Prospects for joint GW and GRB observations

Prospects for joint gravitational wave and short gamma-ray burst observations

J. Clark11affiliation: School of Physics and Astronomy, Cardiff University, Cardiff, UK 22affiliation: University of Massachusetts Amherst, Amherst, MA 01003, USA , H. Evans11affiliation: School of Physics and Astronomy, Cardiff University, Cardiff, UK , S. Fairhurst11affiliation: School of Physics and Astronomy, Cardiff University, Cardiff, UK , I. W. Harry11affiliation: School of Physics and Astronomy, Cardiff University, Cardiff, UK 33affiliation: Department of Physics, Syracuse University, Syracuse NY , E. Macdonald11affiliation: School of Physics and Astronomy, Cardiff University, Cardiff, UK , D. Macleod11affiliation: School of Physics and Astronomy, Cardiff University, Cardiff, UK 44affiliation: Louisiana State University, Baton Rouge, LA 70803, USA , P. J. Sutton11affiliation: School of Physics and Astronomy, Cardiff University, Cardiff, UK , A. R. Williamson11affiliation: School of Physics and Astronomy, Cardiff University, Cardiff, UK
Abstract

We present a detailed evaluation of the expected rate of joint gravitational-wave and short gamma-ray burst (GRB) observations over the coming years. We begin by evaluating the improvement in distance sensitivity of the gravitational wave search that arises from using the GRB observation to restrict the time and sky location of the source. We argue that this gives a 25% increase in sensitivity when compared to an all-sky, all-time search, corresponding to more than doubling the number of detectable gravitational wave signals associated with GRBs. Using this, we present the expected rate of joint observations with the advanced LIGO and Virgo instruments, taking into account the expected evolution of the gravitational wave detector network. We show that in the early advanced gravitational wave detector observing runs, from 2015-2017, there is only a small chance of a joint observation. However, as the detectors approach their design sensitivities, there is a good chance of joint observations provided wide field GRB satellites, such as Fermi and the Interplanetary Network, continue operation. The rate will also depend critically upon the nature of the progenitor, with neutron star–black hole systems observable to greater distances than double neutron star systems. The relative rate of binary mergers and GRBs will depend upon the jet opening angle of GRBs. Consequently, joint observations, as well as accurate measurement of both the GRB rate and binary merger rates, will allow for an improved estimation of the opening angle of GRBs.

1. Introduction

Two distinct classes of gamma ray bursts were first proposed by Kouveliotou et al. (1993), who argued for distinct populations of bursts with different durations and spectral hardness. The separation between short and long bursts is typically taken to be a duration of two seconds. Long GRBs have subsequently been associated with the death of massive stars, based on the localization of afterglows to the star forming regions of galaxies (Bloom et al., 1998) and the association of long GRBs with supernovae (Hjorth et al., 2003). Mergers of binary neutron star (BNS) or neutron star–black hole (NSBH) systems are a strong candidate for the progenitor of short GRBs (Eichler et al., 1989). There is a large amount of evidence in support of the binary merger progenitor hypothesis (Berger, 2014), although nothing definitive.

Binary mergers are also strong emitters of gravitational waves (GW), at frequencies which the LIGO and Virgo detectors have good sensitivity (Thorne, 1987). Consequently, it makes sense to search for gravitational-wave signals originating at a time and sky position consistent with the observed GRB signal. This significantly reduces the size of the gravitational wave parameter space by restricting the time, sky location and component masses of the binary (Williamson et al., 2014). An observed signal would allow for the unambiguous identification of a binary merger origin of a short GRB, providing the strongest possible backing for the favoured progenitor model. To date, numerous searches for gravitational waves associated with short GRBs (Abadie et al., 2012b; Aasi et al., 2014) have been performed with the data from the initial LIGO and Virgo detectors (Abbott et al., 2009; Acernese et al., 2007). Overall, gravitational wave searches have been performed around the time of 80 short GRBs (Aasi et al., 2014), with no evidence of a signal. Given the sensitivity of the detectors (tens of Mpc for BNS and NSBH mergers) and the typical measured redshifts of short GRBs (median z=0.4), this was to be expected. There were, however, two short GRBs, GRB 051103 (Hurley et al., 2010) and GRB070201 (Hurley et al., 2007), whose sky locations overlapped nearby galaxies. The non-detection of gravitational waves associated with these GRBs (Abadie et al., 2012a; Abbott et al., 2008) provided weight to the argument that these events were extra-galactic giant magnetar flares.

The second generation of gravitational wave interferometric detectors, Advanced LIGO (Aasi et al., 2015) and Advanced Virgo (Acernese et al., 2015), are under construction and expected to be operational in 2015 and 2016 respectively, approaching design sensitivities over 3–5 years (Aasi et al., 2013). They will provide a factor of ten increase in sensitivity over a broad range of frequencies, and will therefore be sensitive to binary mergers within a few hundred Mpc for BNS and up to 1 Gpc for NSBH(Abadie et al., 2010b), comparable with the distances to the closest short GRBs. The observation of a gravitational wave signal in coincidence with a GRB is therefore a realistic prospect. Such an observation would firmly establish binary mergers as the progenitors of short GRBs and also allow us to distinguish a BNS or NSBH progenitor in many cases. This paper discusses in detail the prospects of such a joint observation as the advanced detector network evolves towards its design sensitivity.

In order to accurately evaluate the detection prospects, we must evaluate the expected rates of short GRBs in the local universe and the sensitivity and sky coverage of both the GRB and gravitational wave detectors. Numerous studies have provided estimates of the rates of binary mergers and GRBs within the range of the advanced gravitational wave detectors (see, for example Abadie et al. (2010b) or Wanderman & Piran (2015)). Using these, it is possible to estimate the fraction of GRBs which will produce an observable gravitational wave signal. In addition to GRB rates and detector sensitivities, this will also depend upon the masses of the progenitors (and in particular whether they are BNS or NSBH systems) as well as the beaming angle. We obtain an estimate of the rate of joint observations, and show that there is a greatly increased chance of observing a GW around the time of a GRB, in comparison to an arbitrary stretch of data. This increased detection probability allows for a reduction of the detection threshold, to maintain a fixed false detection probability. A timeline for the expected evolution in sensitivity of the advanced LIGO (including LIGO India) and Virgo detectors is given in Aasi et al. (2013). We use this to calculate the expected rate of joint GW-GRB observations as the detectors evolve towards their full sensitivities.

The measure of a GRB redshift provides an additional piece of information, above and beyond the observed time and sky location, that can be used to restrict the parameters of the gravitational wave search. This will be valuable as the vast majority of observed GRBs are expected to be outside the range of the gravitational wave detectors. By incorporating this information into the gravitational wave search, it should be possible to increase the sensitivity of the search to GRBs with measured redshift.

The electromagnetic emission from short GRBs is believed to be be beamed. The jet beaming angle can be measured by observing a jet break in the electromagnetic emission (Sari et al., 1999), and several opening angles have been measured to be less than , with other short GRBs having lower limits up to (Berger, 2014). It will be difficult to constrain the opening angle based on the observation of a gravitational wave signal, due to the low signal to noise ratio of the observations, and an inability to independently measure distance and binary inclination. However, the beaming angle will also affect the relative rates of observed binary mergers and short GRBs: more tightly beamed emission will lead to a lower fraction of binary mergers with observable gamma-ray emission. Consequently, the accurate measurement of both the local GRB rate and the binary merger rate will allow us to infer the (average) beaming angle. Even in the absence of a gravitational wave detection, we can place lower bounds on the short GRB jet opening angle. We also present the expected bounds from early advanced detector runs.

This paper is laid out as follows: in section 2 we discuss GRB observations, briefly reviewing the evidence for the binary merger model of short GRBs and detailing the GRB model we use to obtain our results. In section 3, we discuss targeted gravitational wave searches, providing a brief discussion of previous searches and an evaluation of the sensitivity improvement afforded by a GRB observation. We present the prospects for joint observations in section 4 and in section 5 we discuss the benefits of these observations, focusing on redshift measurements and constraining the opening angle. We end, in section 6, with a discussion.

2. Observations of short GRBs

2.1. GRB satellites and observations

The first GRBs were observed by the Vela satellites (Klebesadel et al., 1973), although it was not until the Burst and Transient Source Explorer (BATSE) instrument on the Compton Gamma Ray Observatory, that they were shown to be of cosmological origin, and classified into two families.

GRB observations were revolutionised by the Swift satellite which, as well as a large area Burst Alert Telescope (BAT), also carries sensitive X-ray and UV-optical telescopes which can be slewed rapidly to observe the burst afterglow (Gehrels et al., 2004). Swift has been operational since 2004, has detected over 800 GRBs to date, including 75 short bursts, and has a field of view of approximately 2 sr. The Swift BAT observes around 10 short GRBs per year111http://swift.gsfc.nasa.gov/. It typically gives localizations with arc minute accuracy, or better if the burst is followed up with the onboard X-ray telescope. Swift’s ability to localise sources rapidly and accurately has enabled the follow-up observation of numerous short burst afterglows, measurements of redshifts and identification of galaxy hosts. The Swift satellite is expected to continue operations until at least 2020.

The Fermi satellite was launched in 2008 and carries on board the Gamma-ray Burst Monitor (GBM) and the Large Area Telescope (LAT)222http://fermi.gsfc.nasa.gov/. It is GBM which provides the broadest sky coverage and is essentially an “all–sky” telescope with a field of view of 9.5 sr (Meegan et al., 2009) Fermi GBM typically observes around 45 short bursts per year, of which only a small fraction are seen in LAT. The Fermi localisation is typically accurate to tens or hundreds of square degrees (von Kienlin et al., 2014), making optical followup of these events challenging. To date, no afterglow from a short GRB observed only by Fermi has been observed, and consequently the redshifts of these bursts are not known. Fermi is currently operational, with its 10 year funding cycle ending in 2018, though it may continue operations further.

The Space-based multi-band astronomical Variable Objects Monitor (SVOM) satellite333http://www.svom.fr/ is a recently approved Chinese-French mission that is scheduled for launch in 2021 (Basa et al., 2008). SVOM will have a similar sky coverage to Swift, and will also carry X-ray, optical and UV telescopes that can be rapidly and automatically slewed to observe afterglows.

Finally, the set of instruments which make up the InterPlanetary Network (IPN)444http://heasarc.gsfc.nasa.gov/W3Browse/all/ipngrb.html , are not dedicated GRB satellites, but instead have GRB monitors on board (Hurley et al., 2003). The majority of satellites in the network are unable to localize the bursts individually but it is possible to localize bursts observed in numerous satellites using triangulation. The sizes and shapes of these error regions vary greatly, depending upon the number of satellites and their locations (more distant satellites greatly improve localization). The IPN provides essentially all sky coverage for GRBs, although, given the sensitivity of the detectors, the GRBs observed tend to be brighter.

There are several key pieces of observational evidence to support the binary merger model for short GRBs. The afterglows of several short GRBs have been observed in both the X-ray (Gehrels et al., 2005) and optical (Hjorth et al., 2005) and consequently localized to galaxies. Despite several low redshift observations, there has been no observation of a supernova associated with any of these events. Additionally, the host galaxies of short GRBs are much more varied than long, with a large fraction of late type galaxies which are not observed as hosts of long GRBs. In the binary merger model, the delay time between formation and merger of the binary can take a wide range of values (Dominik et al., 2012), explaining the range of galaxy types observed as hosts of short GRBs. There is also evidence of “hostless” short GRBs that have been ejected from the galaxy (Fong & Berger, 2013) which arise naturally from supernova kicks imparting a velocity to the binary, coupled with the long delay time to merger. The strongest evidence for the merger model is the observation of a kilonova associated with GRB130603B (Tanvir et al., 2013; Berger et al., 2013). Taken together, these observations provide good evidence for the binary merger model for at least a subset of short GRBs. We note, however, that some fraction may still be mis-classified long GRBs (Bromberg et al., 2013) or soft gamma repeaters in nearby galaxies (Abbott et al., 2008; Hurley et al., 2010).

2.2. The rate of short GRBs

There have been numerous efforts recently to estimate the rate of short GRBs, based primarily on redshift measurements of GRBs observed by Swift (Coward et al., 2012; Wanderman & Piran, 2015; Siellez et al., 2014).555A nice summary of recent rate estimates is provided in Table 4 of (Wanderman & Piran, 2015) Here, we follow Wanderman & Piran (2015), who use the observed GRB populations (and measured redshifts in Swift) in order to derive a luminosity function for GRBs as well as a local rate density.

The energy spectra of short GRBs is modelled, following Band et al. (1993), as a power law decay with exponential cutoff at low energy and a steeper power law at higher frequencies. The parameters used in the Band function are , and . For a GRB at a given distance/redshift, the peak photon count in a detector can be related to the peak luminosity in a straightforward way (Wanderman & Piran, 2015; Regimbau et al., 2015). The detection threshold is taken to be photons per second in the keV band for Swift and photons per second in the keV band for Fermi.

The short GRB luminosity function is taken to be a broken power law, with a logarithmic distribution

(1)

where is the peak luminosity (in the source frame) between keV and MeV, and and give the power law decay below and above the break at .666Other papers use a smaller energy band when defining the luminosity, and this has an impact on the value of , although not on the slopes of the power law components. The other important parameter is the minimum GRB luminosity, which determines the lower cutoff of the luminosity distribution. This is poorly constrained as only nearby low luminosity GRBs would be observable. The minimum luminosity is taken to be erg/s.

The parameters , , are fitted jointly with the short GRB rate. Best fit values are , and erg/s, with a local GRB rate of . The GRB rate evolves with redshift, peaking at .

Other works (Coward et al., 2012; Siellez et al., 2014) take a similar approach to estimating the rate of short GRBs, although the assumptions they make vary. Consequently there is some variation in the rate estimates, but they typically lie in the range with a median rate around . These rates are somewhat lower than earlier estimates based on a smaller sample of GRBs (Guetta & Piran, 2005; Nakar, 2007). For the remainder of this work, we make use of the Band function and luminosity distribution parameters of Wanderman & Piran (2015), but allow for a constant rate per comoving volume between and per Gpc yr. We do not include any variation of GRB rate with redshift as we found it had little impact on the overall results, due to the limited range of the gravitational wave detectors.

Given the evidence for a binary merger progenitor for short GRBs, it is interesting to compare the observed and predicted rates of short GRBs and binary mergers. To do so, we must take into account the beaming of the GRB jet. The evidence for beaming in short GRBs comes primarily from the observation of jet breaks, at which time the material in the jet starts to spread out, leading to a break in the light curve. The observation of such a break can be used to infer the jet’s opening angle (Sari et al., 1999). The observation of a jet break in a number of short GRB afterglows (see for example Fong et al. (2014); Panaitescu (2006); Guelbenzu et al. (2012)) has been used to infer opening angles between and . In others, the lack of an observed break has been used to set a lower limit on the beaming angle. In many cases this leads to a limit of only a few degrees, however, GRB 050724 had no observed break after 22 days, leading to an inferred opening angle of at least . See Berger (2014) for a summary of observations to date.

The rate of observed short GRBs can be related to the all sky rate of binary mergers through

(2)

where is the average jet opening angle of the gamma ray emission, and the factor encodes the fraction of binary mergers which produce a gamma ray burst. The rate of neutron star binary (BNS) mergers, inferred from binary pulsar observations and population synthesis modelling, is taken to lie between and (see Abadie et al. (2010b) and references therein). To date, no neutron star–black hole (NSBH) systems have been observed as binary pulsars, but the rate can still be predicted through population synthesis, constrained by the observations of double neutron star binaries, to be .

Figure 1.— The rate of binary mergers in the local universe. The figure shows the predicted rates of binary neutron star (upper, grey band) andneutron star black hole (lower, blue band) mergers, taken from Abadie et al. (2010b). The shaded regions mark the range of reasonable values, while the dashed lines show the best estimate of the rate. We can also infer the rate of GRB progenitors, given an opening angle, as plotted in red.

In Fig. 1, we compare the observed and predicted rates for short GRBs to those for BNS and NSBH mergers. As has been observed elsewhere (Guetta & Piran, 2005), there is a remarkable concordance between the GRB and BNS rates, with the observed beaming angles compatible with the best guess BNS rate, with a lower rate of BNS mergers requiring larger GRB opening angles.777For this discussion, we have implicitly been assuming that all BNS mergers, produce GRBs, i.e. in (2). There are, however, arguments that only a subset of BNS mergers will produce GRBs, for example Giacomazzo et al. (2012).For NSBH, the rates are not in such good agreement. A jet angle requires an NSBH rate right at the top of the predicted range; alternatively a lower NSBH rate is consistent with a wider opening angle than has been inferred from observations. Furthermore, it is likely that a a reasonable fraction of NSBH mergers will not produce any electromagnetic emission as the NS will be swallowed whole, leaving no material from which to form an accretion disk and, hence, GRB jet (Foucart, 2012; Pannarale & Ohme, 2014). Indeed, in Stone et al. (2013), the fraction, , of NSBH mergers that produce short GRBs is argued to be between and , depending upon black hole mass and spin distributions. Thus, based on rate estimates, it seems unlikely that NSBH mergers can account for all observed short GRBs.

3. Gravitational Wave searches for GRB progenitors

3.1. Implementation of a targeted search

The details of the targeted search for gravitational waves associated with short GRBs are presented in Harry & Fairhurst (2011); Williamson et al. (2014). Given the time and sky location of the GRB, the search is restricted to a six second span of data around the GRB, allowing for a merger time 5 seconds before and 1 second after the time of the GRB. This allows for most realistic delays between the merger and GRB signal. The data from the available gravitational wave detectors is combined coherently, by appropriately time shifting and weighting the data from each detector to account for the known sky location of the source, to produce data streams sensitive to the two gravitational wave polarisations. The data is then searched through matched filtering using a template bank of binary merger waveforms (Blanchet, 2014; Owen & Sathyaprakash, 1999). GRBs observed by Swift are localised to well under a square degree which, for the gravitational wave search means they are effectively localised to a single sky location. For GRBs observed by Fermi GBM or the InterPlanetary Network, the sky location may be poorly constrained by satellite observations, with uncertainties of tens to hundreds of square degrees. In this case, a grid of sky points is searched (Predoi & Hurley, 2012; Williamson et al., 2014). Additional signal consistency tests (Allen, 2005; Harry & Fairhurst, 2011), designed to eliminate glitches which have a different signal morphology than binary merger signals, are used to eliminate spurious events due to non-Gaussian noise. The background is evaluated from data surrounding the six second stretch used in the search and simulated signals are added to the data to evaluate the search sensitivity.

3.2. The benefit of a targeted search

The observation of a short GRB provides a good estimate of the merger time, sky location and (possibly) distance of a potential binary merger signal. This significantly reduces the parameter space of a follow-up gravitational wave search and consequently allows for a reduction in the detection threshold (Dietz et al., 2013; Kelley et al., 2013; Chen & Holz, 2013). We evaluate in detail the sensitivity improvement afforded by the GRB observation. In contrast to most previous studies, we will make use of the results obtained from searches on real data and make use of the results of previous analyses (Abadie et al., 2012c; Williamson et al., 2014).

To investigate the impact of reducing the parameter space for GRB searches, we will deliberately avoid the question of first gravitational wave detection — where a “5-sigma” observation may well be required (Abadie et al., 2012c). Instead, we consider a later observation for which we might require a specific false positive rate: i.e. a limit on the fraction gravitational wave observations are spurious. In that case, the threshold for announcing a detection is tied to the true signal rate. Since neither the GRB or BNS rates are known with great accuracy, for this discussion we will adopt with the “realistic” rates of for BNS mergers and for short GRBs.

A detailed evaluation of the expected rate of BNS observations is provided in Aasi et al. (2013). There, a false rate of one event per century is chosen, corresponding to a signal to noise ratio of 12 in the advanced detectors. When the advanced LIGO and Virgo detectors are operating at design sensitivity, the expected rate of observed BNS mergers is 20 per year. Thus the threshold corresponds to a false positive rate of 1 in 2000.

To obtain a comparable SNR threshold for the GRB search, we need to evaluate both the expected foreground and background around the time of a short GRB. Using the results of Williamson et al. (2014) we estimate a background rate of 1 in 1,000 for event with an SNR above 8 in the GRB search, with the background decreasing by a factor of 100 for a unit increase in SNR:888The analysis in Williamson et al. (2014) was performed for the initial LIGO and Virgo detectors and, assuming that GRB emission is beamed and the jet is perpendicular to the plane of the binary, we obtain a background of 1 in above SNR of 8. However, we must include a trials factor due to requiring a larger template bank for the advanced detectors (Owen & Sathyaprakash, 1999) and consequently we (somewhat conservatively) increase the background by a factor of 100 as was done in Aasi et al. (2013)

(3)

Next, we must determine the probability of any given GRB occurring at a low enough redshift that the gravitational wave signal would be observable by the advanced LIGO and Virgo network. The sky and binary orientation averaged sensitivity of the network is . However, it is natural to assume that the GRB jet is beamed perpendicular to the plane of the binary’s orbit (see e.g. Piran et al. (2013)). The gravitational wave signal is also (weakly) beamed in this direction: the amplitude for a face on signal is a factor of 1.5 greater than the orientation averaged amplitude.999The sensitivity of a detector to binary mergers is typically quoted in two different ways: either the range — the sky and orientation averaged sensitivity; or the horizon — the maximal sensitivity, for binaries which are directly overhead the detector and face on. The horizon distance is a factor of 2.26 greater than the range. Here, we are assuming all sources are face on, but still averaging over sky positions. It turns out that the averaging over orientation and sky give the same factor, so performing just one average increases the sensitivity by . The gravitational wave beaming is rather weak and the amplitude falls off slowly with opening angle. Even with opening angles up to the mean amplitude is only reduced by 5% from the face on case (Dietz et al., 2013). Thus, the nominal sensitivity for GRB signals in the advanced detector network is 300 Mpc, rather than 200 Mpc for signals of arbitrary orientation. The sensitive distance scales inversely with the SNR threshold, i.e.

(4)

There are around 50 short GRBs observed annually (10 by Swift BAT and 45 by Fermi GBM, of which several are observed by both instruments). Assuming a local GRB rate of , we would expect around one event per year to be detected at a distance of 500 Mpc or less, taking into account detector sensitivities, sky coverage and live times. Thus, the chance of any GRB occurring within a distance can be approximated as

(5)

We have ignored the impact of detector sensitivity since, assuming the GRB model in the previous section, the majority of GRBs within this range would be observed by Swift or Fermi if they were in the field of view. This is broadly consistent with the observed redshifts from Swift, where the smallest of 30 measurements is z=0.12, corresponding to a distance of . Obviously, this relationship will break down at larger distances where cosmological effects, variation of the intrinsic GRB rate and detection efficiencies all become significant.

In the GRB search, the chance of a noise event giving an SNR above 9.1 is . At this SNR, the sky averaged sensitivity to face on BNS mergers is Mpc so, from equation (5), there is a 1% chance of the gravitational wave signal from a short GRB being observable. This gives a false positive rate of 1 in 2000 as desired. Therefore, the observation of a GRB allows us to lower the threshold in a gravitational wave search by 25% while maintaining a fixed false positive rate. We note that neither the astrophysical rate of BNS or GRBs nor the noise background of the advanced detectors are known at this time. Nonetheless, the predicted increase in sensitivity of the GRB search is relatively robust. The observed background for the BNS and GRB searches is very similar in nature and, in particular, both show the same, rapid rate of falloff at large SNR. Thus, changes in the required detection confidence will affect both searches in the same way.

Reducing the detection threshold by 25% will more than double the number of detectable signals. In other words, less than half of gravitational wave signals associated with GRBs will be detected based on the gravitational wave signal alone — it is only with a joint search that makes use of the GRB observation that these additional signals will be seen.

It is instructive to ask why the detection threshold can be lowered by 25% for the GRB search. The answer is twofold. First, the expected rate of signals is significantly higher in the data around the time of a GRB. In equation (5), we gave the probability of there being an observable signal in the 6 seconds of data around the time of a GRB, as a function of the sensitive distance. Within the nominal GRB range of 300 Mpc (at SNR 12), there is a 1 in 250 chance of observing a signal associated to the GRB. Meanwhile, for an arbitrary six seconds of data, assuming a BNS rate of , there is a 1 in 150,000 chance of observing a signal associated to a BNS merger. Thus, assuming that BNS are GRB progenitors, it is around a thousand times more likely that we observe a signal within the 6 seconds around a GRB than in an arbitrary six seconds of data. In addition, the GRB background is further reduced because searching a small time window makes a fully coherent search feasible (Harry & Fairhurst, 2011), and this increases the sensitivity relative to the all sky search (Babak et al., 2013). These factors combine to lead to the 25% reduction in threshold that can be achieved by the search.

4. Expected Rate of joint observations

The first advanced detector observing runs of the are expected in late 2015, with sensitivity reaching the design specification towards the end of the decade. At design sensitivity, the aLIGO range for BNS mergers will be and for AdV, . Around 2022, a third LIGO detector in India is expected to begin observing with comparable sensitivity (Aasi et al., 2013). Given the evolution of the advanced detector sensitivities as well as the results of the previous sections, it is straightforward to evaluate the expectations for joint short GRB–gravitational wave observations in the coming years. We consider three GRB observing scenarios: Swift, Fermi and full sky, full sensitivity coverage. While the latter is, of course, somewhat optimistic, it serves to provide an upper bound on the joint observation rate. For Swift and Fermi, we use the sky coverage and detection thresholds outlined in section 2.1 and, in addition, we assume an 80% detector duty cycle for both detectors due to passage through the South Atlantic Anomaly.

Epoch Run Duration BNS Range (Mpc) Number of GW–GRB detections
LIGO Virgo All Sky Fermi GBM Swift BAT
2015 3 months 40 - 80 - - 0.02 - 0.02 - 0.003
2016–17 6 months 80 - 120 20 - 60 0.004 - 0.2 0.003 - 0.1 - 0.03
2017–18 9 months 120-170 60 - 85 0.02 - 0.8 0.01 - 0.5 - 0.1
2019+ (per year) 200 65 - 130 0.1 - 2 0.07 - 1 0.01 - 0.2
2022+ (per year) 200 130 0.2 - 3 0.1 - 2 0.02 - 0.3
Table 1The expected rate of joint gravitational wave–GRB observations in the upcoming science runs, assuming that the progenitor of every short GRB is a BNS merger. Sensitivities, run durations and BNS rates taken from (Aasi et al., 2013).

The expected rates of short GRB observations, assuming a BNS progenitor, are given in Table 1. For each observing run, a range of possible detector sensitivities is quoted, to take into account the uncertain nature of commissioning and operating the advanced detectors (Aasi et al., 2013). The rate of observed BNS mergers is calculated for a merger rate between and . The range of predicted rates reflects the uncertainty in both the detector sensitivities and the rate of sources. For joint gravitational wave–GRB observations, we take the short GRB rate to lie in the range . As discussed in section 3.2, we allow for a 25% decrease in detection threshold associated with a dedicated GRB search when compared to an all-sky all-time gravitational-wave search. When calculating the Swift and Fermi rates, we use the GRB luminosity distribution and energy spectra described in section 2.1. These thresholds, however, have little effect on the rate as the majority of GRBs within the sensitive range of advanced LIGO and Virgo will have a peak luminosity sufficient to be observed by Swift BAT and Fermi GBM.

The expected number of joint observations in the early advanced LIGO-Virgo science runs is much less than one. However, by the 2017–18 observing run, there is a real chance of a joint observation and, with the network operating at design sensitivity, an excellent chance of joint gravitational wave–GRB observations during an extended science run. We note, however, how critical it is to continue monitoring the sky for GRBs: it is only with the sky coverage provided by Fermi (and the InterPlanetary Network) that we expect to make joint observations.

Figure 2.— The expected rate of observed gravitational wave–GRB signals when the LIGO and Virgo detectors are operating at their design sensitivity. We take the intrinsic short GRB rate to be in the range and assume that BNS are the progenitor source of all short GRBs. The grey region shows the range of expected rates with all-sky GRB coverage. The observed rate increases with a small opening angle as the systems are close to face on and thus have the maximum gravitational wave emission. The blue region shows the expected rate for joint observations with Fermi GBM and the red region for Swift BAT. For preferred opening angles (less than ) we expect to see at least one GRB per year in coincidence with Fermi GBM.

Fig. 2 shows the expected annual rate of joint observations, as a function of GRB opening angle for the 2019+ configuration of Table 1. The dependence of the rate on the GRB opening angle is due to the beaming of the gravitational wave signal: the amplitude for a face on signal is a factor of 1.5 greater than the orientation averaged signal, giving a factor of 3.4 between small opening angles and no beaming. Fig. 3 shows the expected all sky BNS merger rate, as a function of GRB opening angle under the assumption that all BNS mergers produce gamma-ray emission. As discussed in Chen & Holz (2013), there is a crossover point, where we see more gravitational waves associated with GRBs than in an all sky, all time search. This will obviously depend upon the sky coverage and sensitivity of the GRB satellites, but assuming full sky coverage, this occurs around . If the beaming angle is larger than this, the GRB search will detect more signals than the all sky all time search, due to the ability to lower thresholds around the time of observed GRBs. Of course, based on astrophysical measurements of GRB opening angles, this is unlikely to be the case.

Figure 3.— The expected rate of observed BNS signals when the LIGO and Virgo detectors are operating at their design sensitivity. We take the intrinsic GRB rate to be in the range . The rate increases with smaller opening angles as this implies a greater fraction of sources which are not observed as GRBs. The horizontal lines bound the predicted number of observations based upon estimates of BNS rates. At the largest opening angles, only the higher GRB rates are consistent with the BNS predictions.
Epoch Run Duration NSBH Range (Mpc) Number of GW–GRB detections
LIGO Virgo All Sky Fermi GBM Swift BAT
2015 3 months 70 - 130 - - 0.06 - 0.03 - 0.007
2016–17 6 months 130 - 200 30 - 100 0.005 - 0.5 0.003 - 0.3 - 0.07
2017–18 9 months 200 - 280 100 - 140 0.03 - 2 0.02 - 1 0.004 - 0.3
2019+ (per year) 330 110 - 220 0.2 - 6 0.1 - 2 0.02 - 0.5
2022+ (per year) 330 220 0.4 - 10 0.2 - 3 0.03 - 0.7
Table 2The expected rate of joint gravitational wave–GRB observations in the upcoming science runs, assuming that the progenitor of every short GRB is a NSBH merger. Sensitivities and run durations taken from (Aasi et al., 2013), we assume a fiducial NSBH with a neutron star mass of and a black hole mass of .

The expected rates of short GRB observations, assuming a NSBH progenitor are given in Table 2. For NSBH mergers, the masses and spins of the system have a stronger effect upon the expected rates of observation. Higher masses and large, aligned spins result in greater gravitational wave emission increasing the distance to which the sources can be observed. For simplicity, we take the system to be a neutron star of mass and a non-spinning black hole of mass . Following the same procedure as before, we assume that all GRB progenitors are NSBH binaries and use the GRB model discussed in section 2.1 to determine the fraction of GRB signals that are observed by Swift and Fermi. This has a significant impact on the rate of observable signals, particularly in the epochs after 2019.

As we have discussed previously, there is already a tension between the observed GRB rates and predicted NSBH rates. Specifically, as is clear from Fig. 1, for all short GRBs to have an NSBH origin requires a merger rate at the high end of the predicted range, a relatively large GRB opening angle, or both. Additionally, numerical simulations indicate that for a large fraction of NSBH mergers, there will not be sufficient matter in the accretion disk to power a GRB, making the rates even less compatible (Foucart, 2012). Thus, the assumption that all GRBs are due to NSBH mergers seems difficult to accommodate, meaning that the highest rates in Table 2 are not realistic. Nonetheless, even if 15% of GRBs have NSBH progenitors, this would double the expected rate of joint observations. Alternatively, the absence of a joint gravitational wave–GRB observation could be used to limit the fraction of short GRBs which have a NSBH progenitor.

To end this section, we compare our results with other recently published works. Wanderman & Piran (2015) calculate the rate of joint GRB–gravitational wave detections by simply assuming a 300 Mpc range for the advanced LIGO-Virgo network. They obtain a rate of joint Fermi (Swift) observations of () assuming a minimum peak luminosity of erg/s. This is entirely consistent with the rates for BNS in the 2019+ epoch given in Table 1. The fact that they have neglected the directional sensitivity of the gravitational wave network has little impact as essentially all GRBs within the advanced LIGO-Virgo range will be observable by Swift and Fermi. By varying the luminosity threshold, they obtain rates that span the same range as ours. For NSBH systems, they assume a 1 Gpc range for the advanced gravitational wave detectors, compared to our range of 660 Mpc, and consequently obtain a significantly higher rate ( for Fermi and for Swift). Regimbau et al. (2015) have also calculated joint detection rates of gravitational wave–GRB signals. They predict rates of joint observations with Swift of per year for BNS and per year for NSBH. The rates are broadly comparable to those presented here, although the range goes somewhat higher for BNS and lower for NSBH. These differences arise due to different choices of parameters in the Band function, GRB luminosity distribution, and detector thresholds. Additionally, the authors choose a fixed BNS (NSBH) rate of and a range of opening angles between and . With these rates, NSBH signals could only account for a fraction of GRBs. This explains why their numbers are lower than the ones in Table 2 where we have assumed that all GRBs have NSBH progenitors.

5. Benefits of joint observations

Numerous previous papers have discussed the benefits of joint gravitational wave–GRB observations, including: the potential to confirm (or rule out) the binary merger progenitor model (Eichler et al., 1989); measuring the time-delay between the binary merber and the GRB signals to understand jet breakout; the ability to probe GRB jet opening angles (Dietz, 2011; Chen & Holz, 2013); the independent measurement of distance and redshift used as a probe of cosmology (Schutz, 1986; Nissanke et al., 2010). We will not discuss all of these in detail, but will focus on two issues. First, we discuss how the measurement of a GRB redshift may actually assist in the detection of a gravitational wave counterpart. Then, we discuss prospects for measuring or constraining opening angles.

5.1. Detecting a GRB with measured redshift

Figure 4.— The probability of obtaining an event of a given SNR for: noise only (the red, dot-dashed line); a GRB progenitor at an unknown distance (the black, dashed line) and a known distance (the blue, solid line). In this example, we have used the parameters from GRB080905A with a distance of 550 Mpc which gives a signal SNR of 7.7. The top plot shows the probability distribution function, while the bottom plot gives the cumulative probability of observing an event as loud or louder.
Figure 5.— The odds ratio between the signal and noise models. We consider two signal models: a GRB at unknown distance (black, dashed line) and a GRB at a known distance (blue, solid line). In this example, we have used the parameters from GRB080905A with a distance of 550 Mpc which gives an expected SNR of 7.7. The blue curve gives the odds ratio for a BNS signal at that distance, as a function of SNR. The black curve gives the odds ratio for a BNS signal at an unknown distance. At low SNR, knowledge of the distance increases the odds ratio significantly; at higher SNRs it decreases the odds as the observed SNR is no longer consistent with the distance.

The advanced detector network will, on average, be sensitive to a BNS merger associated with a GRB within a distance of 400 Mpc, or . The closest observed short GRB is GRB 080905A with a measured redshift of (Rowlinson et al., 2010). It is interesting to ask whether this GRB could have been observed by the advanced LIGO-Virgo network. While the GRB was at a distance of 550 Mpc, it was at a favourable sky location for the advanced LIGO and Virgo network and a BNS merger associated with this GRB may have been marginally detectable, but only once the redshift information is folded in.

Let us consider the expected distribution of the observed SNR in the gravitational wave search, under three distinct scenarios: no observed gravitational wave signal; a BNS merger signal associated with a short GRB at an unknown distance; a BNS merger signal at 550 Mpc. To obtain the distribution in the absence of a signal, we simply use the empirical estimate provided in Eq. (3). For a signal at 550 Mpc in the direction of GRB080905A, a BNS merger will generate an expected network SNR of 7.7. The expected, maximum SNR observed in the gravitational wave search then follows a non-central with four degrees of freedom (Harry & Fairhurst, 2011) overlaid on the noise background given in equation (3). Finally, for a GRB with unmeasured redshift, we use the distance distribution as given in Eq. 5, i.e. signals distributed uniformly in at low redshift, with only a small probability of the GRB occurring within the LIGO-Virgo sensitive range.

In figure 4, we show the probability distribution for the SNR of the gravitational wave event under these three scenarios. The figure shows both the probability distribution as well as the cumulative probability of observing an event above a given SNR. In this example, the knowledge of the distance greatly increases the chance of observing a signal with a moderate SNR. For example, the chance of observing an event with SNR due to noise alone is around 1%, if there is a BNS merger at unknown distance it is 3% whilst when the distance is known to be 550 Mpc it is 60%.

One way to visualise the benefit of a redshift measurement is through the odds ratio: the ratio of the signal probability to the noise probability. This is plotted in Fig. 5. For an observed SNR above 7.5, the signal model is favoured over the noise by a factor of 10, increasing to 100 at SNR of 8. Even at these low SNRs, this would be an interesting event. However, if the distance is not known, a larger SNR (of 8.5 or 9) is required before the signal model is strongly favoured over the noise. Thus, if this GRB had occurred during the advanced detector era, there is a real chance that measuring the redshift would make the difference between identifying a gravitational wave candidate and not.

5.2. Constraining the jet opening angle

A joint gravitational wave-GRB observation would provide a measurement of the binary’s inclination angle and, consequently, would provide a constraint on the jet opening angle of GRBs. However, the majority of observed gravitational wave signals are likely to be weak, with an SNR of ten or less, and this will make accurate parameter recovery difficult. Accurate measurement of the binary inclination angle is further complicated by the fact that it is highly degenerate with the distance, particularly when the signal is close to face on. Specifically, the overall amplitude of the two polarisations scale as and and, at SNR of 10, we would expect to measure these amplitudes with roughly a 10% accuracy.

For a face on signal (with ), the two amplitudes are equal. They differ by 1% for an inclination angle of and by 10% for an inclination of . Thus, while the gravitational wave observation will constrain opening angles, it is most likely to limit the angle to be . In the case where the redshift, and hence distance , is known there will still be a % uncertainty in corresponding to a constraint on the opening angle of . Even for the loudest signals, we are faced with an uncertainty in the Hubble constant of 1% and a likely instrumental calibration error of 1% or more (Abadie et al., 2010a) making it difficult to constrain the opening angle to less than .

Epoch Run Duration BNS Range (Mpc) limit on GRB opening angle ()
LIGO Virgo BNS NSBH
2015 3 months 40 - 80 - 0 - 3 0 - 6
2016–17 6 months 80 - 120 20 - 60 1 - 8 3 - 15
2017–18 9 months 120-170 60 - 85 3 - 15 7 - 35
Table 3The expected bounds on GRB opening angle during the early advanced LIGO-Virgo observing runs. These results assume that no gravitational wave signal is observed and use the observed GRB rate to infer the minimum jet opening angle consistent with the lack of gravitational wave detection. The results in the two columns assume that all GRBs are either BNS (first column) or NSBH (2nd). In both cases, the range quoted takes into account both the uncertainty in the detector performance in these runs as well as the uncertainty in the local rate of GRBs.
Figure 6.— The expected upper limit on BNS rate for the planned observing runs, assuming that no GW events are observed: 2015 in black; 2016–17 in blue and 2017-18 in green. The ranges correspond to the uncertainties in detector sensitivity as detailed in Table 1. The expected rate of GRB progenitors as a function of the GRB opening angle is overlaid in red. Assuming that all GRBs correspond to BNS, we can read off the lower limit on opening angle that would be obtained at the end of each run.

It is more likely that the observed populations of short GRBs and binary mergers will allow us to restrict the opening angle of GRB jets. It is clear from Figure 1 how this would be done: if the permitted range of BNS merger rates can be reduced from three orders of magnitude to a factor of two, then the GRB opening angle will be highly constrained. First, consider the case where the early science runs do not yield a gravitational wave detection. We assume that, in the absence of a detection, the loudest event is consistent with background and estimate the expected rate upper limit as — where is the volume searched and the time. (Brady & Fairhurst, 2008). From this, we can read off the smallest opening angle consistent with the rate upper limit by re-arranging equation (2) to obtain

(6)

Thus the tightest limit on is given by assuming the maximum BNS rate, i.e. right at the upper limit, and an of unity, i.e that all BNS mergers produce GRBs. In Fig. 6 we plot the expected upper limits, in the absence of a gravitational wave detection, from the early observing runs. The bands here correspond to the uncertainties in detector sensitivities as given in Fig. 1. For example, at the end of the 2016-17 run, the lack of a detection can place a constraint on the GRB opening angle between and depending upon the detector sensitivity achieved and the assumed GRB rate. Thus, even in the absence of an observation, we are starting to impact measurements from GRB observations. In Table 3, we summarise the results for BNS signals, and also for NSBH. In both cases, we are assuming that all GRBs are produced by one particular type of merger. This, of course, is unrealistic. Although we cannot know the fraction of GRBs which have a BNS or NSBH progenitor, we might reasonably assume that all GRB progenitors are mergers. Since the sensitivity to BNS mergers is less than NSBH, the conservative limit comes from assuming that all mergers are BNS. Alternatively, it is possible to make reasonable assumptions of priors for the various parameters and then marginalise over them to obtain a distribution for the opening angle (Clark et al., 2014).

Of course, we hope to observe gravitational waves from binary mergers. Even a handful of observations will provide a measurement of the rate within a factor of two, which will correspond to a much tighter horizontal band on figure 1. If, for example, the rate is then this will restrict the GRB opening angle to be between about and .

6. Conclusion

We have presented in detail the expectations for gravitational wave observations associated with short gamma ray bursts in the coming years. The evidence for a binary merger progenitor of short GRBs is strong, and we have focused on this scenario. By making use of the known time and sky location of the source, we have argued that it is appropriate to lower the gravitational wave search threshold by around 25% relative to the all sky, all time search. This decrease in threshold will double the number of gravitational wave events that can be detected in association with a GRB. We have also demonstrated that if the redshift is measured (and is within the sensitive range of gravitational wave network), the detection threshold can be further reduced. Using this threshold reduction and the expected evolution of gravitational wave detector sensitivities given in Aasi et al. (2013), a joint gravitational wave–GRB observation is possible in the 2015 and 2016-17 observing runs, but unlikely. However, as the detectors approach their design sensitivity the rate of joint observations increases and could be one or two per year for a BNS progenitor and even higher if the majority of GRBs have NSBH progenitor. This, of course, depends critically upon the continued operation of wide-field of view GRB satellites, such as Fermi, as well as the continued operation of the InterPlanetary Network.

The joint observation of gravitational wave and GRB signals will be a major milestone in understanding short GRBs, and will finally prove (or disprove) the binary merger progenitor scenario. It will also shed light on the GRB central engine by probing delays between the signals and a bound on the jet opening angle. We have argued that the measurement of binary inclination from gravitational wave observations will have large uncertainties, due in part to the degeneracy with the measurement of distance, and is unlikely to constrain the opening angle tightly. However, an accurate measurement of the populations of both short GRBs and binary neutron star (or neutron star-black hole) mergers will allow us to constrain the opening angle. Even in the early advanced detector runs, we will be able to place lower bounds on the beaming angle of short GRBs that will confront current observations.

The authors would like to thank the participants of the Royal Society International Scientific Seminar on Gravitational Waves and Gamma Ray Bursts for useful discussions. In addition, they thank Ray Frey and Valeriu Predoi for comments on the manuscript. This work was supported by the UK Science and Technologies Funding Council through grant ST/L000962/1, the Royal Society and the US National Science foundation through grants PHY-0847611, PHY-1205835, PHY-0955773 and AST-1333142.

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