Probabilistic Association of Transients to their Hosts (PATH)
Kshitij Aggarwal, Tamás Budavári, Adam T. Deller, Tarraneh Eftekhari, Clancy W. James, J. Xavier Prochaska, Shriharsh P. Tendulkar
DDraft version February 23, 2021
Typeset using L A TEX twocolumn style in AASTeX63
Probabilistic Association of Transients to their Hosts (PATH)
Kshitij Aggarwal ,
1, 2
Tam´as Budav´ari ,
3, 4
Adam T. Deller , Tarraneh Eftekhari , Clancy W. James , J. Xavier Prochaska ,
8, 9 and Shriharsh P. Tendulkar
10, 11 Department of Physics and Astronomy, West Virginia University, Morgantown, WV 26506, USA Center for Gravitational Waves and Cosmology, West Virginia University, Chestnut Ridge Research Building, Morgantown, WV, USA Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD, USA Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD, USA Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn, VIC 3122, Australia Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA 02138, USA International Centre for Radio Astronomy Research, Curtin University, Bentley, WA 6102, Australia University of California - Santa Cruz, 1156 High St., Santa Cruz, CA, USA 95064 Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan Department of Astronomy and Astrophysics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai,Maharashtra, 400005, India National Centre for Radio Astrophysics, Pune University Campus, Post Bag 3, Ganeshkhind, Pune, Maharashtra, 411007, India (Received February 23, 2021; Revised February 23, 2021; Accepted February 23, 2021)
Submitted to ApJABSTRACTWe introduce a new method to estimate the probability that an extragalactic transient source is asso-ciated with a candidate host galaxy. This approach relies solely on simple observables: sky coordinatesand their uncertainties, galaxy fluxes and angular sizes. The formalism invokes Bayes’ rule to calculatethe posterior probability P ( O i | x ) from the galaxy prior P ( O ), observables x , and an assumed model forthe true distribution of transients in/around their host galaxies. Using simulated transients placed inthe well-studied COSMOS field, we consider several agnostic and physically motivated priors and offsetdistributions to explore the method sensitivity. We then apply the methodology to the set of 13 fastradio bursts (FRBs) localized with an uncertainty of several arcseconds. Our methodology finds nineof these are securely associated to a single host galaxy, P ( O i | x ) > .
95. We examine the observed andintrinsic properties of these secure FRB hosts, recovering similar distributions as previous works. Fur-thermore, we find a strong correlation between the apparent magnitude of the securely identified hostgalaxies and the estimated cosmic dispersion measures of the corresponding FRBs, which results fromthe Macquart relation. Future work with FRBs will leverage this relation and other measures from thesecure hosts as priors for future associations. The methodology is generic to transient type, localizationerror, and image quality. We encourage its application to other transients where host galaxy associ-ations are critical to the science, e.g. gravitational wave events, gamma-ray bursts, and supernovae.We have encoded the technique in Python on GitHub: https://github.com/FRBs/astropath.
Keywords:
Galaxies: ISM, star formation – stars: general – Radio bursts – magnetars INTRODUCTIONTransient phenomena offer terrific potential to exploreastrophysical processes on the smallest scales and in the
Corresponding author: J. Xavier [email protected] most extreme conditions. This includes spectacular ex-plosions, intense magnetic fields, gravity in the stronglimit, and the structure of dark matter. Given the veryshort time-scales, the majority of these events are linkedto compact objects, e.g. neutron stars and black holes(e.g. Fishman & Meegan 1995; Gal-Yam 2019; Cordes &Chatterjee 2019), and therefore they give unique insight a r X i v : . [ a s t r o - ph . H E ] F e b FRB Association Team to the processes that generate and destroy these exoticbodies.The three-dimensional location of transient sources isa critical aspect influencing the interpretation of theirnature, allowing measured properties to be translatedinto absolute energetics and determining the nature oftheir environment. While many transient sources can bereasonably well-localised on the sky, depending on thenature of the discovery instrument (and any sufficientlyprompt follow-up), the third dimension of distance is of-ten challenging to obtain. For some transient phenom-ena — supernovae, the afterglows of gamma-ray bursts(GRBs) — spectra of their electromagnetic emission canbe used to identify their redshift (e.g. Blondin & Tonry2007; Fynbo et al. 2009). Many other transients, how-ever, encode no direct measure of the source redshift.This includes the enigmatic fast radio bursts (FRBs)whose dispersion measures (DMs) imply a cosmologicalorigin (Lorimer et al. 2007), yet do not provide a pre-cise redshift estimate (e.g. McQuinn 2014; Prochaska &Zheng 2019). Another example includes short-durationGRBs whose afterglows are too faint to record a highS/N spectrum for precise redshift estimation (e.g. Fonget al. 2010).When a redshift cannot be measured based on thetransient source itself, a suitable alternative is to asso-ciate the transient event with a galaxy and then measurethe galaxy redshift (e.g. Tendulkar et al. 2017). Thispresumes, of course, that the transient is generated in(or at least near) a galaxy – a reasonable assumptionfor compact objects which, aside from exotic and un-proven phenomena such as primordial black holes, areborn in the dense regions of galaxies. Some progenitorsof transient sources may travel considerable distancesfrom their birth sites, for instance via ‘kicks’ during for-mation events or disruptions of stellar multiples. Formost such cases, however, offsets of up to tens kpc orseveral arcseconds on the sky can be expected for distantevents (e.g. Fong & Berger 2013).The process of associating a transient to its hostgalaxy is a non-trivial exercise. Primarily, this is in-fluenced by the uncertainty in the transient localiza-tion combined with the relatively high surface densityof galaxies on the sky, meaning that the allowed regionfor the transient site can potentially overlap with multi-ple galaxies. The initially unknown origins of many suchphenomena — including the transients of our interest,FRBs — further compounds the problem by introduc-ing an uncertainty in the characteristic offset of a sourcefrom the centre of a galaxy.To date, host associations for transient sources havefocused on the probability of chance association. Bloom et al. (2002) introduced the concept of a chance proba-bility P c to ascertain the likelihood that a given galaxywas a coincident association to a transient event. Byinference, a galaxy with a very low P c value might beconsidered the host, while galaxies with P c ∼ • Be driven by simple observables (which are definedbelow). • Assign a posterior probability to every candidategalaxy in consideration. • Develop an extendable framework that can evolveas the field matures. This includes incorporationof additional observational constraints and priors. • Accommodate transients both in the local (hun-dred Mpc) and very distant universe. • Allow for insufficient data, e.g. the non-detectionof the host galaxy due to imaging depth.In the following, we strive to limit the analysis to theseeasily attainable, direct observables:1. The transient localization (RA, DEC, uncertaintyellipse): α FRB , δ FRB , (cid:15) FRB .2. The apparent magnitudes of the galaxy candi-dates: m i .3. The candidate galaxy coordinates: α i , δ i .4. The angular size of the galaxy candidates: φ i .Future work will consider additional observables (e.g.the FRB DM) and also priors based on “secure” hostassociations.This paper introducing the probabilistic associationof transients to hosts (PATH) is organized as follows:section 2 briefly reviews the historical approaches to as-sociations, section 3 introduces our new method, sec-tion 4 defines the priors adopted for our FRB analysis,section 5 presents analysis of simulated transients, and RB host galaxy associations astropy . Figure 1.
Cutout g -band image (VLT/FORS2) centeredon FRB 180924 (small red circle indicates the 5 σ local-ization). There are four extended sources with separation θ < (cid:48)(cid:48) marked as candidates for the host galaxy of theFRB. These are labeled (just above them, except for thebrightest) by their θ , angular size φ , apparent magnitude m ,and the chance probability P c of an association.2. HISTORICAL: CHANCE PROBABILITYA standard approach to associating transients to theirhost galaxies is through assessing the chance probabil-ities P c of galaxies being located close to the transientposition. We reintroduce the formalism for evaluating P c here, propose a new variant, and comment furtheron its application and limitations.2.1. Formalism
Figure 1 shows a VLT/FORS2 g -band cutout image(30 (cid:48)(cid:48) × (cid:48)(cid:48) ) of the field surrounding FRB 180924 withits localization marked by a red circle. As describedby Bannister et al. (2019), this localization lies ≈ (cid:48)(cid:48) from the centroid of galaxy DES J214425.25 − (cid:48)(cid:48) of two additional galaxies. At their red-shifts (Bannister et al. 2019), the galaxies all have pro-jected separations of less than 40 kpc, i.e. separationsless than the estimated radii of their halos. Therefore,while one may be predisposed to assign FRB 180924 to the brighter and closer galaxy, one should also entertainthe possibility that the FRB occurred in the stellar haloof one of the others.The chance probability approach is powerfully simple:estimate the Poisson probability of finding one or moregalaxies as bright or brighter within an effective searcharea A around the FRB, with A determined from the an-gular size φ , the separation θ , and the FRB localizationuncertainty σ FRB . Namely, one defines the probabilityof a chance coincidence P c ( θ eff , m ) = 1 − exp( − ¯ N ) , (1)where ¯ N is the average number of sources in A . It isgiven by ¯ N ( θ eff , m i ) = πθ Σ( m i ) , (2)where Σ( m ) is the angular surface density of galaxies onthe sky with magnitude m ≤ m i , and θ eff is the effectivesearch radius ( A = πθ ). For the former, we adopt thegalaxy number count distribution of Driver et al. (2016),while the latter quantity bears some arbitrariness.Previous works have considered several definitions for θ eff . We introduce yet another more conservative onehere: the quadrature sum of all three angular quantitieswith semi-arbitrary weightings, θ eff = (cid:113) σ + θ + 4 φ . (3)Adopting these, we estimate P c = 0 .
01 forDES J214425.25 − P c > . − P c (e.g. FRB 181112; Prochaska et al.2019), one has no means to favor one over the other.Last, the P c formalism does not naturally allow one tointroduce additional observational measures as evidence(e.g. DM). Together, these considerations motivate ourdevelopment of a full probabilistic treatment.2.2. Nuisances and Nuances
There are several aspects of the P c analysis that re-quire further definition. For completeness, we describethese here although P c does not formally enter into thenew formalism. First, one requires measurements of thegalaxy centroids and angular size. We advocate a non-parametric approach owing to the complexity of galaxy FRB Association Team morphology. In the following, we adopt the centroidingalgorithm encoded in the photutils software package,and use the semimajor axis sigma parameter to esti-mate the angular size. In the following, we will refer it as a image as it matches the definition of that parameterin the more widely used sextractor package.Another issue is Galactic extinction. FRBs are de-tected across the sky including on sightlines that showlarge Galactic extinction ( E ( B − V ) > . r -band, which provide a trade-off betweenthe mapping of stellar content and extinction. We fur-ther recommend an image depth of m r = 25 . σ ), cor-responding to the limiting magnitude for spectroscopy,and sufficient for probing 0 . L ∗ galaxies at z ∼ . (cid:48)(cid:48) seeing, anda > (cid:48) field of view for background estimation.Similarly, source detection should employ a non-parametric approach to accommodate galaxies of vary-ing morphologies. For consistency with this work, werecommend use of the a image parameter to estimatethe angular size of sources, and note that all galaxiesshould be corrected for Galactic extinction prior to ap-plying the Bayesian formalism.The formalism of eq.(1) and (2) ignores galaxy clus-tering. Since matter in the Universe is not uniformlydistributed, the probability of observing either no galax-ies, or a large number of them, in proximity to a randomdirection is enhanced compared to the probability of ob-serving one or a few. Tunnicliffe et al. (2014) show thatincluding clustering decreases the probability of a nearbyrandom galaxy by 25–50% in the case of a random di-rection. Our primary concern however is not whether ornot all the observed galaxies are merely chance coinci-dences, but rather which of the observed galaxies is thetrue host. Clustering is discussed further in this con-text in section 4.2. For now, we remark that given thatFRBs truly are associated with galaxies, clustering actsto increase, not decrease, the probability of a chance as-sociation, since the FRB observation has preferentiallyselected a direction of the Universe in which there is acluster of matter.Furthermore, in this work we ignore for the sake ofsimplicity the ellipticity of the prospective host galax- ies. Our method will be readily adaptable however tosuch ellipticity, or indeed arbitrarily complex functions,since the approach described below does not rely on anyparticular functional forms. PROBABILISTIC APPROACHAssociation of transients to galaxies is not like theusual cross-identification for which probabilistic meth-ods have been in place for over a decade (Budav´ari &Szalay 2008). Matching stars and galaxies typically in-volves asking whether a set of detections (across separateexposures, instruments, telescopes) are of the same ce-lestial object. If they are, their true (latent) directionswould have to be the same; see more in the review byBudav´ari & Loredo (2015).In strong contrast with that, FRBs are presumed tosimply originate from within (or at least near to) galax-ies, hence their true direction should not be requiredto coincide with the center of a galaxy. While this isadmittedly a small difference for the faintest galaxies,resolved extragalactic source are expected to yield dif-ferent results. Here we consider a general scenario wherethe shape of galaxies can be incorporated along with ageometric model about from where FRBs would origi-nate within or around galaxies.3.1.
General Formalism
Since FRBs are sparse on the sky, we can study themseparately, which also simplifies the following descrip-tion of our approach. Let us consider a catalog of galax-ies across the entire sky and a single FRB that eitherbelongs to one of the many catalog objects (its hostgalaxy) or it does not, i.e. its host is not detected ornot included in the catalog. If U is the event that theFRB’s host is unseen, and O i is the event that the FRBis from galaxy i , their prior probabilities must add upto one, P ( U ) + (cid:88) i P ( O i ) = 1 , (4)as there are no other possibilities. The single scalarquantity P ( U ) encodes all the complications that arisefrom the difference in the radial selection functions ofthe catalog and the FRB instruments. For now, we as-sume its value to be known, but note that it could beinferred in a hierarchical fashion when considering mul-tiple FRBs. Also, one could assume a uniform prior forall observable O i as the simplest possible scenario thatessentially ignores any additional information about thegalaxies, e.g., magnitude, color, redshift. We leave suchconsiderations to a future work.Given a vector x representing all measured propertiesof the detected FRB, we ask what the posterior proba-bilities P ( U | x ) and P ( O i | x ) are for all i . Using Bayes’ RB host galaxy associations P ( U | x ) = P ( U ) p ( x | U ) p ( x ) , (5)where p ( x | U ) is the probability density of the FRB prop-erties given the host is unseen. From hereon, we consider x to represent only the measured FRB direction. With-out constraints, x could be anywhere on the sky, henceit is natural to assume a uniform (isotropic) distributionwith a value of 1 (cid:14) π . Similarly, the posterior probabilityfor object i is P ( O i | x ) = P ( O i ) p ( x | O i ) p ( x ) , (6)where p ( x | O i ) is the probability density function (PDF)of x given that the FRB comes from galaxy i . With data x , this is the marginal likelihood of O i , which includesthe galaxy geometry and the uncertainty of the FRBdirection. This key component of the approach is dis-cussed in depth in the next paragraph. The normalizingconstant must be p ( x ) = P ( U ) p ( x | U ) + (cid:88) i P ( O i ) p ( x | O i ) (7)to guarantee that these posteriors also add up to one, P ( U | x ) + (cid:88) i P ( O i | x ) = 1 . (8)3.2. Marginal Likelihoods
Let the 3-D unit vector ω represent the true and un-known direction of the FRB on the sky. Given thatit comes from a particular galaxy, the direction ω hasto point somewhere near the host. The function p ( ω | O i )captures the physical and geometric model for the FRBsspecific to galaxy i , e.g., taking into account its type,distance, orientation, etc.The observed FRB direction x is a measurement of ω with known uncertainty, represented by the localiza-tion error function, L ( x − ω ). Given p ( ω | O i ), p ( x | O i )is calculated by integrating over the ω model directionsto obtain the marginalized likelihood of the associationhypothesis O i , p ( x | O i ) = (cid:90) dω p ( ω | O i ) L ( x − ω ) , (9)which now accounts for all possibilities in various FRBorigins as well as the astrometric uncertainty in themeasurement. If a galaxy is unresolved, p ( ω | O i ) maybecome the Dirac- δ , and p ( x | O i ) is just the astromet-ric uncertainty, as it would be the case for matchingpoint sources. Calculating the above quantity for all i completes the framework, which now provides posteriorprobabilities via eq. 6. 3.3. Limited Field of View
Previously we assumed a galaxy catalog over the entiresky, but catalogs typically have more limited footprints.It is interesting to think about a scenario when the fieldof view Ω is smaller but still large enough not to missany possible counterparts to the FRB in question. Intu-itively, galaxies very far away from the FRB should nothave any effect on the association analysis.Smaller sky coverage would mean fewer observedgalaxies N c , which in turn affect the P ( O i ) priors asthere are fewer galaxies to choose from. Going with theprevious uniform assumption, eq.(4) implies P ( O i ) = 1 − P ( U ) N c , (10)which captures the dependence.Looking now back at Bayes’ rule in eq.(5), the sky cov-erage seems to affect the P ( O i | x ) posteriors, too. For-tunately, this is not the case. The denominator p ( x )changes in accord due to the scaling in p ( x | U ), which isuniform over the field of view, p ( x | U ) = 1 Ω ( x )Ω (11)where the 1 Ω ( x ) is the indicator function that takes thevalue 1 if x is within the field of view and 0 otherwise,and the denominator Ω normalizes the PDF. The frame-work not only matches common-sense expectations, butprovides for more efficient computation where only can-didates within close proximity of FRBs are considered.Given the plethora of deep, wide-field imaging sur-veys, it is possible to consider very large Ω for eachFRB in the analysis. In practice, however, sensible andconservative assumptions for p ( ω | O i ) will greatly limitthe list of viable candidates O i . To ease the analysis,we adopt for Ω the union of the area encompassing allgalaxies within 10 half-light radii and the 99.9% FRBlocalization. As emphasized in the previous section, itis only necessary to consider a large enough area to becertain to include all possible hosts. PRIORS AND ASSUMPTIONS4.1.
Undetected Prior P ( U )It is difficult to a priori assign a prior P ( U ) to theprobability that the FRB host is undetected. Undoubt-edly, P ( U ) is related to the depth of the imaging and the(unknown) source distance, i.e. redshift . In the follow-ing, we consider an arbitrarily assigned value, and we Future work may use the observed DM to inform the host dis-tance and P ( U ). FRB Association Team advocate a low value, based on the paucity of our dataand the set of confidently assigned associations reportedto date (e.g. Heintz et al. 2020). In the analyses thatfollow, we typically assume P ( U ) = 0 (Occam’s razor!)and discuss the impacts of increasing it.4.2. Candidate priors P ( O )For the set of host candidates O i , absent any assump-tions on the distance or the typical separations of FRBsfrom their host galaxies, any galaxy on the sky couldbe a viable candidate. Given the plethora of poten-tial models for FRB progenitors and the limited existingconstraints, we are motivated to consider (for now) sim-ple approaches to the prior for the galaxy candidates.The most agnostic approach is to assign an “identical”prior to every galaxy in consideration, i.e. eq. 10.Inspired by the chance probability calculations ap-proach described in §
2, we introduce an additional priorbased on P c . Specifically, we consider a prior that in-versely weights by Σ( m ). P ( O i ) ∝ m i ) . (12)For this “inverse” prior, brighter candidates have higherprior probability according to their number density onthe sky. The normalization of these priors is set by eq. 4.In section 5 we also briefly consider two other priors with P ( O i ) ∼ / Σ( m i ) φ (inverse1) and P ( O i ) ∼ / Σ( m i ) φ (inverse2). We adopt the prior in Equation 12 in partbecause of its simpler form and also because of the re-sults of simulated experiments (Section 5).Figure 2 illustrates P ( O i ) for the two models for twoexample cases – (a) FRB 180924 and (b) FRB 190523– where we assume P ( U ) = 0. For this illustration,we have restricted the analysis to galaxies within 10 (cid:48)(cid:48) ofthe FRB, which captures all of the viable candidates. Asexpected, the inverse priors for FRB 180924 significantlyfavor the brighter galaxy closest to the FRB. Perhapsless intuitively, the inverse priors favor the most distant(yet brightest) source near FRB 190523. This motivatesthe inclusion of the next ingredient — the offset function p ( ω | O i ). 4.3. Offset function p ( ω | O i )The p ( ω | O i ) function for the probability of the trueangular offset from the galaxy is unknown yet requiredfor the analysis. In our formalism, we develop priorsfor p ( ω | O i ) based solely on the angular offset θ betweenthe galaxy centroid and ω , and also normalized by theobserved galaxy’s angular size φ . This simultaneouslyaccounts for galaxies of different intrinsic size and dif-fering observed size owing to their distance. As the predominance of models associate FRBs to stel-lar sources or compact objects (AGN at the very cen-ters of galaxies are currently disfavored; Bhandari et al.2020a), one might expect the FRB events to track thestellar light. While we wish to remain largely agnostic tothe underlying distribution of offsets, we are physicallymotivated to presume p ( ω | O i ) decreases with increas-ing θ . We assert this despite the fact that geometri-cal considerations do favor large ω , e.g. a model whereFRBs occur with identical probability anywhere in a cir-cular galaxy will have p ( ω | O i ) ∝ ω until one reachesthe “edge” of the galaxy. Therefore, a uniform prior p ( ω | O i ) = 1 /πθ is formally one that assumes FRBsoccur proportional to the galaxy radius. The other twomodels considered are a core model: p ( ω | O i ) = 12 πφ [ θ max /φ − log( θ max /φ + 1)] 1( θ/φ ) + 1 , (13)which implies an approximately 1 /r weighting, and an exponential model p ( ω | O i ) = 12 πφ [1 − (1 + θ max /φ ) exp( − θ max /φ )] exp[ − θ/φ ] , (14)which assumes an underlying exponential distribution.All of these functions are normalized to unity when in-tegrating to θ max , ignoring the curvature of the sky.For all of the p ( ω | O i ) priors, we assert a maximumoffset θ max = 6 φ ; this is especially important for theuniform prior. This value is arbitrary and was chosento be large enough to accommodate prevailing modelsof FRBs without being too conservative.Applying an arbitrary cutoff to the exponential distri-bution is not strictly necessary, but we keep it for sim-plicity and consistency, and demonstrate in Section 6.3that the results for an exponential distribution are in-sensitive to this choice.Figure 3 shows the offset functions, normalized tohave the same total probability. Clearly the exponen-tial model favors FRBs located in the inner regions ofgalaxies. For comparison to the offset distributions ofknown transients, we note that long GRBs appear highlyconcentrated in the inner regions of their hosts relativeto Type Ib/c and IIn supernovae, which occur preferen-tially near their host half-light radii (Lunnan et al. 2015;Blanchard et al. 2016). Conversely, short GRBs exhibitsignificant offsets from their host centers, indicative ofprogenitors born in compact object mergers (Fong &Berger 2013). RB host galaxy associations Figure 2.
Illustration of the two approaches to candidate priors P ( O ) assumed in this manuscript: identical and inverse.The former assumes an identical prior for every galaxy with θ i < φ or within the 99.9% localization error of the FRB. Thelatter adopts the inverse of the estimated chance probability P c of these galaxies which is equivalent to asserting the prior isproportional to the integrated probability that the other sources are all chance coincidences. In practice, we treat the galaxies as “round”, i.e. ig-noring for now any ellipticity. Future works will advancethis aspect. SIMULATIONSTo explore the formalism introduced here, we havegenerated Monte Carlo simulations designed to faith-fully reproduce the FRB experiment. We describe thisfirst and then detail the results.5.1.
Sandboxes
Our Monte Carlo approach leverages the public cata-log of the
HST /COSMOS field (Scoville et al. 2007b,a)which provides over 1 million sources at high spatial res-olution and faint fluxes (median AB magnitude m r ≈ . ≈ z < . a image ) and apparent magnitudes for the catalog,restricted to the sources labeled as galaxies. We generate a series Monte Carlo realizations of FRBs(referred to as a sandboxes, or SBs), using the followingrecipe:1. Define the true distribution of FRBs from theirhost galaxies p ( ω | O i )and magnitude m r .2. Define a sample of potential host galaxies basedon m r .3. Define the distribution of localization errors forFRBs ( σ FRB ).4. Draw N FRB galaxies from the parent sample ofpotential host galaxies without duplicates.5. Set the true FRB positions according to p ( ω | O i ).6. Offset the FRBs to an observed coordinate accord-ing to σ FRB .7. Consider catalogue galaxies within 30 (cid:48)(cid:48) to repre-sent an image.
FRB Association Team
Figure 3.
Three offset functions p ( ω | O i ) considered forthe underlying angular distribution of FRBs relative to theirhost galaxies, normalized by the galaxy’s half-light radius φ . Each is normalized to have identical integrated area to amaximum assumed offset of θ max = 6 φ . Figure 4.
Distribution of angular sizes ( a image parame-ter) versus apparent magnitude m r for all of the galaxies inthe COSMOS catalog (Scoville et al. 2007a). For sandbox 5 (SB-5), 10% of the FRBs were ran-domly placed in the COSMOS field, i.e without a hostgalaxy. We plan to use this sandbox to evaluate theperformance of the framework when the host galaxiesare unseen. Also, as COSMOS is a deep survey, we gen-erate a magnitude-limited catalog of galaxies for each
Table 1.
SandBoxes
Label p ( ω | O i ) N FRB
Sample σ FRB
Catalog Filter( (cid:48)(cid:48) )SB-1 U (0 , φ ) 100,000 - 1 -SB-2 U (0 , φ ) 46,699 m r = [20 , U (0 . , m r ≤ core m r = [20 , U (0 . , m r ≤ exponential m r = [20 , U (0 . , m r ≤ a U (0 , φ ) 50,000 m r = [20 , U (0 . , m r ≤ a See text for details regarding the FRB selection for this sandbox. sandbox (last column of Table 1) on which we run theBayesian framework. In the following sub-section, wediscuss results for five sandboxes (focusing primarily onSB-1) with the parameters described in Table 1.
Table 2 . Sandbox Analysis
Sandbox P ( O ) P ( U ) p ( ω | O i ) θ max / φ f(T+secure) TPSB-1 inverse 0 exp 6 0.33 0.96SB-1 inverse1 0 exp 6 0.30 0.99SB-1 inverse2 0 exp 6 0.32 1.00SB-1 identical 0 uniform 6 0.22 1.00SB-1 inverse 0.05 exp 6 0.24 0.96SB-2 inverse 0 uniform 2 0.86 1.00SB-3 inverse 0 core 6 0.58 1.00SB-4 inverse 0 exp 6 0.68 0.99SB-5 inverse 0.10 exp 6 0.58 0.98 Analysis and Results
We now analyze the sandboxes listed in Table 1 witha variety of priors and assumed p ( ω | O i ) functions (thatgenerally do not match the true p ( ω | O i )). Table 2 liststhe various priors assumed for each analysis performed.Figure 5a shows the posteriors for the candidates usingthe fiducial sandbox (SB-1) and listed in row 1 of Table 2(also referred to as the adopted prior set; Table 3). Sincemost of the candidates defined in step 7 are very far fromthe offset FRB position, we restrict results to the ≈ P ( O i | x ) > .
01. This distribution ismulti-modal, with the overwhelming majority of recov-ered P ( O i | x ) ≈ P ( O i | x ) ≈ ≈
35% of the FRBshave a high probability, P ( O i | x ) > .
95 which we adopt
RB host galaxy associations Figure 5.
Analysis of SB-1: (a) PDF for the posteriorprobabilities for all of the candidates with P ( O i | x ) > . P ( O i | x ) < P ( O i | x ) > ≈
35% of the sourceshas P ( O i | x ) > .
95, which we define as secure. as a “secure” association. Adopting such an arbitraryvalue to define ‘secure’ is useful for including/excludingcandidates for subsequent analyses that rely on know-ing the correct FRB host, e.g. that by Macquart et al.(2020). However, we emphasise that it is in-general bet-ter to consider all host associations as uncertain, with different levels of certainty according to the obtainedposteriors.
Figure 6.
The points are equal number (10,000) binsof FRBs according to the maximum posterior probability oftheir host candidates for SB-1. For each set of 10,000 we de-termined the fraction of correct assignments if one adoptsthe candidate with max P ( O i | x ) as the host. The one-to-one line indicates a perfectly calibrated algorithm. Thecolors indicate different choices for the prior P ( O ): (blue) P ( O ) ∝ / Σ( m ), inverse; (orange) P ( O ) ∝ / Σ( m ) φ , in-verse1; (green) P ( O ) ∝ / Σ( m ) φ , inverse2. Our adoptedprior (inverse) is relatively well normalized in that P ( O i | x )yields an accurate estimate of the fraction of FRBs correctlyassigned to their host galaxy. Figure 6 evaluates, in 10 bins of equal number ofFRBs, the maximum P ( O i | x ) assigned to a candidatefor each FRB and the percentage of correct associationsassuming this is the host. The different colors indicatedifferent choices for P ( O ). Our adopted “inverse” priorappears well calibrated in that P ( O i | x ) yields an accu-rate estimate of the fraction of FRBs correctly assignedto their host galaxy.Figure 8 shows another set of results but for moredifferent choices of priors (Table 2). The remarkablyclose correspondence between the two quantities indi-cates the posterior is well-calibrated, at least for thispairing of sandbox and prior set. We also show resultsfor prior sets where we assume P ( U ) = 0 .
05 and forthe conservative prior set (Table 3). Each of these as-signs systematically lower values to the true host galaxyyielding a higher percentage of correct cases at lowermaximum P ( O i | x ).To characterise the behaviour of our method underdifferent simulated truths (i.e., sandboxes) and differentpriors, Table 2 lists the fraction f of all FRBs which0 FRB Association Team are correctly identified, i.e. the true (T) host is securelyidentified ( P ( O i | x ) > . ± . f ( T + secure)), with theanalysis method on a given sandbox having a secondaryeffect (variation of ± . f ( T + secure) is much higher for the SB-4 analysis thanfor SB-2 and SB-3, despite all three using an assumed p ( ω | O i ) of equal shape to the true p ( ω | O i ), and beingotherwise identical. However, while the SB-4 analysisassumes p ( ω | O i ) to be uniform out to θ = 6 φ , the truedistribution is fully contained within 2 φ , unlike SB-2and SB-3. We thus conclude that the dominant deter-minant of the fraction of securely (and hence, correctly)identified FRB hosts, in the case that the true host isobserved, is the fraction of FRBs lying in close proximityto their hosts, irrespective of other considerations. Wefind it especially reassuring that results are not highlysensitive to the analysis method, i.e. that our formalismyields greater sensitivity to the physical truth than toour choice of reasonable priors.What about unseen hosts? If we use P ( U ) = 0, astypically assumed in this work, then P ( U | x ) = 0 al-ways, and the method will tend to assign the highestposterior P ( O i | x ) to the closest galaxy regardless of dis-tance. Using the 10% of hostless FRBs from SB-5, theconservative and adopted prior sets from Table 3 findsecure associations for the majority of FRBs (55% and62% respectively). However, the typical radial offset forthese secure associations is very large. In Figure 7, weshow the cumulative distribution of θ/φ for such can-didates. The probability of the most likely candidatebeing close to the FRB is small, with 10% or less ofsuch falsely identified hosts having θ/φ <
6. The dis-tributions for secure associations is almost identical tothat from non-secure hosts. We conclude that measur-ing a small θ/φ is a strong discriminant against unseenhosts irrespective of p ( U ).Buoyed by these results, we now proceed to applyPATH to real FRB observations. REAL FRB ANALYSIS AND RESULTSInformed by the results of the previous section, weproceed to apply the formalism to all of the published,well-localized FRBs. We discuss results for two sets of
Figure 7.
Cumulative distribution of θ/φ for candidategalaxies in SB-5 where the true host is unseen, using con-servative (green) and adopted (blue) priors. Lower panel isa zoom-in on the region related to our adopted maximumoffset.
Table 3.
Prior Sets
Set P ( O ) P ( U ) p ( ω | O i ) θ max / φ Conservative identical 0 uniform 6Adopted inverse 0 exp 6 priors — conservative and adopted — as summarized inTable 3. We refer to the first as conservative because allgalaxies within θ max are given equal prior.6.1. FRB Host Candidates
Central to the analysis is the identification and analy-sis of galaxy candidates in imaging data. The first step— source identification — is the most challenging andthe most subjective. For every image, sources near the
RB host galaxy associations Table 4.
FRBs Analyzed
FRB α FRB δ FRB (cid:15) a (cid:15) b (cid:15) PA Filter(deg) (deg) ( (cid:48)(cid:48) ) ( (cid:48)(cid:48) ) (deg)FRB121102 82.99458 33.14792 0.10 0.10 0.0 GMOS N iFRB180916 29.50313 65.71675 0.00 0.00 0.0 GMOS N rFRB180924 326.10523 -40.90003 0.11 0.09 0.0 VLT FORS2 gFRB181112 327.34846 -52.97093 3.25 0.81 120.2 VLT FORS2 IFRB190102 322.41567 -79.47569 0.54 0.47 0.0 VLT FORS2 IFRB190523 207.06500 72.46972 4.00 1.50 340.0 LRIS RFRB190608 334.01987 -7.89825 0.26 0.25 90.0 VLT FORS2 IFRB190611 320.74546 -79.39758 0.67 0.67 0.0 GMOS S iFRB190614 65.07552 73.70674 0.80 0.40 67.0 LRIS IFRB190711 329.41950 -80.35800 0.40 0.31 90.0 GMOS S iFRB190714 183.97967 -13.02103 0.36 0.22 90.0 VLT FORS2 IFRB191001 323.35155 -54.74774 0.17 0.13 90.0 VLT FORS2 IFRB200430 229.70642 12.37689 1.07 0.30 0.0 LRIS I
Note — (cid:15) a , (cid:15) b , (cid:15) PA define the total 1 σ error ellipse for the FRB localizationData are taken from Ravi et al. (2019); Day et al. (2020); Law et al. (2020);Tendulkar et al. (2017); Marcote et al. (2020); Heintz et al. (2020). Figure 8.
Same as Figure 6 but for different prior assump-tions as described in the legend. detection limit are subject to the precise methodology:background subtraction, thresholding, pixel grouping,and deblending. After experimenting with the routinesencoded in the photutils package, we settled on thefollowing key parameters: npixels = 9, deblend =True, xy kernel = (3 , Gaussian2Dkernel , nsig = 3 . (ker-nel), nsig = 1 . background = (50 , filter size = (3 , DETECT MINAREA = 9 and
DETECT THRES = 1 . photutils parameters. Images are filtered with the de-fault convolution kernel ( default.conv ). To recoverblended sources (see e.g., FRB 180924 below), we set DEBLEND MINCONT = 0 . ≈ ≈ photutils and SExtractor method-ologies may be driving these discrepancies; namely,while SExtractor uses a multi-thresholding deblendingtechnique, photutils utilizes a combination of multi-thresholding and watershed segmentation. Further-more, the default.conv convolution kernel is equivalentto a 3 × photutils parameters wouldlead to the non-detection of the fainter sources. Thishighlights the subjectivity of source identification thatcan affect the final results.The source identification packages offer an assessmentof the source shape (e.g. ellipticity and size) which canbe used to used to select and then ignore Galactic stars.For the analysis that follows, we have simply clippedbright stars according to their apparent magnitudeswhen necessary.2 FRB Association Team
Figure 9.
Segmentation images for the sources in 30 (cid:48)(cid:48) cutouts around FRB 180924 (left) and FRB 190523 (right).The multitude of sources (several erroneous) at the top ofthe FRB180924 image is due to artefacts from a very brightstar.
Provided with the segmentation map, one may per-form aperture photometry and estimate φ from the de-rived elliptical apertures. All of the measurements forthe galaxy candidates are provided in Table 5.6.2. FRB Assignments
We then applied the PATH framework. The pri-mary results are displayed in Figure 10 which sum-marizes the P ( O i | x ) values for the candidates in eachfield for each set of priors. We find very similar re-sults for the two prior sets with one obvious excep-tion: FRB 180924. In this case, there are two galaxieswith separation θ < θ max from this precisely localizedFRB. These are treated similarly by the conservativeapproach. We demonstrate below, however, that a uni-form p ( ω | O i ) function with θ max = 6 φ is disfavored bythe data. Imposing the exponential offset model yields ahigher P ( O i | x ) for the primary candidate. Furthermore,if we allow for the great difference in apparent magni-tude by invoking the inverse P ( O ) prior, the posterior P ( O i | x ) raises to near unity for the host reported byBannister et al. (2019).Based on the results from analysis of mock fields ( § P secure = 0 .
95 abovewhich we consider a host association to be highly secure.The results in Figure 10 indicate that nine of the FRBsare associated to a single galaxy with P ( O i | x ) > . P ( O i | x ) ≈ . − . z > . P ( O i | x ) < P secure . Given theterrific scientific value of probing such halos with FRBs(Prochaska et al. 2019), one may need to introduce addi-tional criteria/priors to confidently pursue this science.The next, non-secure FRB association is FRB 190523whose larger localization error incorporates several can-didates. The analysis, however, does favor the pur-ported host reported by Ravi et al. (2019). Third isFRB 190614 which lies near ( (cid:46) (cid:48)(cid:48) ) two faint galaxieswith unknown redshifts (Law et al. 2020). As the FRBwith the highest DM and therefore the highest presumedredshift of the sample, this result emphasizes the likelychallenges of associating high- z FRBs to galaxies. Inparticular, given the host itself is likely faint, the inci-dence of additional, chance associations with compara-ble P ( O i | x ) will be higher. Last is FRB 191001 whichsits next to two bright galaxies known to have a commonredshift (Bhandari et al. 2020b). Therefore, the redshiftof the FRB is secure but the host offset and its inter-nal properties (e.g. stellar mass) are currently based onthe assumption that the closer galaxy is the host and,indeed, it exhibits a 3 × higher P ( O i | x ) value for theadopted prior set.6.3. Towards Additional Priors
Having established a set of nine secure, P ( O i | x ) >P secure = 0 .
95, host associations we may test the as-sumed p ( ω | O i ) functions imposed in the analysis. Fig-ure 11 shows the offset distribution for the secure hostsfor the three p ( ω | O i ) priors of the analysis ( § p ( ω | O i ) could in-clude/exclude FRBs as being secure. The figure alsoshows the values of θ/φ derived for all candidates fromthe full set of FRBs, where we have weighted the θ/φ value of each candidate by P ( O i | x ). Overall, the posteri-ors lend reasonable credibility to the set of p ( ω | O i ) func-tions. On the other hand, a comparison of the secure dis-tribution with the priors yields a one-sided Kolmogorov-Smirnov probability P KS (cid:46) . θ max = 6 φ at > p ( ω | O i ) function that favors a central concentra-tion for FRB locations. Additionally, such small valuesof θ/φ are unlikely when the true host galaxy is unseen(see Figure 7), being <
3% for the seven secure hostswith θ/φ < .
5. Since all FRBs have most likely can-didates with θ/φ <
5, we conclude that no more thanone of the FRBs considered can have an unseen host( p (cid:46) . RB host galaxy associations Figure 10.
Posterior probabilities P ( O i | x ) for the most likely host (solid bars) and all other candidates (open bars) with P ( O i | x ) > P secure = 0 .
95, as a function of prior set (green = conservative; black=adopted). With P secure = 0 .
95 as theprobability for a secure association, there are currently nine FRBs satisfying this criterion using the adopted prior set. Thenon-secure hosts occur for a variety of reasons as described in the text.
Encoded in every FRB is its dispersion measure (DM),the path integral of free electrons along the sightlineweighted by the cosmological scale factor. The first ≈ cosmic withthe point-size proportional to P ( O i | x ). For DM cosmic ,we adopt a simple estimation:DM cosmic = DM FRB − DM MW , ISM −
100 (15) with DM MW , ISM the estimated ISM dispersion measure(Cordes & Lazio 2002) and the factor of 100 DM unitsaccounts for the Galactic halo and the host galaxy (seeProchaska & Zheng 2019; Macquart et al. 2020). The lo-cus of data exhibits a clear correlation reflecting the de-crease in observed galaxy flux with increasing distance.Converting the DM cosmic estimates to redshift , wemay convert a given L r luminosity to m r ; this is illus-trated as the black curve in Figure 13 where we assumeda fiducial L = L ∗ / P ( m r | DM) toinclude in the analysis. This will, however, be subject toscatter in DM cosmic (Macquart et al. 2020), DM host , andthe intrinsic luminosities of the host galaxy population(Figure 12). It will also be subject, however, to theS/N considerations that affect any prior related to DM(James et al. 2020). https://github.com/FRBs/FRB/blob/main/docs/nb/DM cosmic.ipynb FRB Association Team
Figure 11.
The solid histogram shows the distribution of separations for the secure host galaxies, in units of their angularsize ( φ ). The gray histogram is for all of the candidate galaxies but weighted by their posterior probabilities P ( O i | x ). Theseresults were derived by assuming the adopted prior set (Table 3) and by varying the offset function p ( ω | O i ), as labeled in eachpanel. Overplotted on the histogram is the offset function both before (semi-transparent) and after convolving with the FRBlocalization error (solid). The data rule out the uniform offset function that extends to θ max = 6 φ at ≈ Table 5 . Results for FRB Associations
FRB RA cand
Dec cand θ φ m
Filter P c P ( O ) P ( O | x ) P ( U ) P ( U | x )ConservativeFRB121102 82.9945 33 . . Table 5 continued
RB host galaxy associations Figure 12. ( top ) Scatter plot of apparent magnitudesversus redshift for the nine host galaxies. These show an ex-pected decrease in flux with increasing distance. The dashedline marks the approximate apparent magnitude for an L ∗ galaxy. ( bottom ) Estimated galaxy luminosity relative to thecharacteristic luminosity L ∗ at the host redshift. The securehosts have a median L/L ∗ ≈ / Table 5 continued FRB Association Team
Figure 13.
Using the adopted prior set (Table 3), we show the apparent magnitudes of each candidate with P ( O i | x ) > . P ( O i | x ). These are plotted against DM cosmic , an estimate of the dispersion measure of the FRB due tothe cosmic web (see text for details). The black curve shows the estimated m r for an L = L ∗ / cosmic to redshift (using the Macquart relation). The fact that the m r values track this curve implies an underlying Macquart relation. Table 5 (continued)
FRB RA cand
Dec cand θ φ m
Filter P c P ( O ) P ( O | x ) P ( U ) P ( U | x ) Table 5 (continued)
FRB RA cand
Dec cand θ φ m
Filter P c P ( O ) P ( O | x ) P ( U ) P ( U | x )82.9935 33 . . . . . . . . . . . . Table 5 continued
RB host galaxy associations Table 5 (continued)
FRB RA cand
Dec cand θ φ m
Filter P c P ( O ) P ( O | x ) P ( U ) P ( U | x )29.5084 65 . . . . . . . . . . . . . − . − . − . − . − . − . − . − . − . − . . . . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . . . . . . Table 5 continued FRB Association Team
Table 5 (continued)
FRB RA cand
Dec cand θ φ m
Filter P c P ( O ) P ( O | x ) P ( U ) P ( U | x )FRB190711 329.4194 − . − . − . − . − . − . − . − . − . − . − . − . − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . − . Table 5 continued
RB host galaxy associations Table 5 (continued)
FRB RA cand
Dec cand θ φ m
Filter P c P ( O ) P ( O | x ) P ( U ) P ( U | x )326.1042 − . − . − . − . − . − . − . − . − . . . . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . . . . . . − . − . − . − . − . − . − . − . − . − . − . − . − . . . FUTURE DIRECTIONS AND ANALYSES0
FRB Association Team
This paper and the accompanying code base providea new methodology to make probabilistic associations oftransients to hosts (PATH). While we were motivatedby FRB science, the general framework is agnostic totransient type. Therefore, we anticipate it will be ap-plied to GRBs, GW events, Type Ia SNe, and manyother transients. We stress further that because it isfully probabilistic, the outputs may be coupled to otherlikelihood frameworks developed to constrain, e.g. pro-genitor models or cosmology.Applied to 13 well-localised FRBs, our results identifynine secure host galaxies, with posterior probabilities > .
95 of being the true host. We have shown usinga suite of ‘sandbox’ simulations that this identificationis reliable under a wide range of true FRB host galaxydistributions. This allows a reliable data set to be usedwhen analysing host galaxy properties, or using FRBsfor cosmology. Furthermore, by assigning a quantitativeprobability to individual hosts, we allow even non-securehosts associations to be used for statistical purposes,with appropriate weighting.Using these data-sets, we tentatively identify relationsbetween FRB DMs, and host galaxy redshifts, magni-tudes, and luminosities. Our results disfavour FRBs ashaving large offsets from their host galaxies, and we ex-clude more than one FRB considered as having an un-seen host ( p (cid:46) . ∼
10 FRBs to further refine the priors and offset function; and (ii) expanding the formalism toinclude additional observables (see the Appendix). Thelatter may require obtaining additional data or perform-ing additional analyses (e.g. photo- z estimates) than thesimple flux and angular sizes considered here. One mayalso introduce and test priors motivated by progenitormodels. Last, the analysis can inform observing strate-gies to optimize the probability of a secure associationas a function of anticipated FRB redshift, localizationerror, and imaging quality.ACKNOWLEDGEMENTSWe thank C. Kilpatrick and J. Bloom for helpful dis-cussions. The Fast and Fortunate for FRB Follow-up team acknowledges support from NSF grants AST-1911140 and AST-1910471. A.T.D. is the recipient ofan ARC Future Fellowship (FT150100415). K.A. ac-knowledge support from NSF grant AAG-1714897. TBgratefully acknowledges support from NSF via grantsAST-1909709 and AST-1814778. C.W.J. acknowledgessupport by the Australian Government through the Aus-tralian Research Council’s Discovery Projects fundingscheme (project DP210102103). We thank S. Ryder andL. Marnoch for sharing the reduced VLT/FORS2 im-age around FRB 190608 in advance of publication. Thiswork is partly based on observations collected at the Eu-ropean Southern Observatory under ESO programmes0102.A-0450(A), 0103.A-0101(A), 0103.A-0101(B) and105.204W.001.REFERENCES Bannister, K. W., Deller, A. T., Phillips, C., et al. 2019,Science, 365, 565, doi: 10.1126/science.aaw5903Bhandari, S., Sadler, E. M., Prochaska, J. X., et al. 2020a,ApJL, 895, L37, doi: 10.3847/2041-8213/ab672eBhandari, S., Bannister, K. W., Lenc, E., et al. 2020b,ApJL, 901, L20, doi: 10.3847/2041-8213/abb462Blanchard, P. K., Berger, E., & Fong, W.-f. 2016, ApJ, 817,144, doi: 10.3847/0004-637X/817/2/144Blondin, S., & Tonry, J. L. 2007, ApJ, 666, 1024,doi: 10.1086/520494Bloom, J. S., Kulkarni, S. R., & Djorgovski, S. G. 2002, AJ,123, 1111, doi: 10.1086/338893Budav´ari, T., & Loredo, T. J. 2015, Annual Review ofStatistics and Its Application, 2, 113–139,doi: 10.1146/annurev-statistics-010814-020231 https://github.com/FRBs/astropath Budav´ari, T., & Szalay, A. S. 2008, The AstrophysicalJournal, 679, 301–309, doi: 10.1086/587156Cordes, J. M., & Chatterjee, S. 2019, ARA&A, 57, 417,doi: 10.1146/annurev-astro-091918-104501Cordes, J. M., & Lazio, T. J. W. 2002, arXiv e-prints,astro. https://arxiv.org/abs/astro-ph/0207156Day, C. K., Deller, A. T., Shannon, R. M., et al. 2020,MNRAS, 497, 3335, doi: 10.1093/mnras/staa2138Driver, S. P., Wright, A. H., Andrews, S. K., et al. 2016,MNRAS, 455, 3911, doi: 10.1093/mnras/stv2505Eftekhari, T., & Berger, E. 2017, ApJ, 849, 162,doi: 10.3847/1538-4357/aa90b9Fishman, G. J., & Meegan, C. A. 1995, Annual Review ofAstronomy and Astrophysics, 33, 415,doi: 10.1146/annurev.aa.33.090195.002215Fong, W., & Berger, E. 2013, ApJ, 776, 18,doi: 10.1088/0004-637X/776/1/18
RB host galaxy associations21