Constraints on the z ∼ 5 Star-forming Galaxy Luminosity Function From Hubble Space Telescope Imaging of an Unbiased and Complete Sample of Long Gamma-Ray Burst Host Galaxies

We present rest-frame UV Hubble Space Telescope imaging of the largest and most complete sample of 23 long-duration gamma-ray burst (GRB) host galaxies between redshifts 4 and 6. Of these 23, we present new WFC3/F110W imaging for 19 of the hosts, which we combine with archival WFC3/F110W and WFC3/F140W imaging for the remaining four. We use the photometry of the host galaxies from this sample to characterize both the rest-frame UV luminosity function (LF) and the size–luminosity relation of the sample. We find that when assuming the standard Schechter-function parameterization for the UV LF, the GRB host sample is best fit with α=−1.30−0.25+0.30 and M*=−20.33−0.54+0.44 mag, which are consistent with results based on z ∼ 5 Lyman-break galaxies. We find that ∼68% of our size–luminosity measurements fall within or below the same relation for Lyman-break galaxies at z ∼ 4. This study observationally confirms expectations that at z ∼ 5 Lyman-break and GRB host galaxies should trace the same population and demonstrates the utility of GRBs as probes of hidden star formation in the high-redshift Universe. Under the assumption that GRBs unbiasedly trace star formation at this redshift, our nondetection fraction of 7/23 is consistent at the 95% confidence level with 13%–53% of star formation at redshift z ∼ 5 occurring in galaxies fainter than our detection limit of M 1600Å ≈ −18.3 mag.


INTRODUCTION
Long duration gamma-ray bursts (GRBs) have been theoretically (Paczynski 1986;Woosley 1993) and observationally associated with the deaths of massive stars and specifically with Type Ib/c-BL supernovae (SNe).These SNe result from the core collapse of a progenitor star that has completely lost its hydrogen shell and most-to-all of its helium shell, with the "BL" designation in reference to the fast moving SN ejecta resulting in broad-lined (BL) emission features (Galama et al. 1998;MacFadyen & Woosley 1999;Hjorth et al. 2003;Woosley &Bloom 2006, andHjorth &Bloom 2012;Cano et al. 2017 for recent review).There are two main observational components to a GRB -the initial gamma-ray prompt emission believed to be from dissipation pro-cesses within the GRB jet and the multi-wavelength afterglow powered by the synchrotron emission originating from the jet's deceleration into the local environment (Chevalier & Li 1999;Miceli & Nava 2022).
GRB follow-up and afterglow studies were revolutionized with the launch of the Neil Gehrels Swift Observatory (Swift; Gehrels et al. 2004).The X-ray telescope on board (XRT; Burrows et al. 2005) has the ability to localize the GRB afterglow to few arcsecond precision allowing for ground based observations.As long-duration GRBs are known to predominantly occur within the half-light radius and within the bright, starforming regions of their host galaxies (Fruchter et al. 2006;Svensson et al. 2010;Blanchard et al. 2016;Lyman et al. 2017), this precise afterglow-enabled localization Sears et al.
often allows for robust host identification.The extreme luminosity (∼ 10 53 erg s −1 ) of the GRB makes them observable to cosmological distances, with the currently most distant GRB 090429B photometrically estimated to have z = 9.4 (Cucchiara et al. 2011).
High-redshift (z > 3) star-forming galaxies are primarily identified using the Lyman break technique in which the wavelength of the Lyman break is determined via photometric dropout (Steidel et al. 1996).Studies of star-forming galaxies benefit from large-number statistics and deep observations and, prior to JWST, extend through z ∼ 9 (see e.g., Stark 2016 for a recent review).Surveys from JWST, including early data release and dedicated programs like The Cosmic Evolution Early Release Science Survey (CEERS; Finkelstein et al. 2023), The GLASS JWST Early Release Science Program (GLASS-JWST; Treu et al. 2022), and the JWST Advanced Deep Extragalactic Survey (JADES; Eisenstein et al. 2023) have allowed for analysis of these galaxies to continue to even greater redshift (z ∼ 13).An important characterization of Lyman-break galaxies is the UV luminosity function.This function is a fit to a histogram of these galaxies and allows for an estimate of the percentage of undetectable star-formation through extrapolation of the fit to faint magnitudes.It is well defined at the bright end (M U V < −15 mag) (Finkelstein et al. 2015;Bouwens et al. 2021Bouwens et al. , 2022a;;Harikane et al. 2023a,b;Finkelstein et al. 2023) with the generally assumed Schechter (1976) being fit to measurements from thousands of galaxies.
Observations of Lyman-break galaxies, however, only offer a view of the star formation that can be directly observed and are therefore implicitly biased against faint galaxies.Since the ability to detect a GRB is independent of the luminosity of its host galaxy, and the detection of a GRB implies the existence of a galaxy at that location, GRBs offer a way to characterize faint and otherwise unobserved star formation, such as that which is dust-obscured or intrinsically faint.Constraining the amount of star formation that would otherwise go undetected, especially at high redshift, is key for determining how large a role this star-formation played in reionizing the universe.
In the low-redshift Universe (z < 2) GRB host galaxies have been shown to have smaller sizes, lower masses and lower metallicities than the general star-forming galaxy population (Stanek et al. 2006;Kewley et al. 2007;Levesque et al. 2010;Svensson et al. 2010;Han et al. 2010;Graham & Fruchter 2013;Perley et al. 2013;Palmerio et al. 2019).These biases are thought to be a consequence of the preference for a GRB progenitor to form and explode in low-metallicity environments, with low-metallicity star-forming galaxies being smaller and less massive than the general sample (Mannucci et al. 2010;Palmerio et al. 2019).The nature of this preference, both physical and functional, is still actively debated: some studies have theorized multiple metallicitydependent paths for GRB creation (Trenti et al. 2015), while some have found evidence for a host-galaxy stellar metallicity threshold above which GRBs are rare (i.e., it allows for the possibility of a pocket of lower-Z starformation within a high-Z galaxy).Below this threshold, GRBs seem to trace star formation in an unbiased way (though there is uncertainty on the value of this threshold (Z < Z ⊙ : Perley et al. 2016a; Z < 0.7Z ⊙ : Palmerio et al. 2019).
The bias of the GRBs in host galaxy mass and size is consistent with being largely a by-product of the metalaversion (Perley et al. 2016a), and so, as the average metallicity of the Universe decreases with increasing redshift, the differences in the characteristics of GRB host galaxies as compared to those of actively star-forming galaxies should decrease toward triviality.Indeed, up to z ∼ 4, comparisons of the two galaxy samples have followed this expectation when characterized by the massmetallicity relation (Levesque et al. 2010;Laskar et al. 2011;Graham et al. 2019), the UV luminosity function (Greiner et al. 2015;Schulze et al. 2015), and in direct size and stellar mass measurements (Schneider et al. 2022).Comparisons at higher redshift (z ∼ 6) also support these results but are significantly limited in precision due to the small number of localized GRBs with confirmed redshifts at this redshift range (Tanvir et al. 2012b;McGuire et al. 2016).
In this work, we present new Hubble Space Telescope (HST) observations of the largest complete sample of GRB host galaxies at z ∼ 5, to significantly improve these comparisons at the highest possible redshifts with currently available data.In Section 2, we describe our observations and host identification methods.We present our formalism, modeling, and analysis of the UV luminosity function and size-luminosity relation of the GRB host sample and compare to that of Lyman-break galaxies in Section 3. We conclude with presentation and discussion of our non-detection fraction and its implications toward the amount of undetectable star formation.We use a cosmological model with H 0 = 70 km s −1 Mpc −1 , Ω 0 = 0.3, and Ω Λ = 0.7.Uncertainties are reported as the Gaussian-equivalent one-sigma, unless otherwise stated.

Sample Selection
We define selection criteria for our z ∼ 5 GRB host galaxy sample to minimize selection bias while maximixing completeness.Our initial selection criteria were: • The GRB has a spectroscopic or photometric redshift of 4 < z < 6.
• Deep observations at the GRB location were performed with the Spitzer Space Telescope (Spitzer ; Werner et al. 2004).
• The GRB was detected with Swift prior to mid-2015 (the date is a by-product of the Spitzer requirement) and has a localization ≤ 2 ′′ .
• The line of sight along the GRB direction has low Galactic extinction, E(B − V ) < 0.2 mag.
From this first-round sample, we use the following criteria to determine the final sample: 1.The GRB was included in one of the four following uniform samples: From these criteria, we populate a sample of 19 GRBs for host galaxy follow-up.We add to this a randomized subsample of four GRBs (050505, 060223A, 140304A, and 140311A) of the 10 events which did not pass the final sample criteria.After investigating the selection criteria for each of the uniform samples, these four GRBs had been excluded due to a small Sun hour angle separation, too high of a declination, were not observed with XRT within 10 minutes of the Swift Burst Alert Telescope (BAT; Barthelmy et al. 2005) trigger, or had too low a fluence (S 15−150keV ).These properties, as well as the non-existence of rapid NIR follow-up, have no dependence on the characteristics of the GRB host galaxy and the inclusion of these four GRBs has no effect on the uniformity of our GRB host galaxy sample.

Hubble Space Telescope Imaging
We present new HST /WFC3-IR imaging for 19 galaxies in our sample (ID: 15644, PI: Perley), while the remaining four had archival imaging available, which we detail in the following section.The 19 host galaxies from our program were imaged using the F110W filter: galaxies with redshift z < 4.8 were observed over two orbits (average exposure time, 4900 s), while those with z > 4.8 were observed over three orbits (average exposure time, 7400 s).Across our redshift range, the central rest-frame wavelength of F110W converts to 1650-2260 Å, which samples the rest-frame UV emission.
We use archival imaging for four sources which were previously observed by HST.The host galaxies of GRBs 060223, 060522, and 060927 were also imaged using WFC3/F110W (ID: 11734, PI: Levan) with 3 orbits for the fields of 060223 and 060522 and 5 orbits for the field of 060927.The host of GRB 130606A was imaged using WFC3/F140W (ID: 13831, PI: Tanvir) over 4 orbits.At a redshift of z = 5.913 (Lunnan et al. (2013)), the central wavelength of F140W translates to 2014 Å, which is comparable to the observations of the other objects in the sample.
The reduced (i.e., flat fielded, charge transfer efficiency (CTE) corrected, dark subtracted) and ICRSaligned HST images were downloaded from the Barbara A. Milkulski Archive for Space Telescopes (MAST) (see chapters 2 and 3 of Sahu 2021 for details on this reduction).
To drizzle the HST frames and achieve a resolution past the instrument limitation, we use Astrodrizzle (Gonzaga et al. 2012) with final pixfrac = 0.8 and final scale = 0.065 for consistency with previous GRB host galaxy HST analysis (e.g., Blanchard et al. 2016).

Afterglow Localizations
Our analysis requires robust and accurate GRB localizations in order to identify the host galaxy of each GRB, and for that purpose, when possible, we use imaging of the optical afterglow.We were able to use optical afterglow imaging for all but three sources in our sample.For these three sources with no optical/NIR afterglow imaging available, we use their position as reported from Swift-XRT (GRBs 050803 and 050922B; Goad et al. 2007) or from the Karl G. Jansky Very Large Array (VLA; Perley et al. 2011) (GRB 140304A;Laskar et al. 2014).
Optical afterglow images were collected from the public archives of the Low Resolution Imaging Spectrometer at the W.M. Keck Observatory (Keck-LRIS;Oke et al. 1995), the Gemini-North/South Multi-Object Spectrograph at the Gemini North/South Observatory (GMOS-N/S; Hook et al. 2004), the Palomar 60-inch Telescope at Palomar Observatory (P60; Cenko et al. 2006), Very Large Telescope (VLT), the Rapid Eye Mount Telescope at La Silla Observatory (REM1 ), the Device Optimized for the LOw RESolution (DOLORES, in short LRS2 ) at the Telescopio Nazionale Galileo (TNG), and the Ultraviolet/Optical Telescope onboard Swift (UVOT; Roming et al. 2005).To reduce images from Keck-LRIS, we use the LPIPE pipeline (Perley 2019).When possible, we use the reduction pipelines embedded within the archive services.We otherwise use standard reduction steps such as flat division, bias subtraction, and image stacking.Centroid positions for each afterglow were measured using Source Extractor (Bertin & Arnouts 1996).Imaging and reduction steps for each GRB afterglow are detailed in the Appendix with additional references in Table 1.

Astrometric Alignment
Many of the afterglow images had an initial world coordinate system (WCS) assigned by the data archive.For those that did not, we upload the afterglow image to Astrometry.net (Lang et al. 2010) to get a preliminary WCS assignment.To align the afterglow images to the HST images, we used TweakReg (Gonzaga et al. 2012).In the first alignment attempt, we use a catalogue of Gaia sources within 2 ′ of the afterglow position.If this failed or if there were fewer than 6 catalogue sources in the HST image, we instead used a catalogue of at least 6, but often > 10, matching sources (all of the bright and unsaturated stars and sometimes bright galaxies) between each afterglow and HST image pair.These sources were selected from visual inspection in SAOImageDS9 (DS9; Joye & Mandel 2003).The alignment was deemed successful when common sources were aligned to within approximately 1 HST pixel = 0.065 ′′ .In the case of GRB 060223, there was only one source (a saturated star) in common between the two images, and so we instead aligned each image separately to the Gaia DR2 catalogue.Details on alignment steps for each source are in the Appendix (A).
The afterglow positions found by Source Extractor were then converted from pixel to WCS coordinates for use in host galaxy identification in the corresponding HST image.To quantify the uncertainty on the position of the afterglow, reported in Table 1, we add in quadrature the uncertainty in the afterglow centroid from Source Extractor and the root mean square (RMS) of the astrometric match to the HST image of the host.When optical afterglow imaging was unavailable, we list the uncertainty reported in the literature (GRB 140304A; Laskar et al. 2014) or the Swift-XRT catalogue (GRBs 050803 and 050922B; Goad et al. 2007).All but two afterglows (GRBs 050803 and 050922B, for which only Swift-XRT imaging was available) were localized to better than 0.5 ′′ , with a median localization uncertainty of ∼ 0.06 ′′ .
For all but one case (GRB 050922B), if there was a galaxy coincident within the afterglow uncertainty region, we designate that as the host of the GRB, as lowerredshift GRB afterglows are found near the centers of their host galaxy (Blanchard et al. 2016).Within the afterglow uncertainty region of GRB 050922B, there are two galaxies: a compact source and a merging system.In agreement with Perley et al. (2016b), we designate the merging system as the host of this GRB.The identification of this galaxy as the host is elaborated upon in the next section.If there was no galaxy within the region, we classified this as a non-detection for the host galaxy.Details on the detection classification for each host are in the Appendix, and excerpts of the HST imaging with afterglow positions, their three-sigma uncertainty regions, and host galaxy identifications are shown in Figure 1.

P cc Calculations
We consider the false alarm probability for our claimed host galaxies and nearby sources to our claimed nondetections.The false alarm probability is the chance of an unrelated galaxy being within the measured proximity to the line of sight to the GRB.When the afterglow is well-localized, this probability is largely dependent on the offset from the afterglow and the apparent magnitude of the putative host.We calculate the probability of chance coincidence (P cc ) following methods in Bloom et al. (2002) and using P cc = 1 − e −π×R 2 e ×σ(≤m) .R e is taken to be the maximum of 3 , where σ T IE is the uncertainty in the astrometric tie between the afterglow and galaxy positions, and σ AG is the uncertainty in the afterglow position.R is the offset of the considered galaxy from the center of the afterglow, and R ef f is the half-light radius of this considered galaxy.σ(≤ m) is calculated from summing galaxy number densities below the measured m F 110W in Tables 3 and 4 in Metcalfe et al. (2006).
For our detections, we calculate the P cc for the putative host.Only four of the 16 putative host galaxies had P cc > 0.1.These were the host galaxies of GRBs 050803 (P cc = 0.98), 050922B (P cc = 0.99), 071025 (P cc = 0.12), and 140614A (P cc = 0.21).These four cases include our two GRBs with only Swift-XRT po-sitions available (GRBs 050803 and 050922B) and two sources with the next largest afterglow positional uncertainty.In the cases of 050803 and 050922B, these GRB were included in our sample due to the photometric redshifts of the claimed host galaxies (Perley et al. 2016b), and so we continue analysis with the assumption that these are the host galaxies of these GRBs.We repeated our analysis in Section 3 treating these hosts as nondetections, and found that the best-fit LF parameters are consistent to within 1 sigma, so our results are not strongly sensitive to the uncertainty in these host associations.In the other two cases, these were the only sources within the afterglow uncertainty region, and so we classify them as the host galaxy of their respective GRB.Details on each P cc are in the Appendix.
For our non-detections, we calculate the P cc for all sources detected by Source Extractor within a 5 ′′ × 5 ′′ box centered on the afterglow position reported in Table 1.Only two of the 21 nearby sources in the 5 ′′ fields of our non-detections had P cc < 0.1.These two sources (one each in the fields of GRBs 060927 and 100219A) were confirmed to have a lower redshift than each respective GRB and are therefore not the host galaxies.The galaxy in the field of GRB 060927 was detected in R-band VLT imaging (Basa et al. 2012) and has a redshift z < 4, which is incompatible with the spectroscopic afterglow redshift of z = 5.467 reported in Ruiz-Velasco et al. (2007).The galaxy in the field of GRB 100219A was spectroscopically confirmed to have z = 0.217 in Cenko et al. (2010a), which is incompatible with the spectroscopic afterglow redshift of z = 4.667 for GRB 100219A (Selsing et al. 2019).Because all other detected candidate host galaxies have P cc > 0.1, we report the host galaxies of these seven GRBs as non-detections.Details on each P cc are in the Appendix (A).

HST Photometry
We measure apparent magnitudes of all detected GRB host galaxies with Source Extractor using MAG AUTO with PHOT AUTOPARAMS set to the default values of 2.5 and 3.5.This parameter couplet sets the multiplicative factor and minimum Kron radius used in the "auto" measurement and is explained in greater detail in the Source Extractor documentation.3These measurements are reported in Table 2.We convert these apparent magnitudes to absolute UV magnitudes at 1600 Å using the distance modulus and a K-correction, as detailed below.We first aperture correct the apparent magnitudes using the Encircled Energy (EE) tables from STScI. 4 We interpolate the table values for F110W and F140W with a cubic spline to determine the appropriate EE term for the precise KRON RADIUS used by Source Extractor for each galaxy.We then correct these aperture-corrected magnitudes for Galactic dust absorption as reported in the NASA/IPAC Extragalactic Database (NED; Schlafly & Finkbeiner 2011) at the location of the afterglow.We assume a UV spectral slope of β = −2 (see Figure 2 in Wilkins et al. 2013), where f ν ∝ ν −β and then apply a K-correction of −2.5 log 10 (1 + z).The component of the K-correction for the spectral shape is proportional to (2 + β), and therefore vanishes since we assume β = −2.In summary, where D L is the luminosity distance.We report in Table 2 absolute magnitude, M 1600 Å, uncertainties as the uncertainty on the apparent magnitude as reported by Source Extractor with propagation of the redshift uncertainty, when reported.
We report 3σ lower limits on the observed magnitude for sources that are not detected in our images.In each HST image of a non-detected galaxy, we measure the flux within randomly-placed 0.37 ′′ -radius apertures within 6 ′′ of position of the afterglow.This aperture size was chosen as it is the average radius used for the detections.We calculate the median flux within these regions, and we clip any flux measurements with a > 3σ divergence from this value and then recalculate the median.We repeat this 3σ median-clipping until convergence of the median.Three standard deviations above this median value is used as an upper limit for the magnitude of the host galaxy.We then aperture correct these limits using the same methods as were used for the detections and report these final upper limits in Table 3.These galaxy magnitudes, both detections and upper limits, were derived in this way for modeling and comparison of the UV luminosity function (LF) of our sample, which we detail in Section 3. GRB Host Galaxies at z ∼ 5 7 3. DISCUSSION 3.1.Lyman-break Galaxy UV Luminosity Functions We derive the luminosity distribution of our GRB host galaxy sample from Tables 2 and 3. We compare these results to samples of Lyman-break galaxies at z ∼ 5 from Bouwens et al. (2021) and Finkelstein et al. (2015).We elected to not use results from the more recent Bouwens et al. (2022a) due to their choice of functional form for the luminosity function, which deviates from the standard Schechter function by including an additional parameter, δ, that allows for curvature at the faint end (M U V > −16 mag) of the luminosity function.Their formula and best fit parameters result in a divergent luminosity function whose CDF is inherently highly sensitive to the choice of the lower integration limit.Furthermore, since our faintest detected GRB host galaxy is at M U V = −18.1 mag and even the upper limits for our non-detections are not much fainter than this, our data are insensitive to the shape of the faint-end LF and could not place any meaningful limitations on this additional parameter.Indeed, even with their larger sample of 59 z ∼ 5 Lyman-break galaxies, they find δ = 0.07 ± 0.2, to an uncertainty 5× greater than that they find for their α (0.04).The Schechter function parameters from both Bouwens et al. (2021) and Finkelstein et al. (2015) are reported in Table 4.
To meaningfully compare the Lyman-break galaxy luminosity functions of Bouwens et al. 2021 andFinkelstein et al. (2015) to that of our GRB host galaxies (detailed in following sections), we must first account for the GRB production rate.To do so, it is necessary to weight the Lyman-break galaxy luminosity functions by the instantaneous star-formation rate (SFR), as the GRB production rate is expected to be proportional to the SFR.The SFR is proportional to the intrinsic UV luminosity (Kennicutt 1998), and so we can effectively account for GRB selection effects by multiplying the Lyman-break galaxy luminosity function by the intrinsic luminosity of the Lyman-break galaxy.We consider two conversions of the intrinsic to the observed UV luminosity, as the luminosity functions are functions of the observed luminosity.In both cases, we construct the base SFR-weighted Schechter LF (i.e., a predicted GRB host luminosity function) as below: where L * is the characteristic luminosity, L int is the intrinsic luminosity, L obs is the observed luminosity, ϕ * is a normalization parameter, and α is the faint-end slope, as is standard in the Schechter function.In magnitude space, this can be restated as: (2) where f (M obs ) = 0.4 × (M * − M obs ), M int is the intrinsic magnitude, M obs is the observed magnitude, ϕ * * is a normalization parameter, and α is still the faint-end slope.M * is the characteristic magnitude and is defined as , where L 0 is the luminosity of a source with an absolute magnitude of 0.
1.In our first formalism, we assume a luminosityindependent dust-contribution where the intrinsic luminosity, L int , is linearly proportional to the observed luminosity, L obs .Here, L int ∝ L obs ∝ 10 −0.4×M obs .
2. Our second formalism is one where we assume a non-linear luminosity-dependent dustcontribution.We make this assumption because we expect more massive galaxies to have more dust.We construct this formalism from the following two relations from Overzier et al. ( 2011)5 and Bouwens et al. (2014), respectively: where A 1600 is the extinction at 1600 Å and β is defined as: Since A 1600 cannot be negative, this results in a piecewise luminosity function of the form of Eq.
(1) where now  Positions of both the afterglow and host are reported in Tables 1 and 2, respectively.North is up and East is to the left.

and
L int ∝ 10 −0.4×(1.25×Mobs +4.39) for M obs ≤ 17.3 (6) 3. Our third formalism is one where we again assume a non-linear luminosity-dependent dust contribution, however we substitute for Eq. ( 3) an estimation of the same relation from Meurer et al. (1999): We refer to our second and third formalisms as "O11" and "M99," respectively, in reference to the choice of the A 1600 (β) formulation (i.e., the choice of Eq. (3) or Eq. ( 7)).

GRB Host UV Luminosity Function
We use Bayesian hierarchical modelling to constrain the parameters of the luminosity distribution of the GRB host galaxies.We assume the galaxies follow a SFR-weighted Schechter function (see Eq. 1) with a faint-end magnitude limit of M U V = −14.1 mag (this arbitrary magnitude choice converts to a convenient value in our luminosity units and is well below our detection threshold in all cases, although we find that our results are not statistically sensitive to this precise choice).We used weakly informative Gaussian priors of µ α = −1.6,σ α = 1.0 and µ log L * = 10, σ log L * = 1 for the α and logL * parameters, respectively.The model selfconsistently included both the detections and the seven upper limits: the luminosity for each of these 23 objects was a free parameter in the model, and hence each has a corresponding posterior distribution.We symmetrized the uncertainties for each measurement, conservatively selecting the greater of the two, though we find that our results are also not sensitive to this choice.Four chains were run per model with at least 100,000 samples per chain after warm-up, which ensured negligible MCMC standard errors for all parameters of interest.In the final model runs, there were no divergences, and the chains for all parameters mixed well, with the convergence diagnostic R = 1.We complete this process three times, once each for our different considerations of the SFR-weight on the Lyman-break galaxy LF as described in the previous section.To model these luminosity distributions of the GRB host galaxies, we use the Stan software as implemented in version 2.26.13 of the RStan package (Stan Development Team 2023).
We show the posteriors and best-fit SFR-weighted Schechter functions for the L-independent and O11 weightings in Figure 2.These best-fit α and M * parameters, along with their 1σ uncertainties, are provided in Table 4. Best fit Schechter function fit parameters to the GRB host data and the Lyman-break galaxy data sets.The fits to the GRB hosts were measured from our RStan program, while fits to the Lyman-break galaxies were copied from Bouwens et al. (2021) and Finkelstein et al. (2015).The same is true for the parameters from Finkelstein et al. (2015).These Lyman-break galaxy fits are consistent to within 2σ to our L-independent weighting as well.The slightly better agreement of the Lyman-break galaxy LFs to the O11 formalism is expected, as this formalism offers a more realistic estimate of the intrinsic extinction at z ∼ 5. Along this parametric comparison, there is no evidence of disagreement between the GRB host galaxy sample and the Lyman-break galaxy samples.While the differences between the galaxy samples are not statistically significant, the best fits to the GRB host galaxies have a shallower α and a fainter M * .With a larger GRB host galaxy sample, if these parameter differences were to become statistically significant, the move toward a shallower α and a fainter M * would indicate that Lyman-break galaxy LFs over-predict the amount of faint star-formation.
We construct a cumulative distribution function (CDF) of the GRB host galaxy luminosity function by using Kaplan-Meier estimation (Kaplan & Meier 1958) on our observed magnitudes and upper limits.We plot this CDF in Fig. 3.We qualify the uncertainty on this CDF by plotting also a subset of the CDFs created from random draws of the modeled magnitude sets.In this figure we also show the CDFs of the UV LFs from Bouwens et al. (2021) and Finkelstein et al. (2015) with the different extinction assumptions.
To measure the likelihood of inconsistency between the Lyman-break galaxy and metallicity-biased GRB host galaxy luminosity distributions to that of the observed z ∼ 5 GRB host galaxy distribution, we use a log-rank test.This test was chosen because for its applicability to distributions including censored data (i.e., our upper limits), unlike commonly used statistical tests, like a Kolmogorov-Smirnov (Massey 1951) or an Anderson-Darling test (Stephens 1974).We report the p-value corresponding to the calculated χ 2 statistic for each test in Table 5.This p-value is the probability of achieving the χ 2 test statistic, and so since we consider a 2σ threshold, we accept p < 0.05 as confirmation for the null hypothesis that the compared samples are pulled from different distributions.To complete these tests, we use survdiff within the survival package in R (Therneau 2023; Terry M. Therneau & Patricia M. Grambsch 2000; R Core Team 2021).
With p-values all above p = 0.05, we find no evidence for inconsistency between our O11 and M99 SFRweighted Lyman-break galaxy luminosity distributions and our derived GRB host galaxy luminosity distribution.We do, however, find 2σ disagreement (though 3σ agreement) between our L-independent SFR-weighting for both the B21a and F15 Lyman-break galaxy luminosity distributions and that of our GRB host galaxies.These results imply that if GRBs are to trace starformation, either the L-independent extinction correction is an incorrect assumption for the distribution of dust in z ∼ 5 star-forming galaxies or additional parameters are necessary, perhaps the faint-end slope curvature parameter δ presented in Bouwens et al. (2022a).

Investigating the Metallicity Bias
Lastly, we consider the influence of metallicity in our luminosity function fits.To quantify GRB-production metallicity sensitivity, we consider the UV LF predictions at z = 4.75 from Trenti et al. (2015).Those authors developed a model that considers two GRB progenitor pathways: one that is metallicity-dependent and one that is metallicity independent, which they refer to as "metallicity sensitive" (MS) and "metallicity insensitive" (MI) channels.They quantify the percentage of GRBs originating from a MI pathway with their "GRB efficiency" function, κ(Z).This is defined as: where κ 0 , a, and b are piecewise defined based on galaxy metallicity and take on the same values as in Trenti et al. (2015).In this context, p is what they refer to as the "plateau" parameter and can take on any non-negative value.In the low metallicity (and therefore high z) limit, this MI efficiency function, κ(Z), asymptotically "plateaus" to the value p/(1 + p).While p is explicitly not a probability (and can take on any non-negative value), it is correlated with the percentage of GRBs originating from the MI channel.Across all metallicities and redshifts, when p = 0, it is assumed that GRBs originate exclusively from the MS channel and when p = ∞, it is assumed GRBs originate exclusively from the MI channel.
Positive and finite values of p assume a split of GRB progenitor paths.Trenti et al. 2015 applied their models to the Swift GRB catalogue and to other GRB host galaxy samples (Savaglio et al. 2009;Cucchiara et al. 2015a) and found that p = 0.2 best replicates the redshift evolution of the GRB rate to z ∼ 6.At z ∼ 5, the majority of galaxies have metallicities below the threshold values found in the local universe, so we expect the host galaxy LF to be more consistent with the MI parameterization, p = ∞.
We show in Fig. 4 the four z = 4.75 luminosity functions predicted by Trenti et al. 2015 for different values of p overlaid on our GRB host galaxy LF, and we report the results of log-rank tests between these relations in Tab. 5. We find only the p = 1000 case to be consistent with our LF to within the Gaussian-equivalent Right: A similar plot to that on the left, with the same GRB host galaxy LF, but using as comparison now the z ∼ 5 SFR-weighted UV LF from Finkelstein et al. (2015).The pink luminosity function again shows an assumption where galaxy extinction is luminosity independent, while the luminosity function in dark blue (light blue) assumes the O11 (M99) extinction correction.
2σ.Specifically, we find disagreement with our observations and the p = 0.2 model favored by Trenti et al. (2015).The disagreement of the p = 0.2 model with the high-redshift host galaxy LFs (ours at z ∼ 5 and that at z ∼ 3.5 from Greiner et al. 2015) implies that a different metallicity parameterization for GRB production is necessary.

GRB Host Size Distribution
Observations of Lyman-break galaxies have shown correlation between the UV luminosity and half-light radius (Kawamata et al. 2015;Shibuya et al. 2015;Bouwens et al. 2022b).We present half-light radii (R ef f ) for our 16 detected host galaxies in Table 6 and Figure 5.We first constructed point spread functions (PSFs) for each of our two filters, F110W and F140W.Schneider et al. (2022) found that for a sample of the fields of 42 GRB host galaxies at z ∼ 3 imaged with WFC3/F160W, the constructed PSF had a radius profile stable against the choice of field in which to select stars for the star catalogue but had a S/N dependent on the number of stars selected, increasing with the length of the star catalogue.In their study of GRB host halflight radii at z ∼ 3, Schneider et al. (2022) find that N ∼ 30 is a sufficient length for the catalogue.We apply this finding to our sample and use 33 stars from the fields of GRBs 050814 and 050922B to construct the PSF for F110W.The choice of these fields was mostly arbitrary, however we chose not to use fields crowded with several saturated stars (such as that of GRB 140614A).We had only one GRB field imaged in F140W, and so we select 26 stars from the field of GRB 130606A to construct the PSF for this filter.We use the astropy package EPSFBuilder (Bradley et al. 2023) to generate the two PSFs from these star catalogues.
We measure the half-light radii of our detected GRB host galaxies by fitting a Sérsic light profile with GALFIT (Peng et al. 2010).On our first measurement attempt, we use as guesses the results from Source Extractor with GALFIT able to fit all parameters.If the program was not able to converge all parameters, we try again holding R ef f constant but all other parameters open.If the other parameters converge on this run, we fix the parameters to these new values and allow for GALFIT to  Trenti et al. (2015).Described in detail in Section 3.3, the p parameter is tied to the influence of metallicity on the GRB progenitor path.Across all redshifts, when p = 0, there is a metallicity bias where GRBs cannot be produced in environments with Z > Z⊙, and when p tends to ∞, GRB creation is metallicity insensitive.In Trenti et al. 2015, they report Schechter function LF parameters for four choices of p, which we plot here.Results from log-rank tests between the black, median LF for our GRB host sample and the four metallicity-biased LFs are shown in Table 5.
fit for R ef f on the next run.If the parameters do not converge, or if R ef f does not converge as the only free parameter, we instead try fixing all parameters to the Source Extractor guesses and allowing the program to fit for only R ef f .If there still was no convergence, and there was a second source within 20 pixels of the host, we rerun the program with a second Sérsic profile for the second source.We use the same methods to attempt convergence for both sources.In all cases, if there was sufficient convergence, the residual was visually checked for confirmation of a good fit.We record the R ef f and its uncertainty reported by GALFIT in Table 6.
No runs were successful following this script, meaning either no convergence of R ef f or a visually bad residual, for 3 of our sources (the host galaxies of GRBs 050814, 111008A, and 140311A).For the host galaxies of GRBs 111008A and 140311A, we adopt R ef f upper limits as that reported by Source Extractor.For the host galaxy of GRB 050814, we updated the Source Extractor guesses to our best guesses, with our only change being updating the position angle (PA) from −61 deg to 40 deg.With this update, GALFIT converged all parameters.This fit is elaborated upon in the Appendix entry for GRB 050814.
We compare this sample of GRB host galaxy sizes to Lyman-break galaxy sizes at z ∼ 4 and z ∼ 6 − 8 (Bouwens et al. 2022b) in the form of a size-luminosity relation in Figure 5.Under the assumption that GRBs unbiasedly trace star formation, we expect z ∼ 5 GRB host galaxies to fall in-between the z ∼ 4 and z ∼ 7 relations.Since our smaller sample has a average z = 4.6, if this assumption is to be true, we would expect the GRB host sample to be weighted closer to the z ∼ 4 relation.We find that ∼ 68% (11/16) of our GRB host galaxies fall within or below the 1σ scatter of the z ∼ 4 relation.This supports our claim that at z ∼ 5, Lyman-break and GRB host galaxies trace the same stellar populations.

GRB Host Galaxy Non-Detection Fraction and Implications of Hidden Star Formation
The source of the UV photons needed to reionize the intergalactic medium in the early Universe has been and continues to be an area of very active research (Furlanetto & Mesinger 2009;Robertson et al. 2015;Endsley et al. 2023).One explanation for this process is the UV radiation from massive stars in star-forming galaxies (Madau et al. 1999;Ciardi et al. 2000;Bunker et al. 2004;Finkelstein et al. 2010;Bunker et al. 2010;Finkelstein et al. 2012).Until recently, with the launch of JWST, investigations of the feasibility of this explanation have mostly stopped at z ∼ 8 or have relied on the extrapolation of the characterizations of lower redshift observations of Lyman-break galaxies to higher redshifts and fainter magnitudes (Oesch et al. 2010;Bouwens et al. 2012).Recent JWST -based studies have found discrepancies from lower-z expectations and models, namely the detection of more massive, bright galaxies than expected (Harikane et al. 2023b;Finkelstein et al. 2023).There have been many offered explanations for these discrepancies, including stochastic starformation (Furlanetto & Mirocha 2022;Shen et al. 2023;Mirocha & Furlanetto 2023) and high-efficiency star formation (Dekel et al. 2023).
While using GRBs to test the feasibility of massive star reionization of the Universe is not new (e.g., Tanvir et al. 2019), our complete GRB sample offers the first opportunity to test this feasibility with statistical robustness at a redshift just outside the Epoch of Reionization.From our non-detection fraction, we can estimate the percentage of star-formation that is occurring in galax-ies fainter than our detection limit (i.e., galaxies that are intrinsically faint and galaxies that would otherwise be detected but are dust obscured).These are galaxies that are inherently often missed in star-forming galaxy samples as they are not directly observable.Comparing the direct measurement of the percentage of faint star formation to expectations from Lyman-break galaxy LFs is critical, as faint star-forming galaxies are thought to be important contributors of ionizing photons (McLure et al. 2010).
Under the assumption that GRBs unbiasedly trace star formation at this redshift, using binomial statistics, our non-detection fraction of 7/23 is consistent at the 95%-confidence level with 13-53% of star formation occurring in galaxies fainter than our detection limit of M U V ≈ −18.3 mag.This measurement is unique in that it is independent of the functional form of the luminosity function and offers a non-parametric way to test the consistency of an assumed functional form to an observed quantity.It is shown in Figure 3, that the percentage of undetectable star formation predicted by the SFR-weighted Schechter function Lyman-break galaxy luminosity functions is ∼ 40 ± 5% and ∼ 25 ± 5% when considering L-independent and L-dependent (O11 and M99) host extinction, respectively.The lack of disagreement between all of the Lyman-break galaxy predictions and the GRB host galaxy measurement offers support for the hypothesis that star-forming galaxies are large contributors of ionizing photons in the early universe.

CONCLUSIONS
We present new rest-frame UV HST imaging of a complete sample of 23 long GRB host galaxies at z ∼ 5. From our imaging, we measure UV magnitudes and galaxy sizes.We detect 16 GRB host galaxies and place upper limits on the magnitudes of the remaining 7. Of the 16 detections, we are able to spatially resolve 14 and place upper limits on the sizes of the remaining 2. Through the construction of a UV luminosity function, we find that our GRB host sample is statistically consistent (log-rank test p > 0.05) with that of the star-forming galaxy population at the same redshift, when using reasonable corrections for the intrinsic extinction in star-forming galaxies.When investigating the feasibility of a metallicity-bias model of GRBs from Trenti et al. (2015), we find that our host sample is inconsistent with this model.Assuming a SFRweighted Schechter-function formalism and a GRB rate proportional to the dust-corrected UV luminosity, we find parametric agreement between both α and M * of our best-fits and those from Bouwens et al. (2021) and  Finkelstein et al. (2015), again regardless of our choice of galaxy extinction.We find that 11 of our 16 (∼ 68%) host galaxies fall within or below the 1σ scatter of the luminosity-size relation of z ∼ 4 star-forming galaxies from Bouwens et al. (2014).The lack of disagreement between the luminosity-dependent UV LFs and the sizeluminosity relations between the Lyman-break and GRB host galaxy samples implies that at z ∼ 5, GRBs are unbiased tracers of star formation.Under this well-supported assumption that GRBs are unbiased tracers of star-formation at this redshift, we use our non-detection fraction of 7/23 and binomial statistics to estimate that, at 95% confidence, 13-53% of star formation is undetected in observations of these depths.In other words, we find that up to ∼ 50% (or alternatively, only ∼ 10%) of star formation could be occurring in galaxies with M U V > −18.3 mag.This measurement is complementary to and unique from similar measurements from Lyman-break galaxy surveys since it is insensitive to the parameterization of the luminosity function.This solidifies the importance of GRB afterglow and host galaxy observations as a tool for studies of high-z star-formation.Lyman-break Galaxies z = 4 Lyman-break Galaxies z = 6-8 Detected GRB Host Galaxies  2 and 6.The outlier point on the bottom left is that of the host galaxy of GRB 050814.Relations for Lyman-break galaxies at z = 6 − 8 [magenta] and at z = 4 [teal] are from Bouwens et al. (2022b).
The shaded regions show the the one sigma scatter of each relation.
The sample presented here is the largest and most complete sample of GRB host galaxies at this redshift.It is unlikely that this sample will be surpassed in statistical sensitivity in the near future, due to our biasminimizing selection cuts.One of the selection criteria was a detection cut pre-2015.Since then, there have been 12 additional Swift-detected GRBs with z > 4. If all of these sources were to meet our sample criteria and followed our detection distribution, the addition of these 12 sources would improve our sensitivity by ∼ 40% (i.e., our uncertainty on the Schechter parameters would be reduced by ∼ 40%).While an improvement, this precision is still not better than that from Lyman-break galaxy samples and therefore the inclusion would not result in a significant statistical advance from the analysis performed here.What is needed to improve this analysis is, simply, the detection and follow-up of many more high-redshift GRBs.There are missions, like the Space-based multi-band astronomical Variable Objects Monitor (SVOM; Wei et al. 2016) and the Gamow Explorer (White et al. 2021), planned expressly for such follow-up.The analysis presented here shows directly how results from such missions can be interdisciplinary, improving not GRB science but our understanding of star-forming galaxies as well.
GALFIT was unable to converge on a single Sérsic profile, following our standard methods of using the Source Extractor parameter results as input.We were able to achieve converge after modifying the positional angle (PA) guess from -61 deg to 40 deg, our estimate of the PA of the galaxy.While all parameters converged and the residual image of this solution passed our visual check, the Sérsic index, N , converged to N = 9.97 ± 3.03, which is much larger than expected.We also attempted to fit the galaxy with two Sérsic components and achieved convergence for both profiles, but the residual did not pass our visual check.We chose to complete analysis with the R ef f from the single component solution, R ef f = 1.00 ± 0.11 pixel.
A.5. 050922B GRB 050922B has no afterglow detections reported in the literature but has a photometric host redshift of z = 4.9 +0.3 −0.6 as detailed in Perley et al. (2016b) from i -and z -band GTC/OSIRIS imaging.We detect three sources within the Swift-XRT error circle including the source identified in Perley et al. (2016b).We measure an apparent magnitude of this source of m F 110W = 25.37 ± 0.08 mag.We calculate P cc = 0.44 for this source when using the 90%-confidence Swift-XRT uncertainty as R e .This percent chance coincidence is well above our 10% threshold.When estimating the impact of false host-association contamination in our sample, we also consider the possibility that this is a non-detection with a measured limiting magnitude of m F 110W > 27.85 mag.
A.6. 060206 GRB 060206 is located at z = 4.048 as reported in Fynbo et al. (2006).We were unable to use TweakReg to align the P60 R-band imaging of the afterglow from 2006-02-06 (Ofek et al. 2006) to our HST image due to there being only one sufficiently bright source in common between the two images.We instead align each image separately to the Gaia DR2 catalogue.For this alignment, we consider an uncertainty of approximately one HST pixel = 0.065 ′′ .We detect the afterglow with Source Extractor with a positional uncertainty on the centroid of 0.016 ′′ for a total positional uncertainty of 0.067 ′′ .Within a 0.20 ′′ radius region at the position of the afterglow, we detect a source in our HST image.We calculate a P cc = 0.02 for this galaxy, and we therefore identify it as the host galaxy of this GRB.We measure an apparent magnitude of m F 110W = 27.56 ± 0.22 mag.

A.7. 060223
GRB 060223 has a spectroscopic afterglow redshift of z = 4.406 as reported by Chary et al. (2007).The only afterglow imaging provided in the literature is V -band Swift-UVOT imaging from 2006-02-23 (Blustin et al. 2006).There was only one source (a saturated star) in common between the HST and UVOT images, so we were unable to complete image alignment using TweakReg.Since each image was aligned to Gaia DR2 upon download from their respective archives, we consider the alignment uncertainty to be within one HST pixel = 0.065 ′′ .We add this in quadrature to the afterglow centroid uncertainty measured with Source Extractor of 0.042 ′′ to get a total afterglow positional uncertainty of 0.077 ′′ .The afterglow position and its 3σ (0.23 ′′ ) uncertainty region are coincident with a source in the HST image.We calculate a P cc = 0.06 for this source and identify it as the host galaxy of GRB 060223.We measure an apparent magnitude of this galaxy of m F 110W = 26.63 ± 0.07 mag.For this host galaxy, Blanchard et al. (2016) report a Galactic-extinction corrected magnitude of m F 110W = 26.534± 0.069 mag, which is consistent with our measurement of m F 110W = 26.53± 0.07 mag.

A.8. 060510B
GRB 060510B has a spectroscopic afterglow redshift, z = 4.941, as measured in Price et al. (2007).We align i -band GMOS-N imaging of the afterglow from 2006-05-10 ( Price et al. 2006) to the HST image.We measure a RMS alignment uncertainty of 0.09 ′′ , and we add this in quadrature to the afterglow centroid uncertainty of 0.0062 ′′ measured with Source Extractor for a total afterglow positional uncertainty of 0.09 ′′ .The afterglow position and its 3σ (0.27 ′′ ) uncertainty region are coincident with a source in the HST image.We calculate a P cc = 0.04 for this source and identify it as the host galaxy of this gamma-ray burst.We measure an apparent magnitude of this source of m F 110W = 26.05± 0.06.A.9. 060522 GRB 060522 has a spectroscopic afterglow redshift of z = 5.110 as reported in Chary et al. (2007).We reduced Rband TNG imaging of the afterglow from 2006-05-22 and report a 0.028 ′′ uncertainty on the centroid of the afterglow.
We align this reduced image to the HST image and report an uncertainty of 0.05 ′′ on this astrometric alignment.We sum these uncertainties in quadrature and report a total positional uncertainty of 0.05 ′′ .We do not detect a source within a 0.15 ′′ -radius region centered at this afterglow position in the HST image.We find three sources within a 5 ′′ box centered at the position of the afterglow, and we calculate P cc values for all above 0.2.For this reason we consider the host of this GRB to be a non-detection.We report a limiting magnitude of m F 110W > 27.83 mag.For this source, Blanchard et al. (2016) report a non-detection and an upper limit of m F 110W > 28.9 mag, and Tanvir et al. (2012b) report a non-detection and an upper limit of m F 110W > 28.13 mag.Blanchard et al. (2016) define their 3 sigma upper limits as the magnitude at which sources are detected at 3 sigma significance.The result from Tanvir et al. (2012b) is inconsistent with our upper limit, however they perform forced photometry in a 0.4 ′′ −radius aperture at the afterglow location and also use a 2σ detection threshold.When we apply the same methods, we are able to reproduce their limit.For consistency of our GRB host galaxy sample, we continue with our limit of m F 110W > 27.83 mag.
A.10. 060927 GRB 060927 has a spectroscopic afterglow redshift of z = 5.467 as detailed in Ruiz-Velasco et al. (2007) from VLT/FORS1 spectroscopy.We are unable to find the centroid of the afterglow with Source Extractor due to blending with a nearby galaxy in I -band VLT imaging at 2.6 days post-trigger (Ruiz-Velasco et al. 2007), but the afterglow is visible in DS9 after adjusting the scale and smoothing parameters.We are able to estimate the center of the afterglow to within 0.5 VLT pixels (0.126 ′′ ), and we also report a 0.023 ′′ astrometric uncertainty, resulting in a total positional uncertainty of 0.128 ′′ for the afterglow.There are no sources detected within 0.385 ′′ of this position.Within a 5 ′′ box centered at the position of the afterglow, we find three sources and calculate P cc values for two of them above 0.8.The third source (the only one visible with our scaling choice in Figure 1 and is the blended source in the VLT imaging) had a P cc = 0.11.This nearby source was detected in VLT R-band imaging (Basa et al. 2012) and therefore is at z < 4, and we therefore exclude this source as the possible host galaxy for GRB 060927.For these reasons, we consider the host of this GRB to be a non-detection.We report a limiting magnitude of m F 110W > 27.84 mag.Tanvir et al. (2012b) report a limiting magnitude m F 110W > 28.57, however they perform forced photometry in a 0.4 ′′ − radius aperture at the afterglow location and also use a 2σ detection threshold.When we apply the same methods, we are able to reproduce their limit.For consistency of our GRB host galaxy sample, we continue with our limit of m F 110W > 27.84 mag.
A.11. 071025 GRB 071025 has a photometric afterglow redshift of z = 4.8 ± 0.4 as presented in Perley et al. (2010).To identify the host galaxy of this GRB, we use H -band REM imaging of the afterglow from 2007-10-25.We were successful in using TweakReg to align the afterglow and HST images, despite there being few sources (many of them saturated stars) in common between the fields.We report an alignment RMS uncertainty of 0.22 ′′ and an afterglow centroid uncertainty of 0.14 ′′ for a total positional uncertainty on the afterglow of 0.26 ′′ .We detect one source within 0.78 ′′ of the afterglow position in the HST image, though this source has a calculated P cc = 0.12.We identify only one other source within 5 ′′ of the afterglow position: the bright source in the upper left corner in Figure 1.We measure an apparent magnitude of m F 110W = 23.6573mag and a P cc = 0.24 for this source.Because of the bright magnitude and higher P cc , we elect to identify the first source as the host galaxy of GRB 071025.We report an apparent magnitude of m F 110W = 26.06 ± 0.10 mag for this galaxy.
A.12. 090516A GRB 090516A has a spectroscopic afterglow redshift of z = 4.111 as reported in de Ugarte Postigo et al. (2012).We identify the afterglow in VLT/FORS2 R-band imaging from 2009-05-17 and align this imaging to the HST image of the field of the GRB.We report an astrometric alignment uncertainty of 0.022 ′′ and a centroid positional uncertainty of 0.0058 ′′ for a total positional uncertainty on the afterglow of 0.023 ′′ .This position and its 3σ uncertainty region is directly on a galaxy in the HST image.We calculate a P cc = 0.08 for this source and identity it as the host galaxy of GRB 090516A.We report an apparent magnitude of m F 110W = 25.04 ± 0.07 for this galaxy.This source was also identified by Greiner et al. (2015) and has a reported M U V = −20.99± 0.4 mag, which is consistent with our absolute magnitude of M U V = −21.24± 0.07 mag.et al. 2014) to our HST image and report an uncertainty on this astrometric alignment of 0.35 ′′ .We measure an uncertainty of 0.045 ′′ on the centroid of the afterglow with Source Extractor for a total positional uncertainty on the afterglow of 0.349 ′′ .We detect a source in the HST image within the 3σ uncertainty region.We calculate a P cc = 0.21.While this is above our threshold of P cc = 0.1, we identify this source as the host of this GRB because the source is close to the center of the uncertainty region.We report an apparent magnitude for this galaxy of m F 110W = 26.14 ± 0.09 mag.

Figure 1 .
Figure1.IR imaging from WFC3 of the 23 host galaxies.Each box is 5 ′′ wide.The afterglow position is shown in red with either the 3σ-radius or with the Swift-XRT radius, while the host (when detected) is identified by a cyan 0.5 ′′ -radius region.Positions of both the afterglow and host are reported in Tables1 and 2, respectively.North is up and East is to the left.

Figure 2 .
Figure 2. Top: Correlations and marginalized posterior densities for α and M * in the SFR-weighted UV luminosity function for the L-independent (pink) and O11 (blue) extinction corrections.The indigo lines show the α and M * parameters from Bouwens et al. (2021), while the fuchsia lines show the same parameters for Finkelstein et al. (2015).These parameters are also shown with their uncertainty as crosses in the center panels.1σ and 2σ contours are shown in the 2D histograms.Bottom: The observed cumulative distribution functions (CDFs) of and best fits to the UV luminosity function for GRB host galaxies at z ∼ 5.As indicated in the legend, the model shown in pink assumes a L-independent extinction correction and the model in blue assumes the O11 extinction correction.The black line is the observed GRB host galaxy CDF.The same 10 random draws from the modeled data are shown in silver.

Figure 3 .
Figure 3. Left: Observed cumulative distribution function (CDF) of the UV luminosity function for GRB host galaxies at z ∼ 5 [black] compared to the z ∼ 5 CDFs of the SFR-weighted Lyman-break galaxies from Bouwens et al. (2021) with the luminosity-independent galaxy extinction correction in pink and with the empirical luminosity-dependent galaxy extinction from Bouwens et al. (2015) and Overzier et al. (2011) (O11) in dark blue and the empirical luminosity-dependent galaxy extinction from Bouwens et al. (2015) and Meurer et al. (1999) (M99) in light blue.Uncertainties on the Lyman-break galaxy relations are shown as shaded regions and represent 1σ uncertainty on the luminosity function parameters [light-pink and light-blues].Uncertainty on the GRB host galaxy LF is shown with 10 random pulls of the modeled GRB host galaxy magnitudes [silver].Right: A similar plot to that on the left, with the same GRB host galaxy LF, but using as comparison now the z ∼ 5 SFR-weighted UV LF fromFinkelstein et al. (2015).The pink luminosity function again shows an assumption where galaxy extinction is luminosity independent, while the luminosity function in dark blue (light blue) assumes the O11 (M99) extinction correction.

Figure 4 .
Figure 4. Our z ∼ 5 GRB host galaxy LF as described in Figure 3 and predicted GRB host LFs at z = 4.75 fromTrenti et al. (2015).Described in detail in Section 3.3, the p parameter is tied to the influence of metallicity on the GRB progenitor path.Across all redshifts, when p = 0, there is a metallicity bias where GRBs cannot be produced in environments with Z > Z⊙, and when p tends to ∞, GRB creation is metallicity insensitive.InTrenti et al. 2015, they report Schechter function LF parameters for four choices of p, which we plot here.Results from log-rank tests between the black, median LF for our GRB host sample and the four metallicity-biased LFs are shown in Table5.

Figure 5 .
Figure5.Galaxy size vs. absolute UV magnitude relation for Lyman-break and GRB host galaxies.GRB host measurements are shown as black points and are published in Tables2 and 6.The outlier point on the bottom left is that of the host galaxy of GRB 050814.Relations for Lyman-break galaxies at z = 6 − 8 [magenta] and at z = 4 [teal] are fromBouwens et al. (2022b).The shaded regions show the the one sigma scatter of each relation.

Table 1 .
List of GRBs in our sample and their afterglow localizations.From left to right: GRB name, position and uncertainty of the afterglow (as measured from afterglow imaging or reported in the literature), redshift of the afterglow, filter of the afterglow imaging, and references for the afterglow images (or reported position).Afterglow redshift citations are in the Appendix (A). *

Table 2 .
(Schlafly & Finkbeiner 2011)tections.From left to right: name of the GRB, host centroid position in ICRS, apparent magnitude of the host galaxy as reported from Source Extractor, Galactic extinction from NED(Schlafly & Finkbeiner 2011), and the absolute UV magnitude of the host galaxy as converted using the methods described in Section 2. The uncertainty on the absolute magnitude also accounts for that in redshift. *

Table 3 .
(Schlafly & Finkbeiner 2011)on-Detections.From left to right, apparent magnitudes (as 3σ above sky and encircled energy corrected), Galactic extinction from NED(Schlafly & Finkbeiner 2011), and extinction-corrected absolute magnitudes as converted using methods described in Section 2. When applicable, redshift uncertainty was propagated, and the brighter limit was chosen.

Table 6 .
Host Galaxy Size.From left to right the columns are the name of the GRB, the half-light radius (R ef f ) in pixels, and R ef f in pc.When applicable, redshift uncertainty was propagated.