SEARCH FOR GRAVITATIONAL-WAVE INSPIRAL SIGNALS ASSOCIATED WITH SHORT GAMMA-RAY BURSTS DURING LIGO'S FIFTH AND VIRGO'S FIRST SCIENCE RUN

The binary coalescence model suggests that the time delay between the arrival of a gravitational wave and the arrival of the subsequent electromagnetic burst, referred to as trigger time, is a few seconds. We assessed uncertainties in reported trigger times and quantization in our own analysis along integer second boundaries, finding that these each contribute less than 1 s. For example, when the Swift BAT instrument determines that the count rate has risen above a threshold, it waits for the maximum to pass, checking with a 320  ms cadence (N. Gehrels & D. Palmer 2008, private communication); it reports the start time of the block containing the maximum, rather than making any attempt to identify the start of the burst, and does so with a 320  ms granularity. As another example, there have been reports of sub-threshold precursors to many GRBs (Burlon et al. 2009). For each GRB in our sample, we checked tens of seconds of light curve by eye to look for both excessive difference between the trigger time and the apparent rise time, and also for precursors, but found nothing to suggest that we should correct the published trigger times. The largest timing uncertainty we identified is the delay between the compact merger and the prompt emission of the internal shocks. We search for gravitational-wave signals within an on-source segment of [ − 5, + 1) s around each trigger time for each GRB of interest, feeling that this window captures the physical model with some tolerance for its uncertainties.

Because we believe that a gravitational wave associated with a GRB only occurs in the on-source segment, we use 324 off-source trials, each 6 s long, to estimate the distribution of background due to the accidental coincidences of noise triggers. We also re-analyze the off-source trials with simulated signals added to the data to test the response of our search to signals; these we call injection trials. The actual number of off-source trials included in the analysis varied by GRB, as the trials that overlapped with data-quality vetoes were discarded (Abbott et al. 2009b). To prevent biasing our background estimation due to a potential loud signal in the on-source trial, the off-source segments do not use data within 48 s of the on-source segment, reflecting the longest duration of templates in our bank. Finally, we discard 72 s of data subject to filter transients on both ends of the off-source region. Taking all of these requirements into account, the minimum analyzable time is 2190 s.

X-ray and γがんま-ray instruments identified a total of 212 GRBs during the S5 run, 211 with measured durations; 30 of them have a T₉₀ duration smaller than 2 s. T₉₀ is the time interval over which 90% of all counts from a GRB are recorded. While the T₉₀ classifies a burst as long or short, it is not a definitive discriminator of progenitor systems. In addition to the short GRBs, GRB 051211 (Kawai et al. 2005) and GRB 070714B (Barbier et al. 2007) are formally long GRBs, but they have spectral features hinting at an underlying coalescence progenitor. GRB 061210 is another long-duration burst, but it exhibits the typical short spikes of a short GRB (Cannizzo et al. 2006). This gives a list of 33 interesting GRBs with which to search for an association with gravitational waves from compact binary coalescence.

Around the trigger time of each interesting GRB, we required 2190 s of multiply coincident data. The detectors operated with individual duty cycles of 67%–81% over the span of the S5 and VSR1 runs. Where more than two detectors had sufficient data, we selected the most sensitive pair based on the average inspiral range, because including a third, less sensitive detector does not enhance the sensitivity greatly. The one exception was GRB 070923, described below. In descending order of sensitivity, the detectors are H1, L1, H2, and V1. This procedure yielded 11 GRBs searched for in H1–L1 coincident data, 9 GRBs in H1–H2, and 1 in H2–L1.

In addition to these 21, we analyze GRB 070923 because of its sky location relative to the detectors' antenna patterns. The antenna pattern changes with the location of a source relative to a detector and can be expressed by the response $\sqrt{\vphantom{A^A}\smash{\hbox{${F_+^2+F_\times ^2}$}}}$ in which F₊ and F_× denote the antenna-pattern functions (Allen et al. 2005). A value of 1 corresponds to an optimal location of the putative gravitational-wave source relative to the observatory, while a value of 0 corresponds to a source location that will not induce any strain in the detector. For this particular GRB, the optimal antenna response for Virgo is around 0.7, while those for the two LIGO sites are about half of that (see Table 2), yielding a comparable sensitivity in the direction of GRB 070923 for all three of them. Data from H1, L1, and V1 were analyzed, making this the only GRB involving triple coincidences.

Table 1 lists all 22 target GRBs after applying the selection criteria described in this section. Plausible redshifts have been published for only three of these GRBs, placing them well outside of our detectors' range, but short GRB redshift determinations are in general sufficiently tentative to warrant searching for all of these GRBs.

GRB 070201 is also worth special mention. It was already analyzed in a high-priority search because of the striking spatial coincidence of this GRB with M31, a galaxy only ∼780  kpc from Earth. No gravitational-wave signal was found and a coalescence scenario could be ruled out with >99% confidence at that distance (Abbott et al. 2008a), lending additional support for a soft gamma repeater hypothesis (Ofek et al. 2008). However, because of improvements in the analysis pipeline, we re-analyzed this GRB and report the results in this paper. See Section 3.1.1 for details.

We generated candidates using the standard, untriggered compact binary coalescence search pipeline described in detail in Abbott et al. (2009b). The core of the inspiral search involves correlating the measured data against the theoretical waveforms expected from compact binary coalescence, a technique called matched filtering (Helmstrom 1968). The gravitational waves from the inspiral phase, when the binary orbit decays under gravitational-wave emission prior to merger, are accurately modeled by post-Newtonian approximants in the band of the detector's sensitivity for a wide range of binary masses (Blanchet 2006). The expected gravitational-wave signal, as measured by LIGO and Virgo, depends on the masses (m_NS, m_comp) and spins ($\vec{s}_{\rm NS}, \vec{s}_{\rm comp}$) of the NS and its companion (either NS or BH), as well as the spatial location (αあるふぁ, δでるた), inclination angle ιいおた, and polarization angle ψぷさい of the orbital axis. In general, the power of matched filtering depends most sensitively on accurately tracking the phase evolution of the signal. The phasing of compact binary inspiral signals depends on the masses and spins, the time of merger, and an overall phase.

We adopted a discrete bank of template waveforms that span a two-dimensional parameter space (one for each component mass) such that the maximum loss in signal-to-noise ratio (S/N) for a binary with negligible spins would be 3% (Cokelaer 2007). While the spin is ignored in the template waveforms, we verify that the search can still detect binaries with most physically reasonable spin orientations and magnitudes with only moderate loss in sensitivity. For simplicity, the template bank is symmetric in component masses, spanning the range [2,  40) M_☉ in total mass. The number of template waveforms required to achieve this coverage depends on the detector noise spectrum; for the data analyzed in this paper the number of templates was around 7000 for each detector.

We filtered the data from each of the detectors through each template in the bank. If the matched filter S/N exceeds a threshold, the template masses and the time of the maximum S/N are recorded. For a given template, threshold crossings are clustered in time using a sliding window equal to the duration of the template (Allen et al. 2005). For each trigger identified in this way, the coalescence phase and the effective distance—the distance at which an optimally oriented and optimally located binary, with masses corresponding to those of the template, would give the observed S/N—are also computed. Triggers identified in each detector are further required to be coincident with their time and mass parameters with a trigger from at least one other detector, taking into account the correlations between those parameters (Robinson et al. 2008). This significantly reduces the number of background triggers that arise from matched filtering in each detector independently.

The S/N threshold for the matched filtering step was chosen differently depending on which detectors' data are available for a given GRB. If data from H1 and L1 were analyzed, the threshold for each detector was set to 4.25, reflecting their comparable sensitivity. If data from H1 and H2 were analyzed, the threshold of the latter detector—the less sensitive of the two—was set to 3.5 to gain maximum network sensitivity, while the threshold of the more sensitive detector, H1, was set to 5.5 since any signal seen in H2 would be twice as loud in H1, with some uncertainty. In the single case of analyzing only H2–L1 data (GRB 070707) the threshold was 4.25 for L1 and 3.5 for H2, and for the single case of analyzing data with Virgo (GRB 070923), the threshold was set to 4.25 for all involved detectors (H1, L1, and V1). For comparison, a uniform S/N threshold of 5.5 was used in the untriggered S5 search (J. Abadie et al. 2010, in preparation). We applied two signal-based tests to reduce and refine our trigger sets. First, we computed a χかい² statistic (Allen 2005) to measure how different a trigger's S/N integrand looks from that of a real signal; triggers with large χかい² were discarded. Second, we applied the r² veto (Rodríguez 2007) which discards triggers depending on the duration that the χかい² statistic stays above a threshold. The S/N and χかい² from a single detector were combined into an effective S/N (Abbott et al. 2008b). The effective S/Ns from the analyzed detectors were then added in quadrature to form a single quantity ρろー²_eff which provided better separation between signal candidate events and background than S/N alone. The list of coincident triggers at this stage are then called candidate events.

The distribution of effective S/Ns from background and from signals can vary strongly across the template bank, depending most strongly on the chirp mass, a combination of the two component masses that appears in the leading term of the signal amplitude and phase (Thorne 1987). For this reason, we refine our candidate ranking with a likelihood-ratio statistic, which we compute for every candidate in the on-source, off-source, and injection trials. In short, we define the likelihood ratio L for a candidate c to be the efficiency divided by the false-alarm probability. The efficiency here is the probability of obtaining a candidate as loud or louder than c (by effective S/N) within the same region of template space given a signal in the data. The efficiency is a function of the signal parameters m_comp and D and is marginalized over all other parameters; it is obtained by simply counting across injection trials. The false-alarm probability here is the probability of obtaining a candidate as loud or louder than c in the same region of chirp mass from noise alone; it is obtained by counting across off-source trials.

At the end of the search (i.e., Table 2 and Figure 1), we report a different false-alarm probability. It is the fraction of off-source likelihood ratios larger than the largest on-source likelihood ratio.

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There is another noteworthy difference with respect to untriggered inspiral searches. For background estimation, untriggered searches use coincidences found between triggers from different detectors, to which unphysical time shifts greater than the light-travel time between detector sites have been applied. Unfortunately, H1 and H2, being co-located, share a common environmental noise that is absent from the time-shift background measurement. Being unable to estimate the significance of H1–H2 candidates reliably, the untriggered search examines them with significantly greater reservation and does not consider them at all in upper limit statements on rates. The present search performs its background estimation with unshifted coincidences under the assumption that any gravitational-wave signal will appear only in the on-source trial. Thus, we regain the unconditional use of H1–H2 candidates.

We found no evidence for a gravitational-wave signal in coincidence with any GRB in our sample. We ran the search as described in the previous section and found that the loudest observed candidates in each GRB's on-source segment are consistent with the expectation from its off-source trials. The results are summarized in Table 2, with brief highlights in the following subsections. A graphical comparison of on-source to off-source false-alarm probability is shown in Figure 1.

3.1.1. GRB 070201

3.1.2. GRB 070923

With our null observations and a large number of simulations, we can constrain the distance to each GRB assuming it was caused by a compact binary coalescence with a NS (with a mass in the range [1,  3) M_☉) and a companion of mass m_comp. For a given m_comp range, we used the approach of Feldman & Cousins (1998) to compute regions in distance where gravitational-wave events would, with a given confidence, have produced results inconsistent with our observations. Figure 2 shows the lower Feldman–Cousins distances for the 22 analyzed GRBs at 90% confidence for two illustrative choices for the companion mass range. The values are also listed in Table 2. Because the companion mass range has been divided into equally spaced bins, we report on an "NS–NS" system in which the companion mass is in the range [1,  4) M_☉ and an "NS–BH" system in which the BH has a mass in the range [7,  10) M_☉. The median exclusion distance for an NS–BH system is 6.7 Mpc, and for an NS–NS system it is 3.3  Mpc. These distances were derived assuming no beaming (uniform prior on cos ιいおた). NS–BH distances are typically higher than NS–NS because more massive systems radiate more total gravitational-wave energy. The excluded distance depends on various parameters: the location of the GRB on the sky, the detectors used for the GRB, the noise floor of the data itself, and the likelihood ratio of the loudest on-source candidate event for the GRB.

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We drew the simulations from a distribution in which our marginalized parameters roughly reflect our priors on these astrophysical compact binary systems. In our models, a signal is completely specified by $(m_{\rm NS}, \,m_{\rm comp}, \vec{\,s}_{\rm NS}, \vec{\,s}_{\rm comp},\, \iota,\, \psi,\, t_0,\, D, \,\alpha,\, \delta)$. Of these, we wish to constrain m_comp and D, marginalizing over everything else. We drew the NS mass m_NS uniformly from [1,  3) M_☉; the magnitudes of the NS spins $|\vec{s}_{\rm NS}|$ were half 0 and half uniform in [0,  0.75) (Cook et al. 1994); the magnitudes of the companion's spin $|\vec{s}_{\rm comp}|$ were half 0 and half uniform in [0,  0.98) (Mandel & O'Shaughnessy 2010); the orientations of the spins were uniform in solid angle; the inclination ιいおた of the normal to the binary's orbital plane relative to our line of sight was conservatively chosen to be uniform in cos ιいおた instead of making an assumption about the GRB beaming angle; the polarization angle ϕ was uniform in [0,  2πぱい); the coalescence time t₀ was uniform over the off-source region; the declination δでるた was set to that of the GRB; the right ascension αあるふぁ was also set to that of the GRB, but was adjusted based on t₀ to keep each simulation at the same location relative to the detector as the GRB.

A number of systematic uncertainties enter into this analysis, but amplitude calibration error and Monte Carlo counting statistics from the injection trials have the largest effects. We multiplied exclusion distances by 1.28 × (1 + δでるた_cal), where δでるた_cal is the fractional uncertainty (10% for H1 and H2; 13% for L1; 6% for V1; Marion et al. 2008). The factor of 1.28 corresponds to a 90% pessimistic fluctuation, assuming Gaussianity. To take the counting statistics into account, we stretched the Feldman–Cousins confidence belts to cover the probability ${\rm CL} + 1.28 \sqrt{\hbox{CL} (1-\hbox{CL})/n}$, where CL is the desired confidence limit and n is the number of simulations contained in the (m_comp,  D) bin for which we are constructing the belt.

In addition to the individual detection searches above, we would like to assess the presence of gravitational-wave signals that are too weak to stand out above background separately, but that are significant when the entire population of analyzed GRBs is taken together. We compare the cumulative distribution of the false-alarm probabilities of the on-source sample with the off-source sample. The on-source sample consist of the results of all 22 individual searches, including those for GRBs with known redshifts, and the off-source sample consists of 6801 results from the off-source trials. This number is lower than 22 × 324 because for some GRBs, some off-source trials were discarded due to known data-quality issues.

These two distributions are compared in Figure 1. To determine if they are consistent with being drawn from the same parent distribution, we employ the non-parametric Wilcoxon–Mann–Whitney U-statistic (Mann & Whitney 1947), which is a measure of how different two populations are. Applying the U-test, we find that the two distributions are consistent with each other; if the on-source and off-source significances were drawn from the same distribution, they would yield a U-statistic greater than what we observed 53% of the time. Therefore, we find no evidence for an excess of weak gravitational-wave signals associated with GRBs.