Model Waveform Accuracy Standards for Gravitational Wave Data Analysis

LIGO-P080091-00

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 calibration 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 result 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 gravitational 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 transforms; 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 indistinguishable 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 measure 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 ensure 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. observed that exceeds a predetermined threshold signal- h | i We can re-express this limit, Eq. (5), as a simple condi- to-noise ratio. So errors in evaluating the signal-to-noise tion on the needed phase and amplitude accuracies of the ratio will decrease the detection efficiency. We must de- model waveforms. Let χe and Φe denote real functions termine therefore how errors in the model waveform hm representing the logarithmic amplitude and phase of the degrade the measured signal-to-noise ratio, ρm, relative χe+iΦe 1/2 exact waveform: h e . The model waveform to the optimal signal-to-noise ratio ρ = he he . We e h | i may differ from the exact≡ in both amplitude and phase: introduce a parameter ǫ, referred to as the mismatch in δχ+iδΦ he+δh = hee or to first order δh = (δχ+iδΦ)he. It the gravitational-wave data-analysis literature [20], that is also useful to introduce the normalized waveform hê = measures this signal-to-noise reduction: −1 2 heρ , which satisfies hê hê = 1, where ρ = he he . h | i h | i ρm = (1 ǫ)ρ. (11) Using these quantities we can express the inner product − δh δh in the following way h | i To determine how model waveform errors δh effect ǫ, we write the model waveform as hm = he + δh, and re- 2 δh δh = ρ (δχ + iδΦ) hê (δχ + iδΦ) hê , write Eq. (11) in terms of the definitions of ρ and ρm: h | i D E 2 2 2 2 he he + δh 2 = ρ δχ + δΦ , (6) h | i = (1 ǫ) he he . (12) hh + δ h + δh − h | i h e | e i where the signal-weighted averages of the logarithmic This equation can be simplified by decomposing δh into amplitude and phase errors are defined by, two parts: δh = hˆ δh hˆ and δh = δh δh . This k eh | ei ⊥ − k δhk is proportional to he, while δh⊥ is orthogonal to it 2 in the sense that δh h = 0. Using these expressions δχχ δ hê δχhê , (7) ⊥ e ≡ it is straightforwardh to| derivei the following relationship 2 δΦ δΦhê δΦhê . (8) between the mismatch ǫ and the model waveform error: ≡

δh⊥ δh⊥ We can use these definitions to express Eq. (5) as a simple ǫ = h | i. (13) 2 h h limit on the signal-weighted averages of the logarithmic h e| ei amplitude and phase errors: We have kept only the lowest-order terms in δh (which we assume to be small) in this expression. Eq. (13) shows 2 2 1 δχ + δΦ < . (9) that the mismatch ǫ is proportional to the square of the ρ2 distance between two waveforms, as measured by the noise-weighted inner product. Equation (5) or equivalently Eq. (9) are the basic re- To ensure a high level of detection efficiency while us- quirements on model waveforms for measurement pur- ing an optimal filter based on hm, we must ensure that poses. These requirements are optimal in the sense that the model waveform error δh is small enough to prevent waveforms more accurate than this would not improve ǫ from becoming unacceptably large. Let ǫmax be the scientific measurements, while less accurate waveforms maximum mismatch compatible with our target detec- would degrade some measurements. tion efficiency. In that case Eq. (13) places the following limit on the model waveform error:

2 δh δh < 2ρ ǫmax. (14) h ⊥| ⊥i 2 This inequality, obtained through a different argument, was presented by Stephen Fairhurst in a talk at the “Interplay between When real searches are conducted to detect signals by Numerical Relativity and Data Analysis” Miniworkshop, at the matching to a model waveform, the measured signal-to- KITP, UCSB in January 2008. noise ratio ρm is maximized over different time and phase LIGO-P080091-00 4 offsets of the model waveform. Thus the part of the h e model waveform phase error linear in f (the part that depends on the time and phase offsets) is not relevant ε EFF ε for detection. Strictly speaking then, the inner product FF that appears in Eq. (14) should be interpreted, not as the inner product of Eq. (3), but as the “match” inner ε h h h product that is the Eq. (3) inner product optimized over b MM m b' these time and phase offsets [20]. Equation (14), with suitably interpreted inner product FIG. 3: Solid line illustrates the model-waveform submani- δh δh , gives the optimal condition on the allowed fold, with particular members of a discrete template bank hb h ⊥| ⊥i model waveform error for detection purposes. Unfortu- and hb′ . The model waveform hm has the maximum mismatch nately it is not generally possible to determine what the ǫMM with the template bank waveform hb. The correspond- ing exact waveform he has mismatch ǫFF with hm and total orthogonal part of the waveform error δh⊥ actually is without knowing the exact waveform, so a simpler, eas- effective mismatch ǫEFF = ǫMM + ǫFF with hb. ier to evaluate limit is desirable. We can obtain such a condition by noting that δh δh δh⊥ δh⊥ (where ˆ ˆ h | i ≥ h | i match [20]; and we refer to EFF = he hb = 1 ǫEFF the inner product on the left can be the standard noise- as the “effective fitting factor.” Theh goal| isi to have− any weighted inner product); so a sufficient condition that physical waveform match some model waveform in the ensures the target detection efficiency is template bank with a mismatch that is no greater than 2 δh δh < 2ρ ǫmax. (15) the chosen ǫEFF (for example ǫEFF = 0.035). The value of h | i ǫFF is completely determined by the chosen target ǫEFF, We note that this condition is considerably weaker (de- and the parameter ǫMM which describes the spacing of max pending on the values of ρ and ǫ ) than the limit pre- models in the discrete template bank. Since hˆm is the sented in Eq. (5) to ensure no loss of accuracy in param- best fit model waveform for the exact waveform hê, it eter measurements. We can also transform this limit, follows that the relative waveform vector hê hˆm will be Eq. (15), into a simple expression for the needed accu- orthogonal to any vector tangent to the model-waveform− racy of the amplitude and phase of model waveforms, submanifold at hˆm; thus 0 = hê hˆm hˆb hˆm in the in analogy with those found for measurement accuracy ˆ ˆ h − | − ˆ i in Sec. II A. As before we express the model waveform limit that hb and he are infinitesimally close to hm. Writ- error in terms of logarithmic amplitude and phase er- ing out this orthogonality condition in terms of the mismatch parameters defined above gives rors: δh = (δχ + iδΦ)he. Substituting this into Eq. (15), we arrive at a simple expression for the limit on the ǫFF = ǫEFF ǫMM, (17) signal-weighted averages of the logarithmic amplitude − and phase errors required for detection: a kind of Pythagorean theorem for model-waveform mis- 2 2 matches. The template banks used for compact binary δχ + δΦ < 2ǫmax. (16) searches in initial LIGO are constructed with ǫMM =

The value of the maximum mismatch ǫmax that ap- 0.03 [21, 22, 24], implying that the needed accuracy pears in Eqs. (15) and (16) must be set by the de- of the model waveforms is about ǫFF = 0.005 when mands of the particular data-analysis application. Set- ǫEFF = 0.035. Thus we must choose the maximum mis- ting ǫmax = 0.035 in a search using a single model- match ǫmax, that appears in Eqs. (15) and (16), to be waveform template, for example, would result in a re- ǫmax = ǫFF = 0.005 for LIGO searches. duction in detection rate (for sources that are uniformly 3 distributed in space) of 1 (1 ǫmax) 0.10, a tar- C. Sufficient Conditions get that is often adopted in− LIGO− searches≈ for compact binary inspirals [21–24]. Real gravitational-wave searches are more compli- The model waveform accuracy requirements for mea- cated, and making the appropriate choice of ǫmax is surement, Eq. (9), and detection, Eq. (16), are optimal more subtle. Real searches are generally performed by in the sense that a model waveform failing to meet these matching discrete template banks of model waveforms standards will cause a loss of scientific information. Con- versely, a model waveform having smaller errors than re- with the data. Let hˆb represent one of the (normal- quired will result in no added scientific value. Unfortu- ized) waveforms in the discrete template bank; let hˆm be a (normalized) model waveform whose inner product nately these accuracy requirements are somewhat complicated to evaluate, since they place limits on the signal- with hˆb is the minimum allowed in the given template ˆ weighted amplitude and phase errors, δχ and δ Φ. These bank; and let he denote the (normalized) exact wave- weighted averages must be computed with the frequency- ˆ form that corresponds to hm, see Fig. 3. The quantity domain waveform hm(f) and the detector noise spectrum ˆ ˆ FF = he hm 1 ǫFF is often referred to as the fit- S (f). While model waveforms often scale in a trivial h | i ≡ − n ting factor [25]; MM = hˆ hˆ = 1 ǫMM is the minimal way with the total mass of the gravitational wave source, h m| bi − LIGO-P080091-00 5 the detector noise spectrum does not. So the model wave- In this expression fc is a frequency that characterizes the form errors must be evaluated separately for each mass, particular waveform. For the equal mass binary black- and for each detector noise spectrum. The purpose of hole waveforms shown in Fig. 1 for example, a convenient this section is to construct a set of simpler to apply ac- choice might be fc = 0.08/M which occurs near the point curacy requirements that are nevertheless sufficient to in the spectrum where the two black holes merge. This guarantee that the optimal conditions Eqs. (9) and (16) quantity C¯ is the ratio of the standard signal-to-noise are satisfied. While these sufficient conditions are stricter measure ρ, and the non-standard measure Ae(fc)/n¯. Us- than needed in many cases, we hope that their ease of use ing the definition of C¯ and Eq. (20), we can convert will allow the waveform simulation community to employ Eq. (5) into a simple sufficient condition on the model them on a regular basis. In this section we present three waveform accuracy for measurement, different sufficient conditions that can be applied to the 2 2 2 frequency-domain representations of the waveforms, and max δA max A δΦ C¯ | | + | e | < , (23) one condition that can be applied directly to the time- Ae(fc) Ae(fc) ρ domain waveforms. The simplest sufficient conditions can be obtained by and Eq. (15) into a sufficient condition for detection pur- noting that max δχ δχ and max δΦ δ Φ,asacon- poses, sequence of the definitions| | ≥ of the signal-weighted| | ≥ waveform errors in Eqs. (7) and (8). The following are there- 2 2 max δA max AeδΦ ¯2 fore sufficient conditions which ensure the optimal wave- | | + | | < 2ǫmaxC . (24) Ae(fc) Ae(fc) form standard Eq. (9) is satisfied for measurement, We note that the errors bounded in Eqs. (18) and (19) are 2 2 1 the logarithmic amplitude and phase errors, while those (max δχ ) + (max δΦ ) < 2 , (18) | | | | ρ in Eqs. (23) and (24) are errors relative to the fixed char- acteristic amplitude Ae(fc). The limits in Eqs. (23) and and Eq. (16) is satisfied for detection (24) may therefore be more useful, because they avoid 2 2 the unnecessarily restrictive conditions on max δχ and (max δχ ) + (max δΦ ) < 2ǫmax. (19) | | | | | | max δΦ when those maxima occur at frequencies where the amplitude| | of the waveform is small. We note that the maxima max δχ and max δΦ in these The requirements in Eqs. (23) and (24) have the dis- limits refer to the frequency-domain| | waveform| | errors. advantage, however, of involving the quantity C¯ which These requirements are significantly simpler to apply depends on the details of the waveform and the detec- than the optimal waveform standards because they elim- tor noise spectrum. Nevertheless, this quantity can be inate the detector noise spectrum from the calculation, evaluated once-and-for-all for a given class of waveforms except for its contribution to the signal-to-noise ratio ρ. and a given detector noise spectrum. For example, Fig. 4 While simple to evaluate however, these limits on the illustrates C¯ for the case of non-spinning equal-mass bi- maxima are stronger than necessary, especially if the am- nary black-hole waveforms with fc 0.08/M using the plitude or phase errors are sharply peaked in a narrow Initial LIGO noise curve [26] with a≈ 40 Hz low frequency range of frequencies or the maximum occurs at a fre- cuttoff [1], and the Advanced LIGO noise curve [27] with quency where the amplitude of the wave is very small. a 10 Hz low frequency cuttoff. The scale factor M used A second, sometimes less demanding sufficient condi- ⊙ in this figure is the mass of the sun, M = 2.0 1033 g, tion on the accuracy of model waveforms can be obtained ⊙ which in geometrical time units is M = 4.9 ×10−6 s. by noting that δh δh can be written as, ⊙ × h | i This curve shows that C¯ & 0.6 for the mass range of δh δh = δA δA + A δΦ A δΦ , binary black-hole systems of primary relevance to Initial h | i h | i h e | e i LIGO, 5 M/M 100 [28]. Thus for Initial LIGO it 2 2 ≤ ⊙ ≤ max δA max AeδΦ is sufficient to enforce Eqs. (23) and (24) for the case | | + | | , (20) ≤ n¯ n¯ C¯ 0.6. The mass range for binary black-hole systems≈ of primary relavance for Advanced LIGO extends χe where Ae = e , δA = Aeδχ, and the average detector to, 5 M/M⊙ 400, because the low frequency cuttoff noisen ¯ is defined as, is 10≤ Hz instead≤ of 40 Hz. For Advanced LIGO then, it ¯ ∞ is appropriate to use the minimum value C 0.06 when 1 df enforcing Eqs. (23) and (24). ≈ 2 4 = 1 1 . (21) n¯ ≡ Z0 Sn(f) h | i A third sufficient condition can be obtained by noting that, We can use Eq. (20) to convert Eqs. (5) and (15) into alternate sufficient conditions for model waveform accu- ∞ δh(f) 2df racy by introducing a quantity C¯: δh δh = 4 | | , h | i Z0 Sn(f) ρ n¯ 2 δh(f) 2 C¯ . (22) || || (25) ≡ Ae(fc) ≤ min Sn(f) LIGO-P080091-00 6

and for detection 2 2 δA(f) + AeδΦ(f) 2 10 || || || || < 2ǫmaxC . (31) A (f) 2 C ||| e | These requirements depend only on the average value of the model waveform error δh(f) , and so they are considerably weaker than (and|| in this|| sense are superior 1 to) the conditions in Eqs. (23) and (24). Their biggest drawback is their dependence on the waveform and noise spectrum through the quantity C. It is straightforward to show that C 1 following an argument similar to that C which led to Eq.≤ (25). However the exact value of C will 0.1 depend on the details of the model waveform, including in particular the mass of the gravitational wave source. But C can be evaluated once-and-for-all for a given class of model waveforms and a given detector noise spectrum. M/M For example, Fig. 4 shows a graph of C as a function 0.01 of the mass for non-spinning equal-mass binary black- 10 100 hole waveforms (evaluated with the Initial and the Ad- vanced LIGO noise curves). This figure shows that taking FIG. 4: Solid curves illustrate C¯ and C, as defined in Eqs. (22) min C 0.02 in Eqs. (28) and (29) is sufficient for the bi- ≈ and (27), as functions of the total mass for non-spinning nary black-hole mass range 5 M/M⊙ 400 of primary equal-mass binary black-hole waveforms and the initial LIGO relevance to Advanced LIGO,≤ while taking≤ min C 0.04 noise spectrum. Dashed curves give the same quantities com- ≈ is sufficient for the mass range 5 M/M⊙ 100 of puted with the Advanced LIGO noise curve. primary relevance to Initial LIGO [28].≤ ≤ Our discussion up to this point has focused on the development of accuracy standards for the frequency- where δh(f) is the L2 norm of δh(f), defined as || || domain representations of model waveforms. This ap- ∞ proach simplifies the analysis, and is natural from the δh(f) 2 = 2 δh(f) 2df. (26) LIGO data analysis perspective. However, model wave- || || | | Z0 forms must often be computed in the time domain, e.g. The inequality in Eq. (25) can be converted to sufficient by numerical relativity simulations. While time-domain conditions for the optimal waveform requirements by in- waveforms can be converted to the frequency domain, do- troducing the quantity C, ing so is somewhat delicate and requires making judicious choices in performing the needed Fourier transforms— 2 2 ρ min Sn(f) like choosing appropriate windowing functions. Having C = 2 , (27) time-domain versions of the needed accuracy standards 2 he(f) | | || would therefore make it much easier for the waveform the ratio of the standard signal-to-noise measure ρ to an- simulation community to monitor and deliver waveforms other non-standard measure h (f) / min S (f).Us- of the needed accuracy. The third set of sufficient wave- || e || n ing this definition it is straightforward top convert Eq. (25) form accuracy requirements, Eqs. (28) and (29), can eas- into sufficient conditions for the optimal error require- ily be converted to conditions on the time-domain wave- 2 ments of Eq. (5) for measurement, forms. This can be done because the L norm of a time-domain waveform is identical to the L2 norm of its 2 δh(f) 2 C frequency-domain counterpart, δh(t) = δh(f) , by || || || || || ||2 < , (28) Parseval’s theorem. Thus, the following are the corre- he(f) ρ ||| | sponding time-domain waveform accuracy standards for and Eq. (15) for detection measurement,

2 δh(t) C δh(f) 2 || || < , (32) || || < 2ǫmaxC . (29) h (f) 2 he(t) ρ ||| e | ||| | As with the previous requirements, these can be written and for detection in terms of the amplitude and phase of the frequency- δh(t) || || < √2ǫmax C. (33) domain waveform: for measurement, h (t) ||| e | 2 2 2 δA(f) + AeδΦ(f) C It is not generally possible to decompose the real time- || || || || < , (30) A (f) 2 ρ domain waveform unambiguously into amplitude and ||| e | LIGO-P080091-00 7 phase. Therefore it is not possible to construct mean- order in the two types of errors, we find: ingful time-domain analogs of the amplitude and phase h + δh h = ρ2 + ρ2 (δχ δχ )hˆ hˆ limits given in Eqs. (30) and (31). h e R| mi h m − R e| ei ρ2 + (δχ δχ )hˆ (δχ δχ )hˆ 2 h m − R e| m − R ei 2 III. INCLUDING CALIBRATION ERRORS ρ (δΦ δΦ )hˆ (δΦ δΦ )hˆ , (39) − 2 h m − R e| m − R ei In this section we consider the implications of having −1/2 1 1 hm hm = (δχm δχR)hê hê a detector response function that is not known with ab- h | i ρ − ρh − | i solute precision. First we establish a little notation. Let 1 (δχ δχ )hˆ (δχ δχ )hˆ v(f) denote the direct electronic output of the detector, −ρh m − R e| m − R ei and R(f) the response function that converts the raw 3 ˆ ˆ 2 output v(f) into the inferred gravitational wave signal + (δχm δχR)he he . (40) 2ρh − | i h(f): Combining these using Eq. (38) gives an expression for h(f) = R(f)v(f). (34) the effects of model and calibration errors on the mea- 3 sured signal-to-noise ratio ρm: Let us assume that the measured response function R(f) ρ ρm = ρ (δχm δχR)hê (δχm δχR)hê differs from the correct exact response function Re by − 2h − | − i δR = R Re. This error in the response function will ρ − (δΦm δΦR)hê (δΦm δΦR)hê effect measurements in two ways. First, the response − 2h − | − i of the instrument to a gravitational wave signal he will ρ 2 + (δχm δχR)hê hê . (41) produce an electronic output ve. Using the measured 2h − | i response function R, the signal will be interpreted as the It is illuminating to write this expression in the form, δχR+iδΦR waveform h = Rve = hee , where the logarithmic 1 response function amplitude δχR and phase δΦR errors ρm = ρ (δhm δhR)⊥ (δhm δhR)⊥ , (42) are defined by − 2ρh − | − i where δχR+iδΦR R = Re + δR = Ree . (35) (δh δh ) = δh δh hˆ δh δh hˆ . (43) m − R ⊥ m − R − eh m − R| ei Thus there will be a waveform error, The dependence of the measured signal-to-noise ratio ρm on the waveform errors, δhm and δhR, can be under- δh = h eδχR+iδΦR h , (36) R e − e stood as follows: When the modeling errors are identical to the calibration errors, the measured signal and tem- caused by calibration errors in the instrument. The sec- plate are still identical, and so the measured matched fil- ond effect of calibration error on measurements made tering signal-to-noise ratio is unchanged. When the net with the instrument will be errors in our knowledge of waveform error, δhm δhR, is proportional to the ex- the characteristics of the noise in the detector. In par- − act waveform, he, the measured signal-to-noise ratio ρm ticular the estimated power spectral density of the noise is again unchanged because such errors merely re-scale S will differ from the exact S due to the calibration n e the template, which has no effect on ρm. Thus only net error δR. The estimated power spectral density S will n waveform errors that are orthogonal to he, in the sense be related to S by e of Eq. (43), contribute to losses in ρm. We can use this basic expression, Eq. (42), for the com- 2δχR Sn(f) = Se(f)e . (37) bined effects of model waveform error and calibration error to derive a useful inequality on the signal-to-noise The idea now is to evaluate the effects of errors in ratio: δχm+iδΦm the model waveform, hm = hee , plus the ef- 1 fects of errors in the detector response function R = ρm ρ δhm δhR δhm δhR , δχR+iδΦR ≥ − 2ρh − | − i Ree , on the signal-to-noise ratio of a detected 1 2 gravitational wave signal: ρ hδ δh + hδ δh . (44) 2ρ m m R R ≥ − hph | i ph | ii h + δh h ρ = h e R| mi, (38) m h h 1/2 h m| mi 3 where the inner product is evaluated with respect to the A special case of this expression for the signal-to-noise ratio that included the first-order calibration-error terms was obtained pre- estimated power spectral noise density Sn(f) of Eq. (37). viously by Bruce Allen [29], and an expression that included the The needed calculation is quite messy, so we provide a second-order calibration error terms (and some of the model- few intermediate steps. Keeping terms through second error terms) was obtained previously by Sakanta Bose [30]. LIGO-P080091-00 8

To aid our understanding of this expression, it is useful errors alone. From Eq. (49) we see that this requires 2 to define the ratio of the model waveform error to the (1+ηmin) 2 or equivalently ηmin . 0.4. Reducing ηmin response function error, η: below this value≤ has little effect on the signal-to-noise ratio; while quickly increasing the computational cost of the 2 δhm δhm = η δhR δhR , (45) most accurate model waveforms. So this choice should be h | i h | i close to optimal. which allows us to re-write Eq. (44) as The most recent publicly available calibration data from Initial LIGO (S4), see Figs. 23 and 24 of Ref. [31], 1 2 ρm ρ (1 + η) δhR δhR . (46) has the following limits on the frequency-domain calibra- ≥ − 2ρ h | i 2 2 tion error: 0.03 . δχR + δΦR . 0.09 for the L1 detec- 2 2 If η becomes too small, then the error in the measured tor and 0.06 . δχpR + δΦR . 0.12 for the H1 detector. signal-to-noise ratio ρm is dominated by calibration er- Therefore the appropriatep minimum error requirement, rors, and further reductions in the model waveform error including the effects of calibration errors from Eq. (48), have little effect on our ability to make measurements for Initial LIGO is or detections. The idea then is to place a limit on η 2 2 2 2 which ensures the model waveform is only as accurate as δχ + δΦ ηmin (minχ δ R ) + (min δΦR ) q m m ≥ | | | | it needs to be to achieve the ideal-detector accuracy stan- & 0.012p. (50) dards of Sec. II. It is appropriate therefore to require that η be no smaller than some minimum cutoff: η ηmin. ≥ Model waveform errors smaller than this limit will always In other words, the model waveform error need never be be dominated by calibration errors. made smaller than The magnitude of the calibration error in Initial LIGO is small enough that the calibration requirement, δh δh η2 δh δh . (47) h m| mi ≥ minh R| Ri Eq. (50), is consistent with our basic measurement re- This inequality can also be expressed as a condition on quirements, Eq. (9) or (32), for all but the very strongest the signal-weighted amplitude and phase errors defined sources: ρ & 80. Therefore the presence of calibration er- in Sec. II: ror will not effect the ideal-detector waveform accuracy standards for LIGO measurements, unless an extremely 2 2 2 2 2 strong source is detected. This is not to say that the δχm + δΦm ηmin δχR + δΦR . (48) ≥ presence of calibration error will have no effect on the accuracy of measurements for weaker sources; but the These lower limits, Eqs. (47) or (48), on the model wave- question of exactly how large those calibration error ef- form error must be imposed simultaneously with the ap- fects are in that case must be decided by a somewhat propriate upper limits derived in Sec. II for measurement, more detailed analysis than the one presented here. Eq. (9), or detection, Eq. (16). Calibration error will in- The lower limit on model waveform error that arises terfere with the operation of a detector whenever it is from the presence of calibration error is always consis- impossible to satisfy the ideal-detector accuracy require- tent with the condition on the waveform error needed for ments of Sec. II and the calibration-error lower limits detection, Eq. (16). From Eq. (49), the fractional change simultaneously. in the signal-to-noise ratio caused by calibration error is To determine the appropriate value for ηmin, we note from Eq. (46) that a model waveform having the minimal δρ 2 δhR δhR error η = ηmin could have an effect on the measured (1 + ηmin) h | i. (51) ρ ≤ 2ρ2 signal-to-noise ratio that is as large as,

For Initial LIGO δhR δhR . 0.014 (from the calibra- 1 2 h | i ρm ρ (1 + ηmin) δhR δhR . (49) tion error measurements quoted above) thus the signal- ≥ − 2ρ h | i to-noise ratio ρ would have to be smaller than about 1.7 Setting ηmin = 1 corresponds to one natural choice: mak- (which is below any reasonable statistical threshold for ing the model waveform errors greater than or equal to detection) before the right side of Eq. (51) exceeds the the calibration errors. However, we see from Eq. (49) detection limit ǫmax = 0.005. Thus, the current level of that the signal-to-noise ratio may be degraded by up to LIGO calibration errors are never likely to influence the four times the effect of calibration error alone in this case. waveform accuracy requirements established in Sec. II B So this obvious choice for ηmin is probably too large. for detection. Given the relatively low cost of producing improved model waveforms (compared to the cost of improving the hardware needed to reduce calibration errors), it makes IV. DISCUSSION sense to adopt a stricter standard for ηmin. Another natural choice is to require that ηmin be small enough to In this paper we have developed a set of accuracy stan- ensure that the most accurate model waveforms have no dards for model gravitational waveforms. These stan- larger effect on the signal-to-noise ratio than calibration dards are designed to ensure that model waveforms are LIGO-P080091-00 9

Waveform Error Equation Measurement Detection Diagnostic Numbers Requirement Requirement δΦ 9, 16 1/√2 ρ √ǫmax 10 max δΦ 18, 19 1/√2 ρ √ǫmax | | C max AeδΦ /Ae(fc) 23, 24 C¯/√2 ρ √ǫmax C¯ | | ( AeδΦ f) / Ae(f) 30, 31 C/√2 ρ √ǫmax C || || || || 1.0 hδ (t) / he(t) 32, 33 C/ρ √2ǫmax C || || || || TABLE I: Summary of model waveform accuracy require- C ments for various waveform error diagnostics for measurement 0.1 purposes (column three) and for detection purposes (column four). 0.01 accurate enough to support the parameter measurement M/M and detection needs of the gravitational wave data analy- 5 6 7 sis community—without compromising the scientific con- 10 10 10 tent of the data, and without placing needless demands for accuracy on the waveform modeling community. In FIG. 5: Curves illustrate C¯ and C, as defined in Eqs. (22) and Sec. II we developed an optimal requirement for mea- (27), as functions of the total mass for non-spinning equal- surement purposes in Eq. (9) and for detection purposes mass binary black-hole waveforms and an approximate LISA in Eq. (16). These optimal standards place limits on the noise spectrum. signal averaged amplitude and phase errors, δχ and δΦ respectively, as defined in Eqs. (7) and (8). The first row Waveform Error Equation Measurement Detection of Table I summarizes these accuracy standards (assum- Diagnostic Numbers Requirement Requirement ing the amplitude and phase errors to be comparable): ρ δΦ 9, 16 0.008 0.07 represents the standard signal-to-noise ratio of the wave- max δΦ 18, 19 0.008 0.07 form, and ǫmax represents the maximum signal-to-noise | | max AeδΦ /Ae(fc) 23, 24 0.0005 0.004 mismatch tolerated by a given detection procedure. | | ( AeδΦ f) / Ae(f) 30, 31 0.00016 0.0014 Also listed in Table I are summaries of several suffi- || || || || hδ (t) / he (t) 32, 33 0.0002 0.002 cient conditions on the waveform accuracy developed in || || || || Sec. IIC and described in detail in Eqs. (18), (19), (23), (24), (30), (31), (32), and (33). The table entries for TABLE II: Summary of model waveform accuracy require- phase errors assume that the amplitude and phase errors ments for the Advanced LIGO detector using various wave- are comparable. These sufficient conditions are some- form error diagnostics for measurement purposes (column what stronger than needed, but if satisfied they ensure three) and for detection purposes (column four). the optimal requirements are satisfied as well. They are much simpler to apply. The quantities C¯ and C which appear in some of these conditions, defined in Eqs. (22) and nal is detected by Initial LIGO, then the expectation is (27), compare different signal-to-noise measures of the that a similar signal with ten times that signal-to-noise waveforms. These quantities, which are waveform and ratio will be detected by Advanced LIGO as well.) If detector noise dependent, are illustrated for non-spinning Advanced LIGO template banks are constructed in the equal-mass binary black-hole waveforms in Figs. 4 and 5 same way as those for Initial LIGO, then ǫmax = 0.005 is for the LIGO and LISA detectors, respectively. the appropriate mismatch tolerance for Advanced LIGO To apply these waveform accuracy standards to a par- as well. We also need estimates of the quantities min C¯ ticular detector, we must know the values of the param- and min C for some of the waveform accuracy require- eters that appear in the requirements summarized in Ta- ments. From Fig. 4 we see that the appropriate choices ble I. In particular we will need to know max ρ, ǫmax, for these quantities for binary black-hole systems with min C¯ and min C for that detector. The threshold signal- masses in the range 5 M/M⊙ 400 for the Advanced to-noise ratio for detecting binary black-hole mergers in LIGO detector are min≤C¯ 0.06≤ and min C 0.02. Us- Initial LIGO is about ρ 9 [24]. Advanced LIGO will be ing these parameter estimates,≈ we summarize≈ in Table II about ten times more sensitive≈ than Initial LIGO. There- our current expectations for the model waveform accu- fore, if no binary black-hole system is observed in the racy standards that will be needed for Advanced LIGO (S5) Initial LIGO data, then it is likely that the maxi- data analysis. mum signal-to-noise ratio for such events in the first year The detection requirements listed in the first two rows or two of Advanced LIGO observations will be no larger of Table II should apply equally to Initial and to Ad- than max ρ 90. (Of course, if a binary black-hole sig- vanced LIGO, since these requirements depends only on ≈ LIGO-P080091-00 10

Waveform Error Equation Measurement Detection sume the maximum mismatch parameter is ǫmax = 0.005 Diagnostic Numbers Requirement Requirement as in the LIGO case. Figure 5 illustrates the quantities C¯ δΦ 9, 16 20 1 −4 0.07 and C based on the approximate LISA noise curve con- · −4 structed by Barack and Cutler [35] (cf. their Eq. 30) max δΦ 18, 19 20 1 0.07 −4 | | · −6 using a 10 Hz low frequency cuttoff. It follows that max AeδΦ /Ae(fc) 23, 24 40 1 0.001 ¯ | | · −7 min C 0.02 and min C 0.004 for binary systems with ( AeδΦ f) / Ae(f) 30, 31 70 1 0.0003 ≈ ≈ 5 7 || || || || · −6 total masses in the range 10 < M/M⊙ < 10 . Using hδ (t) / he(t) 32, 33 10 1 0.0004 || || || || · these parameter estimates, we summarize in Table III our current expectations for the model waveform accuracy standards that will be needed for LISA data analy- TABLE III: Summary of model waveform accuracy requirements for the LISA detector for various waveform error di- sis. agnostics for measurement purposes (column three) and for The accuracy requirements for measurement and de- detection purposes (column four). tection that we discuss here provide upper limits on the allowed errors of model waveforms. These upper limits are the only requirements on model waveform accuracy the parameter ǫmax which is determined by the proper- when the detector is ideal, i.e. when the response func- ties of the search template bank and the accepted level tion of the detector is known with absolute precision. In of missed detections rather than the sensitivity of the de- Sec. III we discuss the additional requirements that must tector. The somewhat stronger sufficient conditions that be imposed when the response function has errors. We appear in rows 3-5 of Table II also depend on the con- show that the model waveform error need never be de- stants C¯ and C, which are larger for the Initial LIGO creased to a level below a certain fraction, ηmin 0.4, of ≈ case by factors of 10 and 2 respectively because the ap- the response function error. For Initial LIGO the calibra- prorpriate mass range for Initial LIGO is smaller. The tion error is small enough that it will not effect the ability appropriate Initial LIGO requirements for measurement of the instrument to make detections at all. The effect are less clear. The Initial LIGO detector is about ten of this error on the ability of Initial LIGO to make mea- times less sensitive than the planned Advanced LIGO surements is more complicated: Calibration errors will detector, so the expectation is that the accuracy require- degrade the quality of measurements made on sources ments should be about ten times weaker than those listed with large signal to noise ratios, ρ & 80, and decreasing in Table II. However until a detection is actually made, the model waveform error below ηmin times the response no measurement requirements will be needed at all. function error will not improve these measurements sub- While the timetable and the technical specifications of stantially. For weaker signals, ρ . 80, the calibration the proposed LISA detector are still being developed, we error is likely to degrade the quality of measurements to think it is appropriate to consider here what waveform ac- some extent, but the ideal-detector model-waveform accuracy standards this mission will eventually require from curacy standards should nevertheless be enforced in this the waveform simulation community. The maximum case. signal-to-noise ratio for supermassive binary black-hole observations by LISA is expected to be much larger than the stellar mass black-hole observations made by LIGO. Acknowledgments Supermassive black-hole mergers are expected to occur at a rate of about one merger per year within a sphere We thank Emanuele Berti, Curt Cutler, and Michele that extends to cosmological redshift z = 2 [32, 33]. The Vallisneri for illuminating discussions on the subject of largest signal-to-noise ratio for non-spinning equal-mass gravitational wave data analysis, and its particular ap- binaries located at 15 Gpc (redshift z 2) is about plication to LISA. This research was supported in part max ρ 4 103 for an optimally oriented≈ system with by a grant to Caltech from the Sherman Fairchild Foun- ≈ × 6 total mass 5 10 M⊙ [34]. Thus the requirement on the dation, by NSF grants DMS-0553302, PHY-0601459, and accuracy of model× waveforms for measurement purposes PHY-0652995 to Caltech, by NSF grant PHY-0555628 to will be much more demanding for LISA than for LIGO. If Penn State, and by the Penn State Center for Gravita- the LISA template banks are constructed in the same way tional Wave Physics under NSF cooperative agreement as the LIGO template banks, then it is appropriate to as- PHY-0114375.

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