Utility of galaxy catalogs for following up gravitational waves from binary neutron star mergers with wide-field telescopes
UUtility of galaxy catalogs for following up gravitational wavesfrom binary neutron star mergers with wide-field telescopes
Chad Hanna ∗ Perimeter Institute for Theoretical Physics,Waterloo, Ontario, N2L 2Y5, Canada
Ilya Mandel † and Will Vousden ‡ University of Birmingham, Edgbaston,Birmingham, B15 2TT, United Kingdom a r X i v : . [ a s t r o - ph . H E ] D ec bstract The first detections of gravitational waves from binary neutron star mergers with advancedLIGO and Virgo observatories are anticipated in the next five years. These detections could pavethe way for multi-messenger gravitational-wave (GW) and electromagnetic (EM) astronomy if GWtriggers are successfully followed up with targeted EM observations. However, GW sky localizationis relatively poor, with expected localization areas of ∼ ; this presents a challenge forfollowing up GW signals from compact binary mergers. Even for wide-field instruments, tens orhundreds of pointings may be required. Prioritizing pointings based on the relative probability ofsuccessful imaging is important since it may not be possible to tile the entire gravitational-wavelocalization region in a timely fashion. Galaxy catalogs were effective at narrowing down regionsof the sky to search in initial attempts at joint GW/EM observations. The relatively limited rangeof initial GW instruments meant that few galaxies were present per pointing and galaxy catalogswere complete within the search volume. The next generation of GW detectors will have a ten-foldincrease in range thereby increasing the expected number of galaxies per unit solid angle by a factorof ∼ ∼
10% to a factorof 4 relative to follow-up strategies that do not utilize such catalogs for the scenarios we considered.We determine that catalogs with a 75% completeness perform comparably to complete catalogs inmost cases, while 33%-complete catalogs can lead to lower follow-up success rates than completecatalogs for small fields of view, though still providing an advantage over strategies that do notuse a catalog at all. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] . INTRODUCTION Abadie et al. [1, 2] present the first low-latency searches for gravitational waves that trig-gered electromagnetic (EM) follow-up observations with a ∼ . Nogravitational waves were detected, but gravitational-wave (GW) candidate events consistentwith noise were followed up successfully [3, 4]. Later this decade, a network of advanced GWdetectors including LIGO and Virgo [5, 6] may detect tens of binary neutron star (BNS)mergers per year once at full sensitivity (with a plausible range of one detection in a fewyears to a few hundred detections per year) [7]. Some of these detections may be accom-panied by EM counterparts [e.g., 8–10], summarized below. Several nearly “instantaneous”search methods for GWs from BNS mergers have been proposed, introducing the possibilityof transmitting information about the candidate to EM telescope partners within tens ofseconds of a binary merger [11, 12].BNS mergers are thought to generate several distinct EM counterparts spanning most ofthe EM spectrum; Figure 1 of [8] illustrates the counterpart emission mechanisms. Short,hard gamma ray bursts occur on timescales of < ∼ r -process nucleosynthesis in the merger ejecta has been predicted to peak in the infrared[18]; the first hint of such a kilonova signal has been recently observed [19].Several transient telescope networks exist with wide-field coverage and it is important tounderstand what is the best way to tile pointings within the GW localization region usingwide-field instruments. This question has been addressed partly by Singer et al. [20], whopresent a framework for allocating telescope resources to optimally cover the available skylocalization region. In this work we consider the situation in which only a fraction of this areacan be surveyed in a timely fashion, where it is important to choose the tiles that representthe most likely source location first. Both Singer et al. [20] and Fairhurst [21] focus on theassumption of a uniform-on-the-celestial-sphere prior on the GW source location. However,given the broad GW localization region, pointing might be strongly influenced by a sharplypeaked prior expectation for the signal location. Fairhurst [21] mentions that a galaxy < ∼ < ∼
20 Mpc range for GWs from merging BNSs.At this range, nature provides few galaxies as potential hosts for the merger, correspondingto sharp peaks in the prior probability.The same angular scale will encompass many more galaxies in the advanced GW detectorera and the usefulness of a galaxy catalog prior comes into question. Metzger and Berger[8] suggest that the number of bright galaxies in the localization region will be too largeto improve the prospects of imaging the EM counterpart. For example, GW detections inthe advanced detector era will occur at a median distance of ∼
200 Mpc. A source at thisdistance may be optimistically localized to a sky area of 20 deg and a fractional distanceerror of ∼
30% by GW measurements alone [21, 23–26] with a network of three or moreGWs detectors . The volume defined by this solid angle and distance range will containmore than 500 galaxies brighter than 0 . L ∗ (see Section III) – more than can realisticallybe imaged individually on short timescales.If wide-field instruments are used to tile the GW localization region and the requirementson the speed and depth of the search make it impossible to follow up the entire localizationregion, the question arises of how to prioritize which tiles should be observed. Nuttall andSutton [28] partly address this problem by simulating follow-up searches within 100 Mpc inthe advanced detector era using the Gravitational Wave Galaxy Catalog (GWGC) of Whiteet al. [29]. Individual galaxies are targeted on the basis of a ranking algorithm that accountsfor luminosity and distance to putative host galaxies. Meanwhile, Nissanke et al. [30] providecase studies for the process of detecting a GW event and locating and identifying its EMcounterpart, using galaxy catalogs to eliminate false-positive EM signals. Both studies findthat catalogs can be useful both for locating and identifying an EM counterpart when thereare insufficient resources to point individually at each galaxy.In this work, we revisit the utility of a galaxy catalog in the regime where there are toomany galaxies in the GW localization region to be followed up individually, and observationalconstraints on the speed and depth of the search prevent complete coverage with wide-field With significantly larger uncertainties expected for a two-detector network for the early runs of AdvancedLIGO alone [23, 27].
II. PROBLEM STATEMENT
In this work we are not concerned specifically with identifying host galaxies, but ratherwith choosing the most probable sky regions commensurate with a given FOV by usinggalaxy catalog information. We neglect many of the practicalities considered by Nissankeet al. [30] and Singer et al. [20] (e.g. telescope slew time, limiting depth, day/night observa-tion time, etc.) to isolate the utility of galaxy catalogs on their own merits. We do, however,assess the effect of incompleteness of galaxy catalogs in our method.Throughout this work we will use a blue-band galaxy catalog as a proxy for merger ratedensity. This assumes that the rate of BNS mergers is proportional to the instantaneousmassive star formation rate (with negligible time delays between formation and merger) and5s therefore tracked by blue-light luminosity [31]. On the contrary, observational evidenceindicates that a quarter of short gamma ray bursts occur in elliptical galaxies with no signsof ongoing star formation [32]. However, the choice of color is not critical for the modelingbelow; it is sufficient to assume that we have a catalog that is an accurate tracer of mergerrate density. We discuss the validity of this assumption in Section VI.To model the effect of using galaxy catalogs to assist in EM followup we begin by dividingthe GW localization area, A , into N tiles (assumed to be non-overlapping for simplicity),each representing a telescope FOV P , where N = (cid:100) AP (cid:101) .We define a successful follow-up as a GW-triggered EM transient search in which oneof the tiles selected for imaging contains the GW source. For simplicity, we require only thatthe source reside in one of the tiles, and not that the expected EM counterpart is actually detectable by a given follow-up instrument or distinguishable from background events. Wetherefore assume the transient search to be limited in range only by the capabilities ofthe GW detector network and not by the depth of the follow-up instrument. Consideringthe above assumptions, the probability of success is 1 if all tiles in the sky are searched,regardless of whether the correct transient is identified.We define the success fraction F as the fraction of GW events that are expected to besuccessfully followed up for a given follow-up strategy according to the definition above. Ifone ignores the galaxy distribution in the event localization area A , the relative probabilitythat a GW is in a given tile is uniform amongst the tiles. The success fraction is F = N (cid:88) i N = N N ≡ f, (1)where N is the number of telescope pointings compatible with search speed and depthrequirements, and f is simply the fraction of the GW localization area that is followed up, f = N PA .This should be compared to the case where each tile has a relative probability of con- In fact, the depth to which the available telescopes can detect an EM transient may influence the optimalchoice of follow-up target. For example, there is little point in targeting galaxies that are so distant thatthe transients they might contain would not be detectable by a given telescope. In practice, not all pointings in the sky will have the same likelihood; indeed, in the high signal-to-noiseratio (SNR) limit, the likelihood distribution will have a Gaussian shape. The probability density functionon the sky will be computed through coherent parameter estimation on GW detector data [33]; here, wetreat A as a suitable “effective” area. L i , and a greedy pointingalgorithm is used whereby the brightest tiles are pointed at first, L i ≥ L i +1 for all i : F = 1 L N (cid:88) i L i , (2)where L ≡ (cid:80) N i L i . With the greedy strategy, F ≥ f ; in other words, if GW sources aredistributed according to blue luminosity then using that information never hurts the successfraction.The GW amplitude depends on the inclination and orientation of the source relative tothe line of sight, with the highest detector response for face-on sources. This allows us tocompute the probability that a source in a given galaxy at a known distance and sky locationwould pass a signal-to-noise-ratio detection threshold under the assumption that the binary’sinclination and orientation are isotropically distributed. This probability decreases from ≈ ∼
30% in fractional distance uncertainty for an event at the detectionthreshold [24]. Within this range, we will neglect the detection probability in the galaxyprior, and consider only priors proportional to blue-light luminosity. We expect this to beconservative, since the effective galaxy luminosity with the detection probability included7ould have had greater fluctuations than the absolute luminosity, and, as we will see below,luminosity fluctuations increase the utility of galaxy catalogs.We will thus assume that the detector network is able to localize a source at distance D to within a range ∈ [ D min , D max ], with D min = 0 . D and D max = 1 . D . Combining the solidangle P of a telescope pointing with this range, we can define the pointing volume , i.e., thevolume of each pointing within the measured distance range, as V ≡ π ( D − D ) P Ω where Ω ≈ ,
000 deg is the solid angle of the whole sky. The average luminosity perpointing volume is then given by (cid:104) L i (cid:105) = V ρ L , where ρ L is the average spatial density ofluminosity. We will use a luminosity density ρ L = 0 . L Mpc − , where L is defined as10 times the solar blue-light luminosity L B, (cid:12) [7, 22]. III. LUMINOSITY FLUCTUATIONS
In this section, we incorporate the distribution of intrinsic galaxy luminosity and thecounting fluctuations in the number of galaxies in different pointings into the expecteddistribution of L i . We neglect spatial correlations of galaxies (e.g., due to the presence ofgalaxy clusters – a conservative assumption since greater clustering improves the utility ofgalaxy catalogs, as we shall see shortly) and assume they are homogeneously distributed involume. We model the distribution of galaxies in blue luminosity and volume as a Schechterfunction [34] n ( x ) dx ∝ x α e − x dx , (3)where x ≡ L/L ∗ and n ( x ) dx is the expected number of galaxies per Mpc in the interval[ x, x + dx ]. We use the GWGC [29] within 20 Mpc, where it is complete, to estimate α = − . L ∗ = 2 . L , slightly brighter than the Milky Way’s blue-band luminosity of ∼ . L . We normalize the luminosity function to yield ρ L = 0 . L Mpc − on the interval L ∈ [0 . , L [36]. All following results are based on this Schechter luminosity distribution.Figure 1 shows the luminosity function of the GWGC within 20 Mpc as well as the Schechtermodel.If we assume that a pointing tile has volume V , containing a random integer sample of Similar values of these parameters are quoted in the literature, e.g., α = − . L ∗ = 2 . L [35]; ourconclusions are insensitive to small changes in these parameters. æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ -3 -2 -1 -7 -6 -5 -4 -3 -2 -1 Galaxy luminosity ( L ) N u m b e r d e n s i t y ( L - M p c - ) Schechter æ Catalog
FIG. 1: GWGC catalog luminosity function within 20 Mpc compared to a fit of (3) with L ∗ =2 . L and α = − . galaxies taken from the distribution in (3), then the resulting luminosity L in that volumecan be described by a random variable of mean V ρ L . The results of a direct Monte Carlosimulation of (3) for 100 Mpc and 1000 Mpc are shown in Figure 2. To understand theseresults, one can crudely approximate the Schechter galaxy population as a Poisson scatteringof identical galaxies of “typical” luminosity L ∗ . In this case, the luminosity in a volume V is simply L = nL ∗ , where n is drawn from a Poisson distribution of mean V ρ L /L ∗ .For 100 Mpc and 1000 Mpc pointing volumes, for example, we should expect 0 . ± . ± . ± . L and 20 ± . L , respectively. This closely matches the fluctuations of 2 . ± . L and20 ± . L , respectively, measured via a Monte Carlo simulation of the actual Schechterdistribution.The large variations in tile luminosity – 2 ± L for the 100 Mpc volume – suggestthat there can be substantial advantage to following up the brightest tiles first in a surveywith limited pointings. For lower pointing volumes, the distribution becomes increasinglynon-Gaussian, and its skewness amplifies the advantage of luminosity-directed surveys.9 -1 L ) P r o b a b ili t y m a ss
100 Mpc FIG. 2: Distribution of luminosity drawn from (3) for fixed volumes of 100 Mpc and 1000 Mpc .The means are 2 . L and 20 . L and the standard deviations are 2 . L and 6 . L respectively.These values correspond approximately to a Poisson scattering of galaxies of “typical” luminosity L ∗ . IV. RESULTS: COMPLETE GALAXY CATALOG
We show the success fraction F when using a complete, ideal galaxy catalog as a functionof pointing volume in Figure 3, for four choices of the fraction f ∈ { . , . , . , . } ofthe GW localization region being followed up. Recall from Section III that in the case whereno galaxy catalog is used we would expect on average that F = f . We find in Figure 3 thatin all cases when using the galaxy catalog F > f as we would expect, with the advantageof the catalog being more pronounced for smaller pointing volumes where the variation ofluminosity per pointing is larger.It is useful to apply the results of Figure 3 to a few potential scenarios in order to under-stand the impact that an ideal galaxy catalog would have. The distance at which a singleadvanced LIGO detector is capable of detecting an optimally oriented and located BNSmerger at an SNR of 8 – known as the horizon distance – is ∼
450 Mpc [7]. However, aver-aging over sky locations and orientations, we expect 75% of detections to come from within ∼
250 Mpc, or 50% from within ∼
200 Mpc. For early versions of the advanced LIGO/Virgonetwork, this could be reduced to as little as D ∼
100 Mpc for the 50% percentile [23]. Wetherefore consider the median distance of 200 Mpc as a typical distance to a detection withthe advanced detector network operating at design sensitivity in cases 1 and 2. Meanwhile,10 -2 -1 Pointing volume (Mpc ) Su cce ss f r a c t i o n F f = % f = % f = % f = % FIG. 3: A comparison of the success fraction F relative to the follow-up fraction as a functionof pointing volume. The horizontal grid lines represent follow-up searches that do not use galaxycatalogs, wherein F = f . The large variation in luminosity per tile can cause certain FOVs withinthe GW localization region to be more likely to contain the source, suggesting an obvious pointingpriority in the case where the entire localization region cannot be followed up in a timely fashion. case 3 represents the rarer scenario of a closer source at an estimated distance of 100 Mpc.1. Consider a 10 deg FOV telescope following up with a single pointing a GW sourceestimated to be at a distance of 200 Mpc, with a 100 deg localization region. Thepointing volume to this source, assuming GW observations constrain the distance tobe ∈ [140 , ,
000 Mpc . (For comparison, a 1 deg conical FOV containsa volume of ∼
100 Mpc out to a distance of 100 Mpc.) Without a galaxy catalog wewould expect that the success fraction F = f = 10%. However, Figure 3 shows thatusing a galaxy catalog we might expect to have a ∼
11% success fraction.2. Consider the same GW source as case 2 but with a 1 deg FOV follow-up instrumenthaving 10 pointings. While the overall coverage is still f = 10%, the pointing volumeis reduced to 1500 Mpc , so the fluctuations in luminosity between tiles are moresignificant. As a result, the success fraction improves to ∼ FOV instrument. The sky-localization and dis-tance measurement accuracy improve for high signal-to-noise ratio GW detections.11e therefore consider a 10 deg localization region and a reduced distance uncertaintyrange ∈ [90 , , and the suc-cess fraction is ∼ F = f = 10% successfraction in the absence of a galaxy catalog.These cases are meant as illustrations only. Distances to optimally located and orientedsources may range to 450 Mpc for advanced detectors at design sensitivity. Meanwhile,source in the early phases of advanced detector commissioning, when detectors are sensitivewithin a smaller range, may resemble case 3 in typical distance estimates, but with poorersky localization and distance measurements.While the overall fraction f of the GW localization region is 10% for each of the abovecases, the pointing volumes are respectively ∼ ,
000 Mpc , ∼ , and ∼
60 Mpc .The progressively larger success fractions for each case illustrates how the utility of thecatalog depends on pointing volume.The effectiveness of a given follow-up telescope, as characterized by its FOV P and thenumber of pointings N that can be taken within the allotted time while observing to asufficient depth, may be influenced by the sensitivity of the GW search and the distanceestimate it yields. For case 3, an instrument with a larger FOV might be chosen at theexpense of depth, since the EM signal is expected to be louder. Similarly, the greaterimaging depth required to detect transients at 200 Mpc might mean that fewer pointings areavailable for the more distant sources in cases 1 and 2. V. THE EFFECT OF GALAXY CATALOG COMPLETENESS
The previous discussion assumed that an ideal, complete galaxy catalog was available;however, GWGC [29], is incomplete beyond ∼
30 Mpc, and there are limitations to howcomplete catalogs become at ∼
200 Mpc distances [37]. In practice, catalogs may comprisemany different surveys with different characteristics and selection criteria, and will be in-fluenced by spatially dependent factors such as extinction in the Galactic plane. However,the simplest model, and the one we consider here, is incompleteness from a flux-limitedsurvey. As a simple example, we considered an extremely flux-limited survey that does notresolve galaxies fainter than apparent magnitude ∼ . -3 -2 -1 -7 -6 -5 -4 -3 -2 -1 Galaxy luminosity ( L ) N u m b e r d e n s i t y ( L - M p c - ) CompleteFlux - limited FIG. 4: Comparison of an ideal complete luminosity function (dashed) to a hypothetical flux-limited survey (solid) at apparent magnitude m B = 15 . . L Mpc − . The shape of the luminosity function for a flux-limited catalog is sensitive only to theoverall completeness of the catalog, not the specific range and cut-off magnitude. Therefore,in order to express the follow-up success probability for a flux-limited catalog in termsof pointing volume, which incorporates FOV and depth in a single variable, we fix the completeness of the catalog, rather than the cut-off magnitude. This success probability isplotted in Figure 5 for three choices of completeness: 33%, 75%, and 100%.In our flux-limited survey model for incompleteness, the catalog luminosity functionagrees with a hypothetical complete luminosity function for the most luminous galaxies.It is therefore not surprising that incompleteness in a catalog has little effect on the scenar-ios where only a small fraction of the sky uncertainty region will be followed up, since bothcomplete and incomplete catalogs will tend to agree on the most luminous tiles, which arethe only ones that will be pointed at for small follow-up fractions. For example, in Figure 5,13he line corresponding to a follow-up fraction of f = 0 .
01 is virtually unchanged from thecorresponding line for a complete catalog.When the follow-up fraction is large, incomplete catalogs still yield similar success frac-tions to complete catalogs as long as the pointing volume is also sufficiently large. Of course,at very large pointing volumes F asymptotes to f , as the fields of view become increasinglyuniform due to the very large number of galaxies they contain, and a galaxy catalog ceases tobe useful even when complete. Even at moderate pointing volumes and moderate follow-upfractions, catalog incompleteness is not necessarily a concern if it only leads to missing themany dim galaxies which are nearly homogeneously distributed on the sky. A few brightgalaxies can still dominate the prior and since these are included even in incomplete flux-limited catalogs, the success fraction is still relatively insensitive to completeness. Whenpointing volumes are small, even the dimmest galaxies, which are missed out in incompletecatalogs, contribute to the variability between different fields of view. When follow-up frac-tions are large at small pointing volumes, the success rate asymptotes to the maximumpossible success fraction F → λ + f (1 − λ ), where λ is the catalog completeness , andincompleteness limits catalog utility. This happens for a 33%-complete catalog when thepointing volume is V < ∼
100 Mpc and the follow-up fraction is f > ∼ F is significantly larger than f for a large range of pointingvolumes, suggesting that even a moderately complete (33% complete) catalog is still usefulfor pointing at the sky region hosting the source of a GW transient.As discussed above, an incomplete catalog is most useful when it contains a high frac-tion of intrinsically luminous galaxies at the expense of missing galaxies with low absolutemagnitudes. Therefore, a simple flux limit is an optimistic model of a catalog’s incomplete-ness. If a catalog instead has a more gradual cut-off with apparent magnitude, its utilityfor a given completeness fraction λ could be lower than estimated here. For example, for f = 10%, the largest discrepancy between a 33%-complete flux-limited catalog and GWGCis at V ∼
20 Mpc , where they yield success fractions of F ∼
63% and
F ∼ and is below the Consider a situation in which the number of bright galaxies that enter the catalog is sufficiently smallthat all of them can be followed up with a negligible number of pointings; the remaining allowed pointingswill capture a fraction f of the other fields of view, which have no galaxies in the incomplete catalog andhave a uniform prior density 1 − λ painted across them. V = 1000 Mpc . -2 -1 Pointing volume (Mpc ) Su cce ss f r a c t i o n F f = % f = % f = % f = % FIG. 5: The success fraction F as a function of pointing volume for hypothetical flux-limitedcatalogs of 33% (dashed), 75% (dotted) and 100% (solid) completeness within the pointing volumebeing considered. Incomplete catalogs yield similar success fractions to complete catalogs exceptat small pointing volumes and large follow-up fractions. VI. CONCLUSION AND FUTURE WORK
Electromagnetic follow-up prospects in the advanced GW detector era can be aided by theuse of galaxy catalogs to direct follow-up surveys. The relevance of catalog-directed wide-field follow-ups is limited mostly by the modest spatial fluctuations of luminosity on the skyfor the large three-dimensional localization uncertainty volumes of the advanced-detectornetwork.We have shown in Figures 3 and 5 that the utility of a catalog depends on the vol-ume of individual telescope pointings and on the fractional coverage of the GW localizationarea. Catalogs are therefore most relevant for shallow and narrow follow-up searches, al-though narrow-field instruments are unlikely to follow up a sufficient fraction f of the GWlocalization region for a successful follow-up to be realistic (with the possible exception ofshort-range observations, where individual galaxies could be followed up). Loud, nearby GWtriggers are an obvious scenario where catalogs will be particularly useful. It is possible, forexample, that they confer as much as a four-fold increase in success fraction over a follow-up15hat does not use a catalog; c.f. case 3 in Section III. Similarly, follow-ups from shallowerGW searches – during the early commissioning phases of advanced detectors, for example –will also benefit from the use of catalogs.However, even for sources located at the median 200 Mpc distance expected for detec-tions with advanced-detector networks, we have shown that catalogs are still relevant forsufficiently small telescope fields of view. For example, a catalog might confer as much asa 70% increase in the probability of imaging the EM counterpart relative to a follow-upwithout the benefit of a catalog, as in case 2 of Section V.Realistic, incomplete galaxy catalogs are likely adequate for most follow-up campaigns.Metzger et al. [37] propose that a catalog complete to ∼
75% with respect to B-bandluminosity should be achievable. At f = 10%, a hypothetical flux-limited catalog of thiscompleteness concedes a fraction <
1% of the success fraction from a complete catalog forboth 100 Mpc and 1000 Mpc pointing volumes. Metzger et al. [37] suggest that it will bedifficult to construct a galaxy of more than ∼
33% completeness with respect to K-bandluminosity, a tracer of total mass. Even in this case, the fractional loss of success fractionrelative to a search with a complete catalog is small: 7 .
5% and 5% respectively for 100 Mpc and 1000 Mpc pointings. A. Imaging vs. identifying of the counterpart
Our study focuses on the probability of imaging the EM counterpart to a detected GWsignal – i.e., pointing a telescope so that the EM counterpart is within the field of view – butnot on the probability of detecting and identifying it among background sources. In reality,some telescopes may have trouble observing weak, distant EM counterparts [4].For example, Metzger and Berger [8] suggest that the orphan optical afterglow expected toaccompany a BNS merger at 200 Mpc will have a peak optical brightness as faint as ∼
23 magwhen viewed slightly off-axis: beyond the limiting flux of many telescopes. Even if theyare detected, contamination from background events may make it difficult to pick out thecorrect transient. Identification of GW EM counterparts among false positives is addressedby Nissanke et al. [30]. The detectability of EM counterparts could be further investigatedby considering the capabilities of specific telescopes given the observing requirements ofparticular sources (for example, their peak luminosities, light-curve evolution, etc.).16 . Astrophysical assumptions
We have made a number of assumptions about the astrophysics underlying BNS mergersignals:1.
B-band luminosity of the host galaxy – which traces its star formation rate – is a proxyfor the merger rate.
In fact, if there are long time delays between star formation andbinary merger, the total mass of the host galaxy, traced by K-band luminosity, mightbe the more relevant indicator of merger rate. For example, population synthesismodeling suggests that half of all BNS mergers may take place in elliptical galaxieswith little ongoing star formation [38, 39]. Meanwhile, observational evidence on shortgamma ray bursts indicates that about a quarter of them occur in elliptical galaxies[32], though selection effects associated with the detection of afterglows that allow thehost to be identified could influence this fraction.2.
The completeness of the galaxy catalog is known precisely.
In practice, the complete-ness of the catalog is estimated from the expected spatial luminosity density in thelocal Universe ( ∼ . L Mpc − for blue luminosity). Inaccuracy in the estimatedcompleteness may lead to a less-than-optimal ranking of tiles on the sky. We canaccount for the incompleteness of a catalog by changing the weighting we give toindividual galaxies; if the catalog completeness fraction is λ , then the catalogued lu-minosity of a given pixel, L i , is multiplied by λ when computing the prior, with a priorfraction 1 − λ painted uniformly over the entire GW sky uncertainty region to accountfor the galaxies missed in the catalog.3. Mergers are spatially coincident with host galaxies on the celestial sphere.
Natal kicksaccompanying supernovae that give birth to the neutron-star components of a binary(up to hundreds of km s − [40]) can combine to give a significant velocity to the binaryas a whole. As a result, mergers are distributed at larger distances from the galacticcenter than typical stellar concentrations [32], and galaxies should properly be treatedas extended objects rather than point sources. However, for telescope fields of view oforder a square degree or more and typical source distances of 100–200 Mpc, treatinggalaxy sizes < ∼
100 kpc as point sources will not affect our results. On the otherhand, binaries may be completely ejected from their host galaxies [e.g., 41], and some17raction of the“no-host” short gamma ray bursts [15, 42] may provide evidence for thispopulation of merging ejected binaries, which may be separated by more than a Mpcfrom their host galaxy (but see discussion in [43] and references therein).We suggest a future study of the importance of these effects – given our ignorance – asparameterized priors. One would allow nature to choose a true value of a given parameter(e.g. the relative contribution of blue and red luminosity tracers to merger rates) andattempt to image counterparts from the resulting GW events by ranking tiles according toan assumed parameter value representing our own knowledge. The effects of our ignoranceof the true values of each parameter could thus be described by a matrix in which onedimension represents nature’s choice of prior, and the other our assumed knowledge.
C. Coherent use of galaxy catalogs
Finally, we have investigated the utility of a galaxy catalog when applied to the skylocation posterior obtained from a parameter estimation pipeline. In practice, if a galaxycatalog were to be used for follow-up, it should be applied as a prior during coherent Bayesianparameter estimation [33]. Doing so would make it possible to consistently account for theprobability that a given galaxy hosts the GW source, which depends not only on the galaxyluminosity but also on the distance to the galaxy and the inclination and orientation ofthe binary, which must yield a GW signal amplitude consistent with observations. This isparticularly important when considering the correlations between the recovered GW signalparameters such as inclination and distance. Moreover, using coherent Bayesian parameterestimation would allow complex sky location posteriors could be accurately accounted for.
Acknowledgments
We thank Edo Berger, Marica Branchesi, Walter Del Pozzo, Will Farr, Jonah Kanner,Mansi Kasliwal, Erik Katsavounidis, Luke Kelley, Drew Keppel, Brian Metzger, TrevorSidery, and Alberto Vecchio for useful discussions. CH and IM are grateful to the KavliInstitute for Theoretical Physics and the organizers of the “Chirps, Mergers and Explosions:The Final Moments of Coalescing Compact Binaries” program, supported in part by theNational Science Foundation under Grant No. NSF PHY11-2591. They wish to thank18he participants for many fruitful discussions. This research was supported in part byPerimeter Institute for Theoretical Physics. Research at Perimeter Institute is supportedby the Government of Canada through Industry Canada and by the Province of Ontariothrough the Ministry of Research and Innovation. [1] J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, T. Accadia, F. Acernese,C. Adams, and et al., AAP , A124 (2012), 1109.3498.[2] J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, T. Accadia, F. Acernese,C. Adams, R. Adhikari, C. Affeldt, et al., AAP , A155 (2012), 1112.6005.[3] P. A. Evans, J. K. Fridriksson, N. Gehrels, J. Homan, J. P. Osborne, M. Siegel, A. Beardmore,P. Handbauer, J. Gelbord, J. A. Kennea, et al., ArXiv e-prints (2012), 1205.1124.[4] J. Aasi, J. Abadie, B. P. Abbott, R. Abbott, T. Abbott, M. R. Abernathy, T. Accadia,F. Acernese, and et al., ArXiv e-prints (2013), 1310.2314.[5] G. M. Harry and the LIGO Scientific Collaboration, Classical and Quantum Gravity ,084006 (2010).[6] Virgo Collaboration, Virgo Technical Report VIR-0027A-09 (2009).[7] J. Abadie et al. (LIGO Scientific Collaboration and Virgo Collaboration), Classical and Quan-tum Gravity , 173001 (2010).[8] B. D. Metzger and E. Berger, Astrophys. J. , 48 (2012).[9] L. Z. Kelley, I. Mandel, and E. Ramirez-Ruiz, Phys. Rev. D , 123004 (2013).[10] J. S. Bloom et al., ArXiv e-prints (2009), arXiv:0902.1527.[11] K. Cannon, R. Cariou, A. Chapman, M. Crispin-Ortuzar, N. Fotopoulos, M. Frei, C. Hanna,E. Kara, D. Keppel, L. Liao, et al., The Astrophysical Journal , 136 (2012).[12] J. Luan, S. Hooper, L. Wen, and Y. Chen, Phys. Rev. D , 102002 (2012).[13] E. Nakar, A. Gal-Yam, and D. B. Fox, ApJ , 281 (2006).[14] H. J. van Eerten and A. I. MacFadyen, ApJL , L37+ (2011).[15] E. Berger, ApJ , 1946 (2010).[16] D. A. Perley, B. D. Metzger, J. Granot, N. R. Butler, T. Sakamoto, E. Ramirez-Ruiz, A. J.Levan, J. S. Bloom, A. A. Miller, A. Bunker, et al., The Astrophysical Journal , 1871(2009).
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