Probabilistic Association of Transients to their Hosts (PATH)

Figure 1 shows a Very Large Telescope/FOcal Reducer and low dispersion Spectrograph 2 (VLT/FORS2) g-band cutout image (30'' × 30'') of the field surrounding FRB 180924 with its localization marked by a red circle. As described by Bannister et al. (2019), this localization lies ≈1'' from the centroid of galaxy DES J214425.25−405400.81 and within 5'' of two additional galaxies. At their redshifts (Bannister et al. 2019), the galaxies all have projected separations of less than 40 kpc, i.e., separations less 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 consider the possibility that the FRB occurred in the stellar halo of one of the others.

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The chance probability approach is powerfully simple: estimate the Poisson probability of finding one or more galaxies as bright or brighter within an effective search area A around the FRB, with A determined from the angular size ϕ, the separation θ, and the FRB localization uncertainty σ_FRB. Namely, one defines the probability of a chance coincidence

$$\begin{eqnarray}&&{P}^{c}({\theta }_{\mathrm{eff}},m)=1-\exp (-\bar{N}),\end{eqnarray} \tag{ 1 }$$

where $\bar{N}$ is the average number of sources in A. It is given by

$$\begin{eqnarray}&&\bar{N}({\theta }_{\mathrm{eff}},{m}_{i})=\pi {\theta }_{\mathrm{eff}}^{2}\,{\rm{\Sigma }}({m}_{i}),\end{eqnarray} \tag{ 2 }$$

where Σ(m) is the angular surface density of galaxies on the sky with magnitude m ≤ m_i , and θ_eff is the effective search radius ($A=\pi {\theta }_{\mathrm{eff}}^{2}$). For the former, we adopt the galaxy 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 one here: the quadrature sum of all three angular quantities with semi-arbitrary weightings,

$$\begin{eqnarray}&&{\theta }_{\mathrm{eff}}=\sqrt{4{\sigma }_{\mathrm{FRB}}^{2}+{\theta }^{2}+4{\phi }^{2}}.\end{eqnarray} \tag{ 3 }$$

Adopting these, we estimate P^c = 0.01 for DES J214425.25−405400.81 and P^c > 0.1 for the other sources owing to their fainter magnitudes and larger angular separations. From this perspective, the probability of a chance association is nearly negligible for DES J214425.25−405400.81 and sufficiently large for the other sources that one is inclined to favor the former. However, this technique cannot assign a likelihood to this association, nor even a relative assessment of one source over another. Yet worse, if two or more candidates have low 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 to introduce additional observational measures as evidence (e.g., DM). Together, these considerations motivate our development of a full probabilistic treatment.

There are several aspects of the P^c analysis that require further definition. For completeness, we describe these here, although P^c does not formally enter into the new formalism. First, one requires measurements of the galaxy centroids and angular size. We advocate a nonparametric approach owing to the complexity of galaxy morphology. In the following, we adopt the centroiding algorithm encoded in the photutils software package, and use the semimajor_axis_sigma parameter to estimate the angular size. In the following, we will refer it as a_image as it matches the definition of that parameter in the more widely used SExtractor package.

Another issue is Galactic extinction. FRBs are detected across the sky including on sightlines that show large Galactic extinction (E(B − V) > 0.1 mag). In contrast, the number count analyses have intentionally been derived from high-latitude fields with low Galactic extinction. Therefore, the apparent magnitudes of the galaxy candidates should be corrected for Galactic extinction prior to the probability estimation.

We provide the following set of recommendations to optimize the detection and characterization of both faint and bright sources. Namely, we recommend observations in the r band, which provide a trade-off between the mapping of stellar content and extinction. We further recommend an image depth of m_r = 25.5 (5σ), corresponding to the limiting magnitude for spectroscopy, and sufficient for probing 0.01 L^* galaxies at z ∼ 0.5. Finally, given the subarcsecond accuracy of many FRB localizations, we recommend better than 1'' seeing, and a >1' field of view for background estimation.

Similarly, source detection should employ a nonparametric approach to accommodate galaxies of varying morphologies. For consistency in this work, we recommend the use of the a_image parameter to estimate the angular size of sources, and note that all galaxies should be corrected for Galactic extinction prior to applying the Bayesian formalism.

The formalism of Equations (1) and (2) ignores galaxy clustering. Since matter in the universe is not uniformly distributed, the probability of observing either no galaxies, or a large number of them, in proximity to a random direction is enhanced compared to the probability of observing one or a few. Tunnicliffe et al. (2014) showed that including clustering decreases the probability of a nearby random galaxy by 25%–50% in the case of a random direction. Our primary concern however is not whether or not all the observed galaxies are merely chance coincidences, but rather which of the observed galaxies is the true host. Clustering is discussed further in this context in Section 4.2. For now, we remark that given that FRBs truly are associated with galaxies, clustering acts to increase, not decrease, the probability of a chance association, since the FRB observation has preferentially selected a direction of the universe in which there is a cluster of matter.

Furthermore, in this work we ignore for the sake of simplicity the ellipticity of the prospective host galaxies. Our method will be readily adaptable however to such ellipticity, or indeed arbitrarily complex functions, since the approach described below does not rely on any particular functional forms.

Since FRBs are sparse on the sky, we can study them separately, which also simplifies the following description of our approach. Let us consider a catalog of galaxies across the entire sky and a single FRB that either belongs to one of the many catalog objects (its host galaxy) or it does not, i.e., its host is not detected or not included in the catalog. If U is the event that the FRB's host is unseen, and O_i is the event that the FRB is from galaxy i, their prior probabilities must add up to 1,

$$\begin{eqnarray}&&P(U)+\displaystyle \sum _{i}P({O}_{i})=1,\end{eqnarray} \tag{ 4 }$$

as there are no other possibilities. The single scalar quantity P(U) encodes all the complications that arise from the difference in the radial selection functions of the catalog and the FRB instruments. For now, we assume its value to be known, but note that it could be inferred in a hierarchical fashion when considering multiple FRBs. Also, one could assume a uniform prior for all observable O_i as the simplest possible scenario that essentially ignores any additional information about the galaxies, e.g., magnitude, color, and redshift. We leave such considerations to a future work.

Given a vector x representing all measured properties of the detected FRB, we ask what the posterior probabilities P(U∣x) and $P\left({O}_{i}| x\right)$ are for all i. Using Bayes' rule, the unseen posterior is

$$\begin{eqnarray}&&P(U| x)=\displaystyle \frac{P(U)\,p(x| U)}{p(x)},\end{eqnarray} \tag{ 5 }$$

where p(x∣U) is the probability density of the FRB properties given the host is unseen. From hereon, we consider x to represent only the measured FRB direction. Without constraints, x could be anywhere on the sky, hence, it is natural to assume a uniform (isotropic) distribution with a value of 1/4π. Similarly, the posterior probability for object i is

$$\begin{eqnarray}&&P\left({O}_{i}| x\right)=\displaystyle \frac{P({O}_{i})\,p(x| {O}_{i})}{p(x)},\end{eqnarray} \tag{ 6 }$$

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 includes the galaxy geometry and the uncertainty of the FRB direction. This key component of the approach is discussed in-depth in the next paragraph. The normalizing constant must be

$$\begin{eqnarray}&&p(x)=P(U)\,p(x| U)+\sum _{i}P({O}_{i})\,p(x| {O}_{i})\,\,\end{eqnarray} \tag{ 7 }$$

to guarantee that these posteriors also add up to 1,

$$\begin{eqnarray}&&P(U| x)+\sum _{i}P\left({O}_{i}| x\right)=1.\end{eqnarray} \tag{ 8 }$$

Let the 3D unit vector ω represent the true and unknown direction of the FRB on the sky. Given that it comes from a particular galaxy, the direction ω has to point somewhere near the host. The function p(ω∣O_i ) captures the physical and geometric model for the FRBs specific 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 localization error function, L(x − ω). Given p(ω∣O_i ), p(x∣O_i ) is calculated by integrating over the ω model directions to obtain the marginalized likelihood of the association hypothesis O_i ,

$$\begin{eqnarray}&&p(x| {O}_{i})=\int d\omega \,p(\omega | {O}_{i})\,L(x-\omega ),\end{eqnarray} \tag{ 9 }$$

which now accounts for all possibilities in various FRB origins as well as the astrometric uncertainty in the measurement. If a galaxy is unresolved, p(ω∣O_i ) may become the Dirac-δ, and p(x∣O_i ) is just the astrometric uncertainty, as it would be the case for matching point sources. Calculating the above quantity for all i completes the framework, which now provides posterior probabilities via Equation (6).

Previously we assumed a galaxy catalog over the entire sky, but catalogs typically have more limited footprints. It is interesting to think about a scenario when the field of view Ω is smaller but still large enough not to miss any possible counterparts to the FRB in question. Intuitively, galaxies very far away from the FRB should not have any effect on the association analysis.

Smaller sky coverage would mean fewer observed galaxies N_c , which in turn affect the P(O_i ) priors as there are fewer galaxies to choose from. Going with the previous uniform assumption, Equation (4) implies

$$\begin{eqnarray}&&P({O}_{i})=\displaystyle \frac{1-P(U)}{{N}_{c}},\end{eqnarray} \tag{ 10 }$$

which captures the dependence.

Looking now back at Bayes' rule in Equation (5), the sky coverage seems to affect the $P\left({O}_{i}| x\right)$ posteriors, too. Fortunately, this is not the case. The denominator p(x) changes in accord due to the scaling in p(x∣U), which is uniform over the field of view,

$$\begin{eqnarray}&&p(x| U)=\displaystyle \frac{{1}_{{\rm{\Omega }}}(x)}{{\rm{\Omega }}},\end{eqnarray} \tag{ 11 }$$

where the 1_Ω(x) is the indicator function that takes the value 1 if x is within the field of view and 0 otherwise, and the denominator Ω normalizes the PDF. The framework not only matches common-sense expectations, but provides for more efficient computation where only candidates within close proximity of FRBs are considered.

Given the plethora of deep, wide-field imaging surveys, it is possible to consider very large Ω for each FRB in the analysis. In practice, however, sensible and conservative assumptions for p(ω∣O_i ) will greatly limit the list of viable candidates O_i . To ease the analysis, we adopt for Ω the union of the area encompassing all galaxies within 10 half-light radii and the 99.9% FRB localization. As emphasized in the previous section, it is only necessary to consider a large enough area to be certain to include all possible hosts.

It is difficult to a priori assign a prior P(U) to the probability that the FRB host is undetected. Undoubtedly, P(U) is related to the depth of the imaging and the (unknown) source distance, i.e., redshift. ¹² In the following, we consider an arbitrarily assigned value, and we advocate a low value, based on the paucity of our data and the set of confidently assigned associations reported to date (e.g., Heintz et al. 2020). In the analyses that follow, we typically assume P(U) = 0 (Occam's razor!) and discuss the impacts of increasing it.

For the set of host candidates O_i , absent any assumptions on the distance or the typical separations of FRBs from their host galaxies, any galaxy on the sky could be a viable candidate. Given the plethora of potential models for FRB progenitors and the limited existing constraints, we are motivated to consider (for now) simple 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., Equation (10).

Inspired by the chance probability calculations approach described in Section 2, we introduce an additional prior based on P^c . Specifically, we consider a prior that inversely weights by Σ(m).

$$\begin{eqnarray}&&P({O}_{i})\propto \displaystyle \frac{1}{{\rm{\Sigma }}({m}_{i})}.\end{eqnarray} \tag{ 12 }$$

For this "inverse" prior, brighter candidates have higher prior probability according to their number density on the sky. The normalization of these priors is set by Equation (4). In Section 5, we also briefly consider two other priors with P(O_i ) ∼ 1/Σ(m_i )ϕ (inverse1) and P(O_i ) ∼ 1/Σ(m_i )ϕ² (inverse2). We adopt the prior in Equation (12) in part because of its simpler form and also because of the results of simulated experiments (Section 5).

Figure 2 illustrates P(O_i ) for the two models for two example 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'' of the FRB, which captures all of the viable candidates. As expected, the inverse priors for FRB 180924 significantly favor the brighter galaxy closest to the FRB. Perhaps less intuitively, the inverse priors favor the most distant (yet brightest) source near FRB 190523. This motivates the inclusion of the next ingredient—the offset function p(ω∣O_i ).

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The p(ω∣O_i ) function for the probability of the true angular offset from the galaxy is unknown yet required for the analysis. In our formalism, we develop priors for p(ω∣O_i ) based solely on the angular offset θ between the galaxy centroid and ω, and also normalized by the observed galaxy's angular size ϕ. This simultaneously accounts for galaxies of different intrinsic size and differing observed size owing to their distance.

As the predominance of models associate FRBs to stellar sources or compact objects (active galactic nucleus at the very centers of galaxies are currently disfavored; Bhandari et al. 2020b), one might expect the FRB events to track the stellar light. While we wish to remain largely agnostic to the underlying distribution of offsets, we are physically motivated to presume p(ω∣O_i ) decreases with increasing θ. We assert this despite the fact that geometrical considerations do favor large ω, e.g., a model where FRBs occur with identical probability anywhere in a circular galaxy will have p(ω∣O_i ) ∝ ω until one reaches the "edge" of the galaxy. Therefore, a uniform prior $p(\omega | {O}_{i})=1/\pi {\theta }_{\max }^{2}$ is formally one that assumes FRBs occur proportional to the galaxy radius. The other two models considered are a core model:

$$\begin{eqnarray}&&p(\omega | {O}_{i})=\displaystyle \frac{1}{2\pi {\phi }^{2}[{\theta }_{\max }/\phi -\mathrm{log}({\theta }_{\max }/\phi +1)]}\displaystyle \frac{1}{(\theta /\phi )+1},\end{eqnarray} \tag{ 13 }$$

which implies an approximately 1/r² weighting, and an exponential model

$$\begin{eqnarray}\begin{array}{rcl}p(\omega | {O}_{i}) & = & \displaystyle \frac{1}{2\pi {\phi }^{2}[1-(1+{\theta }_{\max }/\phi )\exp (-{\theta }_{\max }/\phi )]}\\ & & \times \exp [-\theta /\phi ],\end{array}\end{eqnarray} \tag{ 14 }$$

which assumes an underlying exponential distribution. All of these functions are normalized to unity when integrating to ${\theta }_{\max }$, ignoring the curvature of the sky.

For all of the p(ω∣O_i ) priors, we assert a maximum offset ${\theta }_{\max }=6\phi ;$ this is especially important for the uniform prior. This value is arbitrary and was chosen to be large enough to accommodate prevailing models of FRBs without being too conservative. Applying an arbitrary cutoff to the exponential distribution is not strictly necessary, but we keep it for simplicity and consistency, and demonstrate in Section 6.3 that the results for an exponential distribution are insensitive to this choice.

Figure 3 shows the offset functions, normalized to have the same total probability. Clearly the exponential model favors FRBs located in the inner regions of galaxies. For comparison to the offset distributions of known transients, we note that long GRBs appear highly concentrated in the inner regions of their hosts relative to Type Ib/c and IIn SNe, which occur preferentially near their host half-light radii (Lunnan et al. 2015; Blanchard et al. 2016). Conversely, short GRBs exhibit significant offsets from their host centers, indicative of progenitors born in compact object mergers (Fong & Berger 2013).

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In practice, we treat the galaxies as "round," i.e., ignoring for now any ellipticity. Future works will advance this aspect.

Our Monte Carlo approach leverages the public catalog of the Hubble Space Telescope/Cosmic Evolution Survey (HST/COSMOS) field (Scoville et al. 2007b, 2007a), which provides over 1 million sources at high spatial resolution and faint fluxes (median AB magnitude m_r ≈ 25.9). These galaxies are distributed over an ≈1 deg² of sky, which captures a great variety of distributions but not the most extreme events (e.g., z < 0.5 galaxy clusters, nearby galaxies). Figure 4 shows the angular size (a_image) and apparent magnitudes for the catalog, restricted to the sources labeled as galaxies.

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We generate a series Monte Carlo realizations of FRBs (referred to as a sandboxes, or SBs), using the following recipe:

• 1.  
  Define the true distribution of FRBs from their host galaxies p(ω∣O_i ) and magnitude m_r .
• 2.  
  Define a sample of potential host galaxies based on m_r .
• 3.  
  Define the distribution of localization errors for FRBs (σ_FRB).
• 4.  
  Draw N_FRB galaxies from the parent sample of potential host galaxies without duplicates.
• 5.  
  Set the true FRB positions according to p(ω∣O_i ).
• 6.  
  Offset the FRBs to an observed coordinate according to σ_FRB.
• 7.  
  Consider catalog galaxies within 30'' to represent an image.

For sandbox 5 (SB-5), 10% of the FRBs were randomly placed in the COSMOS field, i.e., without a host galaxy. We plan to use this sandbox to evaluate the performance of the framework when the host galaxies are unseen. Also, as COSMOS is a deep survey, we generate a magnitude-limited catalog of galaxies for each sandbox (last column of Table 1) on which we run the Bayesian framework. In the following subsection, we discuss results for five sandboxes (focusing primarily on SB-1) with the parameters described in Table 1.

Table 1. Sandboxes

Label	p(ω∣Oi )	NFRB	Sample	σFRB	Catalog
				('')	Filter
SB-1	${ \mathcal U }(0,2\phi )$	100,000	⋯	1	⋯
SB-2	${ \mathcal U }(0,2\phi )$	46,699	mr = [20, 23]	${ \mathcal U }(0.1,1)$	mr ≤ 23
SB-3	core	46,699	mr = [20, 23]	${ \mathcal U }(0.1,1)$	mr ≤ 23
SB-4	exponential	46,699	mr = [20, 23]	${ \mathcal U }(0.1,1)$	mr ≤ 23
SB-5 a	${ \mathcal U }(0,2\phi )$	50,000	mr = [20, 25]	${ \mathcal U }(0.1,1)$	mr ≤ 25

Note.

^a See text for details regarding the FRB selection for this sandbox.

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We now analyze the sandboxes listed in Table 1 with a variety of priors and assumed p(ω∣O_i ) functions (that generally do not match the true p(ω∣O_i )). Table 2 lists the various priors assumed for each analysis performed.

Table 2. Sandbox Analysis

Sandbox	P(O)	P(U)	p(ω∣Oi )	${\theta }_{\max }$/ϕ	f(T+secure)	TP
SB-1	Inverse	0	Exp	6	0.33	0.96
SB-1	Inverse1	0	Exp	6	0.30	0.99
SB-1	Inverse2	0	Exp	6	0.32	1.00
SB-1	Identical	0	Uniform	6	0.22	1.00
SB-1	Inverse	0.05	Exp	6	0.24	0.96
SB-2	Inverse	0	Uniform	2	0.86	1.00
SB-3	Inverse	0	Core	6	0.58	1.00
SB-4	Inverse	0	Exp	6	0.68	0.99
SB-5	Inverse	0.10	Exp	6	0.58	0.98

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Figure 5(a) shows the posteriors for the candidates using the fiducial sandbox (SB-1) and listed in row 1 of Table 2 (also referred to as the adopted prior set; Table 3). Since most of the candidates defined in step 7 are very far from the offset FRB position, we restrict results to the ≈8% of candidates with $P\left({O}_{i}| x\right)\gt 0.01$. This distribution is multimodal, with the overwhelming majority of recovered $P\left({O}_{i}| x\right)\approx 0$ corresponding to unassociated galaxies and another peak at $P\left({O}_{i}| x\right)\approx 1$ corresponding to secure associations. Figure 5(b) shows the posterior value for the most probable candidate for each of the 100,000 FRBs. For this model and analysis, ≈35% of the FRBs have a high probability, $P\left({O}_{i}| x\right)\gt 0.95$ which we adopt as a "secure" association. Adopting such an arbitrary value to define secure is useful for including/excluding candidates for subsequent analyses that rely on knowing the correct FRB host, e.g., that by Macquart et al. (2020). However, we emphasize that it is in general better to consider all host associations as uncertain, with different levels of certainty according to the obtained posteriors.

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Table 3. Prior Sets

Set	P(O)	P(U)	p(ω∣Oi )	${\theta }_{\max }$/ϕ
Conservative	Identical	0	Uniform	6
Adopted	Inverse	0	Exp	6

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Table 4. FRBs Analyzed

FRB	αFRB	δFRB	a	b	PA	Filter
	(deg)	(deg)	('')	('')	(deg)
FRB 121102	82.99458	33.14792	0.10	0.10	0.0	GMOS_N_i
FRB 180916	29.50313	65.71675	0.00	0.00	0.0	GMOS_N_r
FRB 180924	326.10523	−40.90003	0.11	0.09	0.0	VLT_FORS2_g
FRB 181112	327.34846	−52.97093	3.25	0.81	120.2	VLT_FORS2_I
FRB 190102	322.41567	−79.47569	0.54	0.47	0.0	VLT_FORS2_I
FRB 190523	207.06500	72.46972	4.00	1.50	340.0	LRIS_R
FRB 190608	334.01987	−7.89825	0.26	0.25	90.0	VLT_FORS2_I
FRB 190611	320.74546	−79.39758	0.67	0.67	0.0	GMOS_S_i
FRB 190614	65.07552	73.70674	0.80	0.40	67.0	LRIS_I
FRB 190711	329.41950	−80.35800	0.40	0.31	90.0	GMOS_S_i
FRB 190714	183.97967	−13.02103	0.36	0.22	90.0	VLT_FORS2_I
FRB 191001	323.35155	−54.74774	0.17	0.13	90.0	VLT_FORS2_I
FRB 200430	229.70642	12.37689	1.07	0.30	0.0	LRIS_I

Note. _a , _b , _PA define the total 1σ error ellipse for the FRB localization data are taken from Ravi et al. (2019), Day et al. (2020), Law et al. (2020), Tendulkar et al. (2017), Marcote et al. (2020), and Heintz et al. (2020).

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Figure 6 evaluates, in 10 bins of equal number of FRBs, the maximum $P\left({O}_{i}| x\right)$ assigned to a candidate for each FRB and the percentage of correct associations assuming this is the host. The different colors indicate different choices for P(O). Our adopted inverse prior appears well calibrated in that $P\left({O}_{i}| x\right)$ yields an accurate estimate of the fraction of FRBs correctly assigned to their host galaxy.

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Figure 7 shows another set of results but for more different choices of priors (Table 2). The remarkably close correspondence between the two quantities indicates the posterior is well calibrated, at least for this pairing of sandbox and prior set. We also show results for prior sets where we assume P(U) = 0.05 and for the conservative prior set (Table 3). Each of these assigns systematically lower values to the true host galaxy yielding a higher percentage of correct cases at lower maximum $P\left({O}_{i}| x\right)$.

To characterize the behavior of our method under different simulated truths (i.e., sandboxes) and different priors, Table 2 lists the fraction f of all FRBs that are correctly identified, i.e., the true (T) host is securely identified ($P\left({O}_{i}| x\right)\gt 0.95$) and the fraction of secure identifications that are correct. In all cases, at least 96% of all secure associations find the true host, indicating that our method is trustworthy. The fraction of FRBs expected to have such a secure association however varies significantly, primarily as a function of the sandbox (variation of ±0.3 in $f(T+\mathrm{secure})$), with the analysis method on a given sandbox having a secondary effect (variation of ±0.05). An analysis of SB-1 demonstrates that the choice between different inverse priors has little effect, but produces a higher fraction of secure associations than a uniform prior. Interestingly, $f(T+\mathrm{secure})$ is much higher for the SB-4 analysis than for 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 being otherwise identical. However, while the SB-4 analysis assumes p(ω∣O_i ) to be uniform out to θ = 6ϕ, the true distribution is fully contained within 2ϕ, unlike SB-2 and SB-3. We thus conclude that the dominant determinant of the fraction of securely (and hence, correctly) identified FRB hosts, in the case that the true host is observed, is the fraction of FRBs lying in close proximity to their hosts, irrespective of other considerations. We find it especially reassuring that the results are not highly sensitive to the analysis method, i.e., that our formalism yields greater sensitivity to the physical truth than to our choice of reasonable priors.

What about unseen hosts? If we use P(U) = 0, as typically assumed in this work, then P(U∣x) = 0 always, and the method will tend to assign the highest posterior $P\left({O}_{i}| x\right)$ to the closest galaxy regardless of distance. Using the 10% of hostless FRBs from SB-5, the conservative and adopted prior sets from Table 3 find secure associations for the majority of FRBs (55% and 62% respectively). However, the typical radial offset for these secure associations is very large. In Figure 8, we show the cumulative distribution of θ/ϕ for such candidates. The probability of the most likely candidate being close to the FRB is small, with 10% or less of such falsely identified hosts having θ/ϕ < 6. The distributions for secure associations is almost identical to that from nonsecure hosts. We conclude that measuring a small θ/ϕ is a strong discriminant against unseen hosts irrespective of p(U).

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Buoyed by these results, we now proceed to apply PATH to real FRB observations.

Central to the analysis is the identification and analysis of galaxy candidates in imaging data. The first step—source identification—is the most challenging and the most subjective. For every image, sources near the detection limit are subject to the precise methodology: background subtraction, thresholding, pixel grouping, and deblending. After experimenting with the routines encoded in the photutils package, we settled on the following key parameters: npixels = 9, deblend=True, xy_kernel = (3, 3), Gaussian2Dkernel, nsig = 3. (kernel), nsig = 1.5 (threshold), background = (50, 50), and filter_size = (3, 3), median background.

To test these choices, we independently analyzed the data with the SExtractor package using a standard set of input parameters. Namely, we set DETECT_MINAREA = 9 and DETECT_THRES = 1.5 for consistency with the photutils parameters. Images are filtered with the default convolution kernel (default.conv). To recover blended sources (see, e.g., FRB 180924 below), we set DEBLEND_MINCONT = 0.0001.

With this set of parameters, we recover the centroid positions within ≈2%, while the aperture sizes show scatter up to ≈15%. We note that slight differences between the photutils and SExtractor methodologies may be driving these discrepancies; namely, while SExtractor uses a multi-thresholding deblending technique, photutils utilizes a combination of multi-thresholding and watershed segmentation. Furthermore, the default.conv convolution kernel is equivalent to a 3 × 3 Gaussian kernel with an FWHM = 2, in slight contrast to the Gaussian kernel used above. Nevertheless, we find comparable results using the two methods.

Figure 9 shows the segmentation maps of FRB 180924 and FRB 190523. Note the three, blended sources near the localization of FRB 180924 that are known to be unique galaxies at distinct redshifts (Bannister et al. 2019). An image with shallower depth (e.g., DES-DR1) or a different choice of photutils parameters would lead to the non-detection of the fainter sources. This highlights the subjectivity of source identification that can affect the final results.

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The source identification packages offer an assessment of the source shape (e.g., ellipticity and size), which can be used to used to select and then ignore Galactic stars. For the analysis that follows, we have simply clipped bright stars according to their apparent magnitudes when necessary.

Provided with the segmentation map, one may perform aperture photometry and estimate ϕ from the derived elliptical apertures. All of the measurements for the galaxy candidates are provided in Table 5.

We then applied the PATH framework. The primary results are displayed in Figure 10, which summarizes the $P\left({O}_{i}| x\right)$ values for the candidates in each field for each set of priors. We find very similar results for the two prior sets with one obvious exception: FRB 180924. In this case, there are two galaxies with separation $\theta \lt {\theta }_{\max }$ from this precisely localized FRB. These are treated similarly by the conservative approach. We demonstrate below, however, that a uniform p(ω∣O_i ) function with ${\theta }_{\max }=6\phi$ is disfavored by the data. Imposing the exponential offset model yields a higher $P\left({O}_{i}| x\right)$ for the primary candidate. Furthermore, if we allow for the great difference in apparent magnitude by invoking the inverse P(O) prior, the posterior $P\left({O}_{i}| x\right)$ raises to near unity for the host reported by Bannister et al. (2019).

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Based on the results from analysis of mock fields (Section 5), we adopt a probability threshold ${P}_{\mathrm{secure}}=0.95$ above which we consider a host association to be highly secure. The results in Figure 10 indicate that nine of the FRBs are associated with a single galaxy with $P\left({O}_{i}| x\right)\gt 0.95$ for the adopted prior set (and eight for the conservative set with FRB 180924 the difference).

The nonsecure hosts deserve individual consideration, in part to understand the dependence of the formalism to observed variations on the sky. FRB 181112 shows two bright galaxies near the FRB with the brighter assumed to lie in the foreground (Prochaska et al. 2019). The PATH analysis for the purported host gives $P\left({O}_{i}| x\right)\approx 0.7\mbox{--}0.9$, depending on the choice of priors (Table 3). We emphasize that the majority of FRB sightlines that intersect a massive foreground halo (estimated to be a few percent for FRBs at z > 0.5), will tend to have a maximum $P\left({O}_{i}| x\right)\lt {P}_{\mathrm{secure}}$. Given the terrific scientific value of probing such halos with FRBs (Prochaska et al. 2019), one may need to introduce additional criteria/priors to confidently pursue this science.

The next, nonsecure FRB association is FRB 190523 whose larger localization error incorporates several candidates. The analysis, however, does favor the purported host reported by Ravi et al. (2019). Third is FRB 190614, which lies near (≲2'') two faint galaxies with unknown redshifts (Law et al. 2020). As the FRB with the highest DM and therefore the highest presumed redshift of the sample, this result emphasizes the likely challenges of associating high-z FRBs to galaxies. In particular, given the host itself is likely faint, the incidence of additional, chance associations with comparable $P\left({O}_{i}| x\right)$ will be higher. Last is FRB 191001, which sits next to two bright galaxies known to have a common redshift (Bhandari et al. 2020a). Therefore, the redshift of the FRB is secure, but the host offset and its internal properties (e.g., stellar mass) are currently based on the assumption that the closer galaxy is the host, and indeed, it exhibits a 3× higher $P\left({O}_{i}| x\right)$ value for the adopted prior set.

Having established a set of nine secure, $P\left({O}_{i}| x\right)\gt {P}_{\mathrm{secure}}\,=0.95$, host associations we may test the assumed p(ω∣O_i ) functions imposed in the analysis. Figure 11 shows the offset distribution for the secure hosts for the three p(ω∣O_i ) priors of the analysis (Section 4.3). Note that modifying the choice of p(ω∣O_i ) could include/exclude FRBs as being secure. The figure also shows the values of θ/ϕ derived for all candidates from the full set of FRBs, where we have weighted the θ/ϕ value of each candidate by $P\left({O}_{i}| x\right)$. Overall, the posteriors lend reasonable credibility to the set of p(ω∣O_i ) functions. On the other hand, a comparison of the secure distribution with the priors yields a one-sided Kolmogorov–Smirnov (K-S) probability P_K-S ≲ 0.1 and we rule out the uniform prior to ${\theta }_{\max }=6\phi$ at > 99%. The data appear to favor a p(ω∣O_i ) function that favors a central concentration for FRB locations. Additionally, such small values of θ/ϕ are unlikely when the true host galaxy is unseen (see Figure 8), being <3% for the seven secure hosts with θ/ϕ < 1.5. Since all FRBs have most likely candidates with θ/ϕ < 5, we conclude that no more than one of the FRBs considered can have an unseen host (p ≲ 0.01).

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Encoded in every FRB is its DM, the path integral of free electrons along the sightline weighted by the cosmological scale factor. The first ≈10 FRBs localized have established a firm correlation between DM and redshift, now termed the Macquart relation (Macquart et al. 2020). This relation derives from the ionized plasma that permeates the cosmic web. Because redshift (i.e., distance) affects observed properties, one should consider incorporating the DM into the association analysis. A full and proper treatment, however, requires including the intrinsic luminosities and spectral slopes of FRBs convolved with instrumental sensitivity and even the triggering software (C. James et al. 2021, in preparation). Further, we emphasize that adopting the Macquart relation as a prior would likely require estimating the redshift for every galaxy candidate; this will be intractable for many FRBs.

In lieu of a direct application of the Macquart relation, we propose to leverage the luminosity distribution (Figure 12) together with DM. Figure 13 presents the apparent magnitudes of each galaxy candidate against an estimate of the cosmic DM, DM_cosmic, with the point size proportional to $P\left({O}_{i}| x\right)$. For DM_cosmic, we adopt a simple estimation:

$$\begin{eqnarray}&&{\mathrm{DM}}_{\mathrm{cosmic}}={\mathrm{DM}}_{\mathrm{FRB}}-{\mathrm{DM}}_{\mathrm{MW},\mathrm{ISM}}-100,\end{eqnarray} \tag{ 15 }$$

with DM_MW,ISM the estimated ISM DM (Cordes & Lazio 2002) and the factor of 100 DM units accounts for the Galactic halo and the host galaxy (see Prochaska & Zheng 2019; Macquart et al. 2020). The locus of data exhibits a clear correlation reflecting the decrease in observed galaxy flux with increasing distance.

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Converting the DM_cosmic estimates to redshift ¹³ , we may convert a given L_r luminosity to m_r ; this is illustrated as the black curve in Figure 13 , where we assumed a fiducial L = L*/4 based on Figure 12. The good correspondence between the data and this curve reveals the Macquart relation without having used any direct redshift measurements.

In principle, one could construct a prior P(m_r ∣DM) to include in the analysis. This will, however, be subject to scatter in the DM_cosmic (Macquart et al. 2020), DM_host, and the intrinsic luminosities of the host galaxy population (Figure 12). It will also be subject, however, to the S/N considerations that affect any prior related to DM (James et al. 2020).