Outlook for detection of Gw inspirals by Grb-triggered searches in the advanced detector era
Short, hard, gamma–ray bursts are believed to originate from the coalescence of two neutron stars or a NS and a black hole (BH). If this scenario is correct, then short GRBs will be accompanied by the emission of strong gravitational waves, detectable by GW observatories such as LIGO, Virgo, KAGRA, and LIGO—India. As compared with blind, all–sky, all–time GW searches, externally triggered searches for GW counterparts to short GRBs have the advantages of both significantly reduced detection threshold due to known time and sky location and enhanced GW amplitude because of face–on orientation. Based on the distribution of signal–to–noise ratios in candidate compact binary coalescence events in the most recent joint LIGO—Virgo data, our analytic estimates, and our Monte Carlo simulations, we find an effective sensitive volume for GRB–triggered searches that is 2 times greater than for an all–sky, all–time search. For NS—NS systems, a jet angle , a gamma-ray satellite field of view of 10% of the sky, and priors with generally precessing spin, this doubles the number of NS—NS—-short GRB and NS—BH—-short GRB associations, to 3—4% of all detections of NS—NSs and of NS—BHs. We also investigate the power of tests for statistical excesses in lists of subthreshold events, and show that these are unlikely to reveal a subthreshold population until finding GW associations to short GRBs is already routine. Finally, we provide useful formulas for calculating the prior distribution of GW amplitudes from a compact binary coalescence, for a given GW detector network and given sky location.
We currently sit between the first and second generations of kilometer–scale, ground–based interferometric GW detectors. The first direct detection of GWs will very likely occur before the end of the decade.
The first detected signals will probably be from mergers of neutron star—neutron star (NS—NS), neutron star—black hole (NS—BH), and black hole—black hole (BH—BH) binaries, collectively referred to as compact binary coalescences, or CBCs. NS—NS and NS—BH coalescences are also likely progenitors for most short GRBs (Kouveliotou et al., 1993; Horvath, 2002). It is therefore natural to use detections of short GRBs to trigger searches for GW signatures of compact binary coalescences that occur at the same instant (to within a few seconds) and the same sky location (within the error bars). Such GRB–triggered searches for GWs from CBCs are already being carried out (Abbott et al., 2008; Dietz, 2008; Abadie et al., 2010a).
In this paper, we address several questions regarding GRB–triggered CBC searches. We begin by reviewing recent results from such searches in Sec. I.2 and consider what GW and GRB detectors will be available in the advanced detector era in Sec. II. In Sec. III, we discuss the two most important factors in detectability of GW counterparts of GRBs: namely, enhanced GW amplitude due to preferentially low binary inclination, and the reduced GW detection threshold resulting from knowledge of the GRB’s time. Our primary results appear in Sec. IV, in which we estimate the rate of coincident detections both analytically and via Monte Carlo simulations. A closely related data analysis activity has been the search for a statistical excess of high– signal–to–noise ratio (SNR), but subthreshold, CBC candidate events coincident with short GRBs. In Sec. V we predict the science yield from searching for such statistical excesses, and demonstrate that the extra information will typically be negligible. Finally, in Appendix A we derive a number of useful analytic formulas for describing a detector network’s sensitivity to a CBC at a given sky location (i.e., the location of a GRB).
We note that rates of short GRB—GW coincident detections were also estimated in recent papers by Chen and Holz (2012) and another by Kelley et al. (2012) which appeared when this paper was almost finished. Our method is similar to both of theirs. Our conclusions are qualitatively similar to those of (Chen and Holz, 2012), but are qualitatively different from (Kelley et al., 2012) due to different assumptions and approximations: (Kelley et al., 2012; Chen and Holz, 2012) both assume a Gaussian distribution of outliers, while we base our calculations on the distribution of outliers observed in ’s sixth and Virgo’s second and third science runs (S6/VSR2,3). (Kelley et al., 2012) also adopt a fiducial value for the GRB jet angle that is a factor smaller than ours. (Both values are plausible, given the uncertainties.) Because of our different assumptions about the statistics of the GW search and the GRB jet angle, we derive a GRB—GW detection rate that is times higher than theirs, which lifts it from to (i.e., from almost negligible to interesting). Nissanke et al. (2012) perform a similar investigation but focus on the detectability of optical counterparts to GW triggers rather than of GW counterparts of electromagnetic (EM) triggers. However, their simulations do encompass the targeted, EM–triggered GW search scenario, in their tables and figures denoted by the label Net5b. They predict approximately the same number of GRB—GW detections as we do, because although they assume a steeper reduction in SNR threshold relative to an all–sky GW search, they also assume a smaller GRB jet opening angle.
i.1 Science motivation
The detection of GWs from CBCs will have several scientific implications. The masses of the two compact objects will be determined quite accurately (Cutler and Flanagan, 1994; Finn and Chernoff, 1993). With sufficiently high SNR, the spins of these objects can also be constrained (Poisson and Will, 1995). These measurements, and the overall rates, will provide information on stellar evolution (O’Shaughnessy et al., 2008). The details of the late inspiral and postmerger gravitational waveform will also inform the high-density NS equation of state (Flanagan and Hinderer, 2008; Read et al., 2009). Details of the merger will also permit tests of general relativity in the strong field regime (Will, 2005), tests of local Lorentz invariance (Ellis et al., 2006), and constraints on the graviton mass (Stavridis and Will, 2009; Keppel and Ajith, 2010).
A coincident short GRB—GW detection would prove that at least some GRBs are indeed produced by merger events. Furthermore, it should be possible to determine the redshift of the short GRB’s host galaxy, while the GWs accurately encode the distance to the binary. It has been shown that ten short GRB with redshift measurements could constrain to within 2% assuming a GRB jet angle of (Dalal et al., 2006).
i.2 Recent results
To date, two types of searches for CBC signals associated with short GRBs have been executed: single–event targeted analyses for short GRBs associated with very nearby galaxies (GRBs 070201 and 051103; Abbott et al., 2008; Abadie et al., 2012a), and analyses covering all short GRBs during LIGO and Virgo data–taking epochs (Abadie et al., 2010a, 2012b). None of these analyses found a significant GW candidate, but the results were used to establish lower limits on the distances, assuming CBC progenitors.
GRBs 070201 and 051103 had localizations that significantly overlapped with the galaxies M31 (Andromeda, 770 kpc away; Pal’Shin, 2007; Hurley et al., 2007) and M81 (3.6 Mpc away; Hurley et al., 2010), respectively. However searches in contemporaneous GW data were able to exclude CBCs as the source of those bursts (Abbott et al., 2008; Abadie et al., 2012a).
The 22 short GRBs that occurred during ()’s fifth and Virgo’s first science runs (S5/VSR1) were followed up with CBC searches in the GW data, using analysis methods similar to those for GRBs 070201 and 051103 (Abadie et al., 2010a). The 26 short GRBs that occurred during S6/VSR2,3 were also followed up (Abadie et al., 2012b), using an improved, coherent analysis strategy (Harry and Fairhurst, 2011). No coincidences were found, and lower distance limits on putative CBC counterparts were established. In the more sensitive, S6/VSR2,3 search, the median 90% confidence lower limits for NS—NS and NS—BH binaries were and Mpc, respectively. Both analyses included a test for a subthreshold population excess. For S5/VSR1, a Mann—Whitney test (Mann and Whitney, 1947) was used, while a binomial test was used for S6/VSR2,3. In both cases, the subthreshold populations were found to be consistent with the background.
Ii Detector network roadmap
To make sensible predictions of the outcome of future GRB–triggered GW searches, one needs to know what GRB and GW detectors might be operating in the next decade.
ii.1 Gw detector network roadmap
The U.S. LIGO (Abbott et al., 2009) has recently completed a one–year data–taking period between July 2009 and October 2010, in coincidence with the French—Italian Virgo detector (Acernese et al., 2008).
LIGO is currently upgrading to its advanced detector configurations (Harry and the LIGO Scientific Collaboration, 2010), with the goal of increasing the sensitivity gradually to a factor of 10 compared to the initial configuration and extending seismic–limited sensitivity to lower frequencies. More recently, the U.S. National Science Board has authorized one LIGO detector to be moved to India in order to vastly expand the worldwide detector network’s sky localization capabilities. The earlier attempts to do the same in Australia have been formally abandoned.
Virgo is upgrading to the Advanced Virgo configuration (Acernese et al., 2008), similar to Advanced LIGO in optical layout and sensitivity. The start of the data–taking period with the advanced detectors is foreseen in 2015.
GEO600 is a British-German detector with 600 m arms and advanced optical configurations (Grote, 2010). GEO600’s next–generation configuration will be GEO–HF with a focus on high–frequency sensitivity (Willke et al., 2006). GEO–HF’s high–frequency sensitivity will be the best in the world, which will be useful in parameter estimation (Read et al., 2009). However, low–frequency sensitivity is more important for CBC detection and GEO600’s relatively short arms put it at a significant disadvantage. It will likely not be used for detection searches.
Construction has begun on KAGRA (KAmioka GRAvitational–wave observatory, formerly LCGT) in Japan (Kuroda et al., 2010), which should reach its design sensitivity in 2018. It has 3 km–long arms constructed underground and uses cryogenically cooled sapphire mirrors for test masses. The final detector is expected to detect a NS—NS system at a distance of 240 Mpc with SNR=10 (Uchiyama et al., 2004).
ii.2 Grb detector network roadmap
During the last S6/VSR2,3 science run, the triggers came mostly from the Swift and Fermi missions, with a few from the Interplanetary Network (IPN). IPN detected most of the Swift/Fermi triggers too, but with a much poorer sky localization because only triangulation methods can be used by IPN.
IPN has unfortunately lost its primary funding, but is nonetheless expected to operate during 2015—2020, if perhaps with a smaller number of satellites, still detecting GRBs at these times but with a lower rate (Hurley, 2011). Swift might operate for another 5 years or even longer, since it has no expendables and the spacecraft is in good shape (Burrows, 2011), but the operation depends on NASA funding. The Fermi instruments might also operate until . The –ray Burst Monitor (GBM) instrument on Fermi achieves instantaneous sky coverage of about 70% or 8.8 sr (30% of the sky being occulted by the Earth at the altitude of Fermi’s orbit; Wilson-Hodge et al., 2012), but GRBs detected by the GBM alone are very poorly localized.
Lobster is a proposed NASA mission similar to Swift in strategy, with a wide-field X-ray imager (WFI), narrow-field followup IR telescope (IRT), and slewing apparatus to point the latter (Gehrels et al., 2011). WFI is more sensitive than the Swift Burst Alert Telescope (BAT), but has a smaller field of view (FOV) at 0.5 sr.
The French—Chinese Space–based multi–band astronomical Variable Objects Monitor (SVOM) mission is targeted at a broader scientific target, including answering questions related to GRBs, cosmology, and fundamental physics (Paul et al., 2011). Its main instrument, ECLAIR, is a coded-aperture telescope aimed at a broad energy range of 4—250 keV, with a FOV comparable to that of Swift. The effective detection area is also close to that of Swift, resulting in an expected — GRBs per year, of which —% could be short GRBs. As the ECLAIR telescopes are more sensitive to lower energies compared to BAT, the extended emission and afterglows of GRBs can be observed deeper, resulting in improved GRB locations, and hence in a larger number (50%) of GRBs with redshift measurements (Basa, 2011). The anticipated launch date is 2015—2020.
The South Korean-led Ultra–fast Flash Observatory Pathfinder (UFFO-P) mission intends to catch the rise of GRBs (Grossan et al., 2012). It has been constructed and is anticipated to launch in June 2013. It carries a coded-aperture burst alert telescope similar to Swift’s BAT, sensitive from 15—200 keV and with a FOV of 2 sr. UFFO-P’s headlining feature is that it can repoint in response to a trigger in s using its slewing mirror telescope (SMT), which is a substantial improvement in response time over Swift’s 1 minute to slew the whole spacecraft. Though UFFO-P has a small collecting area and only a small optical telescope for followup, this pathfinder mission already has some discovery potential. The conceived UFFO-100 mission will increase collecting area, replace the SMT with a still faster MEMS micromirror array to redirect its optical path, and add an NIR camera, with the goal of gathering a statistically significant population of rising GRBs.
In conclusion, several missions are expected to operate during the advanced detector area, which are either already operating (like Swift, Fermi and IPN3), are in development (like SVOM and UFFO-P), or planned (like Lobster and UFFO-100). However, given the uncertainty in how many of these missions will ultimately fly, throughout this paper we will assume that during the advanced GW detector era the effective coverage of the combined GRB detector network will be approximately that of Swift’s BAT alone, 1.4 sr (Barthelmy et al., 2005) or about one–tenth of the sky.
Iii Detection prospects
GRBs show strong evidence for collimated, relativistic outflow along a jet. Assuming that the jet is roughly conical, its size is described by the jet angle , from the center to its outer edge. We define to be the fraction of the sky into which gamma rays are launched. For a CBC that emits a single jet, this is . If the CBC emits two jets, presumably in opposite directions, then this fraction doubles to . Collimation reduces the number of CBCs that are observable as short GRB events, since the observer only sees the GRBs for which the Earth lies within the jet.
The Lorentz factor of the beam decreases as it sweeps up external material, and at the point where reaches , the flux decay abruptly steepens due to special relativistic effects. The beaming half-angle can be determined for a given GRB by the time of this “jet break.” However the rapid decay of the late-time lightcurves of short GRBs makes the estimation of difficult. Grupe et al. (2006) places a lower limit of while Burrows et al. (2006) infers the value of to be in the range –. Goldstein et al. (2011) suggest a value in the range between and . Fong et al. (2012) find – in a recent short GRB. Simulations of NS—BH mergers indicate a range of – for binaries of moderate spin while finding – for near-extremal spinning systems (Foucart et al., 2011). In this paper we will adopt as a fiducial value when quoting results; however our analytic estimates and Tables allow the reader to trivially convert the results to other values of .
The GRB beaming angle is presumably not a universal constant, but has a distribution. While our simulations assumed that all GRBs had the same value for , our results on rates in this paper are approximately generalizable by simply replacing by its average value . We emphasize that refers to the average over all short GRBs, not just the detected ones, since the detected population depends on selection effects. (E.g., the detected population is biased towards GRBs with larger values of , at fixed flux.)
iii.2 Reduced search space
The search for a GW CBC signal triggered on an EM counterpart has sensitivity advantages over an all–sky search. Semi–analytic calculations and numerical simulations predict that the majority of the NS matter is accreted within milliseconds to seconds. This has guided GRB–triggered CBC detection efforts to search only a s ‘on–source window’ surrounding a GRB trigger to account for up to a 5–second GRB—GW delay and up to 1 s of uncertainty in the GRB time–of–arrival (TOA) (see sections 2.2 and 5.1 of Abadie et al., 2012b, for references). There are further possible reductions in the searched parameter space due to the known sky location and even by restricting to the space of CBC parameters that allow for tidal disruption outside the innermost stable circular orbit, but in this paper we will neglect them, as they have much less impact than the reduction in observation time.
In this section, we are interested in estimating the reduction in SNR threshold in going from an all-sky search to a GRB–triggered search while keeping constant the false–alarm probability (FAP) at the detection threshold. The first detection is likely to be held to a high standard of , but once detections are routine, the threshold should be lessened considerably. Throughout this paper, we assume that is required.
For 20 short GRBs per year of livetime (), the observation time for GWs is . Assuming that the searches are comparably effective at background rejection, the false–alarm rate (FAR) at a given SNR should be the same, but the FAR at a given SNR, , is reduced by a factor
We estimated the SNR threshold for a GRB–triggered search using the background SNR distribution from the S6/VSR2,3 all–sky search (Abadie et al., 2012c). The all–sky search was a coincident search, in which matched filtering and thresholding were performed on individual detectors and spurious triggers were vetoed by demanding consistent TOAs in multiple detectors. For a targeted, GRB–triggered search, a coherent search is possible in which thresholding is done on the suitably time–delayed and summed SNR for the whole network (Harry and Fairhurst, 2011). False–positive rejection is aided by tests on null streams, which are special linear combinations of the detectors that are insensitive to GWs. Coherent searches are less feasible for all–sky searches because each sky location requires unique time delays. Since in Gaussian noise a coincident search with a two–detector network has the same statistics as a coherent search with any network of (more than one) detectors (Harry and Fairhurst, 2011), we can extrapolate the threshold for a targeted, coherent search from the statistics of an all–sky, coincident search.
For the S6/VSR2,3 all–sky search, Fig. 3 of (Abadie et al., 2012c, data available online at https://dcc.ligo.org/LIGO-P1100034-v19/public) gives the FAR as a function of , a –weighted quadrature sum of the SNR in all of the detectors that has been found to be a useful detection statistic. We assume that this is equivalent to the network coherent SNR , as argued above. From this Figure, we find that during S6/VSR2,3, when . When , . From this, we take as the SNR threshold for an all–sky search and as the SNR threshold for a GRB–triggered search. A triggered search can see times farther than the all–sky search. On the other hand, as we will see, the increased range is somewhat offset by jet collimation and the limited FOV of high–energy satellites.
iv.1 Analytic estimates
Here we provide some simple estimates of the CBC detection rate we expect for triggered searches, compared to the CBC detection rate for untriggered, all–sky searches. The method is the same as in Kochanek and Piran (1993), but our inputs and conclusions are different. We will assume that short GRBs come only from CBCs, and denote by the fraction of CBCs that produce short GRBs. For the CBCs that produce bursts, let be the average solid angle into which gamma rays are launched. Again, the average is over the whole GRB population, not just population of detected GRBs (which is biased towards GRBs with wider beams at fixed flux). For any CBC that produces two beams (presumably in opposite directions), we consider for that burst to be the sum of the solid angles for the two beams. Finally, we define to be the average fraction of the sky that is “covered” by the then-existing GRB detector network. For the advanced GW detectors, most CBCs will be detected at distances Mpc, while most detected short GRBs are much farther away (Gpc), so the GRB accompanying an observed CBC should be detectable as long as the Earth lies within the beam, and the source is within the telescope’s FOV. Then the fraction of CBCs for which we detect a GRB is
We shall adopt as a reasonable fiducial number. This would correspond, e.g., to , and all CBCs emitting two jets of each.
As explained above, by looking for NS binaries that are roughly coincident in time with observed short GRBs, we increase the sensitivity of the search at fixed FAR. Let be the ratio of the GW detection thresholds for blind and GRB–triggered searches, respectively. As explained in Sec. III.2, we adopt as a fiducial value .
The final factor that we need arises because the mergers that we see in gamma rays are presumably the ones for which we are viewing the binary nearly face on (because the gamma rays are presumably beamed perpendicular to the orbital plane.) The quadrupolar pattern of the emitted GWs is also strongest along the direction perpendicular to the orbital plane; the amplitude of on the axis is times stronger than the isotropic detection-averaged value. (It is well know that that rms enhancement is , but what is relevant for determining rates is the cube root of mean-cubed enhancement, which is .) Although is the enhancement factor of an optimally oriented binary, we show in Appendix A that as long as the beam half-angle is , then using is at most a overestimate, which is acceptable for our purposes.
For Advanced LIGO, we will be detecting CBCs in the range Mpc to Mpc. At these distances, we can safely approximate spatial geometry as Euclidean and the density of mergers as uniform. Let be the CBC detection rate of blind, all–sky advanced GW detector searches, and let be the rate for triggered searches. What is the ratio ? The volume in which GRB–triggered CBCs are detectable is larger than for the blind case by the factor , but recall that only a fraction of the CBCs emit detectable gamma rays. Thus,
In other words, of the first detections, we would expect only one to come from the GRB–triggered pipeline; it is correspondingly unlikely that the very first GW detection of a CBC will result from a GRB trigger. This is the same conclusion reached in Chen and Holz (2012) and Kelley et al. (2012).
Of the CBC detections associated with short GRBs, a fraction will be strong enough to be detectable even without the GRB trigger. So the increase in the rate of GW detections, thanks to GRB triggers, will be . This is a small increase in CBC detections, but, given the extra information to be gleaned from the EM counterpart, a non-negligible one. Note that this increase is about times higher than the fiducial increase reported in Kelley et al. (2012). The difference comes mostly from two factors: i) a factor — from our use of non-Gaussian statistics, and ii) a factor — from our wider fiducial beam angle of vs. their , combined with our assumption of two opposite–pointing jets per CBC.
iv.2 Effective sensitive volumes
For a given detector network and mass pair, we would like to compute the relative detection capability of a targeted short GRB search compared to an all-sky search. Following Finn and Chernoff (1993), but generalizing from a single detector to a network, we define an effective sensitive volume , which can be multiplied by a constant rate density to obtain the total detection rate :
where is the Heaviside step function, are the source’s right ascension and declination, and is the distance at which the network registers SNR above its threshold for a given source type. depends on the full specification of a waveform plus the detector noise spectrum and the various projection angles. Eq. 3 is quite convenient for Monte Carlo integration. See Appendix A for network SNR expressions and analytical evaluation of Eq. 3 in the nonspinning case.
To take into account the imperfect duty cycle of the detectors, along with our requirement that at least two detectors be “on” to claim a detection, it is useful to define a double detection volume (2DV), in close analogy to the triple detection rate (3DR) of Schutz (2011). (Our formulation generalizes the treatment in Schutz (2011) since it applies to spinning binaries and removes the assumption of equal detector responses.) For a set of detectors ,
where is the set of subsets of with size 2 or greater and denotes the size of set . is the duty cycle, which we take to be the same for all detectors.
iv.3 Results from Simulations
We evaluate the 2DV (equation 4) via Monte Carlo integration, randomly drawing the sky location and the polarization angles, for both NS—NS (1.4–1.4 ) and NS—BH (1.4–10 ) pairs. We use the Taylor T4 waveform (Boyle et al., 2008), including general spin precession, to determine expected signal amplitudes and thus . We match the distributions of spin amplitude and spin orientation used in the recent GRB 051103 analysis (Abadie et al., 2012a) with NS dimensionless spin drawn uniformly from , BH spin drawn uniformly from , and uniform spin orientations subject to a cut on the tilt angle (the angle between the BH spin axis and NS orbital axis) to be 60. We also adopt their rigid rotation procedure so that it is the total angular momentum that is uniformly distributed within the cone of rather than the orbital angular momentum. Using different values for , we compute the relative improvement in effective sensitive volume 2DV of a triggered search (restricted inclination angle, coherent threshold ) compared to an all–sky search (unrestricted inclination, coincident threshold ).
The 2DVs have been defined such that the detection rate for the all–sky search is , where is the merger rate density in the local Universe and the 2DV assuming a coherent SNR threshold of 11.3 and a source population whose inclination is unrestricted (). Similarly, the number of GRBs associated with a GW detection is given by . The rate for GRBs independently detected by both searches (i.e., with GW network SNR ) is . Our computed sensitive volumes are shown in Table 1. Each value of represents simulations, while the values are simply rescaled from .
Under the assumption that all CBCs produce short GRBs and vice versa, we translate sensitive volumes to detection rates in Table 2 and Fig. 1. We take values of and “realistic” coalescence rates of for NS—NS sources and for NS—BH, as used in Abadie et al. (2010b). The coalescence rates are uncertain by two orders of magnitude, but our relative volumes have much smaller errors, so we show two or more significant figures not to reflect uncertainty but rather to allow for relative comparisons. We have assumed an effective FOV of , independent duty cycle factors of , and a network of LIGO—Hanford, LIGO—Livingston, Virgo, KAGRA (final orientation via Takashi, 2012), and LIGO—India (HLVKI; orientation guess via Schutz, 2011). For all GW detectors, we used publicly available projections of noise spectra at full design sensitivities (LIGO, 2010; Virgo, 2010; KAGRA, 2012). However, we started SNR integration from a lower frequency limit of 40 Hz in order to fit waveforms within the memory limits of the bulk of computers available to us, enabling much greater parallelism, at some cost in realism. These assumptions lead a total annual detection rate of 61 NS—NS signals and 18 NS—BH signals for the all-sky search.
Assuming an effective FOV of for the GRB detectors, and that every CBC produces two opposite jets, each with a fiducial jet angle of , we find that 1.8% of CBCs detected by the all–sky search would be coincident with observed short GRBs. Adding the triggered search increases the number of coincidences to %. Clearly, our Monte Carlo results are in very close agreement with the analytic estimates in Sec. IV.1.
V Sub-threshold statistical excess tests
In a calendar year, there will be some number of searches for GWs associated with short GRBs. Some of these may result in individual detections, meaning that there will be events that survive all vetoes and have SNR above some theshold for confident single-source detection. However one can also consider some lower threshold , and look for a statistical excess of events with SNR between and . What science can be gleaned from those excess events?
We begin by providing a Bayesian perspective on this question. First, setting any threshold and discarding events below that threshold amounts to “throwing away” data, and in the ideal case of arbitrary computing power and a perfectly known distribution of the detection statistic, discarding data weakens the analysis. So in the ideal case, would be set very low. However, in practice, finite resources, an imperfectly known distribution of outliers, and diminishing infomation returns (for lower ) all will push data analysts to a value of not too far below . Probably the most important information encoded in the excess, sub- events is an improved estimate of the event rate (e.g., in units of ). As emphasized in Messenger and Veitch (2012), a proper Bayesian analysis takes into account the value of , and so always gives an unbiased estimate of . Including more events by lowering just “shrinks the error bars.”
Next we provide a crude, back-of-the-envelope argument why, in the regime of rare detection, there will generally not be much information in the sub-threshold excess. In the regime of rare detection, the unique loudest event will have SNR . There are 8 times as many events twice as far away, so we would expect an order of true events with . However the number of background events with SNR between and will be of order (the FAP at ) times , or (assuming a Gaussian distribution of noise events). The standard deviation in this expected number of events is , which swamps the true excess. Of course, this calculation was for one (nonoptimized) value of , but we believe that the general conclusion is robust: as is decreased below , the background rate increases far faster than the rate of true events, so we expect relatively little extra information from the sub–threshold events.
In the next subsection we consider two non-Bayesian statistical excess tests currently used in LIGO searches, and show through simulations that, indeed, the sub- excess typically contains rather little useful information. We believe that a proper Bayesian analysis the sub- events would lead to the same qualitative conclusion, but we leave that calculation for future work.
v.1 Results from simulations
In past LIGO searches, two statistical excess tests were employed: the Mann—Whitney—Wilcoxon test and later the binomial test (Abadie et al., 2010a, 2012b). However we are not aware of any systematic comparison of their detection power, so we compare them here.
We ran a Monte Carlo simulation in which each trial drew 20 GRB on-source candidates and, for each one, off-source trial loudest candidates. Off-source candidates were drawn from the S6/VSR2,3 distribution (see Sec. III.2). Each on-source candidate SNR was the maximum of an off-source trial and a draw from a foreground distribution whose rate scaling could be varied. For each trial, we performed individual direct detection searches on all 20 GRBs before performing a statistical excess search on the set of remaining nondetections. Both direct detection of individual GRBs and statistical excess detection had thresholds set by . Fig. 2 shows the fraction of trials in which the and binomial tests were able to detect an excess versus the average number of direct detections among the simulated GRBs. While the test used all on-source and all off-source candidates, the binomial considered only the loudest four on-source candidates.
We see our expectations confirmed: for both tests, the probability of detecting an excess in the sub-threshold populaton is never larger than several percent. Of the two tests, the binomial one is the more powerful. We see both tests become more powerful as the number of direct detections rises, until it reaches . The decline in efficiency for detections is an artifact of the “rules” of our simulation in that there are always true events, so as more of them exceed , fewer true events are left among the sub-threshold candidates.
Vi Summary and discussion
Based on LIGO—Virgo S6/VSR2,3 data and search parameters for CBC—short GRB searches, we have quantified how much deeper into the noise one can dig with knowledge of the external trigger time; we find a coherent SNR threshold of versus an all–sky coincident threshold of for a detection FAP of . (Chen and Holz, 2012; Kelley et al., 2012) have also estimated the reduction in the SNR threshold assuming Gaussian noise and making different choices in how to fold in EM information. The bleakness of (Kelley et al., 2012) compared to our study is easily understood: if the measured distribution of SNRs falls off less steeply than a Gaussian, then reducing the search volume has a relatively larger effect on the detection threshold (at fixed FAP), and hence on the detection rate. While the advanced detectors will not have the same distribution of high-SNR candidates SNRs as LIGO and Virgo did in S6/VSR2,3, the true distribution seems unlikely to be Gaussian, and the difference between our results and (Chen and Holz, 2012; Kelley et al., 2012) provides some measure of the importance of non–Gaussian backgrounds in assessing the value of triggered searches.
Folding the threshold reduction into a large Monte Carlo simulation, including the effects of short GRB collimation, general spin precession, and advanced GW and GRB detector networks, we have estimated the rate of CBC—short GRB coincident detections. Assuming that all NS—NS systems produce short GRBs with a jet angle of , we find that, relative to just an all–sky search, adding a search triggered by a Swift–like satellite increases the total number of CBC detections by 2%, but more importantly doubles the number of GW—GRB associations. A mission such as Fermi that has times the instantaneous sky coverage of Swift would contribute not quite an increase of % to the total number of CBC detections, because the relatively poor sky localization would permit less of a reduction in the GW detection threshold. Although the calculated enhancement of detection rate is dependent on this and other assumptions, we believe it justifies the effort that is being spent on such triggered searches, given the extra scientific value of multimessenger detections.
The externally triggered GRB searches to date have attempted to detect a population of sub–threshold GWs. We performed simulations that show that typically there will not be a detectable excess until the rate of direct detections of individual sources is already high; hence it is highly unlikely that an excess–population test will provide the first strong evidence for CBCs.
Acknowledgements.The authors thank Alan Weinstein and Michal Was for comments on the manuscript, and Neil Gehrels for updating us on GRB missions. LIGO was constructed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation (NSF) and operates under cooperative agreement No. PHY-0107417. L.S. is supported by the NSF through a Graduate Research Fellowship, while A.D. is supported by NSF Grants No. PHY-1067985 and No. PHY-0757937. C.C.’s work was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract to the National Aeronautics and Space Administration. C.C. also gratefully acknowledges support from NSF grant PHY1068881. This paper has LIGO Document No. LIGO-P1200113-v6.
Appendix A Prior distribution of Gw detector network’s response to a Cbc at a given distance and sky location
In this section we discuss the distribution of a network’s response to GWs from a source at a particular, known sky location and distance, whose orientation is unknown but whose inclination is restricted to be less than a maximum value . When studying an individual GRB, we could treat this as a prior distribution for the strength of the signal received by a particular detector network. In the event of a nondetection, it would allow us to parameterize the excluded distance by the jet opening angle.
Let the frequency domain GW strain received by detector be and the noise power spectral density of detector be . Defining the inner product
and then the sum over all of the detectors,
the coherent detection statistic defined by Harry and Fairhurst (2011) and the incoherent, coincident detection statistic are noncentrally chi-squared distributed with the noncentrality parameter given by . (The coherent detection statistic has 4 degrees of freedom, whereas the coincident statistic has 2 degrees of freedom times the number of detectors.) We will first derive summary statistics of : its minimum, maximum, mode, mean, and root–mean–square (rms). Then, we will derive the full distribution of and study its qualitative features.
Harry and Fairhurst (2011) introduce , the luminosity distance of the source; , an arbitrary fiducial distance; the three angles describing the orientation of the source, being respectively the inclination of the orbital plane to the line of sight, the polarization angle, and the orbital phase at coalescence; and the antenna factors of each detector, and , which are functions of sky location. They also define two waveform quadratures, and , which are nearly orthogonal such that
They define a further three quantities that combine the antenna factors of a network of detectors,
It is easily—though laboriously—shown that the detector response or noncentrality parameter depends only on the antenna factors, distance, , and , through
where and . For the purpose of concise parameterization of , we will also introduce , , and . describes the total sensitivity of the detector network as the weighted sum of squares of all of the antenna factors. To lend interpretation to , we write Eq. (5) as
Then, can be shown to equal the ratio of the difference of the eigenvalues of to their sum. measures the extent to which the network is preferentially sensitive to just one polarization, in a way that is independent with respect to rotations of the detector network’s coordinate system. If the network is equally sensitive to two orthogonal polarizations, then vanishes. If the network is sensitive to only one polarization, then is unity.
Lastly, we define as the distance–independent part of :
Our goal is to study the conditional probability density function (pdf), , of , assuming a fixed sky location and a censored orientation distribution. Under the assumption that the GRB emission is collimated within an angle , the burst can only be seen if the Earth is placed inside this cone. As the probability is the same for random placement anywhere on the surface of this cone, the prior distribution of the inclination angle between the line of sight to Earth and the axis of the outflow is given by
since an area element on the cone is given by . This represents a prior distribution on the direction of the system’s orbital axis that is uniform in solid angle, restricted to polar angles .
Having parameterized the detector network’s response, we proceed to derive the distribution of for a given sky location and luminosity distance, as well as the minimum, maximum, mean, rms, and mode of the distribution. Finally, as an example, we will apply our results to GRB 051103.
a.1 Distribution and summary statistics
The distribution of the detector network’s response and its summary statistics, derived below, are plotted in Fig. 3 for a selection of detector networks and maximum inclination angles.
a.1.1 Minimum and maximum
The minimum value of is obtained when the source is at the maximum inclination ( or ) and when :
The maximum value is obtained when the source is at the minimum inclination ( or ):
The mean response is given by
It is possible to cast the integral over into the form of a complete elliptic integral of the second kind,
The rms response is
The integral over kills the term, leaving
The integral over gives
a.1.4 Full probability distribution
We have the prior distribution of ,
To compute the conditional pdf of , , the first step is to effect a change of variables from to . To do this, we write where and . By forming the Jacobian determinant , we find
Using the inverse rule for derivatives, , we solve for as a function of ,
and then differentiate with respect to ,
Now we may express the conditional pdf that we seek as
The factor of accounts for the two distinct values of that give the same value of . Altogether,
We have to be a little careful about the limits of integration; the quantity in the radical must remain positive. It has a zero at
The lower limit should be
The upper limit should be
Eq. (12) may be evaluated numerically using, for example, Simpson’s rule.
The mode of the distribution occurs when the lower limit of integration ceases to clip against the minimum value, where
This occurs at
a.2 Special case: Unrestricted inclination
If the inclination is unrestricted, or , then the above results simplify to
Neither nor the full pdf simplify much for the unrestricted inclination case.
a.3 Special case: One detector
When the detector network consists of only one detector, , , and , where . With these substitutions,
a.4 Case study: Grb 051103
GRB 051103 was an exceptionally short, hard, and bright burst detected by HETE, Suzaku, and Swift, and localized by IPN to an area consistent with the outer disc of M81 (Hurley et al., 2010). Owing to its brightness and hardness, a giant flare from an extragalactic soft repeater (SGR) was a plausible progenitor. The Hanford 2–km detector (H2) and Livingston 4–km detector (L1) were operating at the time, so a targeted search of the GW data was undertaken. No candidate was detected, but the nondetection excluded a CBC event in M81 as the progenitor (Abadie et al., 2012a). Under the assumption that a CBC progenitor would have produced a collimated jet along the axis of strongest gravitational wave emission, Abadie et al. (2012a) placed 90% confidence lower limits on the distance of a CBC progenitor as function of jet angle. A collimated GRB in M81 was firmly excluded.
As an example, we apply our distribution of detector response to the problem of estimating the exclusion distance as a function of jet opening angle for GRB 051103. If we knew the GW search’s detection efficiency for strictly face–on sources, then using our distribution for we could directly calculate the excluded distance for any jet opening angle and any confidence level. Abadie et al. (2012a) did not publish that detection efficiency, but we can do a qualitatively similar calculation by extrapolating from their 90% exclusion distance for , attempting to reproduce their exclusion distance at other jet opening angles.
GRB 051103 occurred at 3 November 2011 09:25:42 UTC. The H2 and L1 horizon distances (distance at which an optimally oriented face-on CBC would register an amplitude ) at this time for both a 1.4—1.4 NS—NS event and a 1.4—10 NS—BH event are given in Table 3, along with the antenna factors at this time in the direction of M81. For a NS—NS signal, this network has , and for a NS—BH signal, . H2 and L1 had almost the same sensitivity up to a frequency-independent factor of 2, so it is not surprising that the value of is almost the same for both the NS—NS and NS—BH signal models.
In Fig. 4, we plot the 90% exclusion distance as a function of from Fig. 3 of Abadie et al. (2012a). We have superimposed the mode of the detector response distribution, Eq. (15), scaled to match the published exclusion distance at , as a dashed line. The value of at which the cumulative distribution function (CDF) is equal to is shown as a solid line. The inverse CDF agrees well with the NS—NS exclusion distance, but the mode agrees much better with the NS—BH exclusion distance than the inverse CDF. Exact agreement is not expected with either: as we have pointed out, a proper application to calculating exclusion distances would require knowledge of both the prior distribution of and the sensitivity of the GW search to face-on sources as a function of signal amplitude. Furthermore, the analysis of Abadie et al. (2012a) includes a Monte Carlo integration over a range of masses, whereas our analysis fixes canonical choices of the masses.
Appendix B Enhanced Gw amplitude of Grb-triggered sources
In this paper, we assume that the emitted gamma rays are beamed within an angle of the normal to the orbital plane and thus the binary inclination must be less than . Where is small, we approximate the GW amplitude for GRB-triggered sources to be on average times the isotropic detection-averaged amplitude for all CBCs, which is the instantaneous value.
Using intermediate results from Appendix A, the azimuthally averaged detector response to a binary whose orbital plane inclined at angle relative to the observer’s line of sight is proportional to
where and . The distance to which a GW source is detectable scales as , so the number of detectable sources scales as . Thus the detection-averaged amplitude of all the sources that we observe within half-angle is
where and .
Using Eq. 17, one easily shows that as long as the beam half-angle is , then average amplitude enhancement relative to an unrestricted distribution is at most a overestimate.
- Kouveliotou et al. (1993) C. Kouveliotou, C. A. Meegan, G. J. Fishman, N. P. Bhat, M. S. Briggs, T. M. Koshut, W. S. Paciesas, and G. N. Pendleton, Astrophys. J. 413, L101 (1993).
- Horvath (2002) I. Horvath, Astron. & Astrophys. 392, 791 (2002), eprint astro-ph/0205004.
- Abbott et al. (2008) B. Abbott et al. (LIGO Scientific Collaboration), ApJ 681, 1419 (2008), eprint arXiv:0711.1163.
- Dietz (2008) A. Dietz (LIGO Scientific Collaboration), in Proceedings of Gamma Ray Bursts 2007, edited by M. Galasi, D. Palmer, and E. Fenimore (Melville, New York, 2008), pp. 284–288, eprint arXiv:0802.0393v1.
- Abadie et al. (2010a) J. Abadie, B. P. Abbott, R. Abbott, T. Accadia, F. Acernese, R. Adhikari, P. Ajith, B. Allen, G. Allen, E. Amador Ceron, et al., Astrophys. J. 715, 1453 (2010a), eprint 1001.0165.
- Chen and Holz (2012) H.-Y. Chen and D. E. Holz, ArXiv e-prints (2012), eprint 1206.0703.
- Kelley et al. (2012) L.-Z. Kelley, I. Mandel, and E. Ramirez-Ruiz, arXiv:1209.3027 (2012).
- Nissanke et al. (2012) S. Nissanke, M. Kasliwal, and A. Georgieva, ApJ, in press, arXiv:1210.6362 (2012).
- Cutler and Flanagan (1994) C. Cutler and E. Flanagan, Phys. Rev. D 49, 2658 (1994).
- Finn and Chernoff (1993) L. S. Finn and D. F. Chernoff, Phys. Rev. D 47, 2198 (1993), eprint arXiv:gr-qc/9301003.
- Poisson and Will (1995) E. Poisson and C. M. Will, Phys. Rev. D 52, 848 (1995), eprint arXiv:gr-qc/9502040.
- O’Shaughnessy et al. (2008) R. O’Shaughnessy, C. Kim, V. Kalogera, and K. Belczynski, Astrophys. J. 672, 479 (2008), URL http://stacks.iop.org/0004-637X/672/i=1/a=479.
- Flanagan and Hinderer (2008) Éanna É.. Flanagan and T. Hinderer, Phys. Rev. D 77, 021502 (pages 5) (2008), URL http://link.aps.org/abstract/PRD/v77/e021502.
- Read et al. (2009) J. S. Read, C. Markakis, M. Shibata, K. Uryū, J. D. E. Creighton, and J. L. Friedman, Phys. Rev. D 79, 124033 (2009), eprint arXiv:0901.3258.
- Will (2005) C. M. Will, Living Rev. Rel. 9 (2005), eprint gr-qc/0510072.
- Ellis et al. (2006) J. Ellis, N. Mavromatos, D. Nanopoulos, A. Sakharov, and E. Sarkisyan, Astroparticle Phys. 25, 402 (2006), ISSN 0927-6505, URL http://www.sciencedirect.com/science/article/B6TJ1-4JXY1RR-1/2/84c3810841cfe896d01a49e5b006ce61.
- Stavridis and Will (2009) A. Stavridis and C. M. Will, Phys. Rev. D 80, 044002 (2009).
- Keppel and Ajith (2010) D. Keppel and P. Ajith, Phys.Rev. D82, 122001 (2010), eprint 1004.0284.
- Dalal et al. (2006) N. Dalal, D. E. Holz, S. A. Hughes, and B. Jain, Phys. Rev. D 74, 063006 (2006), eprint 0601275.
- Abadie et al. (2012a) J. Abadie, B. P. Abbott, T. D. Abbott, R. Abbott, M. Abernathy, C. Adams, R. Adhikari, C. Affeldt, P. Ajith, B. Allen, et al., Astrophys. J. 755, 2 (2012a), eprint 1201.4413.
- Abadie et al. (2012b) J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, T. Accadia, F. Acernese, C. Adams, R. Adhikari, C. Affeldt, et al., Astrophys. J. 760, 12 (2012b).
- Pal’Shin (2007) V. Pal’Shin, GCN Circ. 6098, 1 (2007).
- Hurley et al. (2007) K. Hurley et al., GCN Circ. 6103, 1 (2007).
- Hurley et al. (2010) K. Hurley, A. Rowlinson, E. Bellm, D. Perley, I. G. Mitrofanov, D. V. Golovin, A. S. Kozyrev, M. L. Litvak, A. B. Sanin, W. Boynton, et al., Mon. Not. Royal Astron. Soc. 403, 342 (2010), ISSN 1365-2966, URL http://dx.doi.org/10.1111/j.1365-2966.2009.16118.x.
- Harry and Fairhurst (2011) I. W. Harry and S. Fairhurst, Phys. Rev. D 83, 084002 (2011), eprint 1012.4939.
- Mann and Whitney (1947) H. B. Mann and D. R. Whitney, Ann. Math. Statist. 18, 50 (1947), URL http://projecteuclid.org/euclid.aoms/1177730491.
- Abbott et al. (2009) B. Abbott et al. (LIGO Scientific Collaboration), Rep. Prog. Phys. 72, 076901 (2009).
- Acernese et al. (2008) F. Acernese et al., Class. Quant. Grav. 25, 184001 (2008), URL http://stacks.iop.org/0264-9381/25/i=18/a=184001.
- Harry and the LIGO Scientific Collaboration (2010) G. M. Harry and the LIGO Scientific Collaboration, Class. Quant. Grav. 27, 084006 (2010), URL http://stacks.iop.org/0264-9381/27/i=8/a=084006.
- Grote (2010) H. Grote (LIGO Scientific Collaboration), Class. Quant. Grav. 27, 084003 (2010).
- Willke et al. (2006) B. Willke, P. Ajith, B. Allen, P. Aufmuth, C. Aulbert, S. Babak, R. Balasubramanian, B. W. Barr, S. Berukoff, A. Bunkowski, et al., Class. Quant. Grav 23, 207 (2006).
- Kuroda et al. (2010) K. Kuroda et al., Classical and Quantum Gravity 27, 084004 (2010), URL http://stacks.iop.org/0264-9381/27/i=8/a=084004.
- Uchiyama et al. (2004) T. Uchiyama et al., Class. Quant. Grav. 21, S1161–S1172 (2004).
- Hurley (2011) K. Hurley, private communication (2011).
- Burrows (2011) D. N. Burrows, private communication (2011).
- Wilson-Hodge et al. (2012) C. A. Wilson-Hodge, G. L. Case, M. L. Cherry, J. Rodi, A. Camero-Arranz, P. Jenke, V. Chaplin, E. Beklen, M. Finger, N. Bhat, et al., Astrophys. J. Supp. 201, 33 (2012), URL http://stacks.iop.org/0067-0049/201/i=2/a=33.
- Gehrels et al. (2011) N. Gehrels, S. D. Barthelmy, and J. K. Cannizzo, Proceedings of the International Astronomical Union 7, 41 (2011).
- Paul et al. (2011) J. Paul, J. Wei, S. Basa, and S.-N. Zhang, Comptes Rendus Physique 12, 298 (2011), eprint 1104.0606.
- Basa (2011) S. Basa, private communication (2011).
- Grossan et al. (2012) B. Grossan, I. H. Park, S. Ahmad, K. B. Ahn, P. Barrillon, S. Brandt, C. Budtz-Jørgensen, A. J. Castro-Tirado, P. Chen, H. S. Choi, et al., ArXiv e-prints (2012), eprint 1207.5759.
- Barthelmy et al. (2005) S. D. Barthelmy et al., Space Science Reviews 120, 143 (2005), eprint astro-ph/0507410.
- Grupe et al. (2006) D. Grupe et al., Astrophys. J. 653, 462 (2006), eprint astro-ph/0603773.
- Burrows et al. (2006) D. N. Burrows et al., Astrophys. J. 653, 468 (2006), eprint astro-ph/0604320.
- Goldstein et al. (2011) A. Goldstein et al. (2011), arXiv:1101.2458 [astro-ph.HE], eprint 1101.2458.
- Fong et al. (2012) W.-f. Fong, E. Berger, R. Margutti, B. A. Zauderer, E. Troja, I. Czekala, R. Chornock, N. Gehrels, T. Sakamoto, D. B. Fox, et al., ArXiv e-prints (2012), eprint 1204.5475.
- Foucart et al. (2011) F. Foucart, M. D. Duez, L. E. Kidder, and S. A. Teukolsky, Phys. Rev. D 83, 024005 (2011), eprint 1007.4203.
- Abadie et al. (2012c) J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, T. Accadia, F. Acernese, C. Adams, R. Adhikari, C. Affeldt, et al., Phys. Rev. D 85, 082002 (2012c), eprint 1111.7314.
- Kochanek and Piran (1993) C. S. Kochanek and T. Piran, Astrophys. J. Lett. 417, L17 (1993), eprint arXiv:astro-ph/9305015.
- Schutz (2011) B. F. Schutz, Class. Quant. Grav. 28, 125023 (2011), URL http://stacks.iop.org/0264-9381/28/i=12/a=125023.
- Boyle et al. (2008) M. Boyle et al., Phys. Rev. D 78, 104020 (2008), eprint 0804.4184.
- Abadie et al. (2010b) J. Abadie, B. P. Abbott, R. Abbott, M. Abernathy, T. Accadia, F. Acernese, C. Adams, R. Adhikari, P. Ajith, B. Allen, et al., Class. Quant. Grav. 27, 173001 (2010b), eprint 1003.2480.
- Takashi (2012) U. Takashi, private communication (2012).
- LIGO (2010) LIGO, Advanced ligo anticipated sensitivity curves (2010), URL https://dcc.ligo.org/T0900288-v3/public.
- Virgo (2010) Virgo, Advanced virgo sensitivity (2010), URL https://wwwcascina.virgo.infn.it/advirgo/.
- KAGRA (2012) KAGRA, Kagra parameters (2012), URL http://gwcenter.icrr.u-tokyo.ac.jp/en/researcher/parameter.
- Messenger and Veitch (2012) C. Messenger and J. Veitch, ArXiv e-prints (2012), eprint 1206.3461.