Search for Gravitational Waves Associated with Gamma-Ray Bursts during the First Advanced LIGO Observing Run and Implications for the Origin of GRB 150906B

In addition to the GRBs in the sample we described above, we also consider GRB 150906B, an event of particular interest due to its potential proximity. It occurred on 2015 September 6 at 08:42:20.560 UTC and was detected by the IPN (Golenetskii et al. 2015; Hurley et al. 2015). At the time of GRB 150906B, the Advanced LIGO detectors were undergoing final preparations for O1. Nonetheless, the 4 km detector in Hanford was operational at that time.

GRB 150906B was observed by the Konus–Wind, INTEGRAL, Mars Odyssey, and Swift satellites. It was outside the coded field of view of the Swift BAT, and, consequently, localization was achieved by triangulation of the signals observed by the four satellites (Hurley et al. 2015). The localization region of GRB 150906B lies close to the local galaxy NGC 3313, which has a redshift of $0.0124$ at a luminosity distance of $54$ Mpc (Levan et al. 2015). This galaxy lies $130$ kpc in projection from the GRB error box, a distance that is consistent with observed offsets of short GRBs from galaxies and with the expected supernova kicks imparted on NS binary systems (Berger 2011). NGC 3313 is part of a group of galaxies, and it is the brightest among this group. Other, fainter members of the group also lie close to the GRB error region, as shown in Figure 1. In addition, there are a number of known galaxies at around 500 Mpc within the error region of the GRB (Bilicki et al. 2013). For the GW search, we use a larger error region with a more conservative error assumption. Follow-up electromagnetic observations of the GRB were not possible due to its proximity to the Sun.

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The Konus–Wind observation of GRB 150906B was further used to classify the GRB (Svinkin et al. 2015). It was observed to have a duration of¹⁵⁰ ${T}_{50}=(0.952\pm 0.036)$ s and ${T}_{90}=(1.642\pm 0.076)$ s, which places it at the longer end of the short GRB distribution. Furthermore, GRB 150906B lies between the peaks of the short/hard and long/soft Konus–Wind GRB distributions in the log T₅₀–log HR₃₂ hardness–duration diagram, where log HR₃₂ is the (logarithm of the) ratio of counts in the $[200,760]$ keV and $[50,200]$ keV bands (Svinkin et al. 2015). Thus, a firm classification of the GRB as either short or long is problematic.

Assuming GRB 150906B originated in NGC 3313 yields an isotropic-equivalent γ-ray energy ${E}_{\mathrm{iso}}\sim {10}^{49}$ erg (Levan et al. 2015). This is consistent with inferred luminosities of short GRBs with measured redshifts (Berger 2011), albeit at the lower end of the distribution of ${E}_{\mathrm{iso}}$ values. Theoretical arguments (Ruffini et al. 2015; Zhang et al. 2015) suggest that the energetics fit better with a more distant system around 500 Mpc, possibly originating from one of the galaxies within the error region.

The modeled search for GWs emitted by NS binary mergers addresses the case of short GRB events. While not all NS binary mergers necessarily lead to a short GRB, this search looks for a GW counterpart to a short GRB event under the assumption that short GRBs are generated by NS binary mergers. In the standard scenario (Eichler et al. 1989; Paczynski 1991; Narayan et al. 1992; Nakar 2007), as the two companions spiral inward together due to the emission of GWs, the NSs are expected to tidally disrupt before the coalescence, in order to create a massive torus remnant in the surroundings of the central compact object that is formed by the binary coalescence. The matter in the torus can then power highly relativistic jets along the axis of total angular momentum (Blandford & Znajek 1977; Rosswog & Ramirez-Ruiz 2002; Lee & Ramirez-Ruiz 2007). This picture is supported by observational evidence (Berger 2011; Berger et al. 2013; Tanvir et al. 2013) and numerical simulations (e.g., Rezzolla et al. 2011; Kiuchi et al. 2015) but has not yet been fully confirmed.

The form of the GW signal emitted by a compact binary coalescence depends on the masses (${m}_{\mathrm{NS}},{m}_{\mathrm{comp}}$) and spins of the NS and its companion (either an NS or a BH), as well as the spatial location and orientation relative to the detector. In the remainder of this section we therefore discuss observational constraints on these properties and our choices regarding them that are folded into our search for BNS and NS-BH progenitors of short GRBs.

Mass measurements of NSs in binary systems currently set a lower bound on the maximum possible NS mass to $(2.01\pm 0.04)$ ${M}_{\odot }$ (Antoniadis et al. 2013). On the other hand, theoretical considerations set an upper bound on the maximum NS star mass to ∼3 ${M}_{\odot }$ (Rhoades & Ruffini 1974; Kalogera & Baym 1996), while the standard core-collapse supernova formation scenario restricts NS birth masses above the 1.1–1.6 ${M}_{\odot }$ interval (Lattimer 2012; Ozel et al. 2012; Kiziltan et al. 2013). Finally, we note that the individual NS masses reported for the eight candidate BNS systems lie in the interval $[1.0,1.49]\,{M}_{\odot }$ (Ozel & Freire 2016).

The fastest spinning pulsar ever observed rotates at a frequency of 716 Hz (Hessels et al. 2006). Assuming a mass of 1.4 ${M}_{\odot }$ and a moment of inertia of 10⁴⁵ g cm², this corresponds to a dimensionless spin magnitude of ∼0.4. The highest measured spin frequency of pulsars in confirmed BNS systems is that of J0737−3039A (Burgay et al. 2003). It is equal to 44 Hz (Kramer & Wex 2009), which yields a dimensionless spin magnitude of ∼0.05 (Brown et al. 2012). Finally, the potential BNS pulsar J1807−2500B (Lynch et al. 2012) with a spin of 4.19 ms gives a dimensionless spin magnitude of ∼0.2, if one assumes a pulsar mass of 1.37 ${M}_{\odot }$ and a moment of inertia $2\cdot {10}^{45}$ g cm².

No observations of NS-BH systems are available to date. Notably, however, a likely NS-BH progenitor has been observed, namely Cyg X-3 (Belczynski et al. 2013). While Advanced LIGO has observed a BH with mass ${36}_{-4}^{+5}$ ${M}_{\odot }$ in a binary BH system (Abbott et al. 2016j), and while stellar BHs with masses exceeding even 100 ${M}_{\odot }$ are conceivable (Belczynski et al. 2014; de Mink & Belczynski 2015), mass measurements of galactic stellar-mass BHs in X-ray binaries are between 5 and 24 solar masses (Ozel et al. 2010; Farr et al. 2011; Kreidberg et al. 2012; Wiktorowicz et al. 2013). X-ray observations of accreting BHs provide a broad distribution of dimensionless spin magnitudes ranging from ∼0.1 to above 0.95 (e.g., Miller & Miller 2014). We remark that BH dimensionless spin magnitudes inferred from observations of high-mass X-ray binaries typically have values above 0.85 and that these systems are more likely to be NS-BH system progenitors (McClintock et al. 2014).

A final property to discuss in the context of GW searches for BNS and NS-BH systems in coincidence with short GRBs is the half-opening angle ${\theta }_{\mathrm{jet}}$ of the GRB jet. Relativistic beaming and collimation due to the ambient medium confine the GRB jet to ${\theta }_{\mathrm{jet}}$. In all cases, we assume that the GRB is emitted in the direction of the binary total angular momentum. The observation of prompt γ-ray emission is, therefore, indicative that the inclination of the total angular momentum with respect to the line of sight to the detectors lies within the jet cone. Estimates of ${\theta }_{\mathrm{jet}}$ are based on jet breaks observed in X-ray afterglows and vary across GRBs. Indeed, many GRBs do not even exhibit a jet break. However, studies of observed jet breaks in Swift GRB X-ray afterglows find a mean (median) value of ${\theta }_{\mathrm{jet}}=6\buildrel{\circ}\over{.} 5$ $(5\buildrel{\circ}\over{.} \,4)$, with a tail extending almost to 25° (Racusin et al. 2009). In at least one case where no jet break is observed, the inferred lower limit is 25° and could be as high as 79° (Grupe et al. 2006). By folding in lower limits on ${\theta }_{\mathrm{jet}}$ for short GRBs without opening angle measurements and the indication that ${\theta }_{\mathrm{jet}}\sim 5^\circ \mbox{--}20^\circ$, which arises from simulations of postmerger BH accretion, Fong et al. (2015) find a median of $16^\circ \pm 10^\circ$ for ${\theta }_{\mathrm{jet}}$.

In light of all these considerations on astrophysical observations, we perform the modeled search described in Section 4.2 for NSs with masses between $1\,{M}_{\odot }$ and $2.8\,{M}_{\odot }$ and dimensionless spin magnitude of 0.05 at most.¹⁵¹ For the companion object, we test masses in the range $1\,{M}_{\odot }\leqslant {m}_{\mathrm{comp}}\leqslant 25$ M_⊙ and dimensionless spins up to 0.999. Additionally, we restrict the NS-BH search space (i.e., ${m}_{\mathrm{comp}}\gt 2.8$ M_⊙) to BH masses and spins that are consistent with the presence of remnant material in the surroundings of the central BH, rather than with the direct plunge of the NS onto the BH (Pannarale & Ohme 2014). This astrophysically motivated cut excludes from our search NS-BH systems that do not allow for a GRB counterpart to be produced, even under the most optimistic assumptions regarding the NS equation of state¹⁵² and the amount of tidally disrupted NS material required to ignite the GRB emission¹⁵³ (Pannarale & Ohme 2014). Finally, we search for circularly polarized signals. As discussed in Williamson et al. (2014), this is an excellent approximation for inclination angles between the total angular momentum and the line of sight up to 30°.

Long GRBs are followed up by the search for unmodeled GW transients described in Section 4.3. When making quantitative statements on the basis of this search, we use two families of GW signal models: circular sine-Gaussian (CSG) and accretion disk instability (ADI) signals. The scenarios that these address are discussed below.

No precise waveform is known for stellar collapse. A wide class of scenarios involves a rotational instability developing in the GRB central engine that leads to a slowly evolving, rotating quadrupolar mass distribution. Semianalytical calculations of rotational instabilities suggest that up to ${10}^{-2}\,{M}_{\odot }{c}^{2}$ may be emitted in GWs (Davies et al. 2002; Fryer et al. 2002; Kobayashi & Meszaros 2003; Shibata et al. 2003; Piro & Pfahl 2007; Corsi & Meszaros 2009; Romero et al. 2010), but simulations addressing the nonextreme case of core-collapse supernovae predict an emission of up to ${10}^{-8}\,{M}_{\odot }{c}^{2}$ in GWs (Ott 2009). With this in mind, we use a crude but simple generic model, that is, a CSG waveform with plus (+) and cross (×) polarizations given by

$$\begin{eqnarray}&&\left[\begin{array}{c}{h}_{+}(t)\\ {h}_{\times }(t)\end{array}\right] & = & \displaystyle \frac{1}{r}\sqrt{\displaystyle \frac{G}{{c}^{3}}\displaystyle \frac{{E}_{\mathrm{GW}}}{{f}_{0}Q}\displaystyle \frac{5}{4{\pi }^{3/2}}}\\ & & \times \,\left[\begin{array}{c}(1+{\cos }^{2}\iota )\cos (2\pi {f}_{0}t)\\ 2\cos \iota \sin (2\pi {f}_{0}t)\end{array}\right]\exp \left[-\displaystyle \frac{{(2\pi {f}_{0}t)}^{2}}{2{Q}^{2}}\right],\end{eqnarray} \tag{ 1 }$$

where the signal frequency f₀ is equal to twice the rotation frequency, t is the time relative to the signal peak time, Q characterizes the number of cycles for which the quadrupolar mass moment is large, ${E}_{\mathrm{GW}}$ is the total radiated energy, r is the distance to the source, ι is the rotation axis inclination angle with respect to the observer, and G and c are the gravitational constant and the speed of light, respectively. The inclination angle ι can be once again linked to observations of GRB jet-opening angles: in the case of long GRBs, these are typically $\sim 5^\circ$ (Gal-Yam et al. 2006; Racusin et al. 2009). All other parameters are largely underconstrained.

In the collapsar model of long GRBs, a stellar-mass BH forms, surrounded by a massive accretion disk. An extreme scenario of emission from a stellar collapse is a "magnetically suspended" ADI (van Putten 2001; van Putten et al. 2004). This parametric model may not be a precise representation of realistic signals, but it captures the generic features of many proposed models. It is characterized by four free astrophysical parameters: the mass and the dimensionless spin parameter of the central BH, the fraction of the disk mass that forms clumps, and the accretion disk mass. Waveform parameters such as duration, frequency span, and total radiated energy can be precisely derived for a given set of astrophysical parameters. As discussed in Section 4.3, we use several combinations of values for the astrophysical parameters in order to cover the different predicted morphologies.

RAVEN (Urban 2016) compares the GW triggers recorded in the low-latency all-sky GW analysis with the given time of a GRB. It provides a preliminary indication of any coincident GW candidate event and its associated significance. The cWB (Klimenko et al. 2016), oLIB (Lynch et al. 2015), GstLAL (Messick et al. 2017), and MBTA (Adams et al. 2016) pipelines perform the blind, rapid all-sky GW monitoring. cWB and oLIB search for a broad range of GW transients in the frequency range of 16–2048 Hz without prior knowledge of the signal waveforms. The GstLAL and MBTA pipelines search for GW signals from the coalescence of compact objects, using optimal matched filtering with waveforms. During O1, MBTA covered component masses of 1–12 ${M}_{\odot }$ with a 5 ${M}_{\odot }$ limit on chirp mass. GstLAL, instead, covered systems with component masses of 1–2.8 ${M}_{\odot }$ and 1–16 ${M}_{\odot }$ up to 2015 December 23; then, motivated by the discovery of GW150914, the analysis was extended to cover systems with component masses of 1–99 M_⊙ and total mass less than 100 M_⊙. Both pipelines limit component spins to $\lt 0.99$ and $\lt 0.05$ for BHs and NSs,¹⁵⁵ respectively (see Abbott et al. 2016c for further details).

GW candidates from these low-latency searches were uploaded to GraceDB and compared to the GRB triggers to find any temporal coincidence in [−600, +60] and [−5, +1] second windows, which correspond to the delay between the GW and the GRB trigger for long and short GRBs, respectively, as discussed in the next two sections. This strategy has the advantage of being very low latency and of requiring little additional computational costs over the existing all-sky searches. RAVEN (Urban 2016) results are available to be shared with LIGO partner electromagnetic astronomy facilities¹⁵⁶ within minutes following a GRB detection.

In the vast majority of short GRB progenitor scenarios, the GW signal from an NS binary coalescence is expected to precede the prompt γ-ray emission by no more than a few seconds (Lee & Ramirez-Ruiz 2007; Vedrenne & Atteia 2009). Therefore, we search for NS binary GW signals with end times that lie in an on-source window of [−5, +1) s around the reported GRB time, as done in previous searches in LIGO and Virgo data (Abadie et al. 2012a; Aasi et al. 2014b). The method we use is described in detail in Williamson et al. (2014) and references therein; the code implementing it is available under Nitz et al. (2016).

The data are filtered in the $30$ Hz–$1000$ Hz frequency interval through a discrete bank of ∼110,000 template waveforms (Owen & Sathyaprakash 1999) that covers NS binaries with the properties discussed in Section 3.1. It is the first time that a short GRB follow-up search used a template bank that includes aligned spin systems (Brown et al. 2012; Harry et al. 2014). The bank is designed to have a 3% maximum loss of signal-to-noise ratio (S/N) due to discretization effects for binaries with spins aligned, or antialigned, to the orbital angular momentum over the parameter space discussed at the end of Section 3.1.

For BNS and NS-BH coalescences, we use point-particle post-Newtonian models that describe the inspiral stage, where the orbit of the binary slowly shrinks due to the emission of GWs. This is mainly motivated by the fact that the merger and postmerger regime (i.e., the GW high-frequency behavior) of these systems differs from the binary BH case. While we do have robust inspiral-merger-ringdown binary BH waveforms (Taracchini et al. 2014; Khan et al. 2016), efforts to obtain accurate, complete waveform models for NS binaries are still underway (Lackey et al. 2014; Bernuzzi et al. 2015; Pannarale et al. 2015; Barkett et al. 2016; Haas et al. 2016; Hinderer et al. 2016). Additionally, to go beyond a point-particle inspiral description, the search would have to cover all feasible NS equations of state, at the expense of a significant increase in its computational costs. Each template is therefore modeled with the "TaylorT4" time-domain, post-Newtonian inspiral approximant (Buonanno et al. 2003), filtered against the coherently combined data, and peaks in the matched filter coherent S/N are recorded. Additional signal consistency tests are used to eliminate the effect of non-Gaussian transients in the data and to generate a reweighted coherent S/N (see Williamson et al. 2014 for its formal definition), which forms the detection statistic (Allen 2005; Harry & Fairhurst 2011).

After the filtering and the consistency tests, the event with the largest reweighted coherent S/N (provided that this is greater than 6) in the on-source window is retained as a candidate GW signal. In order to assess the significance of the candidate, the detector background noise distribution is estimated using data from a time period surrounding the on-source data, when a GW signal is not expected to be present. This is known as the off-source data and is processed identically to the on-source data. Specifically, the same data-quality cuts and consistency tests are applied, and the same sky positions relative to the GW detector network are used. The NS binary search method requires a minimum of $1658$ s of off-source data, which it splits into as many 6 s trials as possible. In order to increase the number of background trials, when data from more than one detector are available, the data streams are time-shifted multiple times and reanalyzed for each time shift. The template that produces the largest reweighted coherent S/N in each 6 s off-source time window is retained as a trigger. These are used to calculate a p-value¹⁵⁷ to the on-source loudest event by comparing it to the distribution of loudest off-source triggers in terms of the detection statistics. The p-value is calculated by counting the fraction of background trials containing an event with a greater reweighted coherent S/N than the loudest on-source event. Any candidate events with p-values below 10% are subjected to additional follow-up studies to determine if the events can be associated with some non-GW noise artifact. Further details on the methods used to search for NS binary signals in coincidence with short GRBs can be found in Harry & Fairhurst (2011) and Williamson et al. (2014).

The efficiency of the NS binary search method for recovering relevant GW signals is evaluated via the addition in software of simulated signals to the data. In order to assess performance, these data are filtered with the same bank of templates used for the search. This provides a means of placing constraints on the short GRB progenitor in the event of no detection in the on-source. All simulated signals are modeled using the "TaylorT2" time-domain, post-Newtonian inspiral approximant (Blanchet et al. 1996). We note that this approximant differs from the one used to build the templates. This choice is designed to account for the disagreement among existing inspiral waveform models in our efficiency assessment (see Nitz et al. 2013 on this topic). Further, this approximant allows for generic spin configurations. We inject three sets of simulated inspiral signals; these correspond to (1) BNS systems with a generic spin configuration, (2) NS-BH systems with a generic spin configuration, and (3) NS-BH systems with an aligned spin configuration. We build both a generic and an aligned spin injection set in the NS-BH case in order to assess the impact of precession on the search sensitivity for rapidly spinning and highly precessing systems (as is the NS-BH case in contrast to the BNS case). The considerations illustrated in Section 3.1 motivate the following choices for the parameters that characterize the three families:

• NS masses: These are chosen from a Gaussian distribution centered at 1.4 M_⊙, with a width of 0.2 ${M}_{\odot }$ and 0.4 ${M}_{\odot }$ in the BNS and the NS-BH case, respectively (Ozel et al. 2012; Kiziltan et al. 2013). The larger width for NS-BH binaries reflects the greater uncertainty arising from a lack of observed NS-BH systems.
• BH masses: These are Gaussian distributed with a mean of 10 ${M}_{\odot }$ and a width of 6 M_⊙. Additionally, they are restricted to being less than 15 M_⊙, because the disagreement between different Taylor approximants dominates beyond this point (Nitz et al. 2013).
• Dimensionless spins: These are drawn uniformly over the intervals $[0,0.4]$ and $[0,0.98]$ for NSs and BHs, respectively. For the two sets with generic spin configurations, both spins are isotropically distributed.
• Tidal disruption: NS-BH systems for which the remnant BH is not accompanied by any debris material are not included in the injected populations (Pannarale & Ohme 2014).
• Inclination angle: This is uniformly distributed in cosine over the intervals $[0^\circ ,30^\circ ]$ and $[150^\circ ,180^\circ ]$.
• Distance: Injections are distributed uniformly in distance in the intervals $[10,300]$ Mpc and $[10,600]$ Mpc for BNS and NS-BH systems, respectively.

When performing the efficiency assessment, we marginalize over amplitude detector calibration errors by resampling the assumed distance of each injected signal with a Gaussian distribution of 10% width (Abbott et al. 2017b, 2016f); the phase errors of $\sim 5^\circ$ have a negligible effect.

Long GRBs are associated with the gravitational collapse of massive stars. While GW emission is expected to accompany such events, its details may vary from event to event. We therefore search for any GW transient without assuming a specific signal shape; this type of search is performed for short GRB events as well. We use the time interval starting from 600 s before each GRB trigger and ending either 60 s after the trigger or at the T₉₀ time (whichever is larger) as the on-source window to search for a GW signal. This window is large enough to take into account most plausible time delays between a GW signal from a progenitor and the onset of the γ-ray signal (Koshut et al. 1995; Aloy et al. 2000; MacFadyen et al. 2001; Zhang et al. 2003; Lazzati 2005; Wang & Meszaros 2007; Burlon et al. 2008, 2009; Lazzati et al. 2009; Vedrenne & Atteia 2009). The search is performed on the most sensitive GW band of $16$–$500$ Hz. Above 300 Hz, the GW energy necessary to produce a detectable signal increases sharply as ∝f⁴ (see Figure 2 of Abbott et al. 2017a), and hence the detector sensitivity is highly biased against high-frequency emission scenarios.

The method used to search for generic GW transients follows the one used in previous GRB analyses (Abadie et al. 2012a; Aasi et al. 2014a, 2014b) and is described in detail in Sutton et al. (2010) and Was et al. (2012). The on-source data for each GRB are processed by the search pipeline to generate multiple time-frequency maps of the data stream using short Fourier transforms with duration at all powers of two between $1/128\,{\rm{s}}$ and 2 s. The maps are generated after coherently combining data from the detectors, taking into account the antenna response and noise level of each detector. The time-frequency maps are scanned for clusters of pixels with energy significantly higher than the one expected from background noise. These are referred to as "events" and are characterized by a ranking statistic based on energy. We also perform consistency tests based on the signal correlations measured between the detectors. The event with the highest ranking statistic is taken to be the best candidate for a GW signal for that GRB; it is referred to as the "loudest event." The strategy to associate a p-value with the loudest event is the same as the one adopted by the NS binary search but with off-source trials of $\sim 660\,{\rm{s}}$ duration.

As for the NS binary search method, the efficiency of this search at recovering relevant GW signals is evaluated by the addition in software of simulated signals to the data. The simulated waveforms are chosen to cover the search parameter space; they belong to three types of signals that embrace different potential signal morphologies: NS binary inspiral signals, stellar collapse (represented by CSGs), and disk instability models (represented by ADI waveforms).¹⁵⁸ In particular, the generic time-frequency excess power method used is equally efficient for descending (ADI) and ascending (NS binary) chirps. Because this paper reports results for NS binaries only when these are obtained with the dedicated, modeled search outlined in Section 4.2, we will limit the discussion to the case of the other two signal families.

• CSG: For the standard siren CSG signals defined in Equation (1), we assume an optimistic emission of energy in GWs of ${E}_{\mathrm{GW}}={10}^{-2}\,{M}_{\odot }{c}^{2}$. As discussed in Section 3, this is an upper bound on the predictions: our conclusions thus represent upper bounds, as we work under the optimistic assumption that every GRB emits ${10}^{-2}\,{M}_{\odot }{c}^{2}$ of energy in GWs. Further, we construct four sets of such waveforms with a fixed Q factor of 9 and varying center frequency (70, 100, 150, and 300 Hz).
• ADI: The extreme scenario of ADIs (van Putten 2001; van Putten et al. 2004) provides long-lasting waveforms that the unmodeled search has the ability to recover. We chose the same sets of parameters used in a previous long-transient search (Abbott et al. 2016b) to cover the different predicted morphologies. The values of the parameters are listed in Table 1. As in previous searches, the clumps in the disk are assumed to be forming at a distance of $100\,$ km from the BH innermost stable circular orbit (Ott & Santamaría 2013), which is the typical distance of the transition to a neutrino opaque disk where the accretion disk is expected to have the largest linear density (Lee et al. 2005; Chen & Beloborodov 2007). This constitutes a deviation from the original model that brings the GW emission from ∼1 kHz to a few hundred Hz, where the detectors are more sensitive, thus providing a reasonable means of testing the ability of the search to detect signals in this frequency band and with amplitudes comparable to the original ADI formulation. We note that in the previous search for long-duration signals associated with GRBs (Aasi et al. 2013), these signals were normalized to obtain ${E}_{\mathrm{GW}}=0.1\,{M}_{\odot }{c}^{2}$. These waveforms are tapered by a Tukey window with 1 s at the start and end of the waveform to avoid artifacts from the unphysical sharp start and end of these waveforms.

Finally, calibration errors are folded into the result by jittering the signal amplitude and time of arrival at each detector, following a wider Gaussian distribution of 20% in amplitude and 20 degrees in phase, as this search used the preliminary Advanced LIGO calibration that had greater uncertainties (Tuyenbayev et al. 2017).

Table 1.  Accretion Disk Instability Waveform Parameters

Waveform	M			Duration	Frequency	${E}_{\mathrm{GW}}$
		χ	$\epsilon$
Label	(${M}_{\odot }$)			(s)	(Hz)	(${M}_{\odot }{c}^{2}$)
ADI-A	5	0.30	0.050	39	135–166	0.02
ADI-B	10	0.95	0.200	9	110–209	0.22
ADI-C	10	0.95	0.040	236	130–251	0.25
ADI-D	3	0.70	0.035	142	119–173	0.02
ADI-E	8	0.99	0.065	76	111–234	0.17

Notes. The first column is the label used for the ADI waveform. The second and third columns are the mass and the dimensionless spin parameter of the central BH. The fourth column, $\epsilon$, is the fraction of the disk mass that forms clumps, and in all cases the accretion disk mass is 1.5 M_⊙. The duration, frequency span, and total radiated energy of the resulting signal are also reported in the remaining columns.

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If NGC 3313 were indeed the host of an NS binary merger progenitor of GRB 150906B, Advanced LIGO should have detected a GW signal associated with the event, given the proximity of this galaxy located at a luminosity distance of $54$ Mpc from Earth. A similar hypothesis was previously tested with the initial LIGO detectors for GRB 051103 and GRB 070201, the error boxes of which overlapped the M81/M82 group of galaxies and M31, respectively (Abbott et al. 2008; Abadie et al. 2012b). In both cases, a binary merger scenario was excluded with greater than 90% confidence, and the preferred scenario is that these events were extragalactic soft-gamma-repeater flares.

The NS binary search described in Section 4.2 found no evidence for a GW signal produced at the time and sky position of GRB 150906B. The most significant candidate event in the on-source region around the time of the GRB had a p-value of 53%.

This null-detection result allows us to compute the frequentist confidence with which our search excludes a binary coalescence in NGC 3313. This confidence includes both the search efficiency at recovering signals as well as our uncertainty in measuring such efficiency. Figure 5 shows the exclusion confidence for BNS and NS-BH systems as a function of the jet half-opening angle ${\theta }_{\mathrm{jet}}$, assuming a distance¹⁵⁹ to NGC 3313 of $54$ Mpc and that the NS binary inclination angle ι between the total angular momentum axis and the line of sight is distributed uniformly in $\cos \iota$ up to ${\theta }_{\mathrm{jet}}$. If we assume an isotropic (i.e., unbeamed) γ-ray emission from GRB 150906B, the possibility of a BNS coalescence progenitor is excluded with $\gtrsim 86$% confidence. Taking a fiducial jet half-opening angle upper limit of 30^◦ (or equivalently a maximum binary inclination angle of this size), the exclusion confidence rises to $\gtrsim 99.7$%. NS-BH systems with isotropic emission are excluded at $\gtrsim 97$% confidence, which rises to $\gtrsim 99.7$% for ${\theta }_{\mathrm{jet}}\leqslant 30^\circ$.

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The increase in exclusion confidence for smaller jet angles is due to the fact that the average amplitude of the GW signal from an NS binary coalescence is larger for systems for which the orbital plane is viewed "face-on" (where the detector receives the flux from both GW polarizations) than for systems viewed "edge-on" (where the detector receives the flux from just one GW polarization); small jet angles imply a system closer to face-on.

To determine the distance up to which we can exclude, with 90% confidence, a binary coalescence as the progenitor of GRB 150906B, we assume beamed emission with a maximum opening angle of 30° and compute the distance at which 90% of injected BNS, generic spin NS-BH, and aligned spin NS-BH signals are recovered louder than the loudest on-source event. The result is shown in Figure 6. BNS systems are excluded with 90% confidence out to a distance of 102 Mpc, while generic and aligned spin NS-BH systems are excluded with the same confidence at 170 Mpc and 186 Mpc, respectively. This is consistent with theoretical arguments based on γ-ray spectrum and fluence that place the progenitor of GRB 150906B at more than 270 Mpc (Ruffini et al. 2015; Zhang et al. 2015), possibly in one of the several known galaxies at around 500 Mpc within the error region (Bilicki et al. 2013).

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