SEARCH FOR GRAVITATIONAL-WAVE BURSTS ASSOCIATED WITH GAMMA-RAY BURSTS USING DATA FROM LIGO SCIENCE RUN 5 AND VIRGO SCIENCE RUN 1

The basic search procedure follows that used in recent LIGO GRB searches (Abbott et al. 2008a, 2008b). All GRBs are treated identically, without regard to their duration, redshift (if known), or fluence. We use the interval from 120 s before each GRB trigger time to 60 s after as the window in which to search for an associated GWB. This conservative window is large enough to take into account most reasonable time delays between a gravitational-wave signal from a progenitor and the onset of the gamma-ray signal. For example, it is much larger than the O(10) s delay of the gamma-ray signal resulting from the sub-luminal propagation of the jet to the surface of the star in the collapsar model for long GRBs (see, for example, Aloy et al. 2000; Zhang et al. 2003; Wang & Meszaros 2007; Lazzati et al. 2009). It is also much longer than the ≲1 s delay that may occur in the binary neutron-star merger scenario for short GRBs if a hypermassive neutron star is formed (see, for example, Liu et al. 2008; Baiotti et al. 2008; Kiuchi et al. 2009). Our window is also safely larger than any uncertainty in the definition of the measured GRB trigger time. The data in this search window are called the on-source data.

The on-source data are scanned by an algorithm designed to detect transients that may have been caused by a GWB. In this search, two algorithms are used: the cross-correlation algorithm used in previous LIGO searches (Abbott et al. 2008b), and X-Pipeline,⁹⁹ a new coherent analysis package (Chatterji et al. 2006; Sutton et al. 2009). The cross-correlation algorithm correlates the data between pairs of detectors, while X-Pipeline combines data from arbitrary sets of detectors, taking into account the antenna response and noise level of each detector to improve the search sensitivity.

The data are analyzed independently by X-Pipeline and the cross-correlation algorithm to produce lists of transients, or events, that may be candidate gravitational-wave signals. Each event is characterized by a measure of significance, based on energy (X-Pipeline) or correlation between detectors (cross-correlation algorithm). To reduce the effect of non-stationary background noise, the list of candidate events is subjected to checks that "veto" events overlapping in time with known instrumental or environmental disturbances (Abbott et al. 2009b). X-Pipeline also applies additional consistency tests based on the correlations between the detectors to further reduce the number of background events. The surviving event with the largest significance is taken to be the best candidate for a gravitational-wave signal for that GRB; it is referred to as the loudest event (Brady et al. 2004; Biswas et al. 2009).

To estimate the expected distribution of the loudest events under the null hypothesis, the pipelines are also applied to all coincident data within a 3 hr period surrounding the on-source data. These data for background estimation are called the off-source data. Their proximity to the on-source data makes it likely that the estimated background will properly reflect the noise properties in the on-source segment. The off-source data are processed identically to the on-source data; in particular, the same data-quality cuts and consistency tests are applied, and the same sky position relative to the Earth is used. To increase the off-source distribution statistics, multiple time shifts are applied to the data streams from different detector sites (or between the H1 and H2 streams for GRBs occurring when only those two detectors were operating), and the off-source data are re-analyzed for each time shift. For each 180 s segment of off-source data, the loudest surviving event is determined. The distribution of significances of the loudest background events, $C({\cal S}_{\rm max})$, thus gives us an empirical measure of the expected distribution of the significance of the loudest on-source event ${\cal S}_{\rm max}^{\rm on}$ under the null hypothesis.

To determine if a GWB is present in the on-source data, the loudest on-source event is compared to the background distribution. If the on-source significance is larger than that of the loudest event in 95% of the off-source segments (i.e., if $C({\cal S}_{\rm max}^{\rm on})\ge 0.95$), then the event is considered a candidate gravitational-wave signal. Candidate signals are subjected to additional "detection checklist" studies to try to determine the physical origin of the event; these studies may lead to rejecting the event as being of terrestrial origin, or they may increase our degree of confidence in it being due to a gravitational wave.

Regardless of whether a statistically significant signal is present, we also set a frequentist upper limit on the strength of gravitational waves associated with the GRB. For a given gravitational-wave signal model, we define the 90% confidence level upper limit on the signal amplitude as the minimum amplitude for which there is a 90% or greater chance that such a signal, if present in the on-source region, would have produced an event with significance larger than the largest value ${\cal S}_{\rm max}^{\rm on}$ actually measured. The signal models simulated are discussed in Section 6.1.

Since X-Pipeline was found to be more sensitive to GWBs than the cross-correlation pipeline (by about a factor of 2 in amplitude), we decided in advance to set the upper limits using the X-Pipeline results. The cross-correlation pipeline is used as a detection-only search. Since it was used previously for the analysis of a large number of GRBs during S2–S4, and for GRB 070201 during S5, including the cross-correlation pipeline provides continuity with past GRB searches and allows comparison of X-Pipeline with the technique used for these past searches.

X-Pipeline is a matlab-based software package for performing coherent searches for GWBs in data from arbitrary networks of detectors. Since X-Pipeline has not previously been used in a published LIGO or Virgo search, in this section we give a brief overview of the main steps followed in a GRB-triggered search. For more details on X-Pipeline, see Sutton et al. (2009).

Coherent techniques for GWB detection (see, for example, Gursel & Tinto 1989; Flanagan & Hughes 1998; Anderson et al. 2001; Klimenko et al. 2005, 2006; Mohanty et al. 2006; Rakhmanov 2006; Chatterji et al. 2006; Summerscales et al. 2008) combine data from multiple detectors before scanning it for candidate events. They naturally take into account differences in noise spectrum and antenna response of the detectors in the network. X-Pipeline constructs several different linear combinations of the data streams: those that maximize the expected signal-to-noise ratio for a GWB of either polarization from a given sky position (referred to as the d₊ and d_× streams), and those in which the GWB signal cancels (referred to as the d_null streams). It then looks for transients in the d₊ and d_× streams. Later, the energies in the d₊, d_×, and d_null streams are compared to attempt to discriminate between true GWBs and background noise fluctuations.

4.2.1. Event Generation

X-Pipeline processes data in 256 s blocks. First, it whitens the data from each detector using linear predictor error filters (Chatterji et al. 2004). It then time-shifts each stream according to the time of flight for a gravitational wave incident from the sky position of the GRB, so that a gravitational-wave signal will be simultaneous in all the data streams after the shifting. The data are divided into 50% overlapping segments and Fourier transformed. X-Pipeline then coherently sums and squares these Fourier series to produce time-frequency maps of the energy in the d₊, d_×, and d_null combinations. Specifically, we define the noise-weighted antenna response vectors $\boldsymbol{f}^{+,{\rm DPF}}$ and $\boldsymbol{f}^{\times,{\rm DPF}}$ for the network, with components

$$\begin{eqnarray} f^{+,{\rm DPF}}_{\alpha }(\theta,\phi,f) & = & \frac{F^{+}_\alpha (\theta,\phi,\psi ^{{\rm DPF}})}{\sqrt{S_\alpha (f)}} \,, \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} f^{\times,{\rm DPF}}_{\alpha }(\theta,\phi,f) & = & \frac{F^{\times }_\alpha (\theta,\phi,\psi ^{{\rm DPF}})}{\sqrt{S_\alpha (f)}} \,. \end{eqnarray} \tag{ 2 }$$

Here, (θ, ϕ) is the direction to the GRB, ψ^DPF is the polarization angle specifying the orientation of the plus and cross polarizations, F⁺_α, F^×_α ∈ [ − 1, 1] are the antenna response factors to the plus and cross polarizations (Anderson et al. 2001, see also Section 6.1), and S_α is the noise power spectrum of detector α. DPF stands for the dominant polarization frame; this is a frequency-dependent polarization basis ψ^DPF(f) such that $\boldsymbol{f}^{+,{\rm DPF}}\cdot \boldsymbol{f}^{\times,{\rm DPF}}= 0$ and $|\boldsymbol{f}^{+,{\rm DPF}}| \ge |\boldsymbol{f}^{\times,{\rm DPF}}|$ (Klimenko et al. 2005). With this choice of basis, the d₊ stream is defined as the projection

$$\begin{equation} d_+ \equiv \frac{\boldsymbol{f}^{+,{\rm DPF}}\cdot \boldsymbol{d}}{|\boldsymbol{f}^{+,{\rm DPF}}|} \,, \end{equation} \tag{ 3 }$$

where $\boldsymbol{d}$ is the set of whitened data streams from the individual detectors. The "signal energy" E₊ ≡ |d₊|² can be shown to be the sum-squared signal-to-noise ratio in the network corresponding to the least-squares estimate of the h₊ polarization of the gravitational wave in the dominant polarization frame. The d_× stream and energy E_× are defined analogously. The sum E₊ + E_× is then the maximum sum-squared signal to noise at that frequency that is consistent with a GWB arriving from the given sky position at that time.

The projections of the data orthogonal to $\boldsymbol{f}^{+,{\rm DPF}}$, $\boldsymbol{f}^{\times,{\rm DPF}}$ yield the null streams, in which the contributions of a real gravitational wave incident from the given sky position will cancel. The null stream energy E_null ≡ |d_null|² should therefore be consistent with background noise. (The definition of the null streams is independent of the polarization basis used.) The number of independent data combinations yielding null streams depends on the geometry of the network. Networks containing both the H1 and H2 interferometers have one null stream combination. Networks containing L1, V1, and at least one of H1 or H2 have a second null stream. For the H1–H2–L1–V1 network there are two independent null streams; in this case we sum the null energy maps from the two streams to yield a single null energy.

Events are selected by applying a threshold to the E₊ + E_× map, so that the pixels with the 1% highest values are marked as black pixels. Nearest-neighbor black pixels are grouped together into clusters (Sylvestre 2002). These clusters are our events. Each event is assigned an approximate statistical significance ${\cal S}$ based on a χ² distribution; for Gaussian noise in the absence of a signal, 2(E₊ + E_×) is χ²-distributed with 4N_pix degrees of freedom, where N_pix is the number of pixels in the event cluster. This significance is used when comparing different clusters to determine which is the "loudest." The various coherent energies (E₊, E_×, E_null) are summed over the component pixels of the cluster, and other properties such as duration and bandwidth of the cluster are also recorded.

The analysis of time shifting, FFTing, and cluster identification is repeated for fast Fourier transform (FFT) lengths of $1/8, 1/16, 1/32, 1/64, 1/128, \ {\rm {and}} \ 1/256$ s, to cover a range of possible GWB durations. Clusters produced by different FFT lengths that overlap in time and frequency are compared. The cluster with the largest significance is kept; the others are discarded. Finally, only clusters with central time in the on-source window of 120 s before the GRB time to 60 s after are considered as possible candidate events.

4.2.2. Glitch Rejection

Real detector noise contains glitches, which are short transients of excess strain noise that can masquerade as GWB signals. As shown in Chatterji et al. (2006), one can construct tests that are effective at rejecting glitches. Specifically, each coherent energy E₊, E_×, E_null has a corresponding "incoherent" energy I₊, I_×, I_null which is formed by discarding the cross-correlation terms (d_αd*_β) when computing E₊ = |d₊|², etc. For large-amplitude background noise glitches the coherent and incoherent energies are strongly correlated, $E\sim I \pm \sqrt{I}$. For strong gravitational-wave signals one expects either E₊ > I₊ and E_× < I_× or E₊ < I₊ and E_× > I_× depending on the signal polarization content, and E_null < I_null.

X-Pipeline uses the incoherent energies to apply a pass/fail test to each event. A nonlinear curve is fitted to the measured distribution of background events used for tuning (discussed below), specifically, to the median value of I as a function of E. Each event is assigned a measure of how far it is above or below the median:

$$\begin{equation} \sigma \equiv (I - I_{\rm med}(E)) / I^{1/2} \,. \end{equation} \tag{ 4 }$$

For (I_null, E_null), an event is passed if σ_null > r_null, where r_null is some threshold. For (I₊, E₊) and (I_×, E_×), the event passes if |σ₊|>r₊ and |σ_×|>r_×. (For the H1–H2 network, which contains only aligned interferometers, the conditions are σ₊ < r₊ and σ_null > r_null.) An event may be tested for one, two, or all three of the pairs (I_null, E_null), (I₊, E₊), and (I_×, E_×), depending on the GRB. The choice of which energy pairs are tested and the thresholds applied are determined independently for each of the 137 GRBs. X-Pipeline makes the selection automatically by comparing simulated GWBs to background noise events, as discussed below. In addition, the criterion I_null ⩾ 1.2E_null was imposed for all H1–H2 GRBs, as this was found to be effective at removing loud background glitches without affecting simulated gravitational waves.

In addition to the coherent glitch vetoes, events may also be rejected because they overlap data-quality flags or vetoes, as mentioned in Section 4.1. The flags and vetoes used are discussed in Abbott et al. (2009b). To avoid excessive dead time due to poor data quality, we impose minimum criteria for a detector to be included in the network for a given GRB. Specifically, at least 95% of the 180 s of on-source data must be free of data-quality flags and vetoes, and all of the 6 s spanning the interval from −5 to +1 s around the GRB trigger must be free of flags and vetoes.

4.2.3. Pipeline Tuning

The cross-correlation pipeline has been used in two previous LIGO searches (Abbott et al. 2008a, 2008b) for GWBs associated with GRBs, and is described in detail in these references. We therefore give only a brief summary of the pipeline here.

In the present search, the cross-correlation pipeline is applied to the LIGO detectors only. (The different orientation and noise spectrum shape of Virgo relative to the LIGO detectors is more easily accounted for in a coherent analysis.) The 180 s on-source time series for each interferometer is whitened as described in Abbott et al. (2008b) and divided into time bins, then the cross-correlation for each interferometer pair and time bin is calculated. The cross-correlation cc of two time series s₁ and s₂ is defined as

$$\begin{equation} {\rm cc} = \frac{\sum _{i=1}^m [s_1(i)-\mu _1][s_2(i)-\mu _2]}{ \sqrt{\sum _{j=1}^m [s_1(j)-\mu _1]^2} \sqrt{\sum _{k=1}^m [s_2(k)-\mu _2]^2 \vphantom{\sum _{j=1}^m}} } \,, \end{equation} \tag{ 5 }$$

where μ₁ and μ₂ are the corresponding means and m is the number of samples in the bin. Cross-correlation bins of lengths 25 ms and 100 ms are used to target short-duration GW signals with durations of ∼1–100 ms. The bins are overlapped by half a bin width to avoid loss of signals occurring near a bin boundary. Each Hanford-Livingston pair of 180 s on-source segments is shifted in time relative to each other to account for the time of flight between the detector sites for the known sky position of the GRB before the cross-correlations are calculated.

The cross-correlation is calculated for each interferometer pair and time bin for each bin length used. For an H1–H2 search, the largest cross-correlation value obtained within the 180 s search window is considered the most significant measurement. For an H1–L1 or H2–L1 search, the largest absolute value of the cross-correlation is taken as the most significant measurement. This was done to take into account the possibility that signals at Hanford and Livingston could be anticorrelated depending on the (unknown) polarization of the gravitational wave.

The results of the search for each of the 137 GRBs analyzed by X-Pipeline are shown in Table 1, the Appendix. The seventh column in this table lists the local probability$p \equiv 1-C({\cal S}_{\rm max}^{\rm on})$ for the loudest on-source event, defined as the fraction of background trials that produced a more significant event (a "−" indicates no event survived all cuts). Five GRBs had events passing the threshold of p = 0.05 to become candidate signals.

Since the local probability is typically estimated using approximately 150 off-source segments, small p values are subject to relatively large uncertainty from Poisson statistics. We therefore applied additional time shifts to the off-source data to obtain a total of 18,000 off-source segments for each candidate which were processed to improve the estimate of p. The five GRBs and their refined local probabilities are 060116 (p = 0.0402), 060510B (0.0124), 060807 (0.00967), 061201 (0.0222), and 070529 (0.0776). (Note that for GRB 070529, the refined local probability from the extra off-source segments was larger than the threshold of 0.05 for candidate signals.)

Considering that we analyzed 137 GRBs, these numbers are consistent with the null hypothesis that no GWB signal is associated with any of the GRBs tested. In addition, three of these GRBs have large measured redshift: GRB 060116 (z = 6.6), 060510B (z = 4.9), and 070529 (z = 2.5), making it highly unlikely a priori that we would expect to see a GWB in these cases. Nevertheless, each event has been subjected to follow-up examinations. These include checks of the consistency of the candidate with background events (such as incoherent energies, and frequency), checks of detector performance at the time as indicated by monitoring programs and operator logs, and scans of data from detector and environmental monitoring equipment to look for anomalous behavior. In each case, the candidate event appears consistent with the coherent energy distributions of background events, lying just outside the coherent glitch rejection thresholds. The frequency of each event is also typical of background events for their respective GRBs. Some of these GRBs occurred during periods of elevated background noise, and one occurred during a period of glitchy data in H1. In two cases, scans of data from monitoring equipment indicate a possible physical cause for the candidate event: one from non-stationarity in a calibration line and another due to upconversion of low-frequency noise in H1.

All but two of the GRBs processed by X-Pipeline are also analyzed by the cross-correlation pipeline. The cross-correlation pipeline produces a local probability for each detector pair and for each bin length (25 ms and 100 ms), for a total of 646 measurements from 135 GRBs. The threshold on the cross-correlation local probability corresponding to the 5% threshold for X-Pipeline is therefore 5%×135/646 ≃ 1%. A total of seven GRBs have p < 1% from cross-correlation: 060306 (0.00833), 060719 (0.00669), 060919 (0.00303), 061110 (0.00357), 070704 (0.00123), and 070810 (0.00119). These results are also consistent with the null hypothesis. Furthermore, none of these GRBs are among those that had a low p value from X-Pipeline. This is further indication that the candidate events detected by each pipeline are due to background noise rather than GWBs. Specifically, X-Pipeline and the cross-correlation pipeline use different measures of significance of candidate events. Whereas a strong GWB should be detected by both, any given background noise fluctuation may have very different significance in the two pipelines.

We conclude that we have not identified a plausible GWB signal associated with any of the 137 GRBs tested.

Gravitational-wave signals from individual GRBs are likely to be very weak in most cases due to the cosmological distances involved. Therefore, besides searching for GWB signals from each GRB, we also test for a cumulative signature associated with a sample of several GRBs (Finn et al. 1999). This approach has been used in Astone et al. (2002, 2005) to analyze resonant mass detector data using triggers from the BATSE and BeppoSAX missions, and more recently in the LIGO search for GRBs during the S2, S3, and S4 science runs (Abbott et al. 2008b).

Under the null hypothesis (no GWB signal), the local probability for each GRB is expected to be uniformly distributed on [0, 1]. Moderately strong GWBs associated with one or more of the GRBs will cause the low-p tail of the distribution to deviate from that expected under the null hypothesis. We apply the binomial test used in Abbott et al. (2008b) to search for a statistically significant deviation, applying it to the 5% × 137 ≃ 7 least probable (lowest p) on-source results in the GRB distribution. Briefly, we sort the seven smallest local probabilities in increasing order, p₁, p₂, ..., p₇. For each p_i, we compute the binomial probability P_⩾i(p_i) of getting i or more events out of 137 at least as significant as p_i. The smallest P_⩾i(p_i) is selected as the most significant deviation from the null hypothesis. To account for the trials factor from testing seven values of i, we repeat the test many times using 137 fake local probabilities drawn from uniform discrete distributions corresponding to the number of off-source segments for each GRB (18,000 for our refined p estimates). The probability associated with the actual smallest P_⩾i(p_i) is defined as the fraction of Monte Carlo trials that gave binomial probabilities as small or smaller. Note that this procedure also automatically handles the case of a single loud GWB.

In addition to the five "candidate" GRBs, extra time-shifted off-source segments were analyzed for the two GRBs with the next smallest local probabilities, GRB 060428B (0.0139) and 060930 (p = 0.0248). (By chance, for both of these GRBs the refined local probabilities from the extra off-source segments are smaller than the threshold of 0.05 for candidate signals, though the original estimates were larger.) Together with the five candidates, this gives the seven refined local probabilities 0.00967, 0.0124, 0.0139, 0.0222, 0.0248, 0.0402, and 0.0776. The associated smallest binomial probability is P_⩾5(0.0248) = 0.259. Approximately 56% of Monte Carlo trials give binomial probabilities this small or smaller, hence we conclude that there is no significant deviation of the measured local probabilities from the null hypothesis. For comparison, Figure 2 shows the distribution of local probabilities for all GRBs, as well as the values that would need to be observed to give only 1% consistency with the null hypothesis.

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Similar results are found when restricting the test to GRBs without measured redshift. In this case, the smallest binomial probability is P_⩾4(0.0248) = 0.252 with 48% of Monte Carlo trials yielding binomial probabilities this small or smaller. Analysis of the cross-correlation local probabilities also shows no significant deviation. Combining the local probabilities from the 25 ms and 100 ms analyses, we find the smallest binomial probability to be P_⩾2(0.00123) = 0.190 with 52% of Monte Carlo trials yielding binomial probabilities this small or smaller.

The antenna response of an interferometer to a gravitational wave with polarization strains h₊(t) and h_×(t) depends on the polarization basis angle ψ and the direction (θ, ϕ) to the source as

$$\begin{equation} h(t) = F_+(\theta,\phi,\psi)h_+(t) + F_{\times }(\theta,\phi,\psi)h_\times (t) \,. \end{equation} \tag{ 6 }$$

Here, F₊(θ, ϕ, ψ), F_×(θ, ϕ, ψ) are the plus and cross antenna factors introduced in Section 4.2, see Anderson et al. (2001) for explicit definitions.

A convenient measure of the gravitational-wave amplitude is the root-sum-square amplitude,

$$\begin{equation} h_{\rm rss} = \sqrt{\int (|h_+(t)|^2 + |h_\times (t)|^2) dt}. \end{equation} \tag{ 7 }$$

The energy flux (power per unit area) of the wave is (Isaacson 1968)

$$\begin{equation} F_{\rm GW} = \frac{c^3}{16\pi G} \langle (\dot{h}_{+})^2 + (\dot{h}_{\times })^2 \rangle \,, \end{equation} \tag{ 8 }$$

where the angle brackets denote an average over several periods. The total energy emitted assuming isotropic emission is then

$$\begin{equation} E_{{\rm GW}}^{{\rm iso}} = 4\pi D^2 \int dt \, F_{\rm GW} \,, \end{equation} \tag{ 9 }$$

where D is the distance to the source.

The forms of h₊(t) and h_×(t) depend on the type of simulated waveform. It is likely that many short GRBs are produced by the merger of neutron-star–neutron-star or black-hole–neutron-star binaries. The gravitational-wave signal from inspiralling binaries is fairly well understood (Blanchet 2006; Aylott et al. 2009). Progress is being made on modeling the merger phase; recent numerical studies of the merger of binary neutron-star systems and gravitational-wave emission have been performed by Shibata et al. (2005), Shibata & Taniguchi (2006), Baiotti et al. (2008), Baiotti et al. (2009), Yamamoto et al. (2008), Read et al. (2009), Kiuchi et al. (2009), and Rezzolla et al. (2010). Preliminary explorations of the impact of magnetic fields have also been made by Anderson et al. (2008), Liu et al. (2008), and Giacomazzo et al. (2009). The mergers of black-hole–neutron-star binaries have been studied by Shibata & Uryū (2006), Shibata & Uryū (2007), Shibata & Taniguchi (2008), Etienne et al. (2008), Duez et al. (2008), Yamamoto et al. (2008), Shibata et al. (2009), Etienne et al. (2009), and Duez et al. (2010). For other progenitor types, particularly for long GRBs, there are no robust models for the gravitational-wave emission (see, for example, Fryer et al. 2002; Kobayashi & Meszaros 2003; van Putten et al. 2004; Ott 2009 for possible scenarios). Since our detection algorithm is designed to be sensitive to generic GWBs, we choose simple ad hoc waveforms for tuning and testing. Specifically, we use sine-Gaussian and cosine-Gaussian waveforms:

$$\begin{eqnarray} h_+(t+t_0) &=\, h_{+,0} \sin (2 \pi f_0 t) \exp \left({\displaystyle {-(2\pi f_0 t)^2 \over 2Q^2}}\right) \,, \end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray} h_{\times }(t+t_0) &=\, h_{\times,0} \cos (2 \pi f_0 t) \exp \biggl ({\displaystyle {-(2\pi f_0 t)^2 \over 2Q^2}}\biggr) \,, \end{eqnarray} \tag{ 11 }$$

where t₀ is the central time, f₀ is the central frequency, h_+,0 and h_×,0 are the amplitude parameters of the two polarizations, and Q is a dimensionless constant which represents roughly the number of cycles with which the waveform oscillates with more than half of the peak amplitude. For Q ≳ 3, the root-sum-squared amplitude of this waveform is

$$\begin{equation} h_{\rm rss} \approx \sqrt{{\displaystyle {Q(h_{+,0}^2 + h_{\times,0}^2) \over 4 \pi ^{1/2} f_0}}} \end{equation} \tag{ 12 }$$

and the energy in gravitational waves is

$$\begin{equation} E_{{\rm GW}}^{{\rm iso}} \approx \frac{\pi ^2c^3}{G} D^2 f_0^2 h_{\rm rss}^2 \,. \end{equation} \tag{ 13 }$$

Using these waveforms for h₊(t) and h_×(t), we simulate circularly polarized GW waves by setting the sine-Gaussian and cosine-Gaussian amplitudes equal to each other, h_+,0 = h_×,0. To simulate linearly polarized waves, we set h_×,0 = 0.

The peak time of the simulated signals is distributed uniformly through the on-source interval. We use Q = 2^3/2π = 8.9, a standard choice in LIGO burst searches. The polarization angle ψ for which h₊, h_× take the forms in Equations (10) and (11) is uniform on [0, π), and the sky position used is that of the GRB (fixed in right ascension and declination). We simulate signals at discrete log-spaced amplitudes, with 500 injections of each waveform for each amplitude.

Early tests of the search algorithms used the central frequencies f₀ = (100, 150, 250, 554, 1000, 1850) Hz, and both linearly and circularly polarized injections. The final X-Pipeline tuning (performed after implementation of an improved data-whitening procedure) uses 150 Hz and 1000 Hz injections of both polarizations.

Our upper limit on gravitational-wave emission by a GRB is h^90%_rss, the amplitude at which there is a 90% or greater chance that such a signal, if present in the on-source region, would have produced an event with significance larger than the largest actually measured. There are several sources of error, both statistical and systematic, that can affect our limits. These are calibration uncertainties (amplitude and phase response of the detectors, and relative timing errors), uncertainty in the sky position of the GRB, and uncertainty in the measurement of h^90%_rss due to the finite number of injections and the use of a discrete set of amplitudes.

To estimate the effect of these errors on our upper limits, we repeat the Monte Carlo runs for a subset of the GRBs, simulating all three of these types of errors. Specifically, the amplitude, phase, and time delays for each injection in each detector are perturbed by Gaussian-distributed corrections matching the calibration uncertainties for each detector. The sky position is perturbed in a random direction by a Gaussian-distributed angle with standard deviation of 3 arcmin. Finally, the discrete amplitudes used are offset by those in the standard analysis by a half-step in amplitude. The perturbed injections are then processed, and the open-box upper limit produced using the same coherent consistency test tuning as in the actual open-box search. The typical difference between the upper limits for perturbed injections and unperturbed injections then gives an estimate of the impact of the errors on our upper limits.

For low-frequency injections (at 150 Hz), we find that the ratio of the upper limit for perturbed injections to unperturbed injections is 1.03 with a standard deviation of 0.06. We therefore increase the estimated upper limits at 100 Hz by a factor of 1.03  + 1.28 × 0.06 = 1.10 as a conservative allowance for statistical and systematic errors (the factor 1.28 comes from the 90% upper limit for a Gaussian distribution). The dominant contribution is due to the finite number of injections. For the high-frequency (1000 Hz) injections the factor is 1.10 + 1.28 × 0.12 = 1.25. In addition to finite-number statistics, the calibration uncertainties are more important at high frequencies and make a significant contribution to this factor. All limits reported in this paper include these allowance factors.

The upper limits on GWB amplitude and lower limits on the distance for each of the GRBs analyzed are given in Table 1 in the Appendix. These limits are computed for circularly polarized 150 Hz and 1000 Hz sine-Gaussian waveforms. We compute the distance limits by assuming the source emitted E^iso_GW = 0.01 M_☉c² = 1.8 × 10⁵² erg of energy isotropically in gravitational waves and use Equation (13) to infer a lower limit on D. We choose E^iso_GW = 0.01 M_☉c² because this is a reasonable value one might expect to be emitted in the LIGO–Virgo band by various progenitor models. For example, mergers of neutron-star binaries or neutron-star–black-hole binaries (the likely progenitors of most short GRBs) will have isotropic-equivalent emission on the order of (0.01–0.1) M_☉c² in the 100–200 Hz band. For long GRBs, fragmentation of the accretion disk (Davies et al. 2002; King et al. 2005; Piro & Pfahl 2007) could produce inspiral-like chirps with (0.001–0.01) M_☉c² emission. The suspended accretion model (van Putten et al. 2004) also predicts an energy emission of up to (0.01–0.1) M_☉c² in this band. For other values of E^iso_GW the distance limit scales as D ∝ (E^iso_GW)^1/2.

As can be seen from Table 1, the strongest limits are on gravitational-wave emission at 150 Hz, where the sensitivity of the detectors is best (see Figure 1). Figure 3 shows a histogram of the distance limits for the 137 GRBs tested. The typical limits at 150 Hz from the X-Pipeline analysis are (5–20) Mpc. The best upper limits are for GRBs later in S5–VSR1, when the detector noise levels tended to be lowest (and when the most detectors were operating), and for GRBs that occurred at sky positions for which the detector antenna responses F₊, F_× were best. The strongest limits obtained were for GRB 070429B: h^90%_rss = 1.75 × 10⁻²² Hz^−1/2, D^90% = 26.2 Mpc at 150 Hz. For comparison, the smallest measured redshift in our GRB sample is for 060614, which had z = 0.125 (Price et al. 2006) or D ≃ 578 Mpc (Wright 2006). (Though GRB 060218 at z = 0.0331 (Mirabal et al. 2006) occurred during S5, unfortunately, the LIGO-Hanford and Virgo detectors were not operating at the time.)

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A GRB of particular interest is 070201. This short-duration GRB had a position error box overlapping M31 (see Mazets et al. 2008, and references therein), which is at a distance of only 770 kpc. An analysis of LIGO data from this time was presented in Abbott et al. (2008a). GRB 070201 was included in the present search using the new X-Pipeline search package. Our new upper limits on the amplitude of a GWB associated with GRB 070201 are h^90%_rss = 6.38 × 10⁻²² Hz^−1/2 at 150 Hz, and h^90%_rss = 27.8 × 10⁻²² Hz^−1/2 at 1000 Hz. These are approximately a factor of 2 lower than those placed by the cross-correlation algorithm. For a source at 770 kpc, the energy limit from Equation (13) is E^iso_GW = 1.15 × 10⁻⁴ M_☉c² at 150 Hz. While about a factor of 4 lower than the GWB limit presented in Abbott et al. (2008a), this is still several orders of magnitude away from being able to test the hypothesis that this GRB's progenitor was a SGR in M31 (Mazets et al. 2008).