain in order to take detector sensitivity into account. See [203Jump To The Next Citation Point] for a recent review.

It has been emphasized [202] that the direct use of time domain errors obtained in numerical simulations can be deceptive in assessing the accuracy standards for model waveforms to be suitable for gravitational-wave data analysis. For this reason, the frequency domain accuracy requirements have been translated into requirements on the time domain L2 error norms, so that they can be readily enforced in practice [203Jump To The Next Citation Point, 204Jump To The Next Citation Point, 201].

There are two distinct criteria for waveform accuracy: (i) Insufficient accuracy can lead to an unacceptable fraction of signals to pass undetected through the corresponding matched-filter; (ii) the accuracy affects whether a detected waveform can be used to measure the physical properties of the source, e.g., mass and spin, to a level commensurate with the accuracy of the observational data. Accuracy standards for model waveforms have been formulated to prevent these potential losses in the detection of gravitational waves and the measurement of their scientific content.

For a numerical waveform with strain component h(t), the time domain error is measured by

||δh|| ℰ0 = -----, (84 ) ||h ||
where δh is the error in the numerical approximation and ||F||2 = ∫ dt|F (t)|2, i.e., ||F || is the L 2 norm, which in principle should be integrated over the complete time domain of the model waveform obtained by splicing a perturbative chirp waveform to a numerical waveform for the inspiral and merger.

The error can also be measured in terms of time derivatives of the strain. The first time derivative corresponds to the error in the news

||δReN || ||δImN || ℰ1(ReN ) = ---------, ℰ1 (ImN ) = --------- (85 ) ||ReN || ||ImN ||
and the second time derivative corresponds to the Weyl component error
ℰ2(ReΨ ) = ||δRe-Ψ-||, ℰ2(Im Ψ ) = ||δIm--Ψ||. (86 ) ||Re Ψ|| ||Im Ψ ||

In [203Jump To The Next Citation Point], it was shown that sufficient conditions to satisfy data analysis criteria for detection and measurement can be formulated in terms of any of the error norms ℰk = (ℰ0,ℰ1,ℰ2), i.e., in terms of the strain, the news or the Weyl component. The accuracy requirement for detection is

√ ------ ℰk ≤ Ck 2𝜖max, (87 )
and the requirement for measurement is
ℰk ≤ Ck ηc. (88 ) ρ
Here ρ is the optimal signal-to-noise ratio of the detector, defined by
∫ ∞ 4|ˆh(f )|2 ρ2 = --------df; (89 ) 0 Sn (f)
Ck are dimensionless factors introduced in [203Jump To The Next Citation Point] to rescale the traditional signal-to-noise ratio ρ in making the transition from frequency domain standards to time domain standards; 𝜖max determines the fraction of detections lost due to template mismatch, cf. Equation (14) of [204]; and ηc ≤ 1 corrects for error introduced in detector calibration. These requirements for detection and measurement, for either k = 0,1,2 conservatively overstate the basic frequency domain requirements by replacing Sn (f) by its minimum value in transforming to the time domain.

The values of Ck for the inspiral and merger of non-spinning equal-mass black holes have been calculated in [203] for the advanced LIGO noise spectrum. As the total mass of the binary varies from 0 → ∞, C0 varies between .65 > C0 > 0, C1 varies between .24 < C1 < .8 and C2 varies between 0 < C2 < 1. Thus, only the error ℰ1 in the news can satisfy the criteria over the entire mass range. The error in the strain ℰ0 provides the easiest way to satisfy the criteria in the low mass case M < < M ⊙ and the error in the Weyl component ℰ2 provides the easiest way to satisfy the criteria in the high mass case M > > M ⊙.

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