Use and abuse of the model waveform accuracy standards

Accuracy standards have been developed to ensure that the waveforms used for gravitational-wave data analysis are good enough to serve their intended purposes. These standards place constraints on certain norms of the frequency-domain representations of the waveform errors. Examples are given here of possible misinterpretations and misapplications of these standards, whose effect could be to vitiate the quality control they were intended to enforce. Suggestions are given for ways to avoid these problems.

Figure 1

Shown is $R$, the ratio of the proper frequency-domain waveform-error measure to the simple time-domain maximum measure, for a range of masses and values of the frequency parameter $\lambda$. When $R>1$ the accuracy standards could be violated if the simple maximum error measure were used instead of the proper frequency-domain measure.

Figure 2

Example of a time-domain waveform error $g(t)$ and an envelope function $G(t)$ that satisfies $G(t)\ge |g(t)|$.

Figure 3

Frequency-domain representations of the model-waveform error and envelope functions $g(f)$ and $G(f)$, whose time-domain representations appear in Fig. 2. The envelope function $G(f)$ obviously fails to provide a local bound on the frequency-domain error $g(f)$.

Figure 4

Curves illustrate $C$, the ratio of the standard signal-to-noise measure $\rho$ to a nonstandard measure defined in Eq. (29), as a function of the total mass for nonspinning equal-mass binary black-hole waveforms. Dashed curve is based on the initial LIGO noise spectrum [7]; solid curve is based on an advanced LIGO noise curve [6].

Figure 5

Curves illustrate $C$, the ratio of the standard signal-to-noise measure $\rho$ to a nonstandard measure defined in Eq. (29), as a function of the total mass for nonspinning equal-mass binary black-hole waveforms. Various curves use gravitational waveforms containing different numbers of gravitational-wave cycles, indicated by the parameter $N_{\mathrm{cycles}}$.

Figure 6

Curves illustrate $\tilde{C}$, the ratio of the standard signal-to-noise measure $\rho$ to another nonstandard measure defined in Eq. (32), as a function of the total mass for several nonspinning equal-mass binary black-hole waveforms. This quantity is used to enforce calibration accuracy standards for gravitational-wave detectors.

Figure 7

Curves illustrate $\overline{R}$, as defined in Eq. (38) for waveform errors of the form $\delta {\chi }_E=\delta {\Phi }_E=\delta \mu (A_m/\max A_m{)}^p$. The quantity $\overline{R}$ measures the ratio between the time-domain $L^2$ norm measure of the waveform error $\Vert \delta h\Vert /C\Vert h\Vert$, and the maximum time-domain errors $\sqrt{\delta {\mu }_{\chi }^2+\delta {\mu }_{\Phi }^2}$.