LIGO-P080091-00 Model Waveform Accuracy Standards for Gravitational Wave Data Analysis Lee Lindblom1, Benjamin J. Owen2, and Duncan A. Brown3 1 Theoretical Astrophysics 130-33, California Institute of Technology, Pasadena, CA 91125 2 Institute for Gravitation and the Cosmos, and Center for Gravitational Wave Physics, Department of Physics, The Pennsylvania State University, University Park, PA 16802 and 3 Department of Physics, Syracuse University, Syracuse, NY 13244 (Dated: September 3, 2008) Model waveforms are used in gravitational wave data analysis to detect and then to measure the properties of a source by matching the model waveforms to the signal from a detector. This paper derives accuracy standards for model waveforms which are sufficient to ensure that these data analysis applications are capable of extracting the full scientific content of the data, but without demanding excessive accuracy that would place undue burdens on the model waveform simulation community. I. INTRODUCTION The second part, in Sec. III, evaluates the effects of cal- ibration error (i.e., the errors in the measurement of the The purpose of this paper is to derive standards for response function) on the needed accuracy requirements the accuracy of model waveforms sufficient to ensure that for model waveforms. those waveforms are good enough for their intended uses For simplicity, all of our discussion here will be based in gravitational wave data analysis. This is a timely and on the frequency-domain representations of gravitational important subject which has received relatively little at- waveforms h(f), defined as, tention in the scientific literature up to this point. Sev- ∞ eral gravitational wave detectors [1–3] have now achieved h(f) = h(t)e−2πiftdt, (1) a high enough level of sensitivity that the first astrophys- Z−∞ ical observations are expected to occur within the next few years. The numerical relativity community has also where h(t) is the time-domain representation of the wave- matured to the point that several groups are now com- form.1 Since the frequency-domain representation of puting model gravitational waveforms for the inspiral and waveforms is somewhat less familiar to the numerical rel- merger of black hole and neutron star binary systems [4– ativity community, we include Figs. 1 and 2 to illustrate 13]. Beyond the pioneering work of Mark Miller [14] and the frequency-domain amplitude Ah and phase Φh, de- iΦh Stephen Fairhurst [15], however, little effort has gone into fined as h(f) = Ahe , of the waveform for a binary thinking about the question of how accurate these model black-hole system composed of equal-mass non-spinning waveforms need to be. holes. The numerical part of this waveform was produced This paper contributes to this discussion by formulat- by the Caltech/Cornell numerical relativity group [13], ing a set of accuracy standards for model waveforms, suf- and a post-Newtonian model waveform was stitched on ficient to ensure that those waveforms are able to fulfill for the lowest part of the frequency range. The con- the detection and parameter-measurement roles they will stants M and r (used as scale factors in these figures) be required to play in gravitational wave data analysis. are, respectively, the total mass and the effective lumi- The standards presented here are designed to be optimal nosity distance of the binary system. The dashed curves in the sense that waveforms of lesser accuracy would re- represent the stationary-phase approximation to the re- sult in some loss of scientific information from the data, stricted post-Newtonian gravitational waveform for this while more accurate waveforms would merely increase the system [17]. For clarity of presentation, we have re- cost of computing the waveforms without increasing their moved the linear in f part of Φh(f), which corresponds to scientific value in data analysis. Our discussion is done shifting the origin of the time coordinate. We have also here at a fairly abstract level, with the intention that adjusted the constant part of the phase so that Φh(f) these accuracy standards should be applicable to model does not have a zero and can be graphed more conve- waveforms produced by approximate analytical methods niently. These time and phase constants are kinematic (such as post-Newtonian expansions) as well as model parameters, which are not related to the internal dynam- waveforms produced by numerical simulations. ics of the waveforms. Our discussion is divided into two parts: The first part, in Sec. II, assumes that the calibration of the gravita- tional wave detector is perfect. That is, we assume that the response function used to convert the interferometer 1 We follow the convention of the LIGO Scientific Collabora- −2πift output to a gravitational wave signal is known exactly. tion [16] by using the phase factor e in these Fourier trans- forms; most of the early gravitational wave literature and essen- In this ideal detector case, we present simple derivations tially all other computational physics literature use e2πift. This for the needed accuracy of model waveforms for detec- choice does not affect any of the subsequent equations in this tion and separately for parameter measurement purposes. paper. LIGO-P080091-00 2 3 tional wave signal. Second we present in Sec. II B a sim- 10 ple condition on the accuracy of model waveforms needed to ensure a prescribed level of detection efficiency. These 2 accuracy requirements in Secs. IIA and IIB are optimal 10 in the sense that any model waveform violating them would decrease the scientific effectiveness of the detec- 1 2 tor. Unfortunately these conditions depend on the de- 10 r A / M tector’s noise curve in a complicated way, so enforcing h them is somewhat complicated. Therefore we present in 0 Sec. II C a set of simpler conditions that are nevertheless 10 sufficient to guarantee that the optimal conditions are satisfied. While these sufficient conditions are somewhat stronger than needed, they are much simpler to apply; so -1 10 we hope they will be easier for the waveform simulation M f community to adopt and use on a regular basis. -2 10 -3 -2 -1 10 10 10 A. Accuracy Requirements for Measurement The question we wish to address here is, how small FIG. 1: Amplitude Ah of the frequency-domain gravitational waveform for an equal-mass non-spinning binary black-hole must the difference between two waveforms be to ensure system. Solid curve is the numerical waveform model, dashed that measurements with a particular detector are unable curve is the stationary-phase approximation to the restricted to distinguish them? This condition determines how ac- (0th order in amplitude) post-Newtonian waveform. curate a model waveform hm must be to make it indistin- guishable from the exact physical waveform he through any measurement with a particular detector. 3 For simplicity, we will perform our analysis in terms 10 of the frequency-domain representation of the waveforms he(f) and hm(f). Consider the one parameter family of waveforms, Φ h(λ, f) = (1 λ)h (f) + λh (f), h − e m 2 he(f) + λδh(f), (2) 10 ≡ that interpolates between he and hm as λ varies between 0 and 1. We then ask the related question, how accu- rately can a particular gravitational wave detector mea- sure this parameter λ? There exists a well developed the- 1 ory of parameter measurement accuracy for gravitational 10 wave data analysis, discussed for example in Finn [18], M f Finn and Chernoff [19] and Cutler and Flanagan [17]. -3 -2 -1 We construct the noise-weighted inner product he hm 10 10 10 given by h | i ∞ ∗ ∗ FIG. 2: Phase Φh of the frequency-domain gravitational wave- he(f)hm(f) + he(f)hm(f) he hm = 2 df, (3) form for an equal-mass non-spinning binary black-hole sys- h | i Z0 Sn(f) tem. Solid curve is the numerical waveform model, dashed curve is the stationary-phase approximation to the restricted where Sn(f) is the one-sided power spectral density of 3.5-order post-Newtonian waveform. the detector strain noise. In defining this inner product, we use the fact that the Fourier transform of the real gravitational-wave strain h(t) satisfies h(f) = h∗( f), II. IDEAL DETECTOR CASE thus allowing us to define the inner product as an integral− over positive frequencies only. 2 We split the discussion of the ideal detector case into The variance σλ of measurements of the parameter λ three parts. First we present in Sec. II A a simple deriva- is given by the expression tion of the accuracy of model waveforms needed to en- sure no loss of scientific information when the waveform ∂h ∂h σ−2 = = δh δh , (4) is used to measure the physical properties of a gravita- λ ∂λ ∂λ h | i LIGO-P080091-00 3 using Eq. (3.20) of Ref. [18], or Eq. (2.8) of Ref. [17]. B. Accuracy Requirements for Detection If the standard deviation σλ were greater than one (the parametric distance between he and hm), then the two The signal-to-noise ratio for the detection of a signal waveforms would be indistinguishable through any mea- he using an optimal filter constructed from the model surement with the given detector. Thus the condition, waveform hm is given by σλ > 1, or equivalently, ˆ he hm ρm = he hm = h | 1i/2 , (10) δh δh < 1, (5) h | i hm hm h | i h | i ensures that the two waveforms are indistinguishable.2 If cf. Eq. (A24) of Ref. [17]. The question we wish to ad- we consider he to be the exact waveform, and δh to be the dress here is, how accurate must the model waveform hm difference between the model and the exact waveforms, be to ensure no significant loss in the efficiency of detect- then the model waveform will be indistinguishable from ing the signal he? Detections are made when a signal is the exact if and only if δh δh < 1.