PARAMETER ESTIMATION FOR BINARY NEUTRON-STAR COALESCENCES WITH REALISTIC NOISE DURING THE ADVANCED LIGO ERA

The goal of gravitational-wave (GW) astronomy is to learn about the universe through observations of gravitational radiation. This requires not only the ability to detect GWs, but also to infer the properties of their source systems. In this work, we investigate the ability to perform parameter estimation (PE) on signals detected by the upcoming Advanced Laser Interferometer Gravitational-wave Observatory (aLIGO) instruments (Harry 2010; Aasi et al. 2015) in the initial phase of their operation (Aasi et al. 2013b).

Compact binary coalescences (CBCs), the GW-driven inspiral and merger of stellar-mass compact objects, are a prime source for aLIGO and Advanced Virgo (AdV; Acernese et al. 2009, 2015). Binary neutron-star (BNS) systems may be the most abundant detectable CBCs (Abadie et al. 2010). We focus on BNS mergers in this study.

Following the identification of a detection candidate, we wish to extract the maximum amount of information from the signal. It is possible to make some inferences using selected components of the data. However, full information regarding the source system, including the component objects' masses and spins, is encoded within the gravitational waveform, and can be obtained by comparing the data to theoretical waveform models (Cutler & Flanagan 1994; Jaranowski & Krolak 2012). Doing so can be computationally expensive.

PE is performed within a Bayesian framework. We use algorithms available as part of the LALInference toolkit for the analysis of CBC signals. The most expedient code is bayestar (Singer 2014; Singer et al. 2014), which infers sky location from data returned from the detection pipeline. Exploring the posterior probability densities for the parameters takes longer for models where the parameter space is larger or the likelihood is more complicated. Calculating estimates for parameters beyond sky location is done using the stochastic-sampling algorithms of LALInference (Veitch et al. 2015). There are three interchangeable sampling algorithms: LALInference_nest (Veitch & Vecchio 2010), LALInference_mcmc (van der Sluys et al. 2008a; Raymond et al. 2009), and LALInference_bambi (Graff et al. 2012), which we refer to as LALInference for short. These compute waveform templates for use in the likelihood. Using the least computationally expensive waveforms allows for posteriors to be estimated on timescales of hours to days; potentially more accurate estimates can be calculated with more expensive waveforms. In this paper, we discuss what can be achieved using low-latency (bayestar) and medium-latency (LALInference with inexpensive waveforms) PE; a subsequent paper will evaluate what can be achieved on longer timescales using more expensive waveform templates.

With the detection of GWs, it is also possible to perform multi-messenger astronomy, connecting different types of observations of the same event. BNS mergers could be accompanied by an electromagnetic (EM) counterpart (Metzger & Berger 2012). To associate an EM event with a GW signal, it is beneficial to have an accurate sky location: timing information can also be used for EM signals that are independently detected, such as gamma-ray bursts (Aasi et al. 2014b). To provide triggers for telescopes to follow up a GW detection, it is necessary to provide rapid sky localization.

Several large-scale studies investigated the accuracy with which sky position can be reconstructed from observations with ground-based detector networks. The first only used timing information from a multi-detector network to triangulate the source position on the sky (e.g., Fairhurst 2009, 2011). Subsequently, further information about the phase of the gravitational waveform was folded into the timing triangulation (TT) analysis (Grover et al. 2014). The most sophisticated techniques perform a coherent Bayesian analysis to reconstruct probability distributions for the sky location (e.g., Veitch et al. 2012; Nissanke et al. 2013; Kasliwal & Nissanke 2014; Grover et al. 2014; Sidery et al. 2014). Singer et al. (2014) used both bayestar and LALInference to analyse the potential performance of aLIGO and AdV in the first two years of their operation. They assumed the detector noise was stationary and Gaussian. Here, we further their studies (although we use the same analysis pipeline) by using a set of injections into observed noise from initial LIGO detectors recolored (see Section 2.1) to the expected spectral density of early aLIGO.¹⁵ This provides results closer to those expected in practice, as real interferometer noise includes features such as non-stationary glitches (Aasi et al. 2013b, 2014a). Our results are just for the first observing run (O1) of aLIGO, expected in the latter half of 2015, assuming that this occurs before the introduction of AdV. As the sensitivity of the detectors will increase with time, and because the introduction of further detectors increases the accuracy of sky localization (Schutz 2011), these set a lower bound for the advanced-detector era. Estimates for sky-localization accuracy in later observing periods can be calibrated using our results.

PE beyond sky localization, considering the source system's mass, spin, distance and orientation, has been subject to similar studies. The initial investigations estimated PE using the Fisher information matrix (e.g., Cutler & Flanagan 1994; Poisson & Will 1995; Arun et al. 2005). This only gives an approximation of true PE potential (Vallisneri 2008). More reliable (but computationally expensive) results are found by simulating a GW event and analyzing it using PE codes, mapping the posterior probability distributions (e.g., Rover et al. 2006; van der Sluys et al. 2008b; Veitch & Vecchio 2010; Rodriguez et al. 2014). This has even been done for a blind injection during the run of initial LIGO (Aasi et al. 2013a). As with sky localization, general PE can improve with the introduction of more detectors to the network (Veitch et al. 2012).

To be as faithful as possible, our analysis is performed using one of the pipelines intended for use during O1. We make use of the LIGO Scientific Collaboration Algorithm Library (LAL).¹⁶ In particular, we shall make use of GSTLAL,¹⁷ one of the detection pipelines, to search for signals and LALInference for PE on detection candidates.

We begin by describing the source catalog and detector sensitivity curve used for this study in Section 2. In Section 3 we explain how the data is analyzed to produce sky areas and other parameter estimates. Many details from these two sections are shared with the preceding work of Singer et al. (2014), which can be consulted for further information. In Section 4 we present the results of our work. We first discuss the set of events that are selected by the detection pipeline in Section 4.1 (with supplementary information in Appendix A); then we examine PE, considering sky-localization accuracy in Section 4.2, and mass and distance measurements in Section 4.3. We conclude with a discussion of these results in Section 5; this includes in Section 5.1.2 an analysis of estimates for sky localization in later observing periods with reference to our findings. Estimates of the computational costs associated with running bayestar and full LALInference PE are given in Appendix B. A supplementary catalog of results is described in Appendix C, with data available at http://www.ligo.org/scientists/first2years/.

Our main findings are:

• 1.  
  The detection pipeline returns a population of sources that is not significantly different from the input astrophysical population, despite a selection bias based upon the chirp mass.
• 2.  
  Both bayestar and LALInference return comparable sky-localization accuracies (for a two-detector network). The latter takes more computational time (a total CPU time of $\sim {{10}^{6}}\;{\rm s}$ per event compared with $\sim {{10}^{3}}\;{\rm s}$), but returns estimates for more parameters than just location.
• 3.  
  At a given signal-to-noise ratio (S/N), the character of the noise does not affect sky localization or other PE.
• 4.  
  Switching from a detection threshold based upon S/N to one based upon the false alarm rate (FAR) changes the S/N distribution of detected events. A selection based upon FAR includes more low-S/N events (the distribution at high S/Ns is unaffected).
• 5.  
  TT provides a poor predictor of sky localization for a two-detector network; it does better (on average) for a three-detector network when phase coherence is included, but remains imperfect.
• 6.  
  Systematic errors from uncertainty in the waveform template are significant for chirp-mass estimation. Neglecting the mass–spin degeneracy by using non-spinning waveforms artificially narrows the posterior distribution.

For O1, we find that the luminosity distance is not well-measured: the median 50% credible interval (interquartile range) divided by the true distance is 0.38 and the median 90% credible interval divided by the true distance is 0.85. Despite being subject to systematic error, the chirp mass is still accurately measured, with the posterior mean being less than ${{10}^{-3}}\;{{M}_{\odot }}$ from the true value in almost all (96%) cases. We find that the median area of 50% sky localization credible region is $154\;{{{\rm deg} }^{2}}$ and the median area of the 90% credible region is $632\;{{{\rm deg} }^{2}}$; the median searched area (area of the smallest credible region that encompasses the source location) is $132\;{{{\rm deg} }^{2}}$. EM follow-up to BNS mergers in 2015 will be challenging and require careful planning.

Our input data consists of two components: simulated detector noise and simulated BNS signals. We describe the details of these in the following subsections, before continuing with the analysis of the data in Section 3.

2.1. Recolored 2015 Noise

We consider the initial operation of the advanced detectors at LIGO Hanford and LIGO Livingston. The sensitivity is assumed to be given by the early curve of Barsotti & Fritschel (2012), which has a BNS detection range of $\sim 55\;{\rm Mpc}$ (assuming Gaussian noise). This configuration corresponds to the 2015 observing scenario in Aasi et al. (2013b). Figure 1 plots the noise spectral density, the square root of the power spectral density (Moore et al. 2015), as measured during the sixth science (S6) run of initial LIGO,¹⁸ the early aLIGO sensitivity curve, and final aLIGO curve (Shoemaker 2010).

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The noise is constructed from data from the S6 run of initial LIGO (Christensen 2010; Aasi et al. 2014a), recolored to the early aLIGO noise spectral density as was done for Aasi et al. (2014c). We use real noise, instead of idealized Gaussian noise, to try to capture a realistic detector response including transients; however, the S6 noise can only serve as a proxy for the actual noise in aLIGO since the detectors are different. Two calendar months (2010 August 21–October 20) of S6 data were used. The recolored data are constructed using GSTLAL_fake_frames.¹⁹ The recoloring process can be thought of as applying a finite-impulse response filter to whitened noise. The result is a noise stream that, on average, has the same power spectral density as expected for early aLIGO, but contains transients that are similar to those found in S6. Recoloring preserves the non-stationary and non-Gaussian features of the noise, although they are distorted (Aasi et al. 2014c). The recolored noise is the most realistic noise we can construct ahead of having the real noise from aLIGO.

2.2. BNS Events

To accurately forecast sky localization prospects in O1, we run our simulated events through the same data-analysis pipeline as is intended for real data. The results of this pipeline are analyzed in the next section (Section 4). A GW search is performed using GSTLAL_inspiral (Cannon et al. 2010, 2011, 2012, 2013); this is designed to provide GW triggers in real time with ∼10–$100\;{\rm s}$ latency during LIGO–Virgo observing runs. A trigger is followed up for sky localization if its calculated FAR is less than ${{10}^{-2}}\;{\rm y}{{{\rm r}}^{-1}}$, which is roughly equivalent to a network-S/N threshold of $\varrho \gtrsim 12$ (Aasi et al. 2013b).

In using the FAR to select triggers, our method differs from that used in Singer et al. (2014). Since they considered Gaussian noise, which is free of glitches, their FAR would not be representative of those computed using real noise; the FAR calculated with Gaussian noise corresponds to a S/N-threshold that is too low for detection in realistic noise. Therefore, they also imposed a network-S/N cut of $\varrho \geqslant 12$, in addition to the FAR selection. This joint S/N and FAR threshold was found to differ negligibly from an S/N-only threshold: in effect, they select by S/N alone. While this is a small difference in selection criteria, we shall see in Section 4.2 that this has an impact on our sky-localization results.

To recover the GW signal, another PN inspiral approximant, TaylorF2 (Damour et al. 2001, 2002; Buonanno et al. 2009), was used as a template. This is a frequency-domain stationary-phase approximation waveform accurate to 3.5PN order in phase and Newtonian order in amplitude. It does not include the effects of spin, although it can be modified to incorporate these (Mikoczi et al. 2005; Arun et al. 2009; Bohé et al. 2013). We neglect spin as this should not lead to a significant reduction in detection efficiency for systems with low spins (Brown et al. 2012), which we confirm in Section 4.1.2. TaylorF2 does not incorporate as many physical effects as SpinTaylorT4, notably it does not include precession, but is less computationally expensive, permitting more rapid follow-up.

Rapid sky localization is computed using bayestar (Singer et al. 2014). This reconstructs sky position using a combination of information associated with the triggers: the times, phases and amplitudes of the signals at arrival at each detector. It coherently combines this information to reconstruct posteriors for the sky position. bayestar makes no attempt to infer intrinsic parameters such as the BNS masses and, hence, can avoid computationally expensive waveform calculations. The sky-position distributions can be formulated in under a minute (see Appendix B).

Full PE, which computes posterior distributions for sky localization parameters as well as the other parameters for the source system like mass, orientation, and inclination, is performed using LALInference (Veitch et al. 2015). LALInference maps the posterior probability distribution by stochastically sampling the parameter space (e.g., MacKay 2003, chapter 29). There are three codes within LALInference to sample these posterior distributions: LALInference_nest (Veitch & Vecchio 2010), a nested sampling algorithm (Skilling 2006); LALInference_mcmc (van der Sluys et al. 2008a; Raymond et al. 2009), a Markov-chain Monte Carlo algorithm (Gregory 2005, chapter 12), and LALInference_bambi (Graff et al. 2012), another nested sampling algorithm (Feroz et al. 2009) which incorporates a means of speeding up likelihood evaluation using machine learning (Graff et al. 2014). All three codes use the same likelihood and so should recover the same posteriors; consistency of the codes has been repeatedly checked. While the codes produce the same results, they may not do so in the same times, depending upon the particular problem. All the results here were computed with LALInference_nest.

TaylorF2 waveforms were used again in constructing the LALInference posterior. Since these do not exactly match the waveforms used for injection, there may be a small bias in the recovered parameters (Buonanno et al. 2009). Using TaylorF2 is much less computationally expensive than using SpinTaylorT4, in this case a LALInference run takes $\sim {{10}^{6}}\;{\rm s}$ of CPU time (see Appendix B).

4.1. Detection Catalog

We ran sky-localization codes on a set of 333 events recovered from the detection pipeline. We shall compare these to the results of Singer et al. (2014) who used Gaussian noise for the same sensitivity curve. They ran bayestar on a sample of 630 events, but only ran LALInference on a sub-sample of 250 events. We first consider the set of detected events before moving on to examine sky-localization accuracies in Section 4.2, and mass and distance measurement in Section 4.3.

4.1.1. S/N Distribution

Unsurprisingly, the distribution of S/Ns differs between the recolored and Gaussian data sets. This is shown in Figure 2. The recolored S/N distribution includes a tail at low S/N ($\varrho \simeq 10$–12). If we impose a lower threshold $\varrho \geqslant 12$ for the recolored data set, as was done for the Gaussian data set, we find that the S/N distributions are similar. With the shared S/N cut, the distributions agree within the expected sampling error; performing a Kolmogorov–Smirnov (KS) test (DeGroot 1975, Section 9.5) comparing the recolored S/N distribution to the complete (LALInference only) Gaussian S/N distribution returns a p-value of 0.311 (0.110).

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Comparing injections between the recolored and Gaussian data sets, there are 255 events that have been detected in both sets. There are 108 events shared between the recolored data set and the sub-sample of the Gaussian data set analyzed with LALInference. Considering individual events, we may contrast the S/N for recolored noise ${{\varrho }_{{\rm R}}}$ and Gaussian noise ${{\varrho }_{{\rm G}}}$. The ratio of the two S/Ns is shown in Figure 3. Considering the entire population of shared detections, the mean value of the ratio of S/Ns is ${{\varrho }_{{\rm R}}}/{{\varrho }_{{\rm G}}}=0.938\;\pm \;0.006$, showing a small downwards bias as an effect of the differing cutoffs used for the two samples. To limit selection effects that could skew the distribution of the ratio of S/Ns, we can impose an S/N cut of ${{\varrho }_{{\rm R}}}\geqslant 12$. This reduces the number of events detected in both noise sets to 214 using the full Gaussian set and 88 for the LALInference Gaussian sub-sample. There is a small difference between the S/N as calculated with Gaussian noise and with recolored noise. This does not appear to be a strong function of the S/N. However, the scatter in the ratio decreases as S/N increases, approximately decreasing as ${{\varrho }^{-1}}$. This is as expected as the inclusion of random noise realizations in the signal should produce fluctuations in the S/N of order ±1; these fluctuations become less significant for louder events. After imposing the cut $\varrho \geqslant 12$ on both sets, the mean value of the ratio of S/Ns is ${{\varrho }_{{\rm R}}}/{{\varrho }_{{\rm G}}}=0.955\;\pm \;0.006.$ Although there is a small difference in S/Ns, we shall see that this does not impact our PE results.

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4.1.2. Selection Effects

The population of detected events should not match exactly the injected distribution; depending upon their parameters, some systems are louder and hence easier to detect. Here, we look at the selection effects of the most astrophysically interesting parameters: mass and spin. We expect there to be a selection based upon mass, as the component masses set the amplitude of the waveform. We do not expect there to be a dependence upon the spin because the spin magnitude is small, but since we injected with a spinning waveform and recovered with a non-spinning waveform, there could potentially be a selection effect due to waveform mismatch. Checking these distributions confirms the effectiveness of the detection pipeline for this study.

To leading order, the GW amplitude is determined by the ($5/6$ power of the) chirp mass (Sathyaprakash & Schutz 2009)

$$\begin{eqnarray}&&{{\mathcal{M}}_{{\rm c}}}=\frac{{{({{m}_{1}}{{m}_{2}})}^{3/5}}}{{{({{m}_{1}}+{{m}_{2}})}^{1/5}}},\end{eqnarray} \tag{ 1 }$$

where m₁ and m₂ are the individual component masses. We therefore expect to preferentially select systems with larger chirp masses.

Figure 4 shows the recovered distribution of (injected) chirp masses and the injection distribution (which is calculated numerically). We do detect fewer systems with smaller chirp masses (and more with larger chirp masses), as indicated by the curve for the recovered distribution lying below the curve for the injection distribution. However, this selection effect does not alter the overall character of population. The difference is only marginally statistically significant with this number of events (a KS test with the injection distribution yields p-values of 0.315 and 0.068 for the Gaussian and recolored noise respectively). This is consistent with expectations for this narrow chirp-mass distribution; in Appendix A we use a simple theoretical model to predict that we would need $\sim {{10}^{3}}$ detections (or a broader distribution of chirp masses in the injection set) to see a significant difference between the injected and recovered populations. The character of the noise does not influence the chirp-mass distribution (a KS test gives a p-value of 0.999).

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For completeness, in Appendix A we present the distributions for the individual component masses, the asymmetric mass ratio and the total mass. The selection effects on these depend upon their correlation with the chirp mass; the total mass, which is most strongly correlated with the chirp mass, shows the most noticeable difference between injection and detected distributions.

Since we injected with a spinning waveform and recovered with a non-spinning waveform, there could also be a selection bias depending upon the spin magnitude. Figure 5 shows the recovered distribution of (injected) spins. The detected events are consistent with having the uniform distribution of spins used for the injections. We conclude that the presence of spins with magnitudes a ≤ 0.05 does not affect the detection efficiency for BNS systems, in agreement with Brown et al. (2012).

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4.2. Sky-localization Accuracy

The recovered sky positions from bayestar and LALInference appear in good agreement. A typical example of the recovered posterior probability density is shown in Figure 6. This is a bimodal distribution, reflecting the symmetry in the sensitivity of the detectors, which is common (Singer et al. 2014). We use geographic coordinates to emphasize the connection to the position of the detectors. A catalog of results can be viewed online at http://www.ligo.org/scientists/first2years/ (see Appendix C).

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To quantify the accuracy of sky localization, we use credible regions: areas of the sky that include a given total posterior probability. We denote the credible region for a total posterior probability p as ${\rm C}{{{\rm R}}_{p}}$: it is defined as

$$\begin{eqnarray}&&{\rm C}{{{\rm R}}_{p}}\equiv {\rm min} A\end{eqnarray} \tag{ 2 }$$

such that the sky area A satisfies

$$\begin{eqnarray}&&p={{\int }_{A}}d{\bf \Omega }{{P}_{{\Omega }}}({\bf \Omega }),\end{eqnarray} \tag{ 3 }$$

where ${{P}_{{\Omega }}}({\bf \Omega })$ is the posterior probability density over sky position ${\bf \Omega }$ (Sidery et al. 2014). A smaller ${\rm C}{{{\rm R}}_{p}}$ at a given p indicates more precise sky localization.

We also consider the searched area: the area of the smallest credible region that includes the true location, and, hence, the area of the sky that we expect would have to be observed before the true source was found.

The self-consistency of our sky areas can be checked by calculating the fraction of events that fall within the credible region at the given probability. We expect that a fraction p of true sky positions are found within ${\rm C}{{{\rm R}}_{p}}$; that is the frequentist confidence region agrees with our Bayesian credible region (Sidery et al. 2014). Figure 7 shows the fraction of events found within a given ${\rm C}{{{\rm R}}_{p}}$ as a function of p. The distributions are consistent with expectations: performing a KS test with the predicted distribution yields p-values of 0.455 and 0.546 for LALInference and bayestar, respectively. Both LALInference and bayestar produce self-consistent and unbiased sky areas in the presence of recolored noise.

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The recovered sky areas are plotted in Figure 8. This shows the cumulative distribution of areas for ${\rm C}{{{\rm R}}_{0.5}}$, ${\rm C}{{{\rm R}}_{0.9}}$ and searched areas A_* as recovered from LALInference and bayestar. We plot both the results using recolored noise and the results using Gaussian noise from Singer et al. (2014). All the results are similar. LALInference produces (marginally) more accurate sky localizations than bayestar, but the rapid code does a successful job of reconstructing the sky position in a much shorter time (see Appendix B for estimates of computation time). The recovered areas are (generally) marginally smaller for LALInference as this makes use of more information and so is expected to perform better (a KS test returns p-values of 0.740 when comparing ${\rm C}{{{\rm R}}_{0.9}}$ for Gaussian noise and 0.181 for recolored noise).

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The difference between the Gaussian and recolored results can be understood as a consequence of the S/N distribution (see Figure 2). The S/N is the dominant factor affecting sky localization. For example, there is no strong correlation between the time delay between detection at the two LIGO sites and the sky-localization accuracy. The inclusion of more low-S/N events means that, on average, the results using recolored noise are worse.

The sky-localization accuracy is expected to scale as $\varrho$⁻². The uncertainty in each direction on the sky scales inversely with the S/N, hence the area scales inversely with the square of the S/N (cf. Fairhurst 2009, 2011). This S/N scaling can be verified by plotting recovered sky areas as a function of $\varrho$ as shown in Figure 9. The recovered areas do show the expected correlation, although there is considerable scatter resulting from the variation in intrinsic parameters.

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We have plotted fiducial best-fit lines with the expected scaling. The fitting was done simply using a naive least-squares method, fitting a straight line to ${\rm log} \varrho$ and ${\rm log} A$ for each sky area A. Allowing the slope of the line to vary from −2 yields negligible change to the fit. There is little difference between the trends for the recolored and Gaussian results, indicating that the variation in the sky-localization accuracies is primarily an effect of the different distribution of S/Ns. There is a small discrepancy between LALInference and bayestar in both cases, but the difference is not significant and is within the uncertainty expected from the scatter of results. The general trend for the sky-localization areas can be approximated as

$$\begin{eqnarray}&&{{{\rm log} }_{10}}\left( \frac{{\rm C}{{{\rm R}}_{0.5}}}{{{{\rm deg} }^{2}}} \right)\approx -2{{{\rm log} }_{10}}\varrho +4.46,\end{eqnarray} \tag{ 4a }$$

$$\begin{eqnarray}&&{{{\rm log} }_{10}}\left( \frac{{\rm C}{{{\rm R}}_{0.9}}}{{{{\rm deg} }^{2}}} \right)\approx -2{{{\rm log} }_{10}}\varrho +5.06.\end{eqnarray} \tag{ 4b }$$

Sky-localization accuracy (at a given S/N) does not appear to be sensitive to the Gaussianity of the noise.

From our fits (4), we can immediately see that the ratio ${\rm C}{{{\rm R}}_{0.9}}/{\rm C}{{{\rm R}}_{0.5}}$ is about ${{10}^{0.6}}\simeq 4$. Considering this ratio for each posterior, the mean value of ${{{\rm log} }_{10}}({\rm C}{{{\rm R}}_{0.9}}/{\rm C}{{{\rm R}}_{0.5}})$ is 0.60 and the standard deviation is 0.07. For comparison, if the posterior were a 1D Gaussian, we would expect the ratio to be ${\rm er}{{{\rm f}}^{-1}}(0.9)/{\rm er}{{{\rm f}}^{-1}}(0.5)\simeq 2.4\simeq {{10}^{0.39}}$, and if it were a 2D Gaussian, the ratio would be ${\rm ln} (1-0.9)/{\rm ln} (1-0.5)\simeq 3.3\simeq {{10}^{0.52}}$ (Fairhurst 2009, 2011). Neither of these agree well. The sky-location posteriors can have complicated shapes, and cannot be accurately modeled by a simple Gaussian description.

To verify that S/N distribution is the dominant cause of difference between the Gaussian and recolored results, we impose a cut on the recolored data set of $\varrho \geqslant 12$ to match the Gaussian set. This reduces the number of events from 333 to 236. The cumulative distribution of sky-localization areas for results with $\varrho \geqslant 12$ are shown in Figure 10. The distributions do overlap as expected: the Gaussian and recolored results are in agreement (a KS test on ${\rm C}{{{\rm R}}_{0.9}}$ gives a p-value of 0.550 when comparing LALInference results between noise realizations and 0.673 for bayestar).

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The key numbers describing the distributions are given in Tables 1 and 2; the former gives the fraction of events with sky-localization areas smaller than fiducial values, and the latter gives median sky-localization areas. Our results are discussed further in Section 5.1.

Table 1.  Fractions of Events With Sky-localization Areas Smaller Than a Given Size

Sky Localization		Gaussian Noise		Recolored Noise		Recolored Noise $\varrho \geqslant 12$
		bayestar	LALInference	bayestar	LALInference	bayestar	LALInference
${\rm C}{{{\rm R}}_{0.5}}\;\leqslant$	$5\;{{{\rm deg} }^{2}}$	—	—	—	—	—	—
	$20\;{{{\rm deg} }^{2}}$	0.02	0.03	0.01	0.02	0.02	0.03
	$100\;{{{\rm deg} }^{2}}$	0.30	0.37	0.21	0.30	0.30	0.41
	$200\;{{{\rm deg} }^{2}}$	0.74	0.80	0.58	0.64	0.76	0.80
	$500\;{{{\rm deg} }^{2}}$	1.00	1.00	1.00	0.99	1.00	1.00
	$1000\;{{{\rm deg} }^{2}}$	1.00	1.00	1.00	1.00	1.00	1.00
${\rm C}{{{\rm R}}_{0.9}}\;\leqslant$	$5\;{{{\rm deg} }^{2}}$	—	—	—	—	—	—
	$20\;{{{\rm deg} }^{2}}$	—	—	—	—	—	—
	$100\;{{{\rm deg} }^{2}}$	0.03	0.04	0.02	0.03	0.03	0.04
	$200\;{{{\rm deg} }^{2}}$	0.10	0.13	0.06	0.08	0.09	0.12
	$500\;{{{\rm deg} }^{2}}$	0.44	0.48	0.31	0.38	0.44	0.52
	$1000\;{{{\rm deg} }^{2}}$	0.98	0.93	0.78	0.80	0.96	0.94
${{A}_{*}}\;\leqslant$	$5\;{{{\rm deg} }^{2}}$	0.03	0.04	0.03	0.04	0.03	0.06
	$20\;{{{\rm deg} }^{2}}$	0.14	0.19	0.12	0.14	0.15	0.16
	$100\;{{{\rm deg} }^{2}}$	0.45	0.54	0.40	0.45	0.47	0.52
	$200\;{{{\rm deg} }^{2}}$	0.64	0.70	0.60	0.60	0.66	0.68
	$500\;{{{\rm deg} }^{2}}$	0.87	0.89	0.82	0.83	0.87	0.89
	$1000\;{{{\rm deg} }^{2}}$	0.97	0.99	0.96	0.95	0.98	0.97

Note. Sky-localization areas used are from this study, using recolored noise, and from Singer et al. (2014), which uses Gaussian noise. Results are quoted for the full catalog of results with recolored noise and have an imposed S/N cut of $\varrho \geqslant 12$ to match the Gaussian catalog. Figures for the 50% credible region ${\rm C}{{{\rm R}}_{0.5}}$, the 90% credible region ${\rm C}{{{\rm R}}_{0.9}}$ and the searched area A_* are included. A dash (—) is used for fractions less than 0.01.

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Table 2.  Median Sky-localization Areas

Sky Localization			Gaussian Noise		Recolored Noise		Recolored Noise $\varrho \geqslant 12$
		bayestar	LALInference	bayestar	LALInference	bayestar	LALInference
Median	${\rm C}{{{\rm R}}_{0.5}}$	$138\;{{{\rm deg} }^{2}}$	$124\;{{{\rm deg} }^{2}}$	$175\;{{{\rm deg} }^{2}}$	$154\;{{{\rm deg} }^{2}}$	$145\;{{{\rm deg} }^{2}}$	$118\;{{{\rm deg} }^{2}}$
	${\rm C}{{{\rm R}}_{0.9}}$	$545\;{{{\rm deg} }^{2}}$	$529\;{{{\rm deg} }^{2}}$	$692\;{{{\rm deg} }^{2}}$	$632\;{{{\rm deg} }^{2}}$	$524\;{{{\rm deg} }^{2}}$	$481\;{{{\rm deg} }^{2}}$
	A*	$123\;{{{\rm deg} }^{2}}$	$88\;{{{\rm deg} }^{2}}$	$145\;{{{\rm deg} }^{2}}$	$132\;{{{\rm deg} }^{2}}$	$118\;{{{\rm deg} }^{2}}$	$88\;{{{\rm deg} }^{2}}$

Note. Sky-localization areas used are from this study, using recolored noise, and from Singer et al. (2014), which uses Gaussian noise. Results are quoted for the full catalog of results with recolored noise and have an imposed S/N cut of $\varrho \geqslant 12$ to match the Gaussian catalog. Figures for the 50% credible region ${\rm C}{{{\rm R}}_{0.5}}$, the 90% credible region ${\rm C}{{{\rm R}}_{0.9}}$ and the searched area A_* are included.

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4.3. Mass and Distance Estimation

Independent of any EM counterpart, GW astronomy is still informative. GW observations allow for measurement of various properties of the source system. Here, we examine the ability to measure luminosity distance and mass (principally the chirp mass of the system).

Accurate mass and distance measurements have many physical applications. Measurement of the chirp-mass distribution can constrain binary evolution models (Bulik & Belczynski 2003). Determining the maximum mass of a neutron star would shed light on its equation of state (e.g., Read et al. 2009), and, potentially, on the existence of a mass gap between neutron stars and black holes (Özel et al. 2010; Farr et al. 2011; Kreidberg et al. 2012). Combining mass and distance measurement, it may be possible to construct a new (independent) measure of the Hubble constant (Taylor et al. 2012). GW observations shall give us unique insight into the properties of BNS systems.

In addition to component masses and the distance to the source, the component spins are of astrophysical importance (e.g., Mandel & O'Shaughnessy 2010). Unfortunately, we cannot estimate the component spins as we are using non-spinning waveform templates. Measurement of the spins will be examined in a future study investigating PE using SpinTaylorT4 waveforms.

4.3.1. Luminosity Distance

Quantifying the precision of distance estimation is simpler than for sky localization as we are now working in a single dimension. The equivalent of a credible region is a credible interval. We denote the distance credible interval for a total posterior probability p as ${\rm CI}_{p}^{D}$. It is defined to exclude equal posterior probabilities in each of the tails; it is given by

$$\begin{eqnarray}&&{\rm CI}_{p}^{D}=C_{D}^{-1}\left( \frac{1+p}{2} \right)-C_{D}^{-1}\left( \frac{1-p}{2} \right),\end{eqnarray} \tag{ 5 }$$

where $C_{D}^{-1}(p)$ is the inverse of the cumulative distribution function

$$\begin{eqnarray}&&{{C}_{D}}(D)=\int _{0}^{D}dD^{\prime} {{P}_{D}}(D^{\prime} )\end{eqnarray} \tag{ 6 }$$

for distance posterior P_D(D). The same symmetric definition for the credible interval was used by Aasi et al. (2013a). A smaller ${\rm CI}_{p}^{D}$ for a given p indicates more precise distance estimation.

The self-consistency of our distance estimates can be verified by calculating the fraction of true values that fall within the credible interval at a given p. This is shown in Figure 11 for results from both the Gaussian and recolored noise results. Both distributions are consistent with expectations (performing a KS test with the predicted distribution yields p-values of 0.168 and 0.057 for the recolored and Gaussian noise respectively). LALInference does return self-consistent distance estimates.

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The cumulative distributions of credible intervals are plotted in Figure 12. We divide the credible interval by the true (injected) distance ${{D}_{\star }}$; this gives an approximate analog of twice the fractional uncertainty. The quantity ${\rm CI}_{p}^{D}/{{D}_{\star }}$ appears insensitive to the detection cut-off (a KS test between ${\rm CI}_{0.9}^{D}/{{D}_{\star }}$ for the recolored and Gaussian results gives a p-value of 0.077). This appears in contrast to the case for sky areas, but the differing S/N distributions are accounted for by scaling with respect to the distance (which is inversely proportional to the S/N). The estimation of the distance, like that for sky areas, does not depend upon the character of the noise.

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Distance estimation is imprecise: the posterior widths are frequently comparable to the magnitude of the distance itself. This is a consequence of a degeneracy between the distance and the inclination (Cutler & Flanagan 1994; Aasi et al. 2013a). The key numbers summarizing distance estimation are given in Tables 3 and 4; the former gives the fraction of events with ${\rm CI}_{p}^{D}/{{D}_{\star }}$ smaller than fiducial values, and the latter gives median values.

Table 3.  Fractions of Events With Fractional Distance Estimate Uncertainties Smaller Than a Given Size

Distance Estimate Uncertainty		Gaussian Noise	Recolored Noise
${\rm CI}_{0.5}^{D}/{{D}_{\star }}\;\leqslant$	0.25	0.04	0.03
	0.50	0.77	0.74
	0.75	0.95	0.93
	1.00	0.98	0.98
	2.00	1.00	1.00
${\rm CI}_{0.9}^{D}/{{D}_{\star }}\;\leqslant$	0.25	—	—
	0.50	—	—
	0.75	0.40	0.35
	1.00	0.70	0.66
	2.00	0.96	0.97

Note. Results using recolored noise and Gaussian noise are included (Singer et al. 2014). Figures for the 50% credible interval ${\rm CI}_{0.5}^{D}$ and the 90% credible interval ${\rm CI}_{0.9}^{D}$ are included; both are scaled with respect to the true distance ${{D}_{\star }}$. A dash (—) is used for fractions less than 0.01.

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Table 4.  Median Distance Credible Intervals (Divided by the True Distance) Using Recolored Noise and Gaussian Noise (Singer et al. 2014)

Distance Estimate Uncertainty		Gaussian Noise	Recolored Noise
Median	${\rm CI}_{0.5}^{D}/{{D}_{\star }}$	0.36	0.38
	${\rm CI}_{0.9}^{D}/{{D}_{\star }}$	0.82	0.85

Note. Figures for the 50% credible interval ${\rm CI}_{0.5}^{D}$ and the 90% credible interval ${\rm CI}_{0.9}^{D}$ are included.

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4.3.2. Chirp Mass

The chirp mass should be precisely measured as it determines the GW phase evolution. We again use the credible interval to quantify measurement precision; the chirp-mass credible interval ${\rm CI}_{p}^{{{\mathcal{M}}_{c}}}$ is defined equivalently to its distance counterpart in (5).

The fraction of true chirp masses that fall within ${\rm CI}_{p}^{{{\mathcal{M}}_{c}}}$ at a given p is plotted in Figure 13. Neither the results calculated using Gaussian noise nor those using recolored noise fit our expectations: the posteriors are not well calibrated. However, the two sets of results are entirely consistent with each other (a KS test between the two gives a p-value of 0.524), indicating that the PE is not affected by the noise. There appears to be a systematic error in our posterior distributions of the chirp mass.

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The discrepancies between our posterior estimates for the chirp masses and their true values are a consequence of our use of non-spinning TaylorF2 waveform templates. This has two effects. First, by using a non-spinning waveform, we do not explore the degeneracy between mass and spin (Cutler & Flanagan 1994; van der Sluys et al. 2008b; Baird et al. 2013). This results in an artificially narrow marginalized posterior for mass parameters such as the chirp mass. In effect, we are pinning the spin to be zero, which is information we should not have a priori. Second, we have used a template that does not exactly match the injected waveform (SpinTaylorT4). The small difference in approximants results in a mismatch in estimated parameters (Buonanno et al. 2009; Aasi et al. 2013a). Since the posterior on the chirp mass is narrow, because it is intrinsically well-measured and because we have not included degeneracy with spin, even a small difference in templates is sufficient to offset the posterior from the true chirp mass by a statistically significant amount.

To examine the offset between the estimated and true chirp masses, we plot in Figure 14 the difference between the posterior mean $\overline{{{\mathcal{M}}_{{\rm c}}}}$ and the true value ${{\mathcal{M}}_{\star }}$ divided by the standard deviation of the posterior ${{\sigma }_{{{\mathcal{M}}_{{\rm c}}}}}$. Using the median in place of the mean, or ${\rm CI}_{0.68}^{{{\mathcal{M}}_{{\rm c}}}}/2$ in place of ${{\sigma }_{{{\mathcal{M}}_{{\rm c}}}}}$, gives only a small quantitative difference. Over this narrow mass range, the offset is not a strong function of the chirp mass. The offset is a combination of both error introduced by the presence of noise and theoretical error from the mismatch between the injected waveform and template waveforms (Cutler & Vallisneri 2007). If only the former were significant, we would expect the mean offset to be zero, and the typical scatter of offsets to be of order of the posterior's standard deviation. Neither of these is the case. The average scaled offset $(\overline{{{\mathcal{M}}_{{\rm c}}}}-{{\mathcal{M}}_{\star }})/{{\sigma }_{{{\mathcal{M}}_{{\rm c}}}}}$ across the recolored (Gaussian) data set is −1.3 ± 0.1 (−0.9 ± 0.1). This shows that there is a systematic error. However, it is not as simple as just systematically underestimating the chirp mass; there is a large scatter in the offsets, the standard deviation of the scaled offset for the recolored (Gaussian) data set is 2.07 ± 0.08 (2.09 ± 0.09). This is consistent with our expectation that the mass–spin degeneracy should broaden the posterior; these results imply that the posterior should be a factor of ∼2 wider (cf. Poisson & Will 1995).

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While the theoretical error is important in determining the accuracy to which we can infer the chirp mass, it does not completely dominate the noise error. To illustrate the scale of the errors, we plot distribution of the 50% and 90% credible intervals in Figures 15(a) and (b), and the absolute magnitudes of the offsets in Figure 15(c). For a well calibrated posterior, we would expect the offset to be smaller than ${\rm CI}_{0.9}^{{{\mathcal{M}}_{{\rm c}}}}/2$ (${\rm CI}_{0.5}^{{{\mathcal{M}}_{{\rm c}}}}/2$) in approximately 90% (50%) of events. Figure 13 shows that this is not the case, that we do have systematic error. Figure 14 confirms this and shows that the theoretical error is of a comparable size to the noise error. In Figure 15, we see that the presence of theoretical error does not radically affect the distribution of offsets. The median value of the offsets are $(2.6\times {{10}^{-4}})\;{{M}_{\odot }}$ and $(2.4\times {{10}^{-4}})\;{{M}_{\odot }}$, and the median values of ${\rm CI}_{0.5}^{{{\mathcal{M}}_{{\rm c}}}}/2$ are $(1.2\times {{10}^{-4}})\;{{M}_{\odot }}$ and $(1.3\times {{10}^{-4}})\;{{M}_{\odot }}$ for the recolored and Gaussian data sets respectively; the theoretical error approximately doubles the total uncertainty on the chirp mass. The key numbers summarising the distributions are given in Tables 5 and 6, which give the fraction of events with uncertainties smaller than fiducial values and the median uncertainties respectively.

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Table 5.  Fractions of Events with Chirp-mass Estimate Errors Smaller than a Given Value

Chirp-mass Estimate Error		Gaussian Noise	Recolored Noise
${\rm CI}_{0.5}^{{{\mathcal{M}}_{{\rm c}}}}\;\leqslant$	$(5\times {{10}^{-5}})\;{{M}_{\odot }}$	—	—
	$(1\times {{10}^{-4}})\;{{M}_{\odot }}$	0.05	0.03
	$(2\times {{10}^{-4}})\;{{M}_{\odot }}$	0.34	0.33
	$(5\times {{10}^{-4}})\;{{M}_{\odot }}$	0.89	0.88
	$(1\times {{10}^{-3}})\;{{M}_{\odot }}$	1.00	0.99
	$(2\times {{10}^{-3}})\;{{M}_{\odot }}$	1.00	1.00
${\rm CI}_{0.9}^{{{\mathcal{M}}_{{\rm c}}}}\;\leqslant$	$(5\times {{10}^{-5}})\;{{M}_{\odot }}$	—	—
	$(1\times {{10}^{-4}})\;{{M}_{\odot }}$	—	—
	$(2\times {{10}^{-4}})\;{{M}_{\odot }}$	0.01	0.02
	$(5\times {{10}^{-4}})\;{{M}_{\odot }}$	0.29	0.29
	$(1\times {{10}^{-3}})\;{{M}_{\odot }}$	0.77	0.79
	$(2\times {{10}^{-3}})\;{{M}_{\odot }}$	1.00	0.97
$\left| \overline{{{\mathcal{M}}_{{\rm c}}}}-{{\mathcal{M}}_{\star }} \right|\;\leqslant$	$(5\times {{10}^{-5}})\;{{M}_{\odot }}$	0.09	0.11
	$(1\times {{10}^{-4}})\;{{M}_{\odot }}$	0.20	0.21
	$(2\times {{10}^{-4}})\;{{M}_{\odot }}$	0.42	0.41
	$(5\times {{10}^{-4}})\;{{M}_{\odot }}$	0.83	0.79
	$(1\times {{10}^{-3}})\;{{M}_{\odot }}$	0.98	0.96
	$(2\times {{10}^{-3}})\;{{M}_{\odot }}$	1.00	1.00

Note. Results using recolored noise and Gaussian noise are included (Singer et al. 2014). Included are figures for the 50% credible interval ${\rm CI}_{0.5}^{{{\mathcal{M}}_{{\rm c}}}}$ and the 90% credible interval ${\rm CI}_{0.9}^{{{\mathcal{M}}_{{\rm c}}}}$, which only include statistical error from the noise, and for the posterior mean offset relative to the true chirp mass $|\overline{{{\mathcal{M}}_{{\rm c}}}}-{{\mathcal{M}}_{\star }}|$, which includes both noise error and theoretical error. A dash (—) is used for fractions less than 0.01.

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Table 6.  Median Chirp-mass Credible Intervals and Posterior Estimate Offset Using Recolored Noise and Gaussian Noise (Singer et al. 2014)

Chirp-mass Estimate Error		Gaussian Noise	Recolored Noise
Median	${\rm CI}_{0.5}^{{{\mathcal{M}}_{{\rm c}}}}$	$(2.6\times {{10}^{-4}})\;{{M}_{\odot }}$	$(2.5\times {{10}^{-4}})\;{{M}_{\odot }}$
	${\rm CI}_{0.9}^{{{\mathcal{M}}_{{\rm c}}}}$	$(6.4\times {{10}^{-4}})\;{{M}_{\odot }}$	$(6.4\times {{10}^{-4}})\;{{M}_{\odot }}$
	$\left| \overline{{{\mathcal{M}}_{{\rm c}}}}-{{\mathcal{M}}_{\star }} \right|$	$(2.4\times {{10}^{-4}})\;{{M}_{\odot }}$	$(2.6\times {{10}^{-4}})\;{{M}_{\odot }}$

Note. Included are figures for the 50% credible interval ${\rm CI}_{0.5}^{{{\mathcal{M}}_{{\rm c}}}}$ and the 90% credible interval ${\rm CI}_{0.9}^{{{\mathcal{M}}_{{\rm c}}}}$, and the posterior mean offset relative to the true value $|\overline{{{\mathcal{M}}_{{\rm c}}}}-{{\mathcal{M}}_{\star }}|$.

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Furthermore, Figure 15 shows that the (in)ability to measure the chirp mass is not significantly influenced by the character of the noise or the detection threshold used (a KS test comparing the ${\rm CI}_{0.9}^{{{\mathcal{M}}_{{\rm c}}}}$ and $|\overline{{{\mathcal{M}}_{{\rm c}}}}-{{\mathcal{M}}_{\star }}|$ distributions between the Gaussian and recolored data sets gives p-values of 0.805 and 0.507 respectively). The latter is a consequence of both thresholds recovering equivalent chirp-mass distributions (Figure 4).

It should be possible to incorporate knowledge of theoretical waveform error into PE by marginalizing out the uncertainty. This can be done using parametric models for the uncertainty if a specific form of the waveform error is suspected, or non-parametrically if we wish to be agnostic. The effect of folding in this additional uncertainty is to broaden the posteriors and possibly shift their means; doing so should make posterior estimates consistent with the true values.

While we cannot correctly reconstruct the posterior distribution for the chirp mass, the error in the estimate is still small. We can measure the chirp mass accurately, even though we are affected by systematic error.

4.3.3. Component Masses

The chirp mass is a combination of the component masses; in some cases it can be used to infer whether the source is a BNS or a binary black-hole system (Hannam et al. 2013; Vitale & del Pozzo 2014), but the component masses are of greater interest. The mass–spin degeneracy affects our ability to construct accurate estimates for the individual masses. Since we have already seen a systematic error in the chirp mass, we expect an analogous (larger) phenomenon here.

We are again working in two dimensions, so we use credible regions to quantify PE precision. The mass-space credible region ${\rm CR}_{p}^{{{m}_{1}}-{{m}_{2}}}$ is defined analogously to its sky-area counterpart in (2); it is easier to compute as we do not have to contend with the spherical geometry of the sky or with as intricate posterior distributions. We plot in Figure 16 the fraction of injected masses that fall within ${\rm CR}_{p}^{{{m}_{1}}-{{m}_{2}}}$ at a given p. As for the chirp mass, the posterior is not well calibrated, approximately 40% (38% for results with recolored noise and 42% for Gaussian) of the true component masses lie altogether outside the range of the estimated posterior, but the two sets of results are consistent with each other (performing a KS test gives a p-value of 0.969). We cannot accurately reconstruct the component masses using our non-spinning waveforms.

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To give an indication of the scale of the uncertainty in m₁–m₂ space, we plot the 90% credible region in Figure 17. Since our estimates for the component masses are inaccurate, with many true values lying outside the posterior, ${\rm CR}_{p}^{m1-m2}$ is a lower bound on the typical scale for measurement accuracy. This does not reflect how well we can actually measure the component masses; to produce accurate estimates, we must include the mass–spin degeneracy which broadens the posterior.

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It is apparent that a statement regarding measurement of component masses must wait until an analysis is done using waveforms that include spin. We will return this question in a future publication.

5.1. Observing Scenarios

Having determined the sky-localization accuracy expected for O1, we now use our results to compare with current predictions for observing scenarios in the advanced-detector era. In Section 5.1.1 we consider the two-detector network of O1. In Section 5.1.2 we extend our discussion to consider predictions for sky localization in subsequent observing runs using a three-detector network.

5.1.1. Two-detector Sky-localization Accuracy

Prospects for sky localization in the advanced-detector era are specified by Aasi et al. (2013b). This states that any events detected in 2015 would not be well localized. This has been shown to not be the case (e.g., Nissanke et al. 2011; Kasliwal & Nissanke 2014; Singer et al. 2014). We see that while only a small fraction of events have well-localized sources, this fraction is non-zero. The 90% credible region is almost always smaller than ${{10}^{3}}\;{{{\rm deg} }^{2}}$. The 2015 observing scenario of Aasi et al. (2013b) does not give any figures for potential sky-localization accuracy, but we can now be specific using the results of this work.

The sky-localization figures currently included in Aasi et al. (2013b) are calculated using TT (Fairhurst 2009, 2011). This is a convenient means of predicting sky-localization accuracy; it is not a method used to reconstruct the sky-position posterior of detected signals. For a two-detector network, triangulation predicts an unbroken annulus on the sky. The area of this ring linearly scales with the uncertainty on the timing measurement, which is inversely proportional to the S/N. Our results show that, when using a coherent Bayesian approach, the recovered sky area is not (always) a ring (see Figure 6), and the area scales inversely with the square of the S/N (Raymond et al. 2009). Hence, TT is a poor fit in this case.

In Figure 18 we plot the ratio of the predicted credible region calculated using TT, to the actual credible region calculated using LALInference PE. We include predictions from both standard TT and also TT including phase coherence (Grover et al. 2014). The former method estimates timing accuracy (and hence the width of the sky annulus) as a function of the S/N and detector bandwidth.²⁰ The latter method introduces the requirement of phase consistency between detectors, which can significantly aid source localization. These effects are modeled via a correction factor, whose value depends on how marginalization over polarization is taken into account. Here, we use the larger of the two correction factors proposed in Grover et al. (2014), their Equation (16), although the degeneracy between phase and polarization means that the correction factor is probably too large for the two-detector network. The time and phase method does better, but neither technique does a good job at matching the true localization: both are too pessimistic. Agreement worsens at higher S/N as a consequence of the different S/N scalings. We cannot naively use TT to predict sky-localization accuracy for a two-detector network.

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We have found that sky areas recovered during O1 are likely to be hundreds of square degrees. Covering such a large area to sufficient depth to detect the most plausible EM counterparts ($r\gtrsim 22$–$26\;{\rm mag}$; Metzger & Berger 2012; Barnes & Kasen 2013; Metzger et al. 2015) is challenging for current EM observatories (Kasliwal & Nissanke 2014); furthermore, posterior distributions for the sky location are commonly multimodal or feature long, narrow arcs making them awkward to cover. It will be necessary to carefully consider how to most efficiently point telescopes to maximize the probability of observing a counterpart; using galaxy catalogs could be one means of increasing this chance (Nuttall & Sutton 2010; Nissanke et al. 2013; Hanna et al. 2014; Fan et al. 2014; Bartos et al. 2015).

5.1.2. Three-detector Sky-localization Accuracy

For 2016 onwards, we expect that AdV would also be in operation. The addition of a third detector should significantly improve sky-localization accuracy (Singer et al. 2014).

Aasi et al. (2013b) give figures for sky-localization accuracies in the three-detector era. In 2016, Aasi et al. (2013b) predict that 2% (5–12%) of BNS detections shall be localized within $5\;{{{\rm deg} }^{2}}$ ($20\;{{{\rm deg} }^{2}}$) at 90% confidence. These values are calculated from TT. Ideally, we would like to compare these to results using Bayesian PE using recolored noise, but performing three-detector PE runs for later observing periods is outside the range of this study. However, we have demonstrated that the properties of the noise do not impact sky-localization accuracies, provided that the chosen detection threshold yields similar S/N distributions in all cases. Consequently, we can use the three-detector, Gaussian-noise LALInference results of Singer et al. (2014) as a reference. For comparison, they find that 2% (14%) of events have ${\rm C}{{{\rm R}}_{0.9}}$ smaller than $5\;{{{\rm deg} }^{2}}$ ($20\;{{{\rm deg} }^{2}}$). PE with LALInference provides more optimistic sky-localization accuracies than TT.

In Figure 19 we compare the three-detector results of Singer et al. (2014) to the equivalent results calculated using TT. These results are for 2016, assuming the mid noise curve of Barsotti & Fritschel (2012) for the aLIGO detectors, and the geometric mean of the high and low bounds of the early curve of Aasi et al. (2013b) for the Virgo interferometer. Both triangulation and PE produce sky areas that scale with ${{\varrho }^{-2}}$, such that their ratio shows no significant trend with S/N, although the scatter seems to decrease as S/N increases.

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Comparing the entire population of points, we can calculate average values, which are given in Table 7. We consider the logarithm of the ratio, which should be ${{{\rm log} }_{10}}(1)=0$ for perfect agreement. The median ${{{\rm log} }_{10}}({\rm CR}_{0.5}^{{\rm TT}}/{\rm CR}_{0.5}^{{\rm PE}})$ using only time of arrival is 0.61, in complete agreement with the findings of Grover et al. (2014); using time and phase, the median value is 0.13. The TT and PE results have different ratios ${\rm C}{{{\rm R}}_{0.9}}/{\rm C}{{{\rm R}}_{0.5}}$. The mean value of ${{{\rm log} }_{10}}({\rm CR}_{0.9}^{{\rm PE}}/{\rm CR}_{0.5}^{{\rm PE}})$ is approximately 0.64 and the standard deviation is 0.13; again (see Section 4.2), this does not fit well with a Gaussian model. The 90% credible regions for triangulation and PE are in better agreement with each other, with the time-and-phase triangulation average areas consistent with those from LALInference. The time-and-phase method produces a reasonable estimate when averaged over the entire population. However, for individual events there is large scatter because TT models are purely predictive and do not take into account the actual data realization.

Table 7.  Average Values of the Logarithm of the Ratio of Credible Regions Calculated Using TT to Those Calculated from PE ${{{\rm log} }_{10}}({\rm CR}_{p}^{{\rm TT}}/{\rm CR}_{p}^{{\rm Full}})$

Triangulation Method	p	Mean	Median	Standard Deviation
Time only	0.5	$0.53$	0.61	0.39
	0.9	$0.42$	0.55	0.49
Time and phase	0.5	$0.05$	0.13	0.39
	0.9	−0.07	0.07	0.49

Note. TT results are calculated using just time of arrivals (Fairhurst 2009, 2011) and by also including phase coherence (Grover et al. 2014). PE results with Gaussian noise are calculated from the posteriors returned by LALInference (Singer et al. 2014).

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Despite the good average agreement, there is a large tail of events at low S/Ns where credible regions are too small, and the results suggest that at high S/Ns the credible regions may be too large; this may introduce errors when considering the sub-populations of the best localized or worst localized events (or if the distribution of events is significantly different from that considered here). Given all these findings, we can be confident that the TT results of Aasi et al. (2013b) are overly pessimistic.

There remains one further caveat before we can state that the sky-localization accuracies of Aasi et al. (2013b) should be revised to give better results. We have seen that using a realistic FAR cut allows us to detect signals with $\varrho \lt 12$. These low-S/N results shift the distribution of sky-localization accuracies, such that the performance appears worse. Thus, while we can be confident that the events currently included should have a better accuracy than assumed for Aasi et al. (2013b), the total population of detectable events is potentially larger than previously estimated, and may include some low-S/N events with poorer localization.

5.2. Summary

This work was supported by the Science and Technology Facilities Council. P.B.G. acknowledges NASA grant NNX12AN10G. S.V. acknowledges the support of the National Science Foundation and the LIGO Laboratory. J.V. was supported by STFC grant ST/K005014/1. LIGO was constructed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation and operates under cooperative agreement PHY-0757058.

Results were produced using the computing facilities of the LIGO DataGrid including: the Nemo computing cluster at the Center for Gravitation and Cosmology at the University of Wisconsin–Milwauke under NSF Grants PHY-0923409 and PHY-0600953; the Atlas computing cluster at the Albert Einstein Institute, Hannover; the LIGO computing clusters at Caltech, and the facilities of the Advanced Research Computing @ Cardiff (ARCCA) Cluster at Cardiff University. We are especially grateful to Paul Hopkins of ARCCA for assistance.

Some results were produced using the post-processing tools of the plotutils library at http://github.com/farr/plotutils, and some were derived using HEALPix (Gorski et al. 2005).

This paper is has been assigned LIGO document reference LIGO-P1400232. It contains some results originally included in LIGO technical report LIGO-T1400480.

In Section 4.1.2, we examined selection effects of the detection pipeline. In particular, we looked at the detected distribution of chirp masses, as this sets the GW amplitude. The magnitude of the selection effect depends on the details of the chirp-mass distribution, but can be estimated using a simple model. For low-mass signals whose inspiral spans the sensitive band of the detector, the amplitude of the waveform is proportional to $\mathcal{M}_{{\rm c}}^{5/6}$ (Sathyaprakash & Schutz 2009). The sensitive volume is proportional to the cube of this, or $\mathcal{M}_{{\rm c}}^{5/2}$. Suppose that half of the injections are made at a chirp mass of $\overline{{{\mathcal{M}}_{{\rm c}}}}-\delta {{\mathcal{M}}_{{\rm c}}}$ and the other half at a chirp mass value of $\overline{{{\mathcal{M}}_{{\rm c}}}}+\delta {{\mathcal{M}}_{{\rm c}}}$, with $\delta {{\mathcal{M}}_{{\rm c}}}\ll \overline{{{\mathcal{M}}_{{\rm c}}}}$. Then the expected fraction of higher-mass systems among all detected systems is

$$\begin{eqnarray}&&{{\mathcal{F}}_{{\rm high}}}=\frac{{{\left( \overline{{{\mathcal{M}}_{{\rm c}}}}+\delta {{\mathcal{M}}_{{\rm c}}} \right)}^{5/2}}}{{{\left( \overline{{{\mathcal{M}}_{{\rm c}}}}+\delta {{\mathcal{M}}_{{\rm c}}} \right)}^{5/2}}+{{\left( \overline{{{\mathcal{M}}_{{\rm c}}}}-\delta {{\mathcal{M}}_{{\rm c}}} \right)}^{5/2}}}\simeq \frac{1}{2}+\frac{5}{4}\frac{\delta {{\mathcal{M}}_{{\rm c}}}}{\overline{{{\mathcal{M}}_{{\rm c}}}}}.\end{eqnarray} \tag{ A1 }$$

If N detections are made in total and the selection effects played no role, the expected number of detections from the higher-mass set would be $N/2$ with a standard deviation of $\sqrt{N}/2$. However, in our model, there is a predicted excess of $5\;N\delta {{\mathcal{M}}_{{\rm c}}}/(4\overline{{{\mathcal{M}}_{{\rm c}}}})$ high-mass detections because of selection effects. Consequently, we expect to have xσ confidence in observing a selection effect on chirp mass, where

$$\begin{eqnarray}&&x=\frac{5\sqrt{N}}{2}\frac{\delta {{\mathcal{M}}_{{\rm c}}}}{\overline{{{\mathcal{M}}_{{\rm c}}}}}.\end{eqnarray} \tag{ A2 }$$

We can estimate $\overline{{{\mathcal{M}}_{{\rm c}}}}$ from the mean of the chirp-mass distribution, and $\delta {{\mathcal{M}}_{{\rm c}}}$ from the standard deviation; for our injections set, $\delta {{\mathcal{M}}_{{\rm c}}}/\overline{{{\mathcal{M}}_{{\rm c}}}}\approx 0.06$. For the Gaussian data set N = 250, and so we expect to observe selection effects at only the ∼2-σ confidence level; the actual measurements are roughly consistent with this. For such a narrow chirp-mass distribution, $\gtrsim {{10}^{3}}$ detections are needed to confidently observe the selection effects.

While the chirp mass is of prime importance to GW astronomers (it is their most precisely determined mass parameter), other combinations of mass are of interest in other contexts. Parameters which are correlated with the chirp mass are also subject to selection effects. However, their significance is proportional to the level of correlation of the parameters with chirp mass; given that selection effects on chirp mass are small, we do not expect statistically significant effects for other mass parameters. Here, we present the distributions of the individual component masses, the asymmetric mass ratio, and the total mass.

The distribution of recovered (injected) component masses is shown in Figure A1. The detected events show a slight over-representation of higher-mass objects, which is the effect of selecting systems with larger chirp masses. The deviation from the injection distribution is small (a KS test with the predicted distribution gives p-values of 0.213 and 0.182 for Gaussian noise, and 0.276 and 0.022 for the recolored noise), but noticeably more significant than for the spins.

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The asymmetric mass ratio is

$$\begin{eqnarray}&&q=\frac{{\rm min} \{{{m}_{1}},{{m}_{2}}\}}{{\rm max} \{{{m}_{1}},{{m}_{2}}\}}.\end{eqnarray} \tag{ A3 }$$

For uniformly distributed m₁ and m₂ between m_min and m_max, the probability density function for q is

$$\begin{eqnarray}&&{{P}_{q}}(q)=\left\{ \begin{array}{ccccccccccccccc} \begin{array}{ccccccccccccccc} \frac{1}{{{({{m}_{{\rm max} }}-{{m}_{{\rm min} }})}^{2}}}\;\left( m_{{\rm max} }^{2}-\frac{m_{{\rm min} }^{2}}{{{q}^{2}}} \right) \\ \end{array} & \frac{{{m}_{{\rm min} }}}{{{m}_{{\rm max} }}}\leqslant q\leqslant 1 \\ 0 & {\rm Otherwise}. \\ \end{array} \right.\end{eqnarray} \tag{ A4 }$$

Integrating this gives a cumulative distribution function of

$$\begin{eqnarray}{{C}_{q}}(q)=\left\{ \begin{array}{ccccccccccccccc} 0 & q\leqslant \frac{{{m}_{{\rm min} }}}{{{m}_{{\rm max} }}} \\ \frac{1}{{{({{m}_{{\rm max} }}-{{m}_{{\rm min} }})}^{2}}}(m_{{\rm max} }^{2}q-2{{m}_{{\rm min} }}{{m}_{{\rm max} }}+\frac{m_{{\rm min} }^{2}}{q}) & \frac{{{m}_{{\rm min} }}}{{{m}_{{\rm max} }}}\leqslant q\leqslant 1. \\ 1 & 1\leqslant q \\ \end{array} \right.\end{eqnarray} \tag{ A5 }$$

Figure A2 shows the recovered distribution of mass ratios as well as the injection distribution given by C_q(q). There is a small difference between the injection and recovered distributions (a KS test with the injection distribution returns p-values of 0.536 and 0.050 for the Gaussian and recolored noise respectively).

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The probability density function for the total system mass, $M={{m}_{1}}+{{m}_{2}}$, is

$$\begin{eqnarray}{{P}_{M}}(M)=\left\{ \begin{array}{ccccccccccccccc} \frac{1}{{{({{m}_{{\rm max} }}-{{m}_{{\rm min} }})}^{2}}}\times \left( M-2{{m}_{{\rm min} }} \right) & \begin{array}{ccccccccccccccc} 2{{m}_{{\rm min} }}\leqslant M\leqslant \;{{m}_{{\rm min} }}+{{m}_{{\rm max} }} \\ \end{array} \\ \frac{1}{{{({{m}_{{\rm max} }}-{{m}_{{\rm min} }})}^{2}}}\times \left( 2{{m}_{{\rm max} }}-M \right) & \begin{array}{ccccccccccccccc} {{m}_{{\rm min} }}+{{m}_{{\rm max} }}\leqslant \;M\leqslant 2{{m}_{{\rm max} }} \\ \end{array} \\ 0 & {\rm Otherwise}. \\ \end{array} \right.\end{eqnarray} \tag{ A6 }$$

Consequently, its cumulative distribution function is

$$\begin{eqnarray}{{C}_{M}}(M)=\left\{ \begin{array}{ccccccccccccccc} 0 & M\leqslant 2{{m}_{{\rm min} }} \\ \frac{1}{{{({{m}_{{\rm max} }}-{{m}_{{\rm min} }})}^{2}}}\left( \frac{{{M}^{2}}}{2}-2{{m}_{{\rm min} }}M+2m_{{\rm min} }^{2} \right) & \begin{array}{ccccccccccccccc} 2{{m}_{{\rm min} }}\leqslant M\leqslant \;{{m}_{{\rm min} }}+{{m}_{{\rm max} }} \\ \end{array} \\ \frac{1}{{{({{m}_{{\rm max} }}-{{m}_{{\rm min} }})}^{2}}}\left( 2{{m}_{{\rm max} }}M-\frac{{{M}^{2}}}{2}+m_{{\rm min} }^{2}-2{{m}_{{\rm min} }}{{m}_{{\rm max} }}-m_{{\rm max} }^{2} \right) & \begin{array}{ccccccccccccccc} {{m}_{{\rm min} }}+{{m}_{{\rm max} }}\leqslant \;M\leqslant 2{{m}_{{\rm max} }} \\ \end{array} \\ 1 & 2{{m}_{{\rm max} }}\leqslant M. \\ \end{array} \right.\end{eqnarray} \tag{ A7 }$$

Figure A3 shows the recovered distribution of total masses as well as the injection distribution given by C_M(M). The distributions are similar to those seen for the chirp mass in Figure 4. This is not surprising, as there is a clear link between the two quantities. We are considering a narrow mass range; individual component masses can be described as ${{m}_{1,2}}={{m}_{{\rm min} }}(1+{{\varepsilon }_{1,2}})$, where ${{\varepsilon }_{1,2}}\leqslant ({{m}_{{\rm max} }}-{{m}_{{\rm min} }})/{{m}_{{\rm min} }}\ll 1$. The total mass is ${{m}_{{\rm min} }}(2+{{\varepsilon }_{1}}+{{\varepsilon }_{2}})$; to first order in ${{\varepsilon }_{1,2}}$, the chirp mass can be described as ${{2}^{-6/5}}{{m}_{{\rm min} }}(2+{{\varepsilon }_{1}}+{{\varepsilon }_{2}})$. Hence, the total mass is approximately proportional to the chirp mass across the range of interest. We preferentially select signals with larger total masses as these produce louder signals, although the difference between the injection and recovered distributions is not too large (a KS test with the injection distribution yields p-values of 0.338 and 0.050 for the Gaussian and recolored noise respectively).

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All the mass distributions show a difference between the injection and detected populations. This is as expected. The difference is small, such that for the numbers of events considered in this study, it is only marginally significant. The difference need not always be negligible; it would become more important when considering a larger population of events, or a set of events with a broader chirp-mass distribution.

To perform rapid sky localization, we require that our analysis pipelines are expeditious. Following a detection, bayestar promptly returns a sky localization, and later LALInference returns estimates of the sky position plus further parameters. Here, we present estimates for the computational time required to run bayestar and LALInference.

All results are specific to a two-detector network. The LALInference results are for a (non-spinning) TaylorF2 analysis: this is the least expensive waveform family and provides medium-latency results. Computational times can be significantly longer using other waveforms. Efforts are being made to optimize and speed up the methods of LALInference (e.g., Canizares et al. 2013, 2015; Farr et al. 2014; Pürrer 2014).

The LALInference PE is slower than the rapid sky localization. Distributions of estimated CPU times for the runs are shown in Figure B1. The LALInference_nest times are calculated from log files. This is not entirely reliable as times may not be recorded for a variety of reasons. In this case, the reported time is a lower bound on the true value. In Figure B1(a) we show the distribution of run times for both the set of all estimated times and the subset excluding those we suspect are inaccurate due to a reported error message. The distributions are consistent with our expectation that the inaccurate times are lower bounds. In Figure B1(c) we show the cumulative distribution of run times using only the more reliable set of estimates. The median (accurately estimated) total CPU time for LALInference_nest is $1.96\times {{10}^{6}}\;{\rm s}=545\;{\rm hr}$ (cf. Veitch et al. 2015) and the median total CPU time for bayestar is $921\;{\rm s}=15.4\;{\rm min}$. Hence, on average, LALInference_nest takes ∼2000 times as much CPU time as bayestar.

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The actual latency of a technique is given by the wall time, not the CPU time. Five CPU processes were used per LALInference_nest run, hence the computational wall time can be estimated as a fifth of the total CPU time. This gives a median approximate wall time of $3.92\times {{10}^{5}}\;{\rm s}=109\;{\rm hr}$. Some processes take longer to finish than others, so this is not an exact means of estimating the time taken for a run to finish. These calculations also neglect time spent idle rather than running, which influences the physical wall time required for a job to complete. In online mode, bayestar is generally deployed in a 32–64-way parallel configuration. This gives a median wall time of $28.8\;{\rm s}$ ($14.4\;{\rm s}$) for a 32-way (64-way) configuration. bayestar provides sky localization $\sim {{10}^{4}}$ times quicker than LALInference, furthermore, none of our bayestar runs would have taken longer than a minute to complete (Singer 2014, chapter 4).

The length of the LALInference run depends upon the desired number of posterior samples. We may characterize the computational speed by the average rate at which independent samples are drawn from the posterior: the total number of (independent, as determined by LALInference) posterior samples divided by the total CPU time. The distribution of sampling speeds is shown in Figure B2. We use speeds calculated using both reliably estimated times and those we suspect might be lower bounds (giving upper bounds for sampling speed) in Figure B2(a), but only the more reliable values in Figure B2(b). The median (accurately estimated) LALInference_nest sampling speed is $4.40\times {{10}^{-3}}\;{{{\rm s}}^{-1}}=15.8\;{\rm h}{{{\rm r}}^{-1}}$ corresponding to one independent posterior sample every $227\;{\rm s}=6.31\times {{10}^{-2}}\;{\rm hr}$ of CPU time (cf. Sidery et al. 2014; Veitch et al. 2015).

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In contract, bayestar computes the likelihood 24,576 times. Its computation speed is thus simply inversely proportional to the total CPU time. The median bayestar computational speed is $26.7\;{{{\rm s}}^{-1}}$ corresponding to one likelihood evaluation every $37.5\;{\rm ms}$ of CPU time. The difference between the LALInference and bayestar computational speeds reflects the difference in the complexities of their likelihood functions.

The medium-latency PE runs, using the current code, finish in a few days. This is much longer than is required for bayestar to produce sky-localization estimates. However, LALInference also provides posterior probability distributions for the other parameters as well as more accurate sky localization than bayestar for three-detector networks (Singer et al. 2014).

Data produced for this study are available, as shown in the following tables. In the table stub, only two example entries are included in these tables. Further details are explained in the appendix of Singer et al. (2014). These tables, along with sky maps, are available online at http://www.ligo.org/scientists/first2years/. Table C1 gives the injected (true) parameters of the 333 simulated signals used for this study. Table C2 gives the detection parameters (the S/Ns, FAR and masses returned by the detection pipeline), and the sky areas calculated by bayestar and LALInference. Table C3 gives quantities related to PE for the chirp mass and distance. The second event listed in these tables is the one used for Figure 6. Table C4 is the counterpart of Table C3, but for the 250 events using Gaussian noise. The events shown in the table stub are the same examples used by Singer et al. (2014).

Table C1.  Simulated BNS Signals of Detected Events for 2015 Scenario Using Recolored Noise (cf. Singer et al. 2014, Table 2)

Event	Sim	MJD$/{\rm d}$	$\alpha /{\rm deg}$	$\delta /{\rm deg}$	$\iota /{\rm deg}$	$\psi /{\rm deg}$	${{\phi }_{{\rm c}}}/{\rm deg}$	$D/{\rm Mpc}$	${{m}_{1}}/{{M}_{\odot }}$	${{m}_{2}}/{{M}_{\odot }}$	a1x	a1y	a1z	a2x	a2y	a2z
IDa	IDb
4532	$899$	55430.10310	$99.9$	−30.8	26	349	118	84	1.25	1.36	−0.04	−0.01	−0.01	$0.01$	$0.00$	−0.00
4572	1243	55430.52510	227.5	−51.7	48	$27$	266	61	1.25	1.33	−0.01	−0.00	−0.04	−0.01	−0.01	−0.00

Note. Given are the event ID and simulation ID which specify the signal; the Modified Julian Date (MJD) of arrival at the geocenter of the signal from last stable orbit; the sky position in terms of the R.A. α and Decl. δ (J2000); the binary's orbital-inclination angle ι; the polarization angle ψ (Anderson et al. 2001, Appendix B); the orbital phase at coalescence ${{\phi }_{{\rm c}}}$; the source distance D; the component masses m₁ and m₂, and the x-, y-, and z-components of the spins a₁ and a₂.

^aThis identifier for detection candidates is the same value as the coinc_event_id column in the GSTLAL output database and the OBJECT cards in sky map FITS headers, with the coinc_event:coinc_event_id: prefix stripped. ^bThis identifier for simulateds signal is the same value as the simulation_id column in the GSTLAL output database, with the sim_inspiral:simulation_id: prefix stripped.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Table C2.  Detections and Sky-localization Areas for 2015 Scenario Using Recolored Noise (cf. Singer et al. 2014, Table 3)

								bayestar			LALInference
Event ID	Sim ID	Network	$\varrho$	${{\varrho }_{{\rm H}}}$	${{\varrho }_{{\rm L}}}$	$m_{1}^{{\rm ML}}/{{M}_{\odot }}$	$m_{2}^{{\rm ML}}/{{M}_{\odot }}$	${\rm C}{{{\rm R}}_{0.5}}/{{{\rm deg} }^{2}}$	${\rm C}{{{\rm R}}_{0.9}}/{{{\rm deg} }^{2}}$	${{{\rm A}}_{*}}/{{{\rm deg} }^{2}}$	${\rm C}{{{\rm R}}_{0.5}}/{{{\rm deg} }^{2}}$	${\rm C}{{{\rm R}}_{0.9}}/{{{\rm deg} }^{2}}$	${{{\rm A}}_{*}}/{{{\rm deg} }^{2}}$	FAR/${{{\rm s}}^{-1}}$
4532	$899$	HL	13.9	10.1	9.5	1.60	1.08	181.76	753.06	186.22	168.57	788.15	153.09	$2.14\times {{10}^{-14}}$
4572	1243	HL	13.2	10.0	8.7	1.73	0.98	227.91	828.23	$44.55$	203.63	920.10	$33.27$	$1.27\times {{10}^{-13}}$

Notes. Given are the event and simulation IDs; the detector network;^a the S/N for the network $\varrho$ and for the Hanford ${{\varrho }_{{\rm H}}}$ and Livingston ${{\varrho }_{{\rm L}}}$ detectors;^b the maximum-likelihood estimates of component masses $m_{1}^{{\rm ML}}$ and $m_{2}^{{\rm ML}}$ as reported by GSTLAL; the sky areas returned by bayestar and LALInference, and the FAR corresponding to the detection. ^aAll detections are for a two-detector Hanford–Livingston (HL) network. ^bThe network S/N is calculated by adding individual detectors in quadrature so ${{\varrho }^{2}}=\varrho _{{\rm H}}^{2}+\varrho _{{\rm L}}^{2}$.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Table C3.  Parameter-estimation Accuracies for 2015 Scenario Using Recolored Noise

Event ID	Sim ID	${{\mathcal{M}}_{\star }}/{{M}_{\odot }}$	${{D}_{\star }}/{\rm Mpc}$	${{\bar{\mathcal{M}}}_{{\rm c}}}/{{M}_{\odot }}$	${\rm CI}_{0.5}^{{{\mathcal{M}}_{{\rm c}}}}/{{M}_{\odot }}$	${\rm CI}_{0.9}^{{{\mathcal{M}}_{{\rm c}}}}/{{M}_{\odot }}$	$\bar{D}/{\rm Mpc}$	${\rm CI}_{0.5}^{D}/{\rm Mpc}$	${\rm CI}_{0.9}^{D}/{\rm Mpc}$
4532	$899$	1.136613	84.2	1.136689	0.000355	0.000795	64.6	25.0	53.3
4572	1243	1.123169	60.7	1.123286	0.000410	0.000901	67.5	26.7	57.7

Note. Given are the event and simulation IDs; the injected (true) chirp mass ${{\mathcal{M}}_{\star }}$ and distance ${{D}_{\star }}$; the posterior mean chirp mass ${{\bar{\mathcal{M}}}_{{\rm c}}}$; the chirp-mass credible intervals ${\rm CI}_{0.5}^{{{\mathcal{M}}_{{\rm c}}}}$ and ${\rm CI}_{0.9}^{{{\mathcal{M}}_{{\rm c}}}}$; the posterior mean distance $\bar{D}$; and the distance credible intervals ${\rm CI}_{0.5}^{D}$ and ${\rm CI}_{0.9}^{D}$. All parameter estimates are calculated by LALInference.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Table C4.  Parameter-estimation Accuracies for 2015 Scenario Using Gaussian Noise

Event ID	Sim ID	${{\mathcal{M}}_{\star }}/{{M}_{\odot }}$	${{D}_{\star }}/{\rm Mpc}$	${{\bar{\mathcal{M}}}_{{\rm c}}}/{{M}_{\odot }}$	${\rm CI}_{0.5}^{{{\mathcal{M}}_{{\rm c}}}}/{{M}_{\odot }}$	${\rm CI}_{0.9}^{{{\mathcal{M}}_{{\rm c}}}}/{{M}_{\odot }}$	$\bar{D}/{\rm Mpc}$	${\rm CI}_{0.5}^{D}/{\rm Mpc}$	${\rm CI}_{0.9}^{D}/{\rm Mpc}$
18951	10807	1.264368	74.8	1.264410	0.000457	0.001017	70.4	23.1	50.6
20342	21002	1.223944	75.0	1.223740	0.000444	0.001034	71.2	26.0	57.3

Note. The columns are the same as in Table C3.

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