A deep survey of short GRB host galaxies over $z\sim0-2$: implications for offsets, redshifts, and environments

A significant fraction (30\%) of well-localized short gamma-ray bursts (sGRBs) lack a coincident host galaxy. This leads to two main scenarios: \textit{i}) that the progenitor system merged outside of the visible light of its host, or \textit{ii}) that the sGRB resided within a faint and distant galaxy that was not detected by follow-up observations. Discriminating between these scenarios has important implications for constraining the formation channels of neutron star mergers, the rate and environments of gravitational wave sources, and the production of heavy elements in the Universe. In this work, we present the results of our observing campaign targeted at 31 sGRBs that lack a putative host galaxy. Our study effectively doubles the sample of well-studied sGRB host galaxies, now totaling 72 events of which $28\%$ lack a coincident host to deep limits ($r$\,$\gtrsim$\,$26$ or $F110W$\,$\gtrsim$\,$27$ AB mag), and represents the largest homogeneously selected catalog of sGRB offsets to date. We find that 70\% of sub-arcsecond localized sGRBs occur within 10 kpc of their host's nucleus, with a median projected physical offset of $5.6$ kpc. Using this larger population, we discover an apparent redshift evolution in their locations: bursts at low-$z$ occur at $2\times$ larger offsets compared to those at $z$\,$>$\,$0.5$. This evolution could be due to a physical evolution of the host galaxies themselves or a bias against faint high-$z$ galaxies. Furthermore, we discover a sample of hostless sGRBs at $z$\,$\gtrsim$\,$1$ that are indicative of a larger high-$z$ population, constraining the redshift distribution and disfavoring log-normal delay time models.

In the primordial formation channel, these large offsets are expected due to a change in velocity (a natal kick) imparted to the system, following mass ejection from the second supernova explosion (Lyne & Lorimer 1994;Hansen & Phinney 1997;Bloom, et al. 1999;Fryer, et al. 1999;Wex, et al. 2000;Hobbs, et al. 2005;Belczynski et al. 2006). Combined with the long merger delay times (10 7 − 10 11 yr) predicted for BNS systems (Zheng & Ramirez-Ruiz 2007;Zemp, et al. 2009), a large natal kick can allow the binary to reach substantial distances and even escape its birth galaxy. However, a binary escaping its galaxy, denoted as physically hostless, is theorized to occur in an extremely low density ( < 10 −4 cm −3 ) intergalactic medium (IGM) environment, making detection of an afterglow unlikely (Panaitescu,et al. 2001;Salvaterra, et al. 2010;Duque et al. 2020). Moreover, by studying their early X-ray afterglow lightcurves, O'Connor et al. (2020) found that 16% of sGRBs are consistent with such low densities, including only a single observationally hostless event (GRB 080503;Perley, et al. 2009). Nevertheless, this does not exclude sGRBs with large offsets from having occurred within the halo's of their host galaxies or within a dense globular cluster environment (Salvaterra, et al. 2010).
An alternative explanation for observationally hostless bursts is that these sGRBs occurred in faint, undetected host galaxies at higher redshifts (i.e., 1 − 2; Berger 2010; Tunnicliffe, et al. 2014). Such high-events suggest progenitors that formed through a primordial channel with short merger delay times (e.g., Andrews & Zezas 2019;Beniamini & Piran 2019), indicating that BNS systems may have formed early enough to pollute the early Universe with heavy metals (Ji et al. 2016a,b;Roederer et al. 2016;Hansen et al. 2017;Safarzadeh & Scannapieco 2017;Beniamini et al. 2018;Safarzadeh et al. 2019;Zevin et al. 2019). Furthermore, our understanding of the environments and formation channels of sGRBs has fundamental implications for inferring the rate of detectable gravitational wave (GW) sources and for the follow-up of their electromagnetic (EM) counterparts, as the quick localization of the EM counterpart depends on inferences (such as, e.g., stellar mass, star formation rate, offset) from the known population of sGRB host galaxies (Nissanke, et al. 2013;Arcavi, et al. 2017;Artale et al. 2020b;Ducoin, et al. 2020) and on targeted searches using catalogs of nearby galaxies (White, Daw & Dhillon 2011;Dalya, et al. 2016;Cook, et al. 2019).
Disentangling between the different scenarios is observationally challenging. Due to the faintness of sGRB afterglows, redshift measurements from afterglow spectroscopy are rarely successful (e.g., de Ugarte Postigo et al. 2014;Agüí Fernández et al. 2021). Therefore, deep imaging and spectroscopic observations from the most sensitive telescopes are required to identify the GRB host galaxy and estimate its distance scale. In this work, we targeted a sample of 31 sGRBs that lack a putative host galaxy with large-aperture telescopes to search for faint, coincident galaxies. Our facilities include: the Lowell Discovery Telescope (LDT), the Keck Observatory, the Gemini Observatory, the Gran Telescopio Canarias (GTC), the Very Large Telescope (VLT), and the Hubble Space Telescope (HST).
The paper is outlined as follows. In §2, we define our sample selection criteria, and the optical and near-infrared (nIR) imaging analysis techniques used in this work. In §3, we describe the methods employed to detect, localize, and compute photometry of the host galaxies, as well as the probabilistic criteria used for host assignment. In §4, we present the results and discuss the demographics of sGRB offsets, host galaxies, and environments. We present a discussion of these results in §5 and conclude in §6. We present a detailed summary of the individual events analyzed in this work in Appendix A.
We adopt the standard Λ-CDM cosmology with parameters 0 = 67.4, Ω = 0.315, and Ω Λ = 0.685 (Planck Collaboration et al. 2018). All confidence intervals are at the 1 level and upper limits at the 3 level, unless otherwise stated. All reported magnitudes are in the AB system, and are corrected for Galactic extinction (Schlafly & Finkbeiner 2011). Throughout the paper we adopt the convention ∝ − − .

Sample selection
The association of a GRB with a host galaxy relies on the accurate localization of its afterglow. Therefore, we consider the sample of short GRBs detected with Swift and localized by the X-ray Telescope (XRT; Burrows, et al. 2005) to arcsecond accuracy. We include both GRBs with a short duration 1 , defined as 90 < 2 s (Kouveliotou, et al. 1993), and GRBs with a temporally extended emission (hereafter sGRBEE), as defined by Norris & Bonnell (2006). GRB classification -As of May 2021, the Swift Burst Alert Telescope (BAT; Barthelmy et al. 2005) has detected 127 short duration GRBs of which 91 (72%) have an X-ray afterglow localization. These X-ray localized events form the basis of our sample. Short duration bursts with soft spectra (i.e., a hardness ratio 50−100 keV / 25−50 keV < 1, where represents the gamma-ray fluence in a given energy range; Lien et al. 2016) or non-negligible spectral lag (Norris & Bonnell 2006) were flagged as "possibly short" (see, e.g., Lien et al. 2016) as some of these events may be produced by collapsar progenitors (see, e.g., GRB 040924, Huang et al. 2005;Soderberg et al. 2006;Wiersema et al. 2008;and 200826A, Ahumada et al. 2021;Rossi et al. 2021;Zhang et al. 2021). In addition, 31% 69% 45% 55% Literature This Work Localization XRT Optical Figure 1. The distribution of short GRB localization methods between X-ray and optical for the sample of 31 events analyzed in this work and the sample of 36 events in Fong et al. (2013).
Other events which display the characteristic features of sGRBEE, such as a spectrally hard initial pulse with negligible spectral lag (Norris & Bonnell 2006), can be more confidently assigned to this class. In total, we identify 32 sGRBEE (including 18 candidate sGR-BEE 2 ) of which 29 (90%) have an X-ray localization. Therefore, our initial sample totals 159 events which are either classical sGRBs ( 90 < 2 s) or sGRBEEs.
GRB localization -Past searches for the host galaxies of short GRBs (e.g., Prochaska et al. 2006;D'Avanzo et al. 2009;Berger 2010;Tunnicliffe, et al. 2014) mainly focused on optically localized events with sub-arcsecond positions ( Figure  1). However, an optically selected sample is potentially subject to multiple observing biases, which can affect the observed redshift and offset distributions. An optical position disfavors small offsets from the host's nucleus (e.g., O'Connor et al. 2021) as the afterglow light can be masked by the glare of the host galaxy, especially in the case of faint short GRB afterglows or dusty environments. In addition it may disfavor events occurring in the low-density environments expected for large-offset GRBs (Panaitescu,et al. 2001;Salvaterra, et al. 2010;Duque et al. 2020;O'Connor et al. 2020).
In order to mitigate potential biases due to an optical selection of the sample, we included all XRT localized events within our follow-up campaign. Although XRT positions typically have larger uncertainties than optical, radio, or Chandra localizations, XRT localized bursts contribute valuable information to the demographics of sGRB host galaxies in terms of redshift, stellar mass, star formation rate, and galaxy type (e.g., Gehrels et al. 2005;Bloom et al. 2006). Hereafter, we consider only the 120 events with at least an X-ray localization, of which 49 (∼40%) also have an optical localization.
Selection criteria -We adopt two additional criteria to build a homogeneous sample of bursts. The first is that the uncertainty on the GRB's localization is < 4 (90% confidence level, hereafter CL) as bursts with a poorer localization can only be securely associated to bright ( 21 mag) galaxies and would not benefit from a campaign of deep optical imaging. This requirement excludes 13 XRT localized events from our sample 3 . We further impose a limit of < 1.5 mag (Schlafly & Finkbeiner 2011) on the Galactic extinction along the GRB sightline in order to eliminate regions where host galaxy searches would be less sensitive 4 . This cut allows us to remove crowded regions along the Galactic plane (| | < 15 • ) where our search would not be meaningful due to chance alignment with foreground stars.
Among the remaining 99 short GRBs matching our criteria (see Figure 2), 43 are associated to a host galaxy, 7 are classified as hostless based on deep ground-based and HST imaging (see, e.g., Berger 2010;, and 49 more events lack evidence of an underlying host galaxy based on the initial ground-based follow-up reported through GCN circulars. The latter group of bursts is the focus of our study. Deep late-time imaging is crucial to determine whether the lack of a candidate host galaxy is due to the shallow depth of the initial ground-based follow-up, a high redshift, or a large angular separation due, for example, to a high natal kick velocity imparted to the progenitor. Observing strategy -As a first step (see Figure 3), we targeted these bursts with the 4.3-m LDT (PIs: Troja, Cenko, Gatkine, Dichiara) and performed deep optical imaging, typically in -band, to search for an underlying host galaxy to depth 25 mag. In the case of a detection, we scheduled the target for multi-color imaging in order to characterize the galaxy's spectral energy distribution (SED) and, if the galaxy's candidate was brighter than ≈ 21 − 22 AB mag, for optical spectroscopy in order to measure its redshift. In total, 30 out of 46 short GRBs (65% of the sample) were followed-up with the LDT from 2014 to 2021 through our programs. Those events which were not observed by LDT were either only visible from the Southern Hemisphere or already had limits comparable to LDT's typical depth ( ∼ 24.5 − 25 mag). In all other cases, we flagged the burst for further deep imaging with large-aperture telescopes. We targeted these sGRBs as part of our programs on the twin 8.1-m Gemini telescopes Figure 3. An outline of the candidate selection process, and follow-up methodology employed in this work in order to locate and identify the host galaxies of short GRBs. Optical spectroscopy was carried out if the candidate host galaxy was brighter than 21 − 22 mag, otherwise multi-color imaging was obtained in order to derive a photometric redshift.
(PI: Troja) and the 10-m Keck-I telescope (PI: Cenko) to search for host galaxies to deeper limits ( 26 − 28 AB mag). These observations were further complemented with public archival data from the 10.4-m GTC, the Keck Observatory, the Gemini Observatory, and HST.
The final sample of events observed through these programs comprises 31 sGRBs (see Table 2) discovered between 2009 to 2020 (14 of which have only an XRT localization). Of these 31 events, about 20% display extended emission. When compared to previous studies of sGRB host galaxies, which included 36 sGRBs discovered between 2005 to 2013 (e.g., Fong et al. 2013), our program doubles the sample of well-studied sGRB environments. A table of the X-ray and gamma-ray properties of sGRBs in our sample is shown in Table  B1.

Optical/nIR Imaging
Due to the isotropic distribution of GRBs on the sky and the multiyear nature of this project, the optical and near-infrared imaging obtained for our sample is heterogeneous and spans a range of observatories, filters, and exposure times. These observations were typically taken months to years after the explosion when contamination from the GRB afterglow is negligible. The majority of our optical observations were carried out by the Large Monolithic Imager (LMI) on the LDT, the Gemini Multi-Object Spectographs (GMOS; Hook et al. 2004) on both Gemini North (GMOS-N) and Gemini South (GMOS-S), the Low Resolution Imaging Spectrometer (LRIS; Oke et al. 1995) at the Keck Observatory, and the Optical System for Imaging and low-Intermediate-Resolution Integrated Spectroscopy (OSIRIS; Cepa et al. 2000) at the GTC. We also include publicly available near-infrared observations obtained with the HST Wide Field Camera 3 (WFC3). A log of observations presented in this work is reported in Table 2.

Lowell Discovery Telescope (LDT)
Observations with the Large Monolithic Imager (LMI) mounted on the 4.3-meter LDT at the Lowell Observatory in Happy Jack, AZ were carried out starting in 2014 as part of a long-term project (PIs: Troja, Gatkine, Dichiara) to study the afterglow and host galaxies of sGRBs. In order to have good visibility, only bursts with declination −30 • were selected. Over 60 sGRBs were observed as part of this program, and results on single events were presented in, e.g., Troja et al. (2016Troja et al. ( , 2018Troja et al. ( , 2019O'Connor et al. (2021); Ahumada et al. (2021). In this work, we present unpublished observations for 22 sGRBs in our sample.
LDT/LMI observations were carried out largely in the -band with a typical exposure of 1200 − 1500 s, chosen to obtain a depth of 24.5 − 25 mag in good observing conditions. However, the true image depth varies depending on the observing conditions at the time of our observations, which span multiple observing cycles across ∼ 7 years. All images were visually inspected and those flagged as poor were re-acquired at a later date. When a candidate host galaxy was detected, we performed additional observations in the , , and bands in order to better characterize the galaxy's SED.
Data were reduced and analyzed using a custom pipeline (Toy et al. 2016) that makes use of standard CCD reduction techniques in the IRAF 5 package including bias subtraction, flat-fielding, sky subtraction, fringe correction, and cosmic ray rejection using Laplacian edge detection based on the L.A.Cosmic algorithm (van Dokkum 2001). Following this image reduction process, the pipeline uses SExtractor (Bertin & Arnouts 1996) to identify sources in each frame, and then the Software for Calibrating AstroMetry and Photometry (SCAMP; Bertin 2006) to compute the astrometric solution. The aligned frames are then stacked using the SWarp software (Bertin, et al. 2002;Bertin 2010). The absolute astrometry of the stacked image was calibrated against the astrometric system of either the Sloan Digital Sky Survey (SDSS; Ahumada et al. 2020) Data Release 16 or the Panoramic Survey Telescope and Rapid Response System Survey (Pan-STARRS1, hereafter PS1; Chambers et al. 2016) Data Release 2, likewise using the combination of SExtractor and SCAMP. The SDSS and PS1 catalogs were further used to calibrate the photometric zeropoint (using SExtractor aperture photometry for the magnitude determination). We selected the SDSS catalog when available, and otherwise used PS1. We ensured that the sources used for both the astrometric and photometric calibrations were isolated point sources by sorting out those which did not pass our selection criteria based on their signal-to-noise ratio (SNR), full width at half-maximum intensity (FWHM), ellipticity, and SExtractor CLASS_STAR parameter.
We made use of tasks within the Gemini IRAF package (v. 1.14) to perform bias and overscan subtraction, flat-fielding, de-fringing, and cosmic ray rejection. The individual frames were then aligned and stacked using the IRAF task imcoadd. We additionally performed sky subtraction using the photutils 6 package to estimate the median sky background after masking sources in the image. The world-coordinate systems were then calibrated against the astrometric systems of SDSS or PS1 using either astrometry.net (Lang et al. 2010)  We additionally performed observations of GRB 151229A with Flamingos-2 (hereafter, F2) at Gemini South in Cerro Pachón, Chile on July 22, 2021. These observations were carried out in the and filters (see Table 2). We reduced and analyzed these data using the DRAGONS 7 software (Labrie et al. 2019). The photometry was calibrated using nearby point-sources in the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006) catalog. We then applied a standard conversion between the Vega and AB magnitude systems.

Keck Observatory
Through our program (PI: Cenko) on the 10-m Keck-I Telescope on Mauna Kea we obtained deep late-time imaging of GRBs 120305A, 120630A, 130822A, and 130912A. The Keck/LRIS observations took place during one half-night on October 25, 2014, and were carried out in both the and filters with exposure times of 3000 s and 2750 s, respectively. Observations of a fifth target (GRB 110112A) were incorrectly pointed by 0.15 deg and do not cover the GRB position (Chris Gelino, Priv. Comm.). Therefore, these data were not included. We complemented our observations with public archival LRIS data for GRBs 110402A, 140516A, 160927A, 170127B, 170728A, and 180805B.
The data were retrieved from the Keck Observatory Archive, and analyzed using the LPipe pipeline (Perley 2019). The pipeline processes raw files through standard CCD reduction techniques (e.g., bias-subtraction, flat-fielding, sky-subtraction, cosmic-ray rejection) to produce fully calibrated and stacked images. The final stacked image's absolute astrometry was calculated based on either the SDSS or PS1 catalogs. We used astrometry.net or the combination of SExtractor and SCAMP outlined in §2.2.1. We found that astrometry.net provided an accurate astrometric solution for sparse fields by making use of the standard stars within the Keck field-of-view. The photometric zeropoints were likewise calibrated using unsaturated SDSS (when available) or PS1 sources.
We additionally include archival infrared imaging obtained with Keck MOSFIRE (McLean et al. 2012) for GRBs 131004A, 151229A, 160601A, 170127B, and 180805B. These data were reduced using the MOSFIRE data reduction pipeline 8 , and calibrated using point sources in the 2MASS catalog. Standard offsets were applied to convert magnitudes into the AB system.

Gran Telescopio Canarias (GTC)
We obtained publicly available images of GRBs 160601A and 160927A (Table 2) taken with the 10.4-m GTC, which is located at the Roque de los Muchachos Obervatory in La Palma, Spain. The observations used the OSIRIS instrument, and were carried out in -band. The data were retrieved from the GTC Public Archive 9 . They were reduced and aligned using standard techniques within the astropy (Astropy Collaboration et al. 2018) software library to perform bias subtraction and flat-fielding. The individual frames were then combined to produce the final reduced image. The absolute astrometric correction was performed using astrometry.net, and the photometric zeropoints were calibrated to SDSS.

Very Large Telescope (VLT)
We analyzed archival images of GRBs 091109B, 150423A, and 150831A (Table 2) obtained with the 8.2-m VLT, operated by the European Southern Observatory (ESO) in Cerro Paranal, Chile. The observations were taken with the FOcal Reducer/low dispersion Spectrograph 2 (FORS2) in -band for GRBs 091109B, 150423A, and 150831A and an additional -band observation for GRB 150831A. The raw images were retrieved from the ESO Science Archive 10 . The data were processed using standard tasks within astropy (similarly to §2.2.4).
The data were processed using standard procedures within the DrizzlePac package (Gonzaga et al. 2012) in order to align, drizzle, and combine exposures. The observations within a single epoch were aligned to a common world-coordinate system with the TweakReg package. The AstroDrizzle software was then used to reject cosmic

Optical Spectroscopy
Bright host galaxies identified through our imaging campaign were targeted for optical spectroscopy in order to constrain their dis-tance scale. These targets include the fields of sGRBs 101224A and 140622A, observed with Keck/LRIS, and sGRBs 180618A and 191031D, observed with Gemini/GMOS-N. We complemented these observations with archival Keck spectroscopic data for sGRBs 110402A, 151229A, 160410A and 180805B as these bursts also match our selection criteria ( §2.1). Our spectroscopic campaign also included the candidate short GRB 060121 for which no visible trace was detected in a deep 3 × 900 s Keck/LRIS exposure. This was likewise the case for the archival Keck spectroscopy of sGRB 151229A. For sGRBs 180618A and 191031D, a weak trace was detected by the Gemini spectroscopic observations, but no obvious emission or absorption features were identified. The log of spectroscopic observations analyzed in this work is provided in Table 3. The Gemini data were reduced and analyzed using the Gemini IRAF package (v. 1.14), whereas Keck/LRIS data were reduced using the LPipe software. The processed spectra are displayed in Figure 4, and the result for each sGRB is reported in Table 3 and described in more detail in Section 4. We note that the optical spectrum obtained for sGRB 160410A is a rare case of afterglow spectroscopy ( Figure  5) as discussed in Agüí Fernández et al. (2021).

METHODS
In order to determine the putative host galaxy for each GRB, we began by identifying all galaxies near the GRB position in our late-time imaging. The source detection and classification (star-galaxy separation) procedure is outlined in Section 3.1. The late-time images were aligned with respect to the afterglow discovery images to precisely determine the host offset from the GRB position, as outlined in §3.2. The host association was then determined through probabilistic arguments based on the observed sky density of galaxies in Section 3.3. The results of our analysis for each GRB are presented in §4.

Source Detection and Classification
Source detection was performed using the SExtractor package after applying a Gaussian filter with a FWHM of 3 pixels 12 . We required that a source consist of a minimum area of 5 pixels at > 1 above the background (DET_THRESH = 1). The source detection was visually inspected to prevent erroneous blending of adjacent sources.
Source photometry was computed using the SExtractor MAG_AUTO parameter, which utilizes Kron apertures. In the case of faint sources, the magnitude was computed using seeing matched aperture photometry with the aperture (MAG_APER) diameter set to the FWHM of the image's point-spread function (PSF). The photometry was calibrated for each instrument as outlined in §2.2. The candidate host galaxy photometry for each GRB is presented in Table  4.
In order to determine whether a detected source could be identified as a galaxy we utilized the SExtractor SPREAD_MODEL parameter. First, we ran SExtractor to identify bright, unsaturated and isolated point-like objects. We selected them based on their SNR, FWHM, CLASS_STAR parameter (> 0.8), and ellipticity (< 0.2). We further imposed FLAGS < 1, which excludes sources that are saturated, blended, or too close to the image boundary. These point-like sources were then passed to PSFEx (Bertin 2011(Bertin , 2013 to estimate the image PSF. This was then fed to SExtractor to estimate the SPREAD_MODEL parameter which, for each detected source, measures the deviation of the source profile from the local normalized image PSF. Point-like sources are characterized by SPREAD_MODEL ≈ 0, whereas extended objects deviate significantly from the local PSF and have SPREAD_MODEL > 0. For sources smaller than the image PSF (e.g., cosmic rays or spurious detections), SPREAD_MODEL < 0. These star-galaxy classifiers become more uncertain for fainter sources, and we considered the classification as inconclusive for sources with SNR 5.

Offset Measurements
In order to precisely localize the GRB with respect to a candidate host galaxy, we utilized relative astrometry to align our late-time images with the afterglow discovery image. In our sample, 14 sGRBs (45%) do not have an optical localization, and we relied on the Swift/XRT enhanced positions (Goad et al. 2007;Evans et al. 2009). The associated errors are assumed to follow Rayleigh statistics (Evans et al. , 2020, and in our work are computed at the 68% level of the Rayleigh distribution. The afterglow positional uncertainty from XRT is therefore derived as ≈ err 90 /1.42 (Pineau et al. 2017), where err 90 is the 90% error typically reported by the Swift team 13 .
The remaining 17 sGRBs (55% of the total sample) have an optical counterpart, and for these bursts we obtained publicly available discovery images from the Ultra-Violet Optical Telescope (UVOT; Roming et al. 2005) on-board Swift, the 8.1-m Gemini North Telescope, the GTC, the VLT, the 4.2-m William Herschel Telescope (WHT), the 3.6m Telescopio Nazionale Galileo (TNG), and the 2-m Liverpool Telescope.
We applied standard procedures for reduction and calibration of these ground-based images, and used SExtractor for afterglow localization. For the Swift/UVOT data (GRBs 110402A, 131004A, and 170728A) we used the uvotimsum task within HEASoft v6.27.2 to co-add multiple exposures. This produces a higher signal-to-noise afterglow detection. The afterglow localization error (statistical) was then determined using the uvotdetect task.
We used SExtractor to identify common point sources in both the late-time and discovery images, and then SCAMP to compute the astrometric solution. The rms uncertainty tie in the offset of astrometric matches between the late-time and afterglow images provides the uncertainty in the sGRBs localization on the late-time image frame, and is included within the determination of the host offset error (Bloom, et al. 2002).
The projected offset is then determined by measuring the distance between the afterglow centroid and the host galaxy's center. The latter is determined as the barycenter of the pixel distribution using the parameters XWIN_IMAGE and YWIN_IMAGE and its uncertainty host is derived by adding in quadrature the positional error in both directions. The parameters XWIN_IMAGE and YWIN_IMAGE are calculated within a circular Gaussian window instead of the isophotal footprint of each object. The Gaussian window function is determined separately for each object based on the circular diameter containing half the object's flux. Therefore, XWIN_IMAGE and YWIN_IMAGE are not affected by detection threshold or irregularities in the background, whereas isophotal centroid measurements take into account only pixels with values higher than the detection threshold. The afterglow centroid and its associated uncertainty AG are determined with SExtractor using the same methodology. The uncertainty in the sGRB offset is computed as = √︃ 2 tie + 2 AG + 2 host (Bloom, et al. 2002;. The offset and uncertainty for each GRB is recorded in Table 4. For each candidate host galaxy, we also determine the half-light radius ( ) as measured by SExtractor (with FLUX_RADIUS = 0.5). This allows us to compute a host-normalized offset (see the discussion in §4.1).

Host Galaxy Assignment
The association of a GRB to a host galaxy relies on probabilistic arguments based on the likelihood of finding a random galaxy near the GRB localization. This is estimated by computing the probability to detect a galaxy of equal magnitude or brighter within a given region on the sky (e.g., Bloom, et al. 2002Bloom, et al. , 2007Berger 2010). If the probability is too high or equivalent for multiple galaxies in the field (see Figure 6), the GRB is considered observationally hostless. Using the methods outlined by Bloom, et al. (2002), the probability of chance coincidence is where is the effective angular offset of the galaxy from the GRB position. For XRT localized GRBs, or those where a galaxy is not detected coincident to the GRB position, the effective angular offset is given by = max 3 , √︃ 2 + 4 2 , where 3 ≈ 1.59× err 90 (see, e.g., Section 4.2 of Pineau et al. 2017). If the GRB has a precise (sub-arcsecond) localization, and lies within the visible light of a galaxy, we adopt = 2 (Bloom, et al. 2002). The quantity ( ) in Equation 1 denotes the number density of galaxies brighter than magnitude based on deep optical and infrared surveys (e.g., the Hubble Deep Field; Metcalfe et al. 2001). For our optical observations, we utilize ( ) based on -band number counts from Hogg, et al. (1997). For infrared observations, we use the H-band (HST/ 160 filter) number counts presented by Metcalfe, et al. (2006); Galametz et al. (2013). The magnitude for each galaxy is corrected for Galactic extinction (Schlafly & Finkbeiner 2011) prior to computing the probability. This is done because the galaxy number counts used in this work (Hogg, et al. 1997;Metcalfe, et al. 2006;Galametz et al. 2013) were derived from observations of high Galactic latitude fields, where the extinction is negligible.
For each sGRB, we computed the probability of chance coincidence for all galaxies identified within 1 of the sGRB position. We require that the putative host galaxy for each sGRB has 0.1 to be considered a robust association, otherwise we deem the sGRB to be observationally hostless. At offsets > 1 , a 0.1 requires an extremely bright galaxy 16 mag, which would not be missed in our imaging. We also note that the largest angular offset reported for a sGRB is ∼ 16 for GRB 061201 (Stratta et al. 2007), which we consider to be observationally hostless based on > 0.1. All events with confident host associations are located at smaller angular offsets. In many cases there are a number of faint extended objects ( 23 mag) at 10 which we remove from our analysis due to their high probability of chance coincidence 0.5. The remaining galaxies in the field are then considered candidate hosts; see Figure 6 for an example finding chart for sGRB 130912A based on deep Keck and HST imaging. We report the results of our search for each sGRB in Appendix A, and their finding charts are displayed in Figures 7 and 8. Sources classified as a galaxy are denoted by G1, G2, G3, etc., by increasing offset from the GRB position, whereas sources which could not be classified are labeled as A, B, C, etc., in the same manner.
The probability of chance coincidence reported for each sGRB (Table 4) is based on -band number counts when possible, but if the galaxy is only detected in redder filters we include this probability instead using the number counts presented by Capak et al. (2007) for the -band and Capak et al. (2004) for the -band.

Galaxy SED Modeling
For those events with well-sampled galaxy SEDs but lacking a spectroscopic redshift, we obtained a photometric redshift by modeling the SED using prospector  . We note that these photometric redshifts were determined based on the assumption that the photometric jump between two filters is due to the 4000 Å break. A large break is indicative of an older stellar population. We adopted a Chabrier (2003) initial mass function (IMF) with integration limits of 0.08 and 120 (imf_type = 1), an intrinsic dust attenuation using the extinction law of Calzetti et al. (2000, dust_type = 2), and a delayed-star formation history (sfh=4). Furthermore, we include nebular emission lines using the photoionization code Cloudy (Ferland et al. 2013). In the cases of sGRBs   (Table 3) did not display bright or obvious emission features. The synthetic SEDs derived from these model parameters were calculated using the flexible stellar population synthesis (FSPS) code (Conroy et al 2009) using WMAP9 cosmology (Hinshaw et al. 2013).
The free model parameters are: the redshift , the total stellar mass formed , the age age of the galaxy, the e-folding timescale , the intrinsic reddening , and the metallicity . These parameters are further used to compute the stellar mass * . We adopt uniform priors in log age , log , log , as in Mendel et al. (2014). The prior on the photometric redshift is uniform between phot = 0 − 3. However, only for sGRBs with a UV detection of their afterglow (e.g., sGRBs 110402A and 140129B; see Appendix A) from Swift, we adopt phot = 0 − 1.5. The fits were performed using the dynamic nested sampling method implemented in the DYNESTY package (Speagle 2020). The best fit model SEDs and the resulting photometric redshift estimates are displayed in Figure 9. The photometric redshifts for these sGRBs are recorded in Table 4, and the stellar mass is reported in their individual sections in Appendix A as well as Table 1. In Table 1 we likewise record the star formation rate (SFR), which is computed as outlined in O'Connor et al. (2021).

RESULTS
In this work, we have analyzed the host galaxies and environments of 31 sGRBs; 17 with a sub-arcsecond position from optical observations and 14 with only an XRT localization (Figure 1). In Figures   Table 1. Results of our prospector SED modeling. We present the photometric redshift, stellar mass, and star formation rate. The SED fits are displayed in Figure 9. Short GRB with extended emission. This GRB also has a spectroscopic redshift = 0.854 determined in this work. 7 and 8, we display a finding chart for each sGRB in our sample. We find that 18 events (see Table 4) are associated to a host galaxy ( < 0.1), while 13 events are deemed observationally hostless. With respect to previous work, we have adopted the threshold previously used by Bloom, et al. (2002) and Berger (2010), whereas other authors have utilized lower thresholds, such as 0.01 (Tunnicliffe, et al. 2014) or 0.05 . We demonstrate below that our choice is robust and ensures a low number of spurious associations.
Based on our host galaxy assignments, we identify a spectroscopic redshift for 5 sGRBs in our sample (sGRBEEs 110402A, 160410A, and 180805B, and GRBs 101224A and 140622A; see Tables 4 and 3). In addition, we derive a photometric redshift for 8 events (sGRBEEs 110402A and 170728B, and GRBs 120630A, 140129B, 151229A, 180618A, 191031D and 200411A; Figure 9 and Table 4). The detailed analysis for each sGRB is reported in Appendix A, and the magnitudes and offsets for the putative host galaxies are presented in Table 4. We estimate the number of spurious galaxy associations in our sample following Bloom, et al. (2002). The probability that all sGRB host galaxies discovered in this work are a chance alignment with the GRB localization is given by where = 18 (the number of host galaxies we associate to sGRBs in this work) and is the probability of chance coincidence for each sGRB computed using Equation 1 based on -band number counts ( §3.3). If we compute false for the optical and X-ray localized samples separately, we obtain false = 3.4 × 10 −15 and 1.4 × 10 −10 , respectively. Moreover, the probability that every galaxy has a real, physical association to these GRBs can be estimated using If we consider again the optical and X-ray localized samples individually we find real = 0.76 and 0.48, respectively. As expected, the galaxy associations for the optically localized sample (Figure 7) are more robust, but even the XRT only sample yields a similar result to the value ( real = 0.48) presented by Bloom, et al. (2002) for their sample of long GRBs. Furthermore, we estimate ∼ 2 − 3 spurious associations out of our sample of 31 events (Bloom, et al. 2002). The spurious associations are likely dominated by the XRT localized events. Based on these probabilistic arguments, we consider the host associations determined in this work to be robust, with minimal contamination due to chance alignment. We now compare the properties of the host galaxies determined in this work to other large samples previously presented within the literature (e.g., Fong et al. 2013;Tunnicliffe, et al. 2014). To do so, we supplement the 31 sGRBs that we analyzed with 41 events (29 sub-arcsecond) from the literature with deep host galaxy searches. Out of these 72 well-studied events, we find that 37 have a spectroscopic redshift, 11 have a photometric redshift, 20 are observationally hostless, and 15 display extended emission.
In order to perform a one-to-one comparison with our homogeneously selected sample, we excluded events from the literature which did not satisfy our selection criteria (specified in §2.1 and Table 3): including < 1.5 mag, AG < 4 , and a Swift/BAT detection of the prompt emission. These criteria exclude a number of sGRBs typically included in other samples: sGRBs 050509B, 060502B, 090621B, 100206A, 161104A, and sGRBEE 061210 are excluded due to the large error (> 4 ) of their XRT localization, sGR-BEE 050724 does not satisfy < 1.5 mag, and sGRBEE 050709 (HETE), sGRBEE 060121 (HETE), and sGRB 070707 (INTEGRAL) are excluded as they were not detected with Swift/BAT.
The probabilities of chance coincidence for X-ray localized sGRBs were recalculated with the XRT enhanced positions derived using HEASOFT v6.28. Different versions of the XRT calibration database and analysis software may change the error radius by up to 50% of its value, and this step ensures that all the X-ray positions are based on the same calibration database (HEASOFT v6.28). The resulting probabilities uniformly adopt the 3 positional error (see §3.3), while in the literature different conventions (e.g., 68% or 90% CL) were sometimes adopted.
Based on this re-analysis, 3 XRT localized events (sGRBs 050813, 061217, and 070729) are found to have candidate hosts with > 0.1, and are hereafter considered observationally hostless. This leaves us with only 9 sGRBs in the literature sample with both an XRT localization and a putative host galaxy (sGRBs 051210, 060801, 080123, 100625A, 101219A, 121226A, 141212A, 150120A, and 160624A). Including the events in this work, this sample doubles to 18 XRT localized events with a putative host. The impact of these XRT events is discussed in §4.1.2.

Sub-arcsecond Localized
We begin by studying the angular offset distribution ( Figure 10; top panel) for 34 sGRBs with sub-arcsecond positions. With a few exceptions, this sample coincides with the sample of optically-localized bursts, which have a typical uncertainty of ∼ 0.2 on their offset. The measured angular offsets range between 0.06 (GRB 090426; Antonelli et al. 2009;Levesque et al. 2010) to 16 (GRB 061201; Stratta et al. 2007), with 70% of the bursts lying < 2 from their putative host galaxy's center. For comparison, GRB 170817A was located at 10.6 (2 kpc) from its galaxy's center Im et al. 2017  . Spectral energy distributions of sGRB host galaxies with photometric redshifts determined in this work. The best fit model spectrum (solid line) and model photometry (squares) describing the galaxy SED is compared to the extinction-corrected photometry (circles). The observed Gemini spectrum, smoothed with a Savitzky-Golay filter, for the host galaxies of GRBs 180618A and 191031D is shown by a solid black line (see Table 3).
We convert angular offsets into projected physical offsets by using the sGRB distance scale, typically derived from the putative host galaxy. For sGRBs without a measured redshift (8 events; ∼ 20% of the sub-arcsecond localized sample), we adopt the median redshift ( §4.3), ≈ 0.5, for sGRBs in our sample 14 . We find that the physical offsets of sGRBs range from 0.4 kpc to 75 kpc with a median of 5.6 kpc ( Figure 10; middle panel, red line). This is slightly larger than the median of 4.5 kpc from  and a factor of 4× larger than the median value for long GRBs (Bloom, et al. 2002;Lyman et al. 2017). This result is consistent with the < 10 kpc median sGRB offset derived by O'Connor et al. (2020), and with the expectations from binary population synthesis of BNS mergers (see, e.g., Fryer, et al. 1999;Bloom, et al. 1999;Belczynski et al. 2006;Church et al. 2011;Mandhai et al. 2021;Perna et al. 2021), although 14 We note that the subset of events without a measured redshift are very unlikely to reside at < 0.5, and are more likely between ∼ 0.5 − 1, where the difference in angular scale is ( = 1.0)/ ( = 0.5) ≈ 1.3. We find that varying the redshift of these events does not significantly affect our results. some modeling efforts predict larger median offsets (Zemp, et al. 2009;Wiggins et al. 2018).
The last quantity to explore is the host-normalized offset, which provides the most uniform comparison between the location of sGRBs with respect to their galaxies ( Figure 10; bottom panel). We find that the median host normalized offset of the entire sGRB sample (sub-arcsecond localized) is / ∼ 1.2 ( Figure 10; bottom panel). However, our dataset includes both high-resolution HST imaging and seeing-limited ground-based observations, and the latter might bias the inferred half-light radii of faint unresolved galaxies to larger values. By performing a homogeneous analysis of the HST dataset only, we derive / ∼ 2, consistent with the value from the literature . For comparison, the median host normalized offset for long GRBs is / ∼ 0.6 (Blanchard, et al. 2016;Lyman et al. 2017).
Furthermore, based on Figure 10, we find that the offset distribution of this sample of sGRBEEs (dark blue lines) is a factor of 3 − 4× further extended than long GRBs (purple lines). A KS test between the two samples yields KS ≈ 0.04 (in both host normalized and physical offset), rejecting the null hypothesis that they are drawn from the same distribution at the ∼ 2 level. This provides additional  Figure 10. Top: Cumulative distribution of angular offsets for all subarcsecond localized sGRBs in our sample (red). We split the sample of all sGRBs into two sub-samples: the sample of sGRBs with 90 < 2 s (cyan) and the remaining 10 events displaying EE (blue). Middle: Cumulative distribution of projected physical offsets for 33 sGRBs with sub-arcsecond localization (red). The offsets of long GRBs (purple) are displayed for comparison (Blanchard, et al. 2016). Bottom: Same as middle panel but for host-normalized offsets.
and independent support to the hypothesis that their progenitors are different from those of long GRBs (Norris & Bonnell 2006;Gehrels et al. 2006;Gal-Yam et al. 2006). Moreover, we find that the offset distributions (angular, physical, and host normalized) for classical sGRBs with 90 < 2 s ( Figure 10; cyan lines) and those displaying EE ( Figure 10; dark blue lines) are consistent with being drawn from the same distributions. The comparison in Figure 10 is made for 24 classical sGRBs and 10 sGRBEEs, all of which have a sub-arcsecond localization. If we include the offsets to the lowest candidate hosts for hostless events (see §4.1.3), increasing the sample sizes to 34 sGRBs and 11 sGRBEEs, we find the same result. This suggests that regardless of whether classical sGRBs and sGRBEEs are created by different progenitor systems, their merger environments are indistinguishable based on these limited number of events.
We also explored whether there was an evolution of the observed offset distribution with redshift. In this analysis, we focus only on events with a measured and secure spectroscopic redshift. In Figure 11, we separate the physical offsets for sub-arcsecond localized GRBs into two distributions with < 0.5 and > 0.5. The median offset for sGRBs at < 0.5 (7.5 kpc) is a factor of ∼ 2× higher than those at > 0.5 (3.2 kpc), despite a KS test supporting that they are drawn from the same distribution ( KS = 0.09). In addition, no sGRBs at > 0.5 have a projected physical offset > 15 kpc, compared to 50% of those at < 0.5. If we perform the same comparison for the host normalized offset distribution (Figure 11), we find that the two samples are again consistent with being drawn from the same distribution with KS =0.25, despite all events at > 5 being located at low redshifts. Although the distributions are similar statistically, the lack of large offsets at > 0.5 is suggestive of a redshift evolution effect. The physical implications of this possible redshift evolution are discussed in §5.1.

Including XRT Localized sGRBs
The previous section focused on sub-arcsecond localized events, however, the majority of sGRBs have only an XRT localization. For the sample of 99 events satisfying our selection criteria ( §2.1), the median error on the XRT enhanced position is ∼ 1.8 . Due to this large uncertainty, often comparable to the measured angular offset, XRT localized events are difficult to include in the offset distribution. Here, we adopt a Bayesian formalism to identify the true distribution of offsets for XRT localized GRBs. Following Bloom, et al. (2002), we assume that the probability density distribution of the GRB's offset from its host galaxy follows a Rice distribution (Wax 1954), denoted by R ( , , ) where and are the shape parameters.
Applying Bayes' theorem, the posterior distribution for the true offset, true , of the GRB from its host galaxy's center given the observed offset, obs , and its uncertainty, , is where the probability density for the likelihood ( obs | true ) is given by the Rice distribution R ( obs , true , ).
The choice of prior distribution, ( true ), can have a significant impact on the unknown posterior. While simple priors may appear to minimize our assumptions on the underlying distribution, we note that they are generally unrealistic. For example, assuming that the GRB has an equal probability of occurring anywhere in a circle surrounding the galaxy's centroid (i.e., uniform probability in area), such that ( true ) ∝ true , preferentially favors larger radii. Whereas both observations of sGRBs ( Figure 10) and models of BNS systems (Bloom, et al. 1999) find that the significant majority of systems form at < 10 kpc. Therefore, we consider two different prior distributions: i) following the observed distribution of physical offsets for subarcsecond localized sGRBs ( Figure 10), and ii) assuming that GRBs form following an exponential profile ( true ) ∝ exp(− true / * ) where * is taken to be the half-light radius of each galaxy. In Figure  12, we refer to these priors as "observed" and "exponential".
We choose to adopt the median value of the posterior distribution ( true | obs ) for each GRB's offset, and include these XRT localized GRBs within the cumulative distribution of sGRB offsets. In Figure 12 we demonstrate how the X-ray localized events impact the offset distribution for the two prior distributions. The "observed" and "exponential" priors only cause a marginal deviation from the sub-arcsecond only distribution. Therefore, based on this analysis, the offsets of X-ray localized events are not inherently different from those with an optical localization.

Including Hostless sGRBs
Up to this point, we have focused on the offset distribution of sGRBs with a confident host galaxy association ( < 0.1). Here, we include in our study 12 sub-arcsecond localized observationally hostless events. For these bursts, we identify the galaxy with the lowest chance probability and measure the offset between the burst position and the galaxy's centroid (Appendix A). Only 2 of these events are located within 10 kpc of their most likely host and, as a result, the median offset for the sample is 26.4 kpc, 5× larger than the value derived in §4.1.1 (see also Figure 13). We further examine the implications of these hostless events in §5.2. Sub-arcsecond X-ray (Observed Prior) X-ray (Exponential Prior) Figure 12. Cumulative distribution of sGRB offsets for the sample of subarcsecond localized events (purple) compared to X-ray localized events for two different priors ( §4.1.2): i) the "observed" prior (yellow) and ii) the "exponential" prior (red). . Cumulative distribution of projected physical offsets for subarcsecond localized sGRB with a putative host (red) and for those which are hostless (blue); the total population is shown in black.

Host Luminosities
In Figure 14, we display the apparent -band magnitude (corrected for Galactic extinction) of sGRB host galaxies plotted against their redshift. By comparing the brightness of these galaxies to a sample of ∼ 30, 000 galaxies from the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey project (CANDELS, Koekemoer, et al. 2011;Grogin, et al. 2011) Ultra Deep Survey (UDS; Galametz et al. 2013), we confirm that the host galaxies of sGRBs trace the brightest galaxies (0.1 − 1.0 * ) at each redshift. In the right panel of Figure 14, we report the -band magnitude of candidate host galaxies without a known redshift, including the lowest candidate host galaxies of observationally hostless events.
We have identified that 4 sub-arcsecond localized observationally hostless events within our sample (e.g., GRBs 150423A, 160408A, 160601A, 160927A) have lowest candidates (see §4.1.3) with faint -band magnitudes ( 24.5 mag; corrected for Galactic extinction). When compared to typical sGRB host galaxies ( Figure 14) this is suggestive of either i) an origin at > 1 or ii) a population of under-luminous sGRB galaxies (< 0.1 * ). Even if under-luminous, these galaxies would have to occur at > 0.5 in order to avoid an unexplained gap in luminosity ( Figure 14) between faint galaxies   Figure 15. Host galaxy -band magnitude versus angular offset for the sample of sGRBs included in this work. We also include GRBs where the galaxy with the lowest probability of chance coincidence has > 0.1 (gray). The shaded gray region marks where > 0.1. and the known bright hosts at low-. We note that there are only a handful of examples of low luminosity (< 0.1 * ) sGRB host galaxies in GRBs 070714B (Cenko et al. 2008), 101219A ), 120804A Dichiara et al. 2021), and 151229A (this work), all of which reside at > 0.5. We observe the same trend in the observationally hostless sample of XRT localized sGRBs (e.g., GRB 140516A, 150831A, 170127B, 171007A, 180727A); there are faint 24.5 mag candidates detected within their XRT localization's, which range from 2.2 − 2.7 (90% CL).
We emphasize that none of these events are located near bright, low-galaxies (none within 60 ) from which they could have been kicked. This is in contrast to other observationally hostless events, such as sGRBs 061201, 090515, and 091109B, where the most likely host galaxy is a bright, low-galaxy at a significant offset. We discuss this further in §5.2.
In Figure 15, we show the -band magnitude of sGRB host galaxies versus the angular offset of the sGRB from its host for both X-ray (diamonds) and optically localized GRBs (circles). The gray shaded region represents the region precluded from a strong host association, due to > 0.1. Based on the distribution of XRT localized events we find that it is difficult to associate a galaxy fainter than > 23.5 to a GRB lacking a precise, sub-arcsecond localization. While the brightest sGRBs may have an X-ray localization (from Swift/XRT) of ∼ 1.4 − 1.5 (90% CL), the majority are less precisely localized to > 2 . As such, the majority of X-ray localized sGRBs are limited to associations with galaxies brighter than < 23.5 mag, decreasing the likelihood of association with galaxies at > 1 (see §5.1).

Redshift Distribution
Our sample consists of 72 well-localized sGRBs (including the subclass of sGRBEEs) observed in homogeneous conditions. Of these, 37 (51%) have a spectroscopic redshift, 11 (16%) a photometric redshift, and 24 (33%) lack a distance measurement. Only three of these redshift measurements come from direct afterglow spectroscopy, whereas the large majority are determined from the putative host galaxy. In Figure 16 (top panel), we display a histogram of the observed redshift distribution. The median value is ≈ 0.5 for the sample of spectroscopic redshifts, and ≈ 0.6 for the combined sample of photometric and spectroscopic redshifts. By adding 4 spectroscopic redshifts at >0.5 and 7 photometric redshifts at >0.4, our work mainly populates the upper tail of the distribution. This shows the importance of deep imaging and spectroscopy, using large aperture 8 − 10m telescopes, in probing the most distant sGRBs and their faint host galaxies. However, only 1 of our events lies at > 1 (Table 4). This is not surprising as our survey is optically-driven and affected by complex selection effects, such as the so-called "redshift desert" (1.4 < < 2.5; also marked in Figure 16) where common nebular emission lines are shifted towards infrared wavelengths. A similar systematic survey of sGRBs at nIR wavelengths would be essential to complement our study and extend the redshift distribution of sGRBs.
The number of distant sGRBs is an important constraint for progenitor models and their delay time distribution (DTD). In Figure  16 (bottom panel), we show the cumulative distribution of sGRB redshifts (including photometric redshifts) compared to predictions based on different DTD models. The two models commonly adopted in the literature are: i) a log-normal distribution (Nakar, et al. 2006;Wanderman & Piran 2015) and ii) a power-law with decay index between ∼ −1 to −1.5 (Hao & Yuan 2013).
A KS test between our distribution and the Nakar, et al. (2006) model yields = 10 −2 , rejecting the null hypothesis that the observed redshift distribution is drawn from their model. The observed distribution appears instead consistent with the power-law DTD models with slope ∼ −1 to −1.5 15 . However, a significant population of bursts with no known redshift exists. Our survey identifies that their likely host galaxies are much fainter than the rest of the sample (Figure 14), and a likely explanation is that these bursts represent a missing population of high-sGRBs. A larger number of > 0.5 events increases the tension with the log-normal DTD models.
In the most extreme case, these would be prompt mergers with a negligible delay time between formation and merger. In Figure  16 we show the implications of this scenario. The dotted black line represents the hypothetical redshift distribution derived assuming that all the bursts with no known redshift follow the SFH of the Universe (Moster, et al. 2013). This sets a lower limit to the true redshift distribution and helps constrain the parameter space allowed by observations. By assuming that sGRB progenitors are described by a single DTD function, the Hao & Yuan (2013) curve is consistent with all the observing constraints. 15 We note that the redshift distribution also depends on the assumptions as to the SFH, gamma-ray luminosity function, detector sensitivity, and minimum delay time, and can therefore be different even for the same DTD.  Figure 16. Top: Histogram of the observed spectroscopic redshifts (purple) for 36 sGRBs matching our selection criteria. We also show a sample of photometric redshifts (blue) for 12 additional events. The gray solid region marks the "redshift desert" between 1.4 < < 2.5. Bottom: Cumulative distribution of sGRB redshifts (black) compared to the expected distribution for several different DTDs (Nakar, et al. 2006;Hao & Yuan 2013;Wanderman & Piran 2015). In these models, represents the delay time. For log-normal distributions, the width of the distribution is given (Nakar, et al. 2006;Wanderman & Piran 2015). The dashed black line represents a lower limit to (< ) assuming ∼ 50% of the population occurs at > 1 with a negligible delay time.

Circumburst Environment
In this section, we explore the consistency between the observed offsets of sGRBs around their galaxies and their inferred circumburst environment based on observations of their afterglows in X-rays. First, we use the onset of the X-ray afterglow from Swift/XRT to set a lower limit to the circumburst density for each of the 31 bursts in our sample (see O'Connor et al. 2020 and our Appendix B). Of these 31 bursts we find that < 33% have a circumburst density consistent with min < 10 −4 cm −3 , setting an upper limit to the fraction of sGRBs in this sample occurring in a IGM-like environment (physically hostless; see Appendix B). Of these potentially low-density events, 5 are observationally hostless (Table B1).
Moreover, we searched for a correlation between the GRB offsets and their high-energy properties. In particular, the ratio of the X-ray flux at 11-hours, ,11 , to the prompt gamma-ray fluence, , is known to probe the circumburst density such that ,11 / ∝ 1/2 (Sari, et al. 1998;Wĳers & Galama 1999;Granot & Sari 2002). This is valid only in the synchrotron slow cooling regime when the cooling frequency lies above the X-ray band, and does not accounts for energy injection from the central engine. Moreover, this quantity ,11 / is independent of distance. In Figure 17, we observe that there is a large scatter in the correlation (see also O'Connor et al. 2020). Although GRBs with small offsets tend to occupy the upper part of the plot, and those with larger offsets the lower part, no trend can be conclusively established.
We find no evidence for a population of bursts in a rarefied environment (i.e., a low ratio of X-ray flux to gamma-ray fluence in comparison to other events at a similar offset. For example, see GRB 211211A, Troja et al. 2022). Instead, we find that observationally hostless sGRBs (e.g., sGRBs 061201, 091109B, 110112A, 111020A, 160601A, 160927A) are not X-ray faint when compared to the overall population, as they all lie above log( ,11 / ) −6.1. While these events have no secure host association, we paired them with their most likely host galaxy to calculate their offsets in Figure 17. However, the X-ray brightness of their afterglows does not support the large offset/low density scenario implied by these galaxy's associations and may suggest that they reside in faint hosts at > 1.

A Redshift Evolution of sGRB Locations
By exploring the distribution of sGRB offsets at < 0.5 and > 0.5 ( Figure 11; top panel), we identified a redshift evolution in the locations of sGRBs around their galaxies. Based on our analysis, there are no events with > 0.5 at physical offsets > 15 kpc, compared to 50% at < 0.5. We examine three possible factors which could be at the origin of the observed trend: i) an evolution of the host galaxy size, ii) an intrinsic property of their progenitors, or iii) an observational bias against dim high-z galaxies.
The increased size of sGRB host galaxies over cosmic time possibly leads to a larger birth radius of the progenitor, and therefore a larger offset. This is consistent with observations of galaxy size evolution following the relation ∝ (1 + ) − with ≈ 0.6 − 1.3 (see, e.g., Dahlen et al. 2007 leading to growth by a factor of ∼ 2× between = 1 and the present. It is not clear if this growth is completely due to a true galaxy evolution effect or an observational bias due to surface brightness dimming with distance. Nonetheless, we show that, when normalized by the host galaxy's size, the two distributions at < 0.5 and > 0.5 move closer to each other ( Figure 11). In particular, for offsets < they seem to track each other well. However, we find that all events with offsets > 5 reside only in low-galaxies.
By correlating the physical offset with host galaxy type (see Figure  18), we find that low-early-type galaxies preferentially host these sGRBs with large spatial offsets. These events are commonly interpreted as highly kicked BNS systems (Behroozi et al. 2014;Zevin et al. 2020) or BNS mergers dynamically formed in globular clusters (Salvaterra, et al. 2010;Church et al. 2011). However, we note that an alternative possibility is that the sGRB progenitors were formed in the extended stellar halo of their galaxy (Perets & Beniamini 2021), and as such do not require large natal kicks. Thus, the large host normalized offsets may be due to the fact that is not a good tracer of the extended stellar halo in early-type galaxies (D'Souza et al. 2014;Huang et al. 2018).
Another physical explanation for this evolution is that systems merging at low redshifts had a longer delay time between formation and merger of the binary, allowing them to travel further distances than those merging at higher redshifts. However, through population synthesis, Perna et al. (2021) found the opposite trend: simulated BNS at high redshift reach a larger distance from their host galaxies. In fact, they found that ∼ 20% of BNS systems in simulated galaxies at = 1 reach offsets > 15 kpc, whereas none have been identified observationally. Future population synthesis modeling, specifically using inferences from observations of Galactic BNS systems Nevertheless, we bear in mind that an alternative scenario to explain the redshift evolution is an observational bias against faint highgalaxies. This bias can most easily be understood based on Figure  14, where the decrease in host galaxy apparent magnitude as a function of redshift is displayed. For instance, above > 1 the majority of galaxies in the universe are fainter than > 23.5 mag, with a significant fraction dimmer than > 25 mag. In order to associate a GRB to such faint galaxies ( Figure 15) requires an offset of 3 (corresponding to 25 kpc, assuming ≈ 1). This condition becomes more stringent if the probability of chance coincidence cutoff threshold is decreased from the 10% value used in this work ( §3.3). For example, adopting a cutoff value of 5%, as used in previous studies , requires an offset 2.2 or, equivalently, 18 kpc, even for sub-arcsecond localized sGRBs. Surprisingly, even a Milky Way-like spiral galaxy at ≈ 1 ( ≈ 23 mag) will have a probability of chance alignment larger than 5% (10%) if the projected physical offset is > 20 (30) kpc (Tunnicliffe, et al. 2014). Therefore, we find that it is unlikely, based on probabilistic grounds, to associate high-sGRBs to galaxies at large physical offsets. This bias may explain, at least in part, the observed redshift evolution of sGRB offsets and should be taken into account when comparing the observed offset distribution to progenitor models.

Number of sGRB
Late-type Early-type Figure 18. Histogram of projected physical offset of sGRBs from their host galaxies. The distribution for late-type galaxies is shown in purple, and earlytype hosts in green (Gompertz et al. 2020;Paterson et al. 2020;O'Connor et al. 2021). We have limited the sample to those with classified galaxy type and an error on their offset of < 20%.

Observationally Hostless Fraction
We have selected a homogenous sample ( §2.1) of short GRBs detected by Swift/BAT of which 72 have a sensitive search for their host galaxy. We identify that ∼ 28% (20 events) of these 72 events are observationally hostless (see Figure 19 for a breakdown of the fraction of events with and without a host separated by their localization). This fraction is higher than the value of 17% reported by Fong et al. (2013). We find that this difference is mainly driven by the larger sample of X-ray localized events studied in our work. Considering only the sample with sub-arcsecond positions, the hostless fraction is 26%, consistent between the two works.
As the fraction of hostless sub-arcsecond localized events is consistent with the full population, we find that our result is not driven by the lower accuracy of X-ray localized events. In fact, in §4.1.2, we demonstrated that the offsets of X-ray localized events are consistent with the locations of sub-arcsecond localized sGRBs ( Figure  12). This suggests that any selection bias against large offsets or low-density environments acts on both samples in the same way.

Interpretation of Hostless Events
We emphasize that there is a lingering ambiguity as to the origin of hostless short GRBs. The main scenarios are that i) the GRB was kicked to a substantial distance from its birth galaxy, such that the probability of chance alignment is large, or ii) the GRB merged in a faint, undetected galaxy at a smaller angular distance. However, the diagnosis for individual events is complicated, and it is difficult to distinguish between these two scenarios. For instance, the hostless sGRBs presented by Berger (2010) are located at a significant offset (30 − 75 kpc) from bright low-galaxies ( < 0.5). However, despite their brightness, the probability of chance coincidence is 10%. Therefore, it is not clear whether these sGRBs are truly associated to these low-galaxies, or whether they reside in faint, undetected

Sub-arcsec Host (34) 47%
Sub-arcsec Hostless (12) 17% XRT Hostless (8)  11% XRT Host (18) 25% Figure 19. Breakdown of the fraction of 72 events considered in this work into those with a putative host galaxy and those that are considered hostless. We have separated these events further based on their localization either with XRT (purple) or to a sub-arcsecond position (blue). The total fraction of hostless events is 28% (11% XRT and 17% sub-arcsecond localized). The total number of hostless events is 20, with 12 of them having a sub-arcsecond localization.
hosts ( > 26 mag). The interpretation has a direct impact on the energetics, redshift ( §4.3), and delay time distributions of sGRBs.
In this work, we have tripled the number of observationally hostless sGRBs (from 7 to 20 events). We find that half of the observationally hostless sGRBs lack any nearby (low-) candidate host. These events are more likely to have exploded in faint 24.5 mag galaxies (see §4.2) that are consistent with 0.1 − 1.0 * galaxies at > 1. We note, however, that an alternative explanation is that these represent a population of low luminosity (< 0.1 * ) galaxies hosting sGRBs at < 1, although this is at tension with the population of well-determined sGRB hosts (0.1 − 1 * ; Berger 2010) and with predictions from population synthesis modeling, which find that BNS systems preferentially form in the most massive (brightest) galaxies (Behroozi et al. 2014;Mapelli et al. 2018;Artale et al. 2019Artale et al. , 2020aAdhikari et al. 2020;Mandhai et al. 2021;Chu et al. 2022).
Previous work in the literature (see, e.g., Berger 2010; Tunnicliffe, et al. 2014) has focused on the likelihood to detect faint galaxies at high-, as opposed to the large probability of chance coincidence even in the event that a galaxy is detected. We find that despite detecting these faint galaxies, they are difficult to confidently associate to the GRB using the standard probability of chance coincidence methodology (Bloom, et al. 2002). This is indicative of an observational bias against faint galaxies (see also §5.1).
We note that a larger population of sGRBs at > 1 implies a steep DTD with an increased fraction of events with short delay times, as deduced based on Galactic BNS systems (Beniamini & Piran 2019). This would further disfavor log-normal DTD models ( §4.3), and support a primordial formation channel for these events.
or the We further explored the sample of observationally hostless events that lie close to low-galaxies. We exploited their highenergy properties to probe their environments ( §4.4), as their circumburst density can be used to constrain their allowed physical offset (O'Connor et al. 2020). Figure 17 shows a weak correlation between X-ray afterglow brightness with the sGRB location, such that a larger offset leads to fainter X-ray emission. The X-ray constraints for hostless events are either too shallow or inconsistent with the observed trend. Although this does not conclusively rule out that these hostless sGRBs could be mergers kicked out into the IGM (physically hostless), it does not offer observational support and leaves their nature undetermined. Rapid and deep X-ray observations with nextgeneration instruments (e.g., the Athena X-ray observatory; Nandra et al. 2013) will be capable of probing X-ray fluxes of ∼ 10 −16 erg cm −2 s −1 within 12 hr of the GRB trigger, and, therefore, will be able to detect the low flux regime of physically hostless sGRBs.
We note that the main factor preserving the ambiguity in interpreting these events is that the distance scale to the sGRB is not known. Therefore, in order to disentangle between faint hosts and large offsets we require better constraints as to the distance to short GRBs. The most critical observational tests are i) rapid afterglow spectroscopy to determine redshift independent of the galaxy association (e.g., GRB 160410A; this work and Agüí Fernández et al. 2021), ii) the conclusive identification of a kilonova, providing indirect evidence of the GRB distance scale Chase et al. 2022), or iii) the advent of next generation GW detectors capable of detecting compact binaries at cosmological distances (Punturo et al. 2010;Dwyer et al. 2015).

CONCLUSIONS
We carried out a systematic study of the host galaxies of 31 short GRBs. This analysis effectively doubles the sample of well-studied sGRB host galaxies, leading to a total of 72 events fitting our selection criteria with sensitive searches for their host. We assign a spectroscopic redshift to 5 of these events, and derive a photometric redshift for 7 others. Based on the results of this study, we present the subsequent findings: (i) The sub-arcsecond localized population of sGRBs has a median projected physical offset of 5.6 kpc (4× larger than for long GRBs; Blanchard, et al. 2016;Lyman et al. 2017), with 70% of events occurring at < 10 kpc from their host's nucleus. (ii) We find that 28% of sGRBs (20 out of 72) lack a putative host galaxy to depth > 26 mag. For half of these hostless bursts, the most likely host is a faint ( > 24.5 mag) galaxy consistent with a high redshift origin ( > 1). (iii) Based on this evidence and the larger sample of 48 redshifts, we have presented improved constraints on the redshift distribution of sGRBs. We find that 20% of sGRBs with known redshift lie above > 1, although this number could be as high as 50% when including the population of events with no known host. The data is inconsistent with log-normal DTDs for their progenitors, and instead favors power-law models with index −1 or steeper. (iv) By correlating the high-energy properties of sGRBs with their locations, we find evidence of a possible trend linking the X-ray brightness to the distance from the host galaxy. We point out that hostless events, if associated to their most likely nearby galaxy, do not follow this trend. Hence, their X-ray brightness does not lend support to their interpretation as mergers in a rarefied medium. (v) We find that sGRBEEs are inconsistent with the offset distribution of long GRBs in both projected physical offset and host normalized offset. This conclusion is reached independently of classical sGRBs. (vi) Lastly, we uncover that the low redshift population of sGRBs is further offset by a factor of 2× from their hosts compared to the sample at > 0.5 with the median value increasing from 3.2 to 7.5 kpc. This redshift evolution can be explained either by a physical evolution in their progenitors or the larger size of low-galaxies. Another possibility is that the apparent redshift evolution is due to a selection bias against faint galaxies that reside at higher redshifts.
We emphasize that while late-time observations alone cannot allow for concrete host associations for events at > 50 (25) kpc past 0.1 (1.0), rapid optical spectroscopy can determine the GRB's distance scale and yield a confident host galaxy assignment. Moreover, rapid and deep optical and infrared observations can lead to the identification of a kilonova, providing an indication of the GRB's distance. These transient are expected to be detectable out to ∼ 1 with both current (James Webb Space Telescope; JWST) and future observatories (e.g., the 39-m Extremely Large Telescope; Gilmozzi & Spyromilio 2007).
In addition, the combination of next generation GW detectors (i.e., Einstein Telescope and Cosmic Explorer; Punturo et al. 2010;Dwyer et al. 2015) with EM observations can allow for confident associations (out to ∼ 4 − 10; Hall & Evans 2019;Singh et al. 2021) as the distance of the GW event can be compared to nearby galaxies. This will allow us to unambiguously distinguish between the large offset scenario and a high-explanation for observationally hostless sGRBs.
Lastly, future infrared observations with HST and JWST will probe lower stellar mass galaxies as a function of redshift ( Figure 6), allowing for more robust limits on the possible faint (high-) galaxies these sGRBs. High resolution observations would also allow for an accurate morphological analysis of the detected hosts, leading to a better understanding of the ratio of early-to late-type galaxies, which yields important information as to the age and formation channels of sGRB progenitors and can illuminate whether events at large offsets are due to kicks or formation in their galaxy's halo.
Gemini Observatory, a program of NSF's OIR Lab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini Observatory partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigación y Desarrollo (Chile), Ministerio de Ciencia, Tecnología e Innovación (Argentina), Ministério da Ciência, Tecnologia, Inovações e Comunicações (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). The HST data (ObsID: 14685) used in this work was obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts. These results also made use of Lowell Observatory's Lowell Discovery Telescope (LDT), formerly the Discovery Channel Telescope. Lowell operates the LDT in partnership with Boston University, Northern Arizona University, the University of Maryland, and the University of Toledo. Partial support of the LDT was provided by Discovery Communications. LMI was built by Lowell Observatory using funds from the National Science Foundation (AST-1005313). This paper makes use of data obtained from the Isaac Newton Group of Telescopes Archive which is maintained as part of the CASU Astronomical Data Centre at the Institute of Astronomy, Cambridge. This work is based on data from the GTC Public Archive at CAB (INTA-CSIC), developed in the framework of the Spanish Virtual Observatory project supported by the Spanish MINECO through grants AYA 2011-24052 and AYA 2014-55216. The system is maintained by the Data Archive Unit of the CAB (INTA-CSIC). Based on observations made with the Liverpool Telescope operated on the island of La Palma by Liverpool John Moores University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias with financial support from the UK Science and Technology Facilities Council. Additionally, this work is based on data obtained from the ESO Science Archive Facility. We additionally made use of Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration et al. 2018).

DATA AVAILABILITY
The data underlying this article will be shared on reasonable request to the corresponding author.     XRT position error reported at 90% CL; optical localization error reported at 1 (68%). Host galaxy magnitude in -band, and computed using -band magnitude (Berger 2010), unless otherwise specified. Redshift from afterglow (AG) spectroscopy. Photometric redshift phot based on prospector (Johnson et al. 2019) modeling of the host galaxy SED. HST/ 110 magnitude, and computed using IR number counts (Galametz et al. 2013).
ℎ computed using -band number counts (Capak et al. 2004). Projected physical offset assuming = 0.5. Projected physical offset assuming = 1.0. The uncertainty on the sGRB's offset is computed at the 68% of the Rayleigh distribution.
We analyzed public archival late-time images of GRB 091109B obtained with the HST/WFC3 in the 110 filter. These observations are not contaminated by a diffraction spike at the GRB localization, which was observed in previous HST/WFC3 imaging ) that set a limit 160 25.0 mag on a coincident galaxy. In this new HST observation, we do not find a coincident source to depth 110 27.2 mag (corrected for Galactic extinction).
However, we identify two previously unresolved sources (source A and G1) within 2 of the GRB position ( Figure 7); all other candidate host galaxies were previously discussed in  and Tunnicliffe, et al. (2014). Source A is offset by 1.0 from the GRB position with magnitude 110 = 27.0 ± 0.3. G1 is offset by 1.4 with 110 = 26.51 ± 0.16. The probability of chance alignment is = 0.21 and 0.27 for source A and G1, respectively. The other host galaxy candidates discussed by  and Tunnicliffe, et al. (2014) are located at larger offsets (∼ 12 − 23 ), but are significantly brighter ( 110 ∼ 18 − 20 mag). We find that each of these sources (Source A and B from Tunnicliffe, et al. 2014, andG1 andG2 from Fong & have > 0.2, based on -band number counts, compared to the previously reported ≈ 0.10 (for both sources) based on galaxy number counts in the optical Tunnicliffe, et al. 2014). In either case, there are multiple galaxies with similar probabilities of chance coincidence, which complicates the host identification. These results confirm that GRB 091109B is observationally hostless.
Here, we present unpublished archival HST/WFC3 imaging obtained on October 13, 2016 in the 110 filter. We uncover multiple extended sources within 5 , which were not detected in previous deep ground based imaging (Magellan/Gemini; Fong et al. 2013) to 25.5 and 26.2 mag. Due to the high density of sources, in Figure 7 we label only the sources with the lowest probability of chance coincidence (source A, G1, and G2). The closest source to the GRB position (source A) is offset by 1.6 and has magnitude 110 = 27.2 ± 0.3 mag, yielding = 0.45. The other nearby candidate hosts are G1 and G2 with offsets of 2.3 and 4.8 and magnitude 110 = 26.25 ± 0.15 and 110 = 24.18 ± 0.07 mag, respectively. These sources likewise have a large ; 0.49 and 0.65 for G1 and G2. We do not identify a source coincident to the optical localization to depth 110 27.3 mag. Thus we consider GRB 110112A to be observationally hostless, in agreement with previous work Tunnicliffe, et al. 2014).
The previous analysis by Fong et al. (2013) identified 15 galaxies within 3 of the GRB position with the two galaxies having the lowest probability of chance coincidence located at 4.8 (G2 in our analysis) and 20 with = 0.4 and 0.5, respectively. Therefore, based on both ground based and HST imaging, GRB 110112A is an outlier among observationally hostless GRBs (e.g., Fong et al. 2013;Tunnicliffe, et al. 2014) as there were no likely host galaxies (i.e., < 0.2) identified. Our analysis represents a confirmation of the observationally hostless classification with deep HST imaging.
O'Connor et al. (2020) derived a lower limit to the density of the GRB's environment min 1.4×10 −3 cm −3 (see their Appendix A). This density is inconsistent with the GRB being physically hostless (see also Figure 7 of O'Connor et al. 2020), and strongly implies the GRB occurred within a galactic environment (either G1, Source A, or a fainter undetected host).
Swift/XRT localized a fading X-ray source, identified as the afterglow, at RA, DEC (J2000) = 13 ℎ 09 36 .58, 61°15 9.2 with accuracy 1.5 (90% CL). Swift also detected the optical afterglow in stacked UVOT exposures with detections in the wh, , 1, and 2 filters, implying a redshift 1.5. We use the stacked UVOT wh-band image to localize the GRB position to RA, DEC (J2000) = 13 ℎ 09 36 .63, 61°15 9.9 with 1-sigma error (statistical) AG = 0.07 , consistent with the afterglow position originally reported by Mundell et al. (2011). The position error does not include the systematic tie uncertainty between UVOT and USNO, as we utilize relative alignment between UVOT and our late-time imaging to derive a precise offset of potential host galaxies from the GRB.
We obtained observations with the Gemini North telescope on August 3, 2020 in the -band, followed by observations in the andbands with the LDT on May 5 and 6, 2021. We further complemented our observations with archival Keck/LRIS images taken on May 27, 2014 in the and filters. Our observations unveiled the presence of three galaxies nearby the GRB position (Figure 7). The first galaxy (G1) is located at 0.91±0.17 from the GRB position with magnitude = 24.13 ± 0.11, = 24.20 ± 0.20, = 23.32 ± 0.09, and = 22.98 ± 0.16 mag. The two other galaxies are located at larger offsets of 6.3 (G2) and 7.4 (G3). G2 has magnitudes = 23.30 ± 0.10, = 22.62 ± 0.08, and = 22.18 ± 0.09 mag, whereas G3 has = 23.19 ± 0.12, = 22.22 ± 0.05, and = 21.21 ± 0.05 mag. No other sources are identified near the GRB position to depth 25.2 mag. The probability of chance coincidence for these galaxies is = 0.03, 0.36, and 0.29 for G1, G2, and G3, respectively. Based on this, we consider G1 the putative host galaxy of GRB 110402A.
We utilized the broadband SED (see Table 2) from the Keck, Gemini, and LDT observations to derive a photometric redshift phot = 0.9 ± 0.1 and a stellar mass log( * / ) = 9.5 +0.4 −0.2 using the prospector software (Johnson et al. 2019) with the methods outlined in O' Connor et al. (2021) and ; see also Appendix 3.4 and Figure 9 for more details. This photometric redshift is consistent with the upper limit to the GRB redshift based on the 2 detection of the afterglow. Additionally, we analyzed Keck/LRIS spectra of G1 taken on May 27, 2014 (Table 3). A faint trace is visible above 7000 Å and we identify a single emission line at 6910 Å which we interpret as [OII] 3727 at = 0.854 ± 0.001. This interpretation is supported by the galaxy SED and the photometric redshift from prospector.
Adopting a redshift = 0.854, we derive a lower limit on the density of the GRB environment, min 4.0 × 10 −4 cm −3 , using the early X-ray lightcurve. This limit is consistent with the GRBs moderate offset from G1, = 7.2 ± 1.3 kpc. We further derive a host-normalized offset of / ∼ 1.3 ± 0.3.
We carried out late-time observations of GRB 130912A on February 25, 2014 with LDT/LMI in -band and on October 25, 2014 with Keck/LRIS in the and filters. We supplement these observations with archival imaging by HST/WFC3 in the F110W filter obtained on January 9, 2017. In the HST imaging we detect three very faint sources at < 3 from the afterglow location which were not previously detected in the ground-based LDT or Keck imaging, see Figures 6 and 7. Sources A and B have magnitudes ∼ 26.8 ± 0.3 and ∼ 26.7±0.3 at offsets 0.7 and 1.2 , respectively. This yields chance probability = 0.08 and 0.21 using -band number counts. The third source, labelled as G1, likewise has a high probability of chance coincidence, ∼ 0.4. We do not find any other sources at the GRB's optical localization to 110 27.0 mag (corrected for Galactic extinction). We note that although there are other field galaxies identified at offsets > 6 these sources have > 0.4. Based on these probabilistic arguments, we consider Source A the host galaxy of GRB 130912A, pending confirmation of the source as a galaxy. Based on the extremely faint nature of Source A, we consider that it likely has a high-origin, and assume = 1 in Table 4 to compute the projected physical offset of 5.6 ± 2.6 kpc.
Based on the early X-ray lightcurve, we derive a lower limit to the density of min 2.1 × 10 −3 cm −3 . This density is consistent with an ISM environment, and suggests that GRB 130912A originated within the confines of a nearby host galaxy.
In order to identify the environment and host galaxy of GRB 131004A, we used archival imaging from Keck/MOSFIRE in the -band and HST/WFC3 in the F110W filter. We note that the field is relatively crowded (Figure 7), with many foreground stars within a few arcseconds of the GRB position. However, we detected an extended source (G1) nearby to the GRB's optical localization. This source has magnitude 110 = 25.58 ± 0.05 mag and its centroid is located at an offset of 0.41 from the GRB position. The probability of chance alignment for G1 is = 0.05. There are a number of other nearby faint sources, which cannot be classified as either stars or galaxies. These are Source A with 110 = 26.6 ± 0.3 mag at 0.6 , Source B with 110 = 26.2 ± 0.2 mag at 2.3 , and Source C with 110 = 25.91 ± 0.13 mag at 3.3 from the optical localization. These sources have a significantly higher probability of chance coincidence compared to G1 with = 0.14, 0.50, and 0.67 for Sources A, B, and C, respectively.
The closest bright galaxy, besides G1, is located at an offset of 7.8 and has magnitude 110 = 21.19 ± 0.01. We refer to this source as G2, and exclude it as a candidate host due to the high probability of chance coincidence ( = 0.22), as well as the fact that it would be odd to detect emission features at such a large offset from the galaxy (∼ 60 kpc at = 0.717).
No other source is found coincident to the GRB localization with a 3 upper limit 110 27.0 mag (corrected for Galactic extinction). Given the emission line features coincident with the GRB position in the optical spectrum (Chornock et al. 2013;D'Elia et al. 2013), we suggest that the GRB originated from a star forming region within G1. At = 0.717, G1 is significantly under-luminous for a sGRB host galaxy (< 0.1 * ), and this may suggest that GRB 131004A is an interloping long GRB (which is also possible given the softness of its prompt gamma-ray emission). The GRB may just appear short due to a tip-of-the-iceberg effect (Moss et al. 2022) (see also Bromberg et al. 2013).
We compute a lower limit to the circumburst density of min 1.5 × 10 −3 cm −3 (see Table B1). We note that the physical offset of the GRB from its host galaxy, assuming the galaxy is the true host and also resides at = 0.717 (Chornock et al. 2013), is 3.1 ± 1.3 kpc. Moreover, the host-normalized offset is / = 1.0 ± 0.4, consistent with the half-light, , radius of its host galaxy (Table 4). These two factors (i.e., density and offset) are consistent with the GRB occurring in an ISM environment within its host galaxy.
We obtained late-time imaging with the LDT/LMI on June 10, 2014, November 3, 2019, and August 6, 2021 covering the filters. At the optical localization, offset by only ∼ 0.5 , we identify an extended galaxy, referred to as G1 (Figure 7). We derive magnitudes = 24.22 ± 0.18, = 23.30 ± 0.09, = 23.37 ± 0.10, and > 23.0 AB mag. This photometry suggests that the 4000 Åbreak occurs in the -band, leading to a photometric redshift estimate between = 0.3 − 0.6. We compute the probability of chance alignment for G1 to be = 0.009 using the -band magnitude. We note that the next closest galaxy candidates are located at offsets > 30 with > 0.25. We can exclude additional nearby galaxies to depth 24.8 mag (corrected for Galactic extinction, see Table 2). Based on this, we consider G1 the putative host of GRB 140129B.
We utilized the broadband SED ( ; see Table 2) to derive a photometric redshift phot = 0.4 ± 0.1 and a stellar mass log( * / ) = 9.1 ± 0.1 using the prospector software. This photometric redshift is consistent with the upper limit to the GRB redshift ( < 1.5) based on the 2 detection of the afterglow. Using the early X-ray afterglow, we compute a lower limit to the circumburst density of GRB 140129B yielding min 1.0 × 10 −3 . This is consistent with the GRB occurring in an ISM environment, as expected based on the small offset of the GRB from its host galaxy. Assuming ∼ 0.5, as suggested by the galaxy's SED, the physical offset of the GRB from G1 is ≈ 3.0±1.0 kpc, and the host-normalized offset is / = 1.0 ± 0.3 (see Table 4).

A1.7 GRB 140930B
GRB 140930B was detected with Swift/BAT and Konus-Wind on September 30, 2014 at 19:41:42 UT. The GRB had a duration 90 = 0.84 ± 0.12 s. Swift/XRT localized the X-ray afterglow to RA, DEC (J2000) = 00 ℎ 25 23 .40, 24°17 41.7 with uncertainty 2.0 . The optical counterpart was localized to RA, DEC (J2000) = 00 ℎ 25 23 .43, 24°17 39.4 (Tanvir et al. 2014). We note that the most up-to-date XRT enhanced position is now shifted away from this optical localization, compared to the originally reported enhanced position (Goad et al. 2014), but that the positions are still consistent at the 99.7% confidence level (assuming the XRT position error follows Rayleigh statistics Evans et al. 2014Evans et al. , 2020. On August 1, 2020, we obtained late-time imaging of the field of GRB 140930B with Gemini GMOS-N in r-band. We supplemented this with early-time Gemini GMOS-N imaging from October 1 and 2, 2014 which was aimed at identifying the GRB afterglow. The afterglow is clearly detected in these early images, but the position is contaminated by the PSF of a saturated, nearby star ( ∼ 13.1 mag). Although the afterglow position is contaminated, we uncover a faint source with magnitude = 23.8 ± 0.2 AB mag at an offset of ∼ 1.4 from the afterglow localization. The probability of chance coincidence for this source is = 0.02. However, due to the PSF of the saturated star we cannot confirm whether this is a foreground star or a galaxy, and, therefore, we refer to this as Source A. Furthermore, we note that in each of these three Gemini images there is a possible extension of Source A to the northwest, but it is not clear based on this data whether this is due to a secondary source underlying the GRB position or a true extension of Source A.
As Source A is also clearly detected in the early Gemini GMOS-N afterglow imaging from October 1 and 2, 2014, we can determine a precise offset (i.e., without a tie uncertainty tie ) from the GRB position of = 1.4 ± 1.1 . Assuming ∼ 0.5, this yields a physical offset of 8.8 ± 0.9 kpc. As there are no other likely hosts for GRB 140930B identified in these Gemini images, we consider Source A to be the candidate host galaxy, although we note that deeper observations are required to determine the extension of Source A and confirm its nature as a galaxy.
Following O'Connor et al. (2020), we further derive a lower limit to the circumburst density of 1.4×10 −3 cm −3 . This implies that the GRB originated from within a dense galactic environment, consistent with the ISM.
We analyzed archival HST/WFC3 imaging obtained on February 3, 2017 in the 110 filter. The field is relatively crowded with many galaxies located at < 8 from the optical localization of GRB 150423A (Figure 7). There are also a few bright (∼ 20 to 21 mag) SDSS galaxies residing at larger offsets 15 with high probabilities of chance coincidence ( 0.3). These SDSS galaxies are not displayed in Figure 7.
The closest source to the GRB position is a faint galaxy (G1) offset by 1.6 with magnitude 110 = 25.3 ± 0.07 mag, yielding = 0.18 using -band number counts (Metcalfe, et al. 2006;Galametz et al. 2013). The other galaxies displayed in Figure 7 are located at offsets of 3.8, 4.7, 6.2, and 7.0 with magnitudes 110 = 22.696±0.007, 22.620±0.006, 23.93±0.03, and 22.85±0.01 for G2, G3, G4, and G5, respectively. These galaxies have a high probability of chance alignment with the GRB position ranging from = 0.15, 0.2, 0.6, and 0.5 for G2, G3, G4, and G5. We further note that the nearby galaxy G2 has a spectroscopic redshift = 0.456 reported by Perley (2015). No coincident source is detected at the GRB position to 110 27.2 mag. We therefore conclude that GRB 150423A is observationally hostless as it is unclear which of these multiple candidates is the true host or whether the BNS system resided within a faint undetected galaxy.
We note that optical spectroscopy of the afterglow starting at ∼22 min set a robust upper limit < 2.5 to the redshift of GRB 150423A (Malesani et al. 2015). The same observation marginally detected an MgII absorption doublet at = 1.394. However, due to the tentative nature of the detection and lack of other evidence, we do not consider this the conclusive redshift of GRB 150423A.
We set a lower limit min 2.6 × 10 −4 cm −3 to the density of the GRB's environment. This suggests that the GRB occurred within a galactic ISM environment, either within one of the nearby candidate galaxies or in a faint galaxy ( < 2.5) which was not detected with the optical and infrared observations presented in this work.
We carried out late-time imaging with the LDT in filters on March 29, 2020. These observations were supplemented by Gemini GMOS-N imaging obtained in r-band on April 8 and 9, 2016. In the Gemini imaging we detect two nearby candidate hosts at offsets 1.6 (source A) and 3.8 (G1), see Figure 7, whereas in our shallower LDT imaging we detect only G1. Source A has magnitude = 25.5 ± 0.2 mag and G1 has magnitude = 23.54 ± 0.10 mag. The probability of chance alignment is = 0.13 and 0.16 for source A and G1, respectively. No source is detected coincident with the optical localization to depth 25.8 mag. As both source A and G1 have similar probabilities of chance association, we consider GRB 160408A to be observationally hostless. Moreover, there are no bright galaxies from which it is likely the GRB was highly kicked.
Using the early X-ray afterglow lightcurve, we set a lower limit of min 1.8 × 10 −4 cm −3 to the circumburst environment of GRB 160408A. This density implies the GRB occurred within a galactic environment.

A1.10 GRB 160410A
At 05:09:48 UT on April 10, 2016, Swift/BAT  and Konus-Wind (Frederiks et al. 2016) triggered on GRB 160410A. The BAT lightcurve displays an initial short, hard pulse with duration 2 s. However, there is a clear extended tail of the burst lasting for tens of seconds. The duration reported in the BAT GRB Catalog (Table B1) is 90 = 96 ± 50 s. In addition, Sakamoto et al. (2016) found that the spectral lag of the initial short pulse is consistent with zero, typical of sGRBEE. The GRB is therefore interpreted as having extended emission. Shortly after the GRB, Swift/XRT localized the X-ray afterglow to RA, DEC (J2000) = 10 ℎ 02 44 .47, 3°28 41.0 with 3.2 uncertainty. A more precise localization of the optical counterpart to RA, DEC (J2000) = 10 ℎ 02 44 .37, 3°28 42.4 was quickly discovered (Yates et al. 2016).
We obtained late-time imaging of the field of GRB 160410A with the LDT/LMI on December 14, 2021 and January 15, 2021 in the and -bands. These observations were supplemented by public archival imaging with Keck/DEIMOS from April 28, 2016. In order to precisely localize the afterglow location in these late-time images, we utilized the initial detection of the optical counterpart by Swift/UVOT . We display a finding chart of the field in Figure 7. No source is identified coincident with the optical localization to depth 24.9, 25.0, and 24.2 AB mag. We note that a deeper constraint on an underlying host of 27.17 (3 ; corrected for Galactic extinction) was presented by Agüí Fernández et al. (2021) based on late-time deep GTC imaging. This is in sharp contrast to the results obtained from optical spectroscopy of the afterglow (see below). However, we note the presence of two bright SDSS galaxies within 60 of the GRB localization with = 18.9 mag at 20 and = 17.8 mag at 35 yielding = 0.11 and 0.14, respectively. Despite the lower compared to other candidates, the projected physical offset from these galaxies at their estimated photometric redshifts of phot = 0.2 and phot = 0.1 is ∼ 69 and 67 kpc, respectively. Furthermore, the photometric redshifts are inconsistent with the measured redshift for GRB 160410A (see below). The probability of chance coincidence for any other extended object at larger offsets is 0.5 due to their faintness ∼ 24 mag. We therefore consider GRB 160410A to be observationally hostless.
We analyzed Keck spectroscopy performed with LRIS on April 10, 2016 targeted at the optical afterglow of GRB 160410A beginning at 84 min after the GRB. The afterglow is detected as a blue continuum from ∼ 3100 -5680 Å with a large number of visible absorption features. The continuum normalized spectrum is displayed in Figure 5. We identify a broad damped Lyman alpha (hereafter, Ly ) absorption feature at obs ∼ 3304 Å, which drives the redshift derivation. In addition, we find a number of absorption features located at obs ≈ 3427, 3547, 3559 and 4146 Å that correspond to [SiII] transitions; see Figure 5. These features, on top of the Ly trough, allow us to derive a redshift = 1.717 ± 0.001. Moreover, we identify absorption features corresponding to two intervening absorbers for which we identify [CIV] at both = 1.444 and = 1.581. In Figure 5, we mark also tentative detections of [SiII] and [SiIV] at = 1.444 and [SiII] and [NII] at = 1.581. The redshifts of these absorbers are consistent with the estimates of Bloom et al. (1997) that the GRB is not residing further than 1.25× the redshift of the intervening system. Our results are consistent with the analyses presented by Selsing et al. (2016), Cao et al. (2016), Selsing et al. (2019), and Agüí Fernández et al. (2021).
The Ly trough provides strong evidence that the GRB originated from within a dense galactic environment with a neutral Hydrogen column density of log( HI /cm −2 ) = 21.3 ± 0.3 (Selsing et al. 2019;Agüí Fernández et al. 2021), see Agüí Fernández et al. (2021) for an in depth discussion of the environment of GRB 160410A. Therefore, GRB 160410A is very unlikely to be physically hostless (i.e., occurring in an IGM-like environment outside of its birth galaxy). This is is contrast to the field of the GRB, for which there are no candidate host galaxies identified to deep limits ( 27.17 mag;Agüí Fernández et al. 2021). This event delivers the first substantial evidence for a sample of short GRBs located in high− galaxies, which are not identified through observational follow-up. Furthermore, the two intervening absorbers at = 1.444 and = 1.581 are likewise not detected in the Keck/DEIMOS or LDT imaging, further emphasizing the possibility of non-detected high− galaxies coincident to short GRBs. We emphasize that deep nIR imaging (e.g., HST, JWST) is crucial to the detection of these galaxies.
As further evidence, we utilized the early X-ray lightcurve in order to derive a lower limit to the circumburst density of min 2.6×10 −3 cm −3 . This value is inconsistent with an IGM-like environment, and provides further evidence that GRB 160410A occurred within a undetected host galaxy at = 1.717.

A1.11 GRB 160525B
At 09:25:07 UT, Swift/BAT triggered and located GRB 160525B ). The short burst had a duration 90 = 0.29±0.05. The XRT localized the X-ray afterglow to an enhanced position RA, DEC (J2000) = 09 ℎ 57 32 .30, 51°12 24.0 with 2.1 uncertainty (90% CL). In an initial finding chart exposure UVOT marginally detected an optical source coincident with the XRT position. The source was located at RA, DEC (J2000) = 09 ℎ 57 32 .23, 51°12 24.9 with uncertainty 0.6 (90% CL). The UVOT detection of the afterglow in the wh filter sets an upper limit of 5 to the redshift of GRB 160525B. We utilize this detection of the optical afterglow for relative astrometry with our late-time images.
We performed optical imaging with the LDT/LMI on January 29, 2020, February 29, 2020, and December 15, 2021 covering wavelengths. We identified a host galaxy candidate coincident with the UVOT localization of GRB 160525B (Figure 7). This galaxy, G1, has magnitudes = 23.27 ± 0.15, = 23.27 ± 0.09, = 23.28 ± 0.18, and = 23.4 ± 0.3 mag. G1 has a probability of chance alignment of = 0.03. In addition to G1, there are a number of other candidate hosts in the field (see Figure 7), including two other faint sources within 7 and two bright SDSS galaxies at offsets of 13 and 21 . No other sources are uncovered nearby the GRB position to depth 24.6 AB mag. The nearby sources, G2 and source A, have magnitudes = 24.2 ± 0.2 and 24.3 ± 0.2 mag with = 0.25 and 0.6. The bright SDSS galaxies have magnitude = 19.43 ± 0.03 and 19.95 ± 0.03 mag for G3 and G4, respectively, yielding = 0.09 and 0.26. Based on the significantly smaller for G1 compared to these other candidates, we consider the coincident galaxy G1 to be the putative host of GRB 160525B.
Using the early X-ray lightcurve, we set a lower limit to the density surrounding the GRB's explosion site of min 6.6 × 10 −3 cm −3 . This density is consistent with the GRB occurring in an ISM environment, which is likely given the very small offset, 0.06 ± 0.25 , of the GRB from its putative host galaxy (G1). We note that as the halflight radius of G1 is ∼ 1.0 the host-normalized offset is likewise 0.06 ± 0.25.
We observed the GRB position with Gemini/GMOS-N on August 1, 2020 to search for underlying galaxies. We supplemented this observation with LDT imaging in the filters, archival GTC/OSIRIS imaging in -band, and archival imaging from Keck/MOSFIRE in the -band. We identify four nearby galaxies with offsets ∼ 4.8 to the West (G1) and North-East (G2), 6.2 to the East (G3), and 6.5 to the South-West (G4) of the GRB position (see Figure 7). Their -band magnitudes are 25.1 ± 0.15 mag (G1), 25.4 ± 0.3 (G2), 22.90 ± 0.05 (G3), and 24.55 ± 0.10 mag (G4). The chance probability, based on -band number counts, for each is 0.4, with the exception of G3 which has = 0.24. However, we note that G2, G3, and G4 are infrared bright, and detected in the Keck/MOSFIRE imaging with magnitudes = 21.55 ± 0.09, 21.50 ± 0.15, and 20.90 ± 0.07 AB mag, respectively. The based on these infrared magnitudes is 0.11, 0.13, and 0.13 for G2, G3, and G4, respectively. This further complicates the host identification for GRB 160601A, as these three galaxies are equally likely hosts and none has < 0.1. As no other sources are identified coincident to the optical localization to depth 25.9 mag, we assign it an observationally hostless classification. Moreover, there are no bright galaxies from which it is likely the GRB was highly kicked.
Based on the early X-ray lightcurve, we set a lower limit to the circumburst density surrounding the GRB of min 1.2×10 −5 cm −3 . We note that this lower limit is consistent with the GRB occurring in either an ISM or an IGM-like environment.
We obtained late-time imaging of GRB 160927A with LDT on May 20, 2018 in r-band and with Gemini GMOS-N on August 1, 2020 in i-band. We supplemented these observations with archival imaging from the GTC in r-band taken on February 23, 2017 and with Keck/LRIS imaging in the filters from October 6, 2018 and September 4, 2019. These late-time images do not resolve any source coincident with the position of the optical afterglow to depth 26.0 AB mag (corrected for Galactic extinction). The closest source to the GRB position (source A in Figure 7) is offset by ∼ 3 with magnitudes = 25.8 +0.3 −0.2 and = 25.6 +0.3 −0.2 mag. This source is too faint for a conclusive star/galaxy classification, although we note it appears marginally extended. The chance probability for Source A is = 0.5. Additionally, there are a number of SDSS galaxies (G1, G2, G3, and G4) within the field at > 9 , but 0.5 for each of them. Due to the lack of putative host galaxy, we consider GRB 160927A to be observationally hostless.
We set a lower limit of min 1.1×10 −4 cm −3 to the density of the GRB's environment based on the early X-ray afterglow. This density is consistent with the GRB occurring within the Virial radius of its host galaxy (O'Connor et al. 2020), and introduces the possibility that this GRB occurred in a faint, undetected galaxy.
We carried out late-time imaging of the field with LDT/LMI on May 21, 2018 in the filters. These data were supplemented by early-time LDT imaging from April 29, 2017 (∼ 1 d post-burst) and archival observations by TNG in i and z from May 1, 2017 (∼ 3 d post-burst). In order to localize the afterglow, we performed image subtraction between these early and late-time images using the HOTPANTS software (Becker 2015). We do not detect the afterglow in either the LDT or TNG images, and instead use the reported position from GROND (Bolmer et al. 2017).
In our late-time LDT imaging, we detect a candidate host galaxy (G1) at offset 1.2 from the afterglow localization. The galaxy has magnitudes > 23.3, = 22.09 ± 0.10, = 21.84 ± 0.15, and = 21.88 ± 0.15 mag; the galaxy is not detected in the -band due to the 4000 Å break. The probability of chance coincidence is = 0.01. We report the detection of another extended galaxy (G2) at offset ∼ 13 with = 21.53 ± 0.07. This galaxy has an 34% probability of chance alignment. There is no source detected at the GRB's optical localization to 23.6 and 23.4 mag (corrected for Galactic extinction). Based on these arguments, we consider G1 the putative host galaxy for GRB 170428A.
The galaxy G1 has a redshift of = 0.454 determined by optical spectroscopy with the GTC (Izzo et al. 2017). At this redshift, the projected physical offset of the GRB from its host is 7.2 ± 1.8 kpc.
The host-normalized offset is / = 1.0 ± 0.3, consistent with the GRB occurring within the half-light radius of G1. We compute a lower limit for the density of the environment surrounding the GRB of min 1.6 × 10 −5 cm −3 .
In order to search for the host galaxy of GRB 170728A, we obtained late-time imaging with the LDT/LMI on January 8, 2019 in -band. Additionally, we retrieved publicly available late-time images from the Keck Observatory (PI: Fong) taken January 14, 2018 in and . In these imaging, we uncover four visually extended sources within 15 of the GRB position. However, the PSF of a nearby, very bright star ( ∼11.8 mag; SDSS) contaminates the GRB localization in each image. No source is detected coincident to the GRB position with a 3 upper limit of 24.7 mag (the shallow limit is due to a diffraction spike from the bright star, and the Galactic extinction, ( − ) = 0.21 mag, in the direction of the burst). For the nearby galaxies, we derive magnitudes = 23. 89 ± 0.12, 23.31 ± 0.15, 23.76 ± 0.13, and 22.76 ± 0.15 mag for G1, G2, G3, and G4, respectively, with offsets of 4.4 , 6.7 , 7.4 , and 14 . We note that the photometry for G3, in particular, is contaminated by the diffraction spike from the bright star. We find a probability of chance coincidence of = 0.23, 0.32, 0.49, and 0.67 for G1, G2, G3, and G4, respectively. Thus, we find that GRB 170728A is observationally hostless. Future observations at a different position angle can provide deeper constraints on an underlying source.
We compute a lower limit to the circumburst density of the GRB's environment, constraining it to be 1.2 × 10 −4 cm −2 . This suggests the GRB originated from within a galactic environment.
We carried out late-time observations with the LDT/LMI on November 3, 2019 and December 7, 2019 covering wavelengths. At the position of the optical counterpart we identify a bright host galaxy (G1) with magnitudes = 23.71 ± 0.06, = 23.06 ± 0.06, = 22.63 ± 0.05, and = 22.33 ± 0.15 mag. The SED suggests that the 4000 Å break occurs between the and -bands, hinting at a photometric redshift in the range ∼ 0.3 − 0.6. The offset of the GRB from this galaxy is 0.8 yielding = 0.014. There are no other nearby galaxy candidates to magnitude 24.6 mag. We note the presence of a catalogued galaxy with magnitude = 20.3 at offset ∼24 , but the = 0.4 (due to the large offset, this galaxy is not displayed in the finding chart). We therefore consider G1 to be the putative host galaxy of GRB 170728B.
We used prospector to model the SED of G1 (Figure 9), and obtain a photometric redshift phot = 0.6 ± 0.1 and a stellar mass log( * / ) = 9.7 ± 0.2. We further derive a density min 7.5 × 10 −4 cm −3 for the GRB environment using the early X-ray afterglow lightcurve. This value is inconsistent with the GRB occurring in an IGM-like environment (i.e., < 10 −4 cm −3 ; O' Connor et al. 2020). We note that the host-normalized offset / = 1.1 ± 0.3 is consisent with the GRB occurring within the half-light radius of G1. Assuming a redshift ∼ 0.64, we compute the physical offset between the GRB and G1 to be ≈ 5.5 ± 1.7 kpc.
We carried out imaging with the LDT/LMI on April 7, 2019, December 7, 2019, and May 5, 2021. We uncovered a faint galaxy at an offset of ∼ 1.6 from the optical localization of the GRB with magnitudes = 23.89 ± 0.12, = 22.92 ± 0.08, = 22.33±0.10, = 22.26±0.12, and 21.5 AB mag. The probability of chance coincidence is = 0.03. Furthermore, we identified three other candidate host galaxies in the vicinity of the GRB: Source A with = 24.5 ± 0.2 at ∼ 1.6 , G2 with = 23.01 ± 0.08 at 4.1 , and G3 with = 22.29 ± 0.06 at 7.4 . The probability of chance coincidence for these sources is 0.08, 0.15, and 0.23 for Source A, G2, and G3, respectively. No other sources are identified near the GRB localization to 24.7 AB mag (corrected for Galactic extinction). Due to the similar probability of chance coincidence for G1 and Source A (0.03 vs. 0.08), we cannot differentiate between which is the more likely host galaxy. However, deeper observations are required to confirm the source classification of Source A, and whether it is a foreground star or a galaxy. Therefore, we tentatively consider G1 the host galaxy of GRB 180805B.
We obtained optical spectroscopy of G1 with Gemini GMOS-N on February 1, 2021. We detect a very weak trace between ∼ 7300 to 9500 Å. There are no obvious emission or absorption features. Therefore, we instead modelled the broadband SED ( ) within prospector. As the spectrum does not show bright emission features, we turned off nebular emission lines within prospector. We found that ≈ 0 provided the best fit to the SED, due to the near flat slope in the filters. Thus, we fixed the intrinsic extinction to = 0 in order to allow for minimization of the likelihood function. The MCMC fit resulted in phot = 0.4 +0.2 −0.1 and a stellar mass log( * / ) = 9.6 ± 0.3 (see Figure 9). At this redshift, the offset of GRB 180618A from G1 is 8.8 ± 1.1 kpc. The host-normalized offset is / = 1.58 ± 0.24.
Using the early X-ray lightcurve, and assuming ≈ 0.4, we identified a lower limit of 4.0 × 10 −3 cm −3 . This supports that the sGRB occurred within an ISM-like environment.

A2.1 GRB 101224A
GRB 101224A was detected with Swift/BAT (Krimm et al. 2010) and A candidate host galaxy (G1) was discovered at the edge of the enhanced XRT position, see Figure 8. This galaxy was previously reported by Nugent & Bloom (2010) and Tunnicliffe, et al. (2014). We derive magnitudes = 22.54 ± 0.06, = 21.99 ± 0.06, = 21.83 ± 0.05, and = 21.78 ± 0.05 AB mag. The probability of chance coincidence for G1 is = 0.11. In addition, we discovered a very faint source, referred to as Source A, within the XRT error circle with magnitude = 24.7 ± 0.2. Three other candidate host galaxies, visible in Figure 8, are uncovered at offsets of 4.5 , 6.4 , and 8.4 . The probability of chance coincidence is > 0.25 for each of these sources. No other sources are identified within the XRT enhanced position to depth 24.9 AB mag (3 , corrected for Galactic extinction). Due to this, no other galaxy will have a lower probability of chance coincidence than G1, even if uncovered in deeper observations, making G1 the most likely host galaxy, despite the higher value. We performed optical spectroscopy of the candidate host galaxy, G1, on October 27, 2014 with Keck/LRIS (see Table 3). The resulting spectrum is displayed in Figure 4. We detect multiple emission lines at obs ≈ 5422, 7067, 7209, 7278, and 9542 Åwhich we associate to the [OII] doublet, H , [OIII] 4960 , [OIII] 5008 , and H transitions at a redshift = 0.4536 ± 0.0004.We note that at this redshift there is a marginal detection of H . Although we cannot classify the galaxy type based on morphology, we suggest that the strong emission features are typical of a late-type galaxy. At this redshift, the offset of GRB 101224A from this galaxy is = 14 ± 17 kpc.
We derive a lower limit, min 3.6 × 10 −5 cm −3 , to the density of the GRBs environment using the early X-ray lightcurve. This density is consistent with an IGM-like environment ( < 10 −4 cm −3 ).

A2.2 GRB 120305A
GRB 120305A was detected with Swift/BAT on March 5, 2012 at 19:37:30 UT (Stratta et al. 2012). The burst displayed a single peak with a fast rise and slower decay. The burst had a duration 90 = 0.10± 0.02 s. A fading X-ray source, identified as the afterglow, was detected at RA, DEC (J2000) = 03 ℎ 10 08 .68, 28°29 31.0 with uncertainty 2.0 . No optical counterpart was identified. The lack of an optical counterpart may be due to the high Galactic extinction = 1.2 mag (Schlafly & Finkbeiner 2011) from the GRB's localization in the direction of a molecular cloud (Planck Collaboration et al. 2016), which also leads to an enhanced background in the XRT localization and throughout the image (see Figure 8). This enhanced background is observed with a consistent pattern in all imaging of this field (e.g., Gemini, Keck, LDT), and leads to a shallower upper limit (see below).
We performed late-time imaging with the LDT in -band on March 6, 2014 and with Keck on October 25, 2014 in the and -bands to search for an underlying galaxy. We further supplemented our imaging with archival Gemini observations taken in -band (PI: Tanvir). We did not discover a source within the XRT enhanced position to depth 24.6 and 24.9 (corrected for Galactic extinction). However, our imaging revealed the presence of three uncatalogued galaxies (G1, G2, and G3) at offsets < 15 , see Figure 8.
The nearest galaxy, G1, has magnitudes = 21.7 ± 0.06, = 21.53 ± 0.04, and = 20.85 ± 0.08 mag. The galaxy is offset by 5.4 from the GRB position, whereas G2 and G3 are fainter ( = 22.4 ± 0.06 and 22.84 ± 0.06 mag) with larger offsets of 9.8 and 12.6 , respectively. The probability of chance coincidence for these galaxies is 0.07, 0.36, and 0.65 for G1, G2, and G3. We therefore consider G1 to be the putative host galaxy for GRB 120305A. We note that G1 has a morphology suggestive of a late-type galaxy. The host-normalized offset is / = 4.6 ± 1.2 (see Table 4). Furthermore, the magnitudes hint at a 4000 Å break around the -band, suggesting a redshift ∼ 0.6 − 0.9.
We derive a lower limit, min 2.0 × 10 −5 cm −3 , to the density of the GRBs environment using the early X-ray lightcurve. This is consistent with the expected density for an IGM-like environment, but does not rule out that the GRB occurred within a higher density galactic environment, such as G1.
Gemini observations were carried out on July 1, 2012 at 0.5 d after the GRB to search for the optical afterglow of GRB 120630A. No afterglow was detected within the XRT enhanced position to depth 25.0 mag. However, in these Gemini images we identify seven nearby candidate host galaxies for GRB 120630A (see Figure 8). We therefore carried out follow-up imaging at late-times with the LDT/LMI on September 5, 2014 in riz and Keck/LRIS on October 25, 2014 in the GR filters to better identify the putative host.
Within the XRT enhanced position we detect two extremely faint sources (Source A and B) which, due to their faintness, we cannot confirm are extended. Source A has magnitudes = 25.7 ± 0.2, = 25.4 ± 0.3, = 24.7 ± 0.3, = 24.8 ± 0.3, whereas Source B has = 25.5 ± 0.2, = 25.5 ± 0.3. Due to the large XRT position error (4.0 ), these sources have a significant probability ( ∼ 0.8) of random alignment with the GRB localization. We therefore exclude these sources as candidate host galaxies. The 3 upper limit to any other source within the XRT position is 25.7, 25.6, 24.9, and 24.9 mag (corrected for Galactic extinction).
The other five sources identified near the GRB position are detected with a high significance, and easily identified as extended galaxies. The brightest of these sources (G1) is located at an offset ∼ 5.8 with magnitude = 22.11 ± 0.03, = 21.42 ± 0.04, = 21.06 ± 0.07, = 20.99 ± 0.05 mag. This galaxy is catalogued in both the PS1 ( ) and CatWISE2020 (Marocco et al. 2020) catalogs. The WISE infrared magnitudes are 1 = 19.48 ± 0.04 and 2 = 19.61 ± 0.08 AB mag. G1 has a significantly lower probability of chance alignment with the XRT position, = 0.07, compared to Sources A and B, especially in the redder filters. In comparison to this source the other candidate host galaxies (G2, G3, G4, G5) in Figure 8, which are much fainter ( 23 mag), have a large 0.4. Therefore, we consider the bright galaxy G1 to be the putative host.
We derive a lower limit, min 9.0 × 10 −5 cm −3 , to the density of the GRBs environment using the early X-ray lightcurve. This is consistent with the expected density for an IGM-like environment, but does not rule out that the GRB occurred within a higher density galactic environment, such as G1.
We obtained late-time imaging with Keck/LRIS in the G and Rbands on October 25, 2014. The field of GRB 130822A is crowded with > 30 sources within 20 in our Keck imaging. There are 8 SDSS galaxies ( ∼ 20.7 − 21.7 mag) within 60 , one of which is significantly brighter than the rest with = 18.13 ± 0.02 mag. We label this bright galaxy at offset 22 as G7. G7 has = 0.08 and redshift = 0.154 ). An even brighter SDSS galaxy (referred to as G12) at = 0.045  resides at 84 offset from the GRB position with = 16.204 ± 0.005 ( = 0.23). In addition to these galaxies, there are a number of 24 mag galaxies at offsets 10 , with 0.8. We also identify 4 faint sources, 25 mag, within 5 of the XRT position (one of which resides inside the 90% localization region; Figure 8). These sources have = 0.25 − 0.5. The 3 upper limit within the XRT position is 25.8 AB mag. Due to its lower probability of chance alignment, we consider G7 as the putative GRB host. We note that the morphology of G7 is a face-on late-type galaxy. The projected offset from the GRB position is 22.0 ± 2.3 , which at = 0.154 corresponds to 61 ± 6 kpc. The host-normalized offset is / = 8.1 ± 0.9. Thus, GRB 130822A represents the largest offset of a sGRB from a late-type galaxy ( Figure  18).
Based on the early X-ray afterglow lightcurve, we set a lower limit to the density of the environment surrounding the GRB of 7.1 × 10 −4 cm −3 . This value is consistent with the GRB occurring in an ISM-like environment. However, we caution that for this GRB re-binning the XRT lightcurve yields two data points with a very steep decay index, hinting that the observed X-ray emission may not be due to the forward shock. In such a case the formalism to constrain the density is not applicable.
We obtained late-time imaging of GRB 140516A with the LDT in -band. This was supplemented with archival Gemini and Keck imaging in and , respectively. The field surrounding the GRB position is sparse, with the exception of a bright foreground star slightly overlapping the XRT position. However, we uncover an extremely faint candidate host galaxy (referred to as Source A) at the edge of the XRT position that is detected in both the Gemini and Keck imaging. Source A has magnitudes 25.0, = 25.9 ± 0.3, and = 23.15 ± 0.20 AB mag, suggestive of a high-origin. The probability of chance coincidence is 0.6 based in the -band magnitude and 0.2 based on the -band. No other source is uncovered in the XRT position to depth 26.1 AB mag, and there are no other nearby candidate galaxies. We note the presence of a bright ∼ 17.5 mag galaxy at an offset of 80 , however, the > 0.3. We, therefore, consider GRB 140516A to be observationally hostless.
Based on the early X-ray afterglow lightcurve, we set a lower limit to the density of the environment surrounding the GRB of 7.3 × 10 −4 cm −3 . This value is consistent with the GRB occurring in an ISM-like environment.
We performed late-time observations with the LDT/LMI on August 6, 2021 in the filters. We identify a nearby galaxy ( Figure  8) uncovered at offset 4.6 with magnitudes = 22.53 ± 0.07, = 22.28 ± 0.07, = 21.84 ± 0.06, and = 21.92 ± 0.20. The probability of chance coincidence for G1 is = 0.08 using the -band magnitude. Another galaxy, G2, is detected at an offset of 7.7 with = 22.66 ± 0.07 yielding = 0.29. In addition, no source is detected within the XRT position to depth 24.1 mag (a previous limit of 25.8 mag was reported by Pandey et al. 2019 using GTC). We note that any source fainter ( 24.1 mag) than this residing with the XRT error circle would have 0.25. These arguments lead us to classify G1 as the putative host of GRB 140622A.
In order to derive the redshift of this galaxy, we carried out optical spectroscopy with Keck/LRIS on October 27, 2014 (see Table 3). The spectrum is displayed in Figure 4. We identified emission lines at obs ≈ 7304 and 9810 Åwhich we associate to the [OII] doublet and [OIII] 5008 , respectively. This yields a redshift = 0.959 ± 0.001, which is consistent with that reported by Hartoog et al. (2014). At this redshift there is a very marginal detection of both and [OIII] 4960 . In our LDT imaging we cannot classify the galaxy type based on morphology, but the emission features are suggestive of a late-type galaxy. At this redshift the offset of the galaxy from the GRB position is 38 ± 17 kpc, towards the high end of the short GRB offset distribution. The host-normalized offset is / = 3.8 ± 1.7.
We derive a lower limit, min 1.8 × 10 −5 cm −3 , to the density of the GRBs environment using the early X-ray lightcurve. This is consistent with the GRB occurring at an offset of ∼ 38 kpc from G1, and does not exclude the association.
We analyzed public archival imaging obtained with Gemini/GMOS-S on July 29, 2020 in i-band, and from VLT/FORS2 in -band and -band from September 1, 2016 and March 7, 2017, respectively. We identify a galaxy within the XRT enhanced position with magnitude = 24.95 ± 0.10 and = 25.1 ± 0.3 mag. Due to its faintness, this source has a ∼ 32% probability of chance alignment with the XRT position. There are no other sources detected within the XRT position to depth 25.6 and 25.6 mag. We identify two other galaxies within 15 of the GRB localization ( Figure 8): G2 has magnitude = 23.45 ± 0.09 at offset 10.9 , and G3 with = 22.14 ± 0.05 at 12.1 These sources have = 0.5 and 0.25 for G2 and G3, respectively. There are no other bright galaxies within 60 of the GRB localization. Consequently, there is no putative host galaxy for GRB 150831A, and we consider the GRB to be observationally hostless.
Using the early X-ray afterglow lightcurve, we set a lower limit of min 2.4 × 10 −5 cm −3 to the circumburst environment of GRB 150831A. This density is consistent with that expected for an IGMlike environment, but does not exclude a higher density.
We carried out late-time imaging of GRB 151229A with the LDT/LMI in the and -bands, Gemini/GMOS-N in -band, Gemini/GMOS-S in -band, and Gemini/Flamingos-2 (hereafter F2) in the and -bands. We supplemented these observations with archival -band imaging with Gemini/GMOS-S (PI: Fong) andband imaging with Keck/MOSFIRE (PI: Terreran). In these observations we uncover an extended source (G1) coincident with the XRT enhanced position. We derive magnitudes = 25.75 ± 0.2, = 25.41 ± 0.15, = 24.47 ± 0.10, = 24.0 ± 0.2, = 23.10 ± 0.18, and = 22.78 ± 0.2 AB mag. We note that the probability of chance coincidence (using the -band magnitude) for this galaxy is large, = 0.25. However, the probability of chance coincidence for G1 based on the redder and magnitudes is significantly lower with ≈ 0.1 − 0.15. Moreover, the field of GRB 151229A is sparse, and no other candidate hosts were identified to depth 26.1 mag. Therefore, we consider G1 as the putative host galaxy of GRB 151229A.
We analyzed archival Keck/LRIS spectroscopy of this galaxy (see Table 3), but did not identify a trace or any emission lines. Instead, we modeled the broadband SED ( ) of G1 within prospector in order to derive a photometric redshift. We found that in order for the code to achieve a good fit to the SED, we had to turn off nebular emission lines within prospector. Finally, we obtain phot = 1.4 ± 0.2 and a stellar mass log( * / ) = 10.3 ± 0.2 (Figure 9). At this redshift, the physical offset of the GRB is 9±9 kpc. We further derive a host-normalized offset of / = 2.5 ± 2.5.
Adopting ≈ 1.4, as suggested by the galaxy's SED, we set a lower limit to the density of the GRBs environment min 1.2 × 10 −1 cm −3 . These limits suggest the GRB occurred within a high density galactic environment, and support the association with G1.

A2.9 GRB 170127B
Swift/BAT triggered and localized GRB 170127B on January 27, 2017 at 15:13:28 UT . The burst was also detected with Fermi/GBM (Veres & Meegan 2017). As seen by BAT, the burst was single pulsed with duration 90 = 0.51 ± 0.14 s.Swift/XRT discovered the X-ray afterglow at RA, DEC (J2000) = 01 ℎ 19 54 .47, −30°21 28.6 with accuracy 2.6 (90% CL). No optical counterpart was uncovered for this GRB. We obtained late-time imaging of GRB 170127B on January 30, 2021 from Gemini South in -band (PI: Troja). We also include in our analysis public archival Gemini South observations in -band (PI: Fong) as well as public archival Keck imaging (LRIS/MOSFIRE; PIs: Miller, Terreran) in the , , , and filters. The field is very sparse, with no bright candidate host galaxies. Nevertheless, in the Keck imaging we identify a faint, extended source (G1 in Figure 8) within the XRT enhanced position, which is not detected in the Gemini images. This source has magnitudes = 25.7 ± 0.2, = 25.5 ± 0.2, = 25.5 ± 0.2, 23.9 and 24.1 AB mag. The probability of chance coincidence using -band number counts is = 0.55. No other source is identified within the XRT position to a 3 upper limit 26.0 mag. We note there are also two faint ( ∼ 24.5 − 25.0 mag) galaxies (G2 and G3), which we refer to as G2 and G3, at offsets ∼ 6 and 9 with a similarly large chance probability = 0.54 and 0.82, respectively. Due to these high probabilities, we find that GRB 170127B is observationally hostless.
Using the early X-ray afterglow lightcurve from Swift/XRT, we set a lower limit to the density of the GRB's environment of min 7.3×10 −4 cm −2 . This density implies that the GRB originated within a galactic environment.
We obtained late-time imaging with LDT on January 9, 2020 inband and the Gemini North telescope on February 1, 2021 in -band. We uncovered two extremely faint sources in our Gemini imaging at the edge of the XRT enhanced position, see Figure 8. Due to their faintness we cannot determine whether these sources are extended. The first source, referred to as Source A, has magnitude = 25.1±0.2 and the second source (Source B) has magnitude = 26 ± 0.4. Source A is also detected in our LDT imaging with = 24.8 ± 0.3, whereas Source B is not detected to depth 24.9 mag. The probability of chance coincidence for either source is quite large, 0.5. Therefore, due to the large XRT localization we cannot confidently associate either source to the GRB. No other sources are detected to 26.1 mag within the XRT localization. In addition, there are no other sources with lower outside of the XRT error circle, leading to an observationally hostless classification as it is not clear if either of these sources is the host. We note that any fainter sources identified in deeper imaging would similarly be difficult to confirm a physical association to GRB 171007A due to the high . Using the early X-ray lightcurve, we derive a lower limit to the circumburst density of 2.0 × 10 −6 cm −3 . We note that this lower limit is not very constraining to the density due to the plateau and early steep decline phase of the X-ray lightcurve, leading us to apply a late time X-ray data point in order to compute the lower limit.
We analyzed public archival observations obtained with Gemini/GMOS-S in the filters. We identify an extremely faint source (Source A) within the XRT error circle with magnitudes = 26.1 ± 0.3, = 25.9 ± 0.3, = 25.5 ± 0.3, and = 25.5 ± 0.3 mag. The probability of chance coincidence for this source is ∼ 0.6. The upper limit to other sources in the XRT position is > 26.1. We detect three other sources within 10 of the XRT position (Figure 8). These sources have > 0.3, and all other galaxies in the field have > 0.5. We, therefore, consider GRB 180727A to be observationally hostless.

A2.12 GRB 180805B
At 13:02:36 UT on August 5, 2018, Swift/BAT ) and Fermi/GBM ) triggered on GRB 180805B. The burst displayed an initial short pulse with duration < 1 s followed by a softer, weak emission for over a hundred seconds. The total duration of the burst detected with BAT is 90 = 122 ± 18 s. This lightcurve displays characteristics common to other sGRBEE, and we therefore classify GRB 180805B as an sGRBEE. The Xray afterglow for this event was localized to RA, DEC (J2000) = 01 ℎ 43 07 .59, −17°29 36.4 with uncertainty 2.1 . There was no optical counterpart discovered for this event.
We obtained late-time imaging of the field of GRB 180805B with the LDT/LMI on January 16, 2021 in -band. We supplemented this with archival Keck imaging obtained with LRIS on September 10, 2018 and September 4, 2019 in , , , and and with MOSFIRE in from October 15, 2019. We uncover four galaxies nearby to the GRB's XRT position, but no source is identified within the XRT localization to depth 26.0, 25.6, 25.4, 24.4, 24.1 AB mag (3 ; corrected for Galactic extinction). These four galaxies surround the GRB localization on all sides, with offsets ranging from 2.8 to 4.2 for G1 and G4, respectively. The brightest galaxy, G3, is located North of the GRB position with magnitudes = 23.46 ± 0.07, = 22.79 ± 0.09, = 22.31 ± 0.12, = 21.99 ± 0.14, and = 21.22 ± 0.15 AB mag. G3 is offset by 3.4 ± 1.0 from the XRT position, yielding a probability of chance alignment of = 0.07. The other galaxies have magnitudes = 24.6 ± 0.2, 25.2 ± 0.2, and 24.5 ± 0.2 yielding = 0.19, 0.36, and 0.33 for G1, G2, and G4, respectively. In addition to these, we note that there is a bright SDSS galaxy ( ∼ 15.5 mag with phot = 0.029 ± 0.006) at an offset of ∼ 90 with = 0.15. Based on these probabilistic arguments we consider G3 to be the putative host galaxy for GRB 180805B.
We analyzed optical spectroscopy of G1 taken with Keck/LRIS on September 10, 2018 in order to identify the redshift of the galaxy. The spectrum is shown in Figure 4. We identified emission lines at obs ≈ 6190, 7210, 8076, 8238, and 8318 Åwhich we associate to the [OII] doublet, H , H , [OIII] 4960 , and [OIII] 5008 , respectively. This yields a redshift = 0.6609 ± 0.0004. In the photometry of G1 we observe the 4000 Åbreak at this redshift.
Based on the early X-ray afterglow lightcurve, we derive a density of min 3 × 10 −6 cm −3 for the environment surrounding GRB 180805B. This is consistent with the projected physical offset, = 25 ± 11 kpc, of G3 from the GRB position. The host-normalized offset is / = 5.6 ± 2.4. We observed the field of GRB 191031D on November 2, 2019 at 1.3 d after the GRB to search for the optical afterglow. No optical source was detected within the XRT position to depth 25.0 mag (Dichiara & Troja 2019). However, we identified two candidate host galaxies for GRB 191031D, see Figure 8. In order to better characterize the galaxy SEDs we carried out additional LDT observations in the filters. The first source, referred to as Source A, is offset by 3.9 from the GRB position and has magnitude = 24.49 ± 0.15 mag. We cannot determine whether or not the source is extended, and this source is not detected in our LDT imaging. The second source (G1) is a clear galaxy with magnitudes = 22.47±0.07, = 21.64±0.05, = 21.2± 0.2, = 21.2 ± 0.3, and = 21.0 ± 0.3. This galaxy is offset by 7.4 from the GRB position. Using -band number counts we derive = 0.12 and 0.3 for G1 and Source A respectively. G1 is also detected in PS1 with smaller errors on the and -band (as at the time of our LDT observations the conditions were extremely poor). We make use of the PS1 magnitudes in our SED modeling (see below). We further note that G1 is also observed in the ALLWISE catalog (Cutri et al. 2021) with magnitudes 1 = 19.60 ± 0.15 and 2 = 20.16 ± 0.30 AB mag. These magnitudes suggest that the 4000 Å break lies above the -band. Therefore, if instead we compute the probability in the redder and filters, where the magnitude is significantly brighter, we find = 0.05 − 0.08. Based on these arguments, we identify G1 as the putative host galaxy of GRB 191031D.
On November 3, 2019, we carried out optical spectroscopy (Table  3) of G1 with Gemini GMOS-N. A trace is visible from ∼ 6400 to 9500 Å, although there are no obvious absorption or emission features. Therefore, we instead modelled the broadband SED ( 1 2) within prospector. As the spectrum does not show bright emission features, we turned off nebular emission lines within prospector. We derive a photometric redshift of phot = 0.5 ± 0.2 and a stellar mass log( * / ) = 10.2 ± 0.2 (see Figure 9). At redshift ≈ 0.5, we set a lower limit to the circumburst density of the GRB min = 7.9 × 10 −4 (see Table B1) using the X-ray lightcurve.
We performed late-time imaging with the Gemini/GMOS-S telescope on January 25, 2021 in the r-band. We identified two potential host galaxies near to the XRT position (see Figure 8). The first source (Source A) lies within the XRT enhanced position, and has magnitude = 25.5 ± 0.3. Due to its faint nature we cannot conclude whether the source is extended. The upper limit to any additional source within the XRT enhanced position is 25.8 mag. The second source (G1) is located at an offset of 4.5 and displays a morphology suggestive of a late-type galaxy. In our Gemini imaging we derive a magnitude = 22.52 ± 0.03 mag. Based on their -band magnitudes, the probability of chance coincidence for these sources is = 0.21 and 0.11 for Source A and G1, respectively. However, G1 is also visible in the DES, Vista Hemisphere Survey (VHS; McMahon et al. 2013), and ALLWISE (Cutri et al. 2021) catalogs with AB magnitudes: = 23.6 ± 0.2, = 22.6 ± 0.1, = 21.9 ± 0.1, = 21.3 ± 0.1, = 20.9 ± 0.2, 1 = 20.0 ± 0.1, and 2 = 20.2 ± 0.3 mag. The probability of chance coincidence for G1 is significantly smaller in these redder filters with = 0.08 using -band number counts (Capak et al. 2004). Based on these probabilistic arguments and the lack of other candidates, we consider G1 to be the putative host galaxy of GRB 200411A.
Additionally, we utilized the broadband SED ( Figure 9) from these archival observations to derive a photometric redshift phot = 0.6 ± 0.1 and a moderate stellar mass log( * / ) = 10.4 ± 0.1 using the prospector software. At this redshift, we derive a lower limit min 2.3 × 10 −4 cm −3 to the circumburst density using the early X-ray lightcurve. Adopting ∼ 0.6, the physical offset of G1 from the GRB position is = 31 ± 8 kpc and the host-normalized distance is / = 3.9 ± 0.9. The gas density at this distance is ∼ 7 × 10 −4 cm −3 , assuming the density profile outlined in O'Connor et al. (2020), which is consistent with the lower limit implied by the early X-ray afterglow.

APPENDIX B: DERIVATION OF THE CIRCUMBURST DENSITY
Following O'Connor et al. (2020), we compute a lower limit to the circumburst density using constraints on the deceleration time of the GRB jet based on early Swift/XRT follow-up. The parameters required to compute the circumburst density min , namely an upper limit to the time of deceleration of the GRB's jet and a lower limit to the peak X-ray flux , are tabulated in Table B1. In order to calculate the density we adopt the fiducial parameters: the fraction of the burst kinetic energy residing in electrons = 0.1 and magnetic fields = 10 −2 , a bulk Lorentz factor Γ = 300, and a gamma-ray efficiency = 0.15. The lower limit on circumburst density is then derived using Equation 17 of O'Connor et al. (2020). We record this value for each GRB in Table B1. Due to the different selection criteria in O'Connor et al. (2020) (i.e., requiring 90 < 0.8 s), 17 events in our sample were not included in their work (i.e., those with extended emission or 0.8 < 90 < 2 s).
We remind the reader that in order for a sGRB to be considered physically hostless (or consistent with the scenario) the density must be < 10 −4 cm −3 (O'Connor et al. 2020). In the case of these lower limits, if min > 10 −4 cm −3 then the sGRB is inconsistent with being physically hostless, whereas a smaller value of min only implies that the sGRB could be physically hostless and is not conclusive one way or another. This paper has been typeset from a T E X/L A T E X file prepared by the author.