Efficiency of nonspinning templates in gravitational wave searches for aligned-spin binary black holes
aa r X i v : . [ g r- q c ] D ec Efficiency of nonspinning templates in gravitational wave searches for aligned-spin binary blackholes
Hee-Suk Cho ∗ Korea Institute of Science and Technology Information, Daejeon 34141, Korea (Dated: November 12, 2018)We study the efficiency of nonspinning waveform templates in gravitational wave searches for aligned-spinbinary black holes (BBHs). We use PhenomD, which is the most recent phenomenological waveform modeldesigned to generate the full inspiral-merger-ringdown waveforms emitted from BBHs with the spins alignedwith the orbital angular momentum. Here, we treat the effect of aligned-spins with a single spin parameter χ .We consider the BBH signals with moderately small spins in the range of − . ≤ χ ≤ . . Using nonspinningtemplates, we calculate fitting factors of the aligned-spin signals in a wide mass range up to ∼ M ⊙ . Wefind that the range in spin over which the nonspinning bank has fitting factors exceeding the threshold of 0.965for all signals in our mass range is very narrow, i.e., − . ≤ χ ≤ . The signals with negative spins can havehigher fitting factors than those with positive spins. If χ = 0 . , only the highly asymmetric-mass signals canhave the fitting factors exceeding the threshold, while the fitting factors for all of the signals can be larger thanthe threshold if χ = − . . We demonstrate that the discrepancy between the regions of a positive and a negativespin is due to the physical boundary ( η ≤ . ) of the template parameter space. In conclusion, we emphasizethe necessity of an aligned-spin template bank in the current Advanced LIGO searches for aligned-spin BBHs.We also show that the recovered mass parameters can be significantly biased from the true parameters. PACS numbers: 04.30.–w, 04.80.Nn, 95.55.Ym
I. INTRODUCTION
Recently, two gravitational wave (GW) signals named asGW150914 and GW151226, were detected by the two LIGOdetectors [1, 2], and these observations indicate that futureobserving runs of the advanced detector network [3–5] willyield more binary black hole (BBH) merger signals [6–9]. De-tailed analyses in the parameter estimation showed that bothsignals were emitted from merging BBHs [2, 10, 11]. Themasses of the two binaries were found to be ∼ and M ⊙ for GW150914 and GW151226, respectively. In particular,the two components of GW150914 are the heaviest stellarmass BHs known to date. On the other hand, the preces-sion effects for both signals were poorly measured, while thealigned-spins ( χ ) were meaningfully constrained. Althoughwe might expect high-spin BHs from the X-ray observations[12], both binaries had small values of χ . The credibleintervals in their parameter estimations were in the range of − . ≤ χ ≤ . .The waveforms emitted from BBHs have three phases: in-spiral, merger, and ringdown (IMR), and the IMR phases ofstellar mass BBHs are likely to be captured in the sensitiv-ity band of ground-based detectors. In the search for BBHs,therefore, we have to use the full IMR waveforms as tem-plates. Over the past decade, two classes of IMR waveformmodels have been developed: effective-one-body models cal-ibrated to numerical relativity simulations (EOBNR) and phe-nomenological models. Since EOBNR is formulated in thetime domain as a set of differential equations, generation ofthose waveforms are computationally much more expensivethan generation of frequency-domain waveforms. Therefore, ∗ Electronic address: [email protected] for the purpose of the GW data analysis, P¨urrer [13, 14] has re-cently built a Fourier-domain reduced order model that faith-fully represents the original EOBNR model [15, 16]. On theother hand, a series of the phenomenological models havebeen developed, and those were also constructed in the fre-quency domain. The first phenomenological model was Phe-nomA [17–19] that was designed to model the IMR wave-forms of nonspinning BBHs, and this model was extended toan aligned-spin system in PhenomB [20] by adding the effec-tive spin parameter χ . The third model was PhenomC [21]that was also designed for aligned-spin BBHs, and extendedto a precessing system in PhenomP [22]. The most recentphenomenological model is PhenomD [23]. This model isalso designed for aligned-spin BBHs but covers much widerranges of mass (up to mass ratios of ) and spin (up to | χ | ∼ . ) than any other phenomenological models. Re-cently, it has been shown that PhenomD can perform verywell for BBH searches, losing less than of the recover-able signal-to-noise ratio [24]. Therefore, we use PhenomDfor the waveforms of aligned-spin BBHs in this work .We study the efficiency of nonspinning waveform templatesin GW searches for aligned-spin BBH signals by investigatingthe fitting factor. The fitting factor is defined as the best-matchbetween a normalized signal and a set of normalized templates[25]. For the GW data analysis purposes, the fitting factor isconsidered to evaluate the search efficiency. Since the detec-tion rate is proportional to ρ / , a FF ≃ . corresponds toa loss of detection rates of ∼ . Similar works have beencarried by several authors in the past few years. Using thepost-Newtonian waveform model, Ajith [26] calculated fitting The recent version of EOBNR reduced order model was also calibrated inwide parameter ranges up to mass ratios of and spins of − ≤ χ i ≤ . [14]. factors for several binaries with masses of M ≤ M ⊙ . Hefound that the spin value of the signal clearly separated thepopulation of binaries producing a poor fitting factor fromthose producing a high fitting factor (see Fig. 11 therein).Dal Canton et al. [27] also calculated fitting factors for BH-neutron star (NS) binaries with masses of M ≤ M ⊙ , andthey also found a clear separation between the two populations(see Fig. 8 therein). In the same paper, the authors showed thatthis behavior was due to the fact that the template parameterspace is physically bounded as η ≤ . (see Figs. 5 and 6therein). A similar work has also been performed by Privitera et al. [28] for BBH systems with M ⊙ ≤ M ≤ M ⊙ and m /m ≤ using the PhenomB waveforms [20]. They foundthat a nonspinning template bank achieved fitting factors ex-ceeding 0.97 over a wide region of parameter space, spanningroughly − . ≤ χ ≤ . over the entire mass range consid-ered in their work (see Fig. 1 therein). Recently, the work of[28] has been extended to higher-mass systems M ≤ M ⊙ by Capano et al. [29] using the EOBNR waveforms [16]. Onthe other hand, several works have used precessing signals totest nonspinning and aligned-spin template banks [30–33].In this work, we revisit the issues on the effectualness ofnonspinning templates for aligned-spin BBH signals. Al-though the template bank used for current Advanced LIGOsearches covers the binary masses up to M ⊙ [34, 35],the previous works have only considered low-mass systems.We therefore extend the study to high-mass systems up to M = 100 M ⊙ and compare our result with those of the pre-vious works. The purpose of this work is to examine the ef-ficiency of a nonspinning bank for aligned-spin signals in awide mass range. To this end we investigate the range in spinover which the nonspinning bank has fitting factors larger than0.965 varying total mass and mass ratio of the signal. II. GW DATA ANALYSIS
In signal processing, if a signal of known shape is buried instationary Gaussian noise, the matched filter can be the opti-mal method to identify the signal. For the GWs emitted frommerging BBHs, since there exist various models that can pro-duce accurate full IMR waveforms, the matched filter can beemployed in the BBH searches. If a detector data stream x ( t ) contains stationary Gaussian noise n ( t ) and a GW signal s ( t ) ,the match between x ( t ) and a template waveform h ( t ) is de-termined by h x | h i = 4Re Z ∞ f low ˜ x ( f )˜ h ∗ ( f ) S n ( f ) df, (1)where the tilde denotes the Fourier transform of the time-domain waveform, S n ( f ) is the power spectral density (PSD)of the detector noise, and f low is the low frequency cutoff thatdepends on the shape of S n ( f ) . In this work, we consider asingle detector configuration and use the zero-detuned, high-power noise PSD with f low = 10 Hz [36]. Using the relationin Eq. (1), the signal-to-noise ratio ρ (SNR) can be determined by ρ = h s | ˆ h i , (2)where ˆ h ≡ h/ h h | h i / is the normalized template. When thetemplate waveform h has the same shape as the signal wave-form s , the matched filter gives the optimal SNR as ρ opt = h s | s i / . (3)If the template has a different shape, the SNR is reduced to ρ = FF × ρ opt , (4)where FF is the fitting factor defined as the best-match be-tween a normalized signal and a set of normalized templates[25].To fully describe the wave function of an aligned-spin BBHsystem, we need 11 parameters except the eccentricity. Thoseare five extrinsic parameters (luminosity distance of the bi-nary, two angles defining the sky position of the binary withrespect to the detector, orbital inclination, and wave polariza-tion), four intrinsic parameters (component masses and spins),the coalescence time t c , and the coalescence phase φ c . How-ever, since the extrinsic parameters only scale the wave am-plitude, and we work with the normalized wave function, wedo not need to consider the extrinsic parameters in our anal-ysis. In addition, the inverse Fourier transform of the matchcan give the output for all possible coalescence times at once,and we can maximize the match over all possible coalescencephases by taking the absolute value of the complex-valuedoutput (see [37] for more details). Therefore, we need onlythe intrinsic parameters ( m , m , χ , χ ) in our analysis, andthose are the input parameters of PhenomD.On the other hand, it is often more efficient to treat the ef-fect of aligned-spins with a single spin parameter rather thanthe two component spins because the two spins are stronglycorrelated [21, 26, 38–40]. For this purpose, the spin effectsin the phenomenological models are parametrized by an ef-fective spin χ : χ ≡ m χ + m χ M . (5)The value of χ can be determined simply by choosing χ = χ = χ in the PhenomD wave function . Thus,our signal waveform is given by h s = h ( m , m , χ ) = h PhenomD ( m , m , χ, χ ) , while the nonspinning templatesare given by h t = h ( m , m ) = h PhenomD ( m , m , , .In this work, we define the overlap P by the match betweenthe signal ˆ h s and the template ˆ h t maximized over t c and φ c : P = max t c ,φ c h ˆ h s | ˆ h t i . (6) PhenomD is parametrized by a normalized reduced effective spin ˆ χ [23],but we can have ˆ χ = χ by choosing χ = χ . Thus, we can have P = 1 if the signal and the template havethe same shapes. Changing the mass parameters of the non-spinning templates, we calculate the two-dimensional overlapsurface as P ( λ ) = max t c ,φ c h ˆ h s ( λ ) | ˆ h t ( λ ) i , (7)where λ denotes the true values of the mass and the spin ofthe signal, and λ denotes the mass parameters of the template.Then, in our analysis the fitting factor corresponds to the max-imum value in the overlap surface: FF = max λ P ( λ ) . (8)On the other hand, in an actual search for BBHs the tem-plate waveforms are discretely placed in the bank, hence thefitting factor can be marginally reduced depending on the tem-plate density. Thus, the effective fitting factor is obtained by FF eff = max h t ∈ bank P ( λ ) . (9)Typically, when one chooses a waveform model for the search,the template bank is constructed densely enough such that themismatch between the templates and the signal does not ex-ceed including the effect of the discreteness of the tem-plate spacing, i.e. − FF eff ≥ . [41, 42]. In this work,however, we want to remove the effect of discreteness on thefitting factor. To do so, we choose sufficiently fine spacingsin the template space defined in the M c − η plane [43, 44]. For example, in order to obtain FF eff for one signal, we re-peat a grid search around λ until we find the crude locationof the peak point in the overlap surface. Next, we estimate thesize of the contour ¯ P ≡ P/P max = 0 . , where P max is themaximum overlap value in that contour (if the recovered massparameters are biased from λ , then P max < ), hence ¯ P cor-responds to the weighted overlap. Finally, we find (almost)the exact location of the peak point by performing a × grid search in the region of ¯ P > . , and the overlap valueat the peak point is regarded as FF .Once a fitting factor is determined through the above proce-dure, we can measure the systematic bias, which correspondsto the distance from the true value λ to the recovered value λ rec : b = λ rec − λ . (10)Typically, the recovered parameters are systematically biasedfrom the true parameters if the incomplete template wave-forms are used. In our analysis, the incompleteness of tem-plates arises from neglecting the spin effect in the wave func-tion. As the efficiency of a template waveform model for thesearch is evaluated by the fitting factor, its validity for the pa-rameter estimation can be examined by the systematic bias. In general, the overlap surface is obtained more efficiently in the parameterspace consisting of the chirp mass ( M c ≡ ( m m ) / /M / ) and thesymmetric mass ratio ( η ≡ m m /M ), so we take into account theparameters M c , η instead of m , m in the overlap calculations. III. RESULT
We choose as our target signals aligned-spin BBHs in theparameter regions of m , m ≥ M ⊙ ( m ≤ m ) , M ≤ M ⊙ and − . ≤ χ ≤ . . The signal waveforms are gen-erated by using PhenomD with χ = χ = χ . We constructa template bank in the M c − η plane with nonspinning wave-forms assuming χ = χ = 0 in PhenomD. The templates areassumed to be placed densely enough so that we can avoid theeffect of the discreteness of the bank. Using the nonspinningtemplates with an aligned-spin signal we calculate the overlapsurface that includes the confidence region, and determine thefitting factor and the systematic bias for the signal. A. Fitting factor
In Fig. 1, we show the fitting factors for all of the BBHsignals. In each panel, the darkest region corresponds tothe signals that cannot achieve the fitting factor exceeding athreshold of 0.965 beyond which a loss of detection rates doesnot exceed ∼ . We find that the signals with negativespins can have higher fitting factors than those with positivespins. If χ = 0 . , only the highly asymmetric-mass signalscan have the fitting factors exceeding the threshold. However,if χ = − . , the fitting factors for all of the signals can belarger than the threshold, and if χ = − . , about two thirdof the signals can have fitting factors exceeding the thresh-old. In particular, if the signal has a small spin in the range of − . ≤ χ ≤ . , the fitting factor can be larger than 0.99 (thelightest region) for all of the signals except those in the highlysymmetric-mass region. The range in spin over which all ofthe signals in our mass range have fitting factors exceeding0.965 is very narrow, i.e., − . ≤ χ ≤ . On the other hand,a few binaries can achieve FF ≥ . in our spin range, andwe show several examples in Fig. 2.In Fig. 3, we also show the fitting factors in the M − η planeusing the same color scales as in Fig. 1. In this figure, we caninterpret the pattern of the fitting factors more easily. In theregion of a negative spin, the fitting factor tends to decreaseas the total mass or the symmetric mass ratio increases. Onthe other hand, in the region of a positive spin, we can see astrong dependence of the fitting factor on the symmetric massratio. In this case, the fitting factors in the symmetric-massregion rapidly decrease with increasing χ , especially, thosewith low masses can drop below the threshold even with thesmall spin of χ = 0 . . In Fig. 4a, we show some examplesthat show highly asymmetric fitting factors between a positiveand a negative spins. We find that if χ > , the fitting factorsuddenly falls off at a certain spin value, and the falling ratetends to slacken for higher-masses.Dal Canton et al. [27] showed that the sudden fall-off of thefitting factor is associated with the physical boundary of thetemplate space. For the (positively) aligned-spin signals, theparameter value of η recovered by the nonspinning templatesincreases as the spin of the signal increases. However, in theparameter space of ( M c , η ), the physical value of η should berestricted to the range of ≤ η ≤ . . Thus, the recov-
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20 40 60 801020304050 m [ M ⊙ ] m [ M ⊙ ] . ≤ FF . ≤ FF < . FF < . χ = − . χ = − . χ = − . χ = − . χ = 0 . χ = 0 . χ = 0 . χ = 0 . FIG. 1: Fitting factors obtained by using nonspinning templates for aligned-spin BBH signals. The spin value of the signal is given in eachpanel. The signals with negative spins can have higher fitting factors than those with positive spins - - Χ FF
85, 565, 555, 1045, 525, 5 m (cid:144) M Ÿ , m (cid:144) M Ÿ FIG. 2: Examples that have the fitting factors larger than 0.965 in thespin range of − . ≤ χ ≤ . . ered value of η cannot exceed 0.25 even though the signal hashigher spins. For example, Fig. 4b shows the recovered η ( η rec ) as a function of χ for the same binaries as in Fig. 4a. Ifthe true value of η is 0.25, η rec is already at the boundary at χ = 0 , hence always equal to 0.25 in the entire range of pos-itive spins. In Fig. 4, we find that the spin value at which the η rec reaches 0.25 is consistent with the one at which the sud-den fall-off of the fitting factor occurs. On the other hand, thepost-Newtonian waveforms are well behaved for < η < . although the unphysical value of η implies complex-valuedmasses. Boyle et al. [45] showed that the fitting factors forhigh-mass systems above ∼ M ⊙ can be significantly im-proved if η is allowed to range over unphysical values. How-ever, such the unphysical masses are not permitted in the phe-nomenological models. B. Comparing with other works
In Fig. 5, we represent the fitting factors in the η − χ plane ina different way. We classify our binaries into low-mass ( M ≤ M ⊙ ), medium-mass ( M ⊙ ≤ M ≤ M ⊙ ), and high-mass ( M ⊙ ≤ M ) systems, and calculate the mean fittingfactors ( FF ) by averaging over M for each system. Note thatsince we assume the minimum mass of m to be M ⊙ , thevalues of η start from 0.09 (top), ∼ . (bottom left), and ∼ . (bottom right), respectively. The range of χ , in which FF ≥ . , becomes smaller as η increases. For the low-masssystems, the fitting factor curves in the region of a positive
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20 40 60 80 100 . ≤ FF0 . ≤ FF < . FF < . χ = − . χ = − . χ = − . χ = − . χ = 0 . χ = 0 . χ = 0 . χ = 0 . M [ M ⊙ ] η FIG. 3: The same fitting factors as in Fig. 1 but described in the M − η plane. spin rapidly drop to zero. Dal Canton et al. [27] also describedthe fitting factors in the same manner for BH-NS binaries withmasses of M ≤ M ⊙ (see Fig. 8 therein), and our result forthe low-mass system shows the pattern of fitting factor similarto their result in the region of a positive spin. However, inthe region of a negative spin, they had poor fitting factors, andthey pointed out that this is because the minimum NS massin the template bank is limited to M ⊙ . In particular, we findthat the overall area with high fitting factors is narrower forhigher-mass systems. That means the nonspinning bank hasworse search efficiency for higher-mass systems.We also describe the fitting factors in the M − χ plane inFig. 6 and compare those with the result of Privitera et al. [28].While Privitera et al. considered low-mass BBHs in the rangeof M ≤ M ⊙ with the initial LIGO PSD [46] assuming f low = 40 Hz, we take into account the higher-mass binariesin the range of M ≥ M ⊙ with the Advanced LIGO PSD[36] assuming f low = 10 Hz. Therefore, our result cannotbe directly compared with their result. However, we find thatthe overall pattern of the fitting factors in our result is sim-ilar to the result of [28] (see, Fig. 1 (a) therein). The top Since we have only few samples in the range of
M < M ⊙ , we do notinclude the results for those binaries in this figure. panel in Fig. 6 shows very asymmetric fitting factors betweenthe regions of a positive and a negative spins. For positivespins, the fitting factor contours gradually increase as the totalmass increases, and this is roughly consistent with the resultof [28]. On the contrary, for negative spins, the range of χ ,in which the signals have high fitting factors, is much largerthan the case for positive spins in the low-mass region, butthat becomes smaller as the total mass increases. We alreadyshowed that the discrepancy between the two spin regions iscaused by the physical boundary of the template space. Tosee this concretely, we select only the symmetric-mass bina-ries with m /m ≤ and show their results in the bottomleft panel in Fig. 6. We find that the discrepancy is morepronounced compared to the result of the top panel. We alsochoose the asymmetric-mass binaries for which η rec does notreach the physical boundary, i.e., η rec < . , and show theirresults in the bottom right panel. As expected, we can seenearly symmetric fitting factors between the regions of a pos-itive and a negative spins. Especially, in this case, most ofthe binaries can have mean fitting factors greater than . .That means, in the nonspinning template search for aligned-spin BBH signals, most of the signals, that have the masses of M ≤ M ⊙ and the spins of − . ≤ χ ≤ . , have highfitting factors exceeding the threshold 0.965 if only the binaryhas the asymmetric masses such that η rec does not reach 0.25. - - Χ FF Η , M @ M Ÿ D (a) - - Χ Η r ec Η , M @ M Ÿ D (b) FIG. 4: Fitting factors and recovered η ( η rec ) for several binaries.When χ > , the fitting factor suddenly falls off at a certain spinvalue (a). η rec cannot exceed the physical boundary 0.25 (b). Thespin value at which η rec reaches 0.25 is consistent with the one atwhich the sudden fall-off of the fitting factor occurs. C. Systematic bias of the recovered parameter
Once a detection is made in the search pipeline, the pa-rameter estimation pipeline conducts post-processing with thedata stream, that contains the GW signal. The purpose of theparameter estimation analysis is to extract the parameters ofa signal with high accuracy [42]. The results of the param-eter estimation are given by the posterior probability densityfunctions for the parameters [10, 47, 48]. Usually the poste-rior probability distribution is sampled by the Markov-chainMonte Carlo or nested sampling methods [48]. However,these algorithms are computationally intensive. In the highSNR limit, the Fisher matrix method can be used to approxi-mate the statistical error in the parameter estimation [49–53](for more details refer to [54] and references therein). χ η . ≤ FF . ≤ FF < . . ≤ FF < . FF < . - - M £ M Ÿ - - M Ÿ £ M £ M Ÿ - - M ³ M Ÿ FIG. 5: Mean fitting factors ( FF ) described in the η − χ plane for thelow-mass (top), medium-mass (bottom left), and high-mass (bottomright) systems, respectively. The mean fitting factor is calculated byaveraging over M . M [ M ⊙ ] χ
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30 40 50 60 (cid:5)(cid:6) (cid:7)(cid:8) (cid:9)(cid:10) - (cid:11) - (cid:12) m (cid:144) m £ . ≤ FF . ≤ FF < . . ≤ FF < . FF < .
30 40 50 60 70 80 90 100 - - Η rec < FIG. 6: Mean fitting factors ( FF ) described in the M − χ plane forall of the binaries with M ≥ M ⊙ (top), symmetric-mass binarieswith m /m ≥ (bottom left), and asymmetric-mass binaries forwhich η rec < . , respectively. The mean fitting factor is calculatedby averaging over η . On the other hand, in the search, parameters of a signalcan also be inferred from the identified template parameters,but the recovered parameters can be significantly biased fromthe true parameters. In this subsection, we show how muchthe recovered parameter is biased depending on the spin ofthe signal. In Fig. 7, we show the fractional bias ( b λ /λ ) asa function of χ . Here, as concrete examples we select sev-eral asymmetric-mass binaries that satisfy η rec < . . Inthe top panel, as χ increases the bias for η also increases,and the dependence of the bias on χ is stronger for a posi-tive spin than a negative spin. On the contrary, in the bot- m /M ⊙ , m /M ⊙ , , , , , - - - - Χ b Η (cid:144) Η ´ - - - - - Χ b M (cid:144) M ´ - - - - - Χ b M c (cid:144) M c ´ FIG. 7: Systematic bias ( b λ /λ ) as a function of χ for η (top), M c (bottom left), and M (bottom right), respectively. tom left panel, the bias for M c decreases with increasing χ ,and that exhibits a similar dependence on χ between a posi-tive and a negative spins. The biases incorporated in the twomass parameters can be well understood by describing thosein terms of a total mass. When the spin is positively alignedwith the orbital angular momentum, the spin-orbit couplingmakes the binary’s phase evolution slightly slower, hence de-lays the onset of the plunge phase, as compared to its nonspin-ning counterpart [55]. On the contrary, in the antialigned case,the phase evolution becomes slightly faster, and the plunge ishastened. Consequently, for a given starting GW frequency, apositively (negatively) aligned-spin increases (decreases) thelength of the waveform, as compared to the nonspinning case.Therefore, positively (negatively) spinning systems can be re-covered by lower (higher) mass nonspinning templates. Weclearly describe this in the bottom right panel, showing thebias for the parameter M as a function of χ . Interestingly, wefind that the systematic bias for M almost linearly depends on χ in our spin range. In addition, all of the results seem to havesimilar fractional biases ( b M /M ) for a given χ even thoughtheir masses are very different.In Fig. 8, we show the fractional biases ( B ≡ b M /M ) forthe signals with the spins of χ = − . , − . , . , and . .The red color indicates a negative bias while the blue color in-dicates a positive bias. We find that the magnitudes of biasesare similar between the red and the blue in the asymmetric-mass region ( η . . ), while those are smaller for the pos-itive spins in the symmetric-mass region ( η & . ). As ex-pected, the difference in the symmetric-mass region is due tothe fact that for the positive spins η rec is restricted by thephysical boundary, and thereby the corresponding M rec hassmaller biases. We also find that the contours B = 30 , − inthe top panels are consistent with the contours B = 15 , − in the bottom panels, and this indicates a linear relation be-tween B and χ . Finally, we find that in the asymmetric-massregion all of the fractional biases are comparable for a given χ independently of the total mass. For example, we have
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20 40 60 80 100 χ = − . M [ M ⊙ ] η χ = 0 . χ = 0 . < B ≤ < B ≤ < B ≤ < B B < − − ≤ B < − ≤ B < − − ≤ B < − χ = − . -10 -15-30 30-15 15-10 10 FIG. 8: Fractional biases for the parameter M ( B ≡ b M /M × ).The biases are similar between a positive (red) and a negative (blue)spins in the asymmetric-mass region ( η . . ). The similaritybetween the contours B = 15 ( − and
30 ( − indicates a linearrelation between B and χ .
15 (30) . B .
17 (35) for χ = − . − . . IV. SUMMARY AND DISCUSSION
We investigated the efficiency of nonspinning templates inGW searches for aligned-spin BBHs. We considered the sig-nals with moderately small spins in the range of − . ≤ χ ≤ . . We employed as our waveform model PhenomD, and weset the spins to zero for the nonspinning waveforms. Usingthe nonspinning templates, we calculated the fitting factorsof the aligned-spin BBH signals in a wide mass range up to ∼ M ⊙ . The results are summarized in Figs. 1 and 3 in the m − m plane and M − η plane, respectively. The signalswith negative spins can have higher fitting factors than thosewith positive spins. If χ = 0 . , only the highly asymmetric-mass signals can have the fitting factors exceeding the thresh-old 0.965. However, if χ = − . , the fitting factors for allof the signals can be larger than the threshold. The discrep-ancy between the regions of a positive and a negative spinis due to the fact that the template parameter space is physi-cally restricted to η ≤ . so that the recovered value of η ( η rec ) cannot exceed 0.25. We demonstrated this by choosingthe asymmetric-mass binaries that satisfy η rec < . , andshowing the nearly symmetric fitting factors for those bina-ries between the two regions. We classified our binaries intolow-mass, medium-mass, and high-mass systems and calcu-lated the mean fitting factor by averaging over M in the η − χ plane, and found that the overall area with high fitting factorsis narrower for higher-mass systems. The mass parameters re-covered by the nonspinning templates are significantly biasedfrom the true parameters of the aligned-spin signals.In this work, we revisited the issues on the effectualnessof nonspinning templates in aligned-spin BBH searches thatwere addressed in several works for low-mass BBHs. Weobtained a similar result to those of the previous works andfound that the nonspinning bank has worse search efficiencyfor higher-mass systems. Overall, we obtained a very narrowrange in spin ( − . ≤ χ ≤ ) over which the nonspinningbank has fitting factors exceeding 0.965 for all of the aligned-spin signals in our mass range. Moreover, the fitting factorsgiven in this work should be a bit lowered if the discretenessof template spacing is considered in our analysis. Therefore,our study demonstrates the ineffectualness of the nonspinningbank and emphasizes the necessity of aligned-spin templatesin the current Advance LIGO searches for aligned-spin BBHs. ACKNOWLEDGMENTS
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