A Fast-cadenced Search for Gamma-Ray Burst Orphan Afterglows with the Deeper, Wider, Faster Programme

The relativistic outflows that produce Long GRBs (LGRBs) can be described by a structured jet model where prompt $\gamma$-ray emission is restricted to a narrow region in the jet's core. Viewing the jet off-axis from the core, a population of afterglows without an associated GRB detection can be predicted. In this work, we conduct an archival search for these `orphan' afterglows (OAs) with minute-cadence, deep ($g\sim23$) data from the Dark Energy Camera (DECam) taken as part of the Deeper, Wider, Faster programme (DWF). We introduce a method to select fast-evolving OA candidates within DWF data that comprises a machine learning model, based on a realistic synthetic population of OAs. Using this classifier, we recover 51 OA candidates. Of these candidates, 42 are likely flare events from M-class stars. The remaining nine possess quiescent, coincident sources in archival data with angular profiles consistent with a star and are inconsistent with the expected population of LGRB host galaxies. We therefore conclude that these are likely Galactic events. We calculate an upper limit on the rate of OAs down to $g<22$ AB mag of 7.46\,deg$^{-2}$yr$^{-1}$ using our criteria and constrain possible jet structures. We also place an upper limit of the characteristic angle between the $\gamma$-ray emitting region and the jet's half opening angle. For a smooth power-law and a power-law with core jet model respectively, these values are $58.3^{\circ}$ and $56.6^{\circ}$, for a power-law index of 0.8 and $75.3^{\circ}$ and $76.8^{\circ}$ for a power-law index of 1.2.


INTRODUCTION
Gamma-ray Bursts (GRBs) are understood to occur as a result of a relativistic outflow of material (or jet) projected from a central engine.Understanding the GRB population requires measuring properties such as their intrinsic energy release, event rates and circumburst medium densities.These properties are degenerate with the jet collimation (Fong et al. 2015).Additionally, constraints on jet geometry also allows us to probe the jet launching mechanism and its interaction with the surrounding medium (Salafia & Ghirlanda 2022).
The traditional, 'top-hat', model for relativistic outflows comprises a jet that is constant in Lorentz factor (Γ) and energy throughout the jet's collimation angle and sharply goes to zero at its edges (Panaitescu & Mészáros 1999;Rhoads 1999;Sari et al. 1999).More complex angular jet profiles were initially proposed to explain the variety of observed energies associated with GRBs, arguing that they possessed a standard energy reservoir (Zhang & Mészáros 2002;Rossi et al. 2002;Ghirlanda et al. 2004).Numerical simulations have provided a mechanism for how these jets would arise.They show that mixing between the jetted material and the cocoon creates an interface layer which results in an angular jet profile exhibiting a gradual decay in energy at the edges (Gottlieb et al. 2020(Gottlieb et al. , 2021;;Salafia & Ghirlanda 2022).
The extremely bright GRB 221009A showed evidence of a structured jet (Williams et al. 2023;Lesage et al. 2023;Frederiks et al. 2023;Kann et al. 2023; LHAASO Collaboration et al. 2023).The shallow evolution of the post-break X-ray afterglow can be explained by emitting material outside the core of the jet.The collimation angle of the core was determined to be many times smaller than angle swept out by the entire jet half-opening angle.Additionally, an unusually small viewing angle was determined for GRB 221009A (LHAASO Collaboration et al. 2023).The small viewing angle and large jet half-opening angle presents a contradiction in the rareity of events similar to GRB 221009A, as one would expect a comparatively large population of similar events observed at larger viewing angles.One possible way of explaining this contradiction is by restricting prompt -ray emission to the core of the jet (O'Connor et al. 2023).
With a viewing angle outside the -ray emission angle, an observer would be unable to detect a GRB.An afterglow, however, would be detectable.These are referred to as 'orphan' afterglows (OAs) (Nakar et al. 2002;Dalal et al. 2002;Huang et al. 2002;Rhoads 2003).If a large fraction of the GRB population posses a shallow jet structure, similar to GRB 221009A, with -ray emission restricted to a small region in the jet's core, we would expect to see a significant population of OAs (O'Connor et al. 2023).Gill & Granot (2023) found that the multi-wavelength observations of GRB 221009A were consistent with a shallow jet structure, characteristic of a weakly magnetised jet.With such a structure, a prominent jet-break should not be observed at late times.It is rare that GRB afterglows are observable long enough for the jet-break to be observed, and among the afterglows with late-time detections, there is a sub-population that do not exhibit a jet-break.This could be a result of a shallow angular jet profile similar to GRB 221009A (O'Connor et al. 2023).Beniamini & Nakar (2019) provide constraints on jet structure using the observed population of prompt GRBs and their counterparts.These constraints, however, break down in a shallow structured jet scenario where prompt -ray emission is restricted to the core, similar to that observed with GRB 221009A.Searches for OAs, therefore, provide an avenue to further constrain GRB jet structure by observing GRBs at viewing angles larger than where prompt -ray emission is detectable (Nakar & Piran 2003).
For this work, we only consider OAs emitted from misaligned structured jets.However, there are a number of scenarios in which OAs are produced.One of which occurs when observing a GRB entirely off-axis from the jet.These are difficult to observe originating from collapsars in optical wavelengths as they are expected to be substantially fainter than the accompanying supernova emission (Kathirgamaraju et al. 2016).
OAs can also emitted from a collapsar if the jet is initially loaded with baryons.The Lorentz factor of such a jet is insufficient to produce a GRB, but peaks at lower energies, resulting in an OA which follows a similar evolution to an on-axis afterglow (Huang et al. 2002).These are commonly referred to as 'dirty fireballs' or 'failed GRBs.' Authors Totani & Panaitescu (2002), Rau et al. (2006) and Ho et al. (2022) conducted searches for OAs in this scenario.
These searches, along with other surveys, have yielded a number of suspected OAs.The earliest discovery of such an event achieved the the Palomar Transient Factory (Law et al. 2009) was PTF11agg (Cenko et al. 2013).In recent years, the Zwicky Transient Facility (Bellm et al. 2019) has dominated in this effort with events such as AT 2019pim (Ho et al. 2022;Perley et al. 2024), AT 2020blt (Ho et al. 2020), AT 2021any (Andreoni et al. 2021) and AT 2021lfa (Ho et al. 2022).
Once an afterglow detection is made without an associated GRB, it becomes important to rule out the existence of prompt -ray emission that was simply not detected by GRB monitors like Swift Burst Alert Telescope (BAT) and Fermi GBM (Barthelmy et al. 2005;Narayana Bhat et al. 2016).The best constraints to-date on GRB emission accompanying an OA candidate was AT2019pim, reported by Perley et al. (2024).They find that prompt, -ray emission accompanying AT2019pim is disfavoured.This presents evidence for the existence of a population of OAs, highlighting the opportunity for the discovery of further OAs.However, OAs observed on-axis are fast evolving and can be observable for mere minutes (Greiner et al. 2008).Therefore, existing surveys with day cadences may be unable to detect a significant fraction of the OA population.The highest cadence data used for an OA search was in Ho et al. (2022) with up to three visits per night.No OA search to-date has been conducted with a cadence on minute timescales.
The Deeper, Wider, Faster Programme (DWF) involves observations with the Dark Energy Camera (DECam) mounted on the CTIO 4m telescope.DWF's observing strategy involves minute cadence observations while reaching deeper ( ∼ 23) than other transient surveys such as ZTF (∼ 21).DWF utilises near real-time data reduction and analysis, designed to identify transients for rapid, spectroscopic follow-up (e.g, Andreoni & Cooke 2017).Although much of the data has been analysed in real-time, conducting late-time analysis allows for a comprehensive search with rigorous rate constraints.This work comprises a search for OAs in 100 nights of archival DECam data across 18 fields, each one covering ∼ 2.1 deg 2 of effective sky area.With this search, we constrain GRB jet structure.Other prominent transient surveys lack either the cadence or depth necessary to probe the population of OAs occurring from misaligned structured jets.
The DWF dataset has been previously used for other works studying minute timescale transients.For example, Andreoni et al. (2020) searched for extragalactic fast transients broadly and provided rate constraints but only analysed 25 nights across five fields, about a quarter of the data that is available as of the writing of this work.Webb et al. (2021) searched a similarly large subset of the data but was targeted towards stellar flares within 500 pc.No work to-date has conducted a search tuned specifically for GRB afterglows on all of the applicable data.This paper is organised as follows: In Section 2, we describe the DWF DECam data used for this work.Section 3 outlines our synthetic population of OAs, used for our search methodology, efficiency calculations and rate estimates.We then describe the machine learning algorithm used to identify OAs in the data and their expected rates in Section 4. In Section 5, we analyse the OA candidates found in the data.A discussion on the implications of a non-detection in the data is detailed in Section 6.This includes constraints on GRB jet structure and prospects for future work.We then conclude in Section 7.

THE DEEPER WIDER FASTER PROGRAMME
To-date, there have been 13 DWF coordinated operational runs (which we denote O1 through O13) spanning from December 2015 to January 2024.Each of these runs lasted for six nights, observing 3-5 fields.Typically, these fields are observed by taking continuous minute-cadenced imaging, 20 s exposures with ∼30 s readout and CCD clear, without dithering and for 1-3 hr per night.We call these 'observing windows.'Our classifier, described in Section 4.1, was designed to ingest well sampled light curves, which comprise the majority of the DWF data, and its efficiency drops for light curves with a small number of exposures.As a result, we exclude observing windows where less than 20 exposures (< 20 min) were taken.We also exclude four DWF observing runs; O8, O9, O11 and O13.This work focuses on data taken with DECam and DWF O8 and O11 used Subaru Hyper Suprime-Cam and KMTNet, respectively.DWF O9 experimented in using an alternate strategy with dithered exposures and a larger number of fields tailored to detect kilonovae and had fewer exposures in each field per night.DWF O13 primarily observed the Large Magellanic Cloud, with crowded fields, less well suited for the discovery of extragalactic transients.Finally, there were two DWF pilot runs that employed a dithering strategy, not compatible with the photometry pipeline used in this search.We use only data from the remaining nine DWF runs (see table 1) which comprise 9033 images and 145 hours of observing time.

Photometric Pipeline
For this search, we use a data processing pipeline that takes in DECam images, calibrated with the NOIRLab community pipeline (Valdes et al. 2014), and outputs light curves for each source in the field.The full details of this pipeline will be presented in a future publication (Freeburn et al. in prep.).
The source extraction software, sextractor is utilised for our photometric pipeline (Bertin & Arnouts 1996).We run adaptive aperture photometry (known as 'MAG_AUTO') in double image mode.Double image mode requires a separate, detection image to identify sources.It then measures photometry, in the science image, at the location of the sources found in the detection image.
The image with the largest value of the full width at half maximum (FWHM) is chosen as the detection image for a given observing window.This corresponds to the image with the most unfavourable seeing conditions and typically shallowest depth.An initial sextractor run is conducted on the science image that comprises a given observing window.Any new sources found during this run that are not present in the detection image, are then injected into the detection image.
We then match the PSF of the detection image to the rest of the observing window's images with hotpants, using Gaussian convolution.Light curves are then obtained by conducting a second sextractor run on the convolved images, using the detection image.The instrumental photometry is then calibrated with photometric catalogs from SkyMapper (Onken et al. 2019) or Pan-STARRS (Kaiser et al. 2010).

A SYNTHETIC POPULATION OF ORPHAN AFTERGLOWS
We consider GRB jets to be described by three collimation angles shown in Figure 1.The angle of the core,   describes the inner region of the jet that is approximately constant in Γ and energy.We restrict the angle at which a GRB is produced to   .  defines the entire angular extent of the jet.The angle at which we view the jet is denoted by   .Viewing the jet outside the core,   >   , will result in an observable OA.With a viewing angle outside the jet's half-opening angle,   >   , OA detection becomes very difficult, as explained in Section 1.In this work, we explore only the scenario where we observe the jet inside its half-opening angle but outside the core,   >   >   .This provides a more luminous and fastevolving population of OAs (Kumar & Granot 2003;Beniamini et al. 2020Beniamini et al. , 2022)).GRB afterglow light curves vary significantly in their evolution.The rise time, peak luminosity, fade rate and the time of the jet break all depend on a number of free parameters.In order to accurately calculate the rates of orphan afterglows we expect to detect for a given jet structure and effectively identify them, it is necessary to use a representative synthetic dataset of afterglows tailored to the DWF data.
In Section 3.1, we describe how we generate a synthetic population of GRBs.For this population, we then model their corresponding afterglows in Section 3.2.In Section 3.3, we inject these afterglows into DWF images.

GRB Population Synthesis
The Swift BAT6 complete sample comprises Swift/BAT LGRB detections with fluxes > 2.6 ph cm −2 s −1 .These LGRBs have a 90 % completeness in  and possess well-defined detection rate in Swift BAT's half-coded region of R Swift ∼ 15 events sr −1 yr −1 (Salvaterra et al. 2012).Ghirlanda et al. (2013a, G13) use the BAT6 sample to generate a synthetic population that reproduces the properties and rates of observed LGRBs.We use the results from G13, to produce a population of LGRBs that are representative of observed LGRB and afterglow fluxes and rates.
G13 assume a standard rest frame energy reservoir of 1.5×10 48 erg and peak energy of 1.5 keV with a GRB formation rate based on Li (2008) and Hopkins & Beacom (2006).T90 values are generated in a log-normal distribution centred at 27.5 s with a dispersion of 0.35, truncated at 2 s.We assume an isotropic distribution of viewing angles which corresponds to a probability density that scales with sin   .Using this formalism, distributions of the jet half-opening angle,   and the initial Lorentz factor of the jet, Γ 0 were fit to the Swift BAT6 complete sample.The relation in Equation 1 was derived using this methodology.The isotropic equivalent energy release,  iso can then be calculated from Γ 0 and   using where   is the total energy release in the observer frame and The resultant synthetic population successfully reproduces the distribution of fluences and  peak values of Swift BAT6 LGRBs.Additionally, modelling the afterglows of these bursts in R-band reproduce the observed flux distribution of Swift BAT6 LGRB afterglows 11hrs post burst as shown in Ghirlanda et al. (2015).

Generating a Sample of Synthetic Afterglows
Afterglows are characterised by a further four parameters: The fraction of the forward shock's thermal energy in electrons and the magnetic field are described by   and   respectively, the electron distribution power-law index,  and the circumburst number density, .These parameters are currently poorly constrained due to degeneracy in predicting afterglow light curves.Ghirlanda et al. (2013bGhirlanda et al. ( , 2015) ) used the same synthetic population to reproduce observed properties of afterglows in radio, optical and X-ray wavelengths.We adopt the same ensemble of parameters in those works (see Table 2), as they are consistent with observed GRB afterglows and are values derived from first-principles simulations of particle acceleration in relativistic shocks (Sironi & Spitkovsky 2011).Ghirlanda et al. (2015) also show that the population synthesis models are able to recover the distribution of observed afterglow Rband fluxes for the Swift BAT6 complete sample.
We use afterglowpy, a python package for generating afterglow light curves (Ryan et al. 2020).Power-law jet models have been found to be consistent with hydrodynamical simulations (Gottlieb et al. 2021) and observations of GRB 221009A's afterglow (Gill & Granot 2023).We therefore investigate both a smooth power-law jet and a power-law with a uniform core in this work.The energy distributions for these jet models are given in Equations 3 and 4  2023).In this model, -ray emission is beamed at an angle of   .The energy of the jet decays as a power-law out to an angle of   according to Eq. 3 and Eq. 4. OAs are therefore detectable with viewing angles, that satisfy   <   <   .respectively: Where the model is normalised with  0 ,  is the angle from the centre of the jet,   is the characteristic width of the distribution and  is the power-law index.We set  0 from Equations 3 and 4 to  iso from Equation 2.
Eq. 3 applies to the jet out to an angle of   , where  () drops to zero.We assume that, for angles larger than   , prompt -ray emission is not detectable.G13 assumed that prompt -ray emission is produced for   <   .Therefore, we take the   values generated in our population synthesis model to be   for generating afterglows.We adopt a uniform distribution of   <   < 90 • .
We draw values  uniformly such that 0 <  < 3, as with the data used for this work, we are insensitive to OAs originating from structured jets with indices larger than  = 3.This is highlighted in Section 4.2.Additionally, simulations favour angular jet structures with 0.7 ⪅  ⪅ 2.8 (Gottlieb et al. 2020(Gottlieb et al. , 2021) ) and GRB 221009A has a measured value of  = 0.8, well within the range of  explored in this work (Gill & Granot 2023).
Each combination of  and   −   can be considered a distinct jet structure, for which an OA rate can be calculated.
We note that, by default, afterglowpy does not model the jet's deceleration phase, which affects the afterglow's rise.It also means that Γ 0 is not taken into account directly (Ryan et al. 2020).However, due to afterglowpy's flexibility and low computational cost, we find that it is optimal for this work.Low Γ 0 events may have an early evolution that departs from our modelling.This may slightly affect our total efficiency in finding these events but we leave this to future work.

Injecting Synthetic Afterglows into DWF Images
Realistic model afterglow light curves, as they would be detected with DWF, are important for both training data and the calculation of detection and classification efficiencies.We inject a population of point sources in consecutive images forming light curves (fakes) from the synthetic sample of afterglows described in Section 3.2 into a representative subset of the DWF images.Our photometry pipeline, described in Section 2.1 is then run on these images to recover the injected source light curves.
The fake sources are modelled using a Moffat profile (Moffat 1969) with a FWHM matching the exposure and stellar point sources in the charge-coupled device (CCD) image in which the fakes are injected.Injected sources are initially calibrated to each image by injecting and recovering five sources with varying instrumental flux values.We present an example of one of these injected afterglows in Figure 2. Figure 3 shows the difference in injected versus recovered flux with magnitude.We see broad agreement between injected and recovered flux up to a magnitude of  ∼ 22.
We inject the fakes both randomly throughout the field and coincident with visible galaxies.Krühler et al. (2015) analyse a sample of LGRB host galaxies from The Optically Unbiased GRB Host (TOUGH) survey (Schulze et al. 2015), BAT6, the Gamma-Ray Burst Optical and Near-Infrared Detector (GROND) (Greiner et al. 2008;Krühler et al. 2011) and the Swift Gamma-Ray Burst Host Galaxy Legacy Survey (SHOALS) (Perley et al. 2016).The vast majority of LGRB hosts in this sample are fainter than 23 AB magnitude in -band.Using our synthetic population of afterglows, we find that DWF's depth and cadence results in a higher median redshift than the sample used in Krühler et al. (2015).We therefore assume that the OAs found in our search will have hosts fainter than 23 AB mag in -band and only consider the randomly distributed fakes in our efficiency calculations.The sources that were injected onto galaxies will be used as part of the training data for completeness and to prevent a bias towards transients without a visible coincident galaxy.

INVESTIGATING ORPHAN AFTERGLOW CANDIDATES
We use the synthetic dataset described in Section 3 to train a machine learning classifier that extracts OA candidates from the DWF data.This is outlined in Section 4.1.In Section 4.2, we then discuss the efficiency of this classifier in probing this theoretical population of OAs and predict how many OAs we expect to find.

Light Curve Classifier
Our photometry pipeline outputs approximately ∼ 10 5 light curves per night per field, with an average of 90 measurements in each light curve.We obtain 1.63 × 10 7 light curves total.To search this large dataset, we use a series of cuts and a machine learning classifier.
The first cut we make on this pool of light curves is based on their variability.We only consider light curves that exhibit significant variability during a single observing window.We use the von Neumann statistic (von Neumann 1941) to measure variability.Here,   describes a series of measurements, equally spaced in time.Sokolovsky et al. (2017) find  −1 is an effective variability indicator for photometric time-series data.A cut of  −1 lc > 0.6 removes 86.8 % of all DWF light curves while retaining 64.7 % of injected afterglows brighter than  = 23 AB mag.Injected afterglows that exhibit low variability, with  −1 lc < 0.6, predominantly fade below our detection threshold before the next observing window, ∼24 hours later.The lack of vital evolutionary information, in this case, would make classification almost impossible.We, therefore, remove light curves from our search if they satisfy  −1 lc < 0.6.For the light curves satisfying  −1 lc > 0.6, we use a sliding window of three detections and use the peak detection from the window with the largest median value to identify the peak,  peak .For light curves with quiescent emission, the quiescent magnitude is identified as the median of all the detections before the peak.If the peak is identified in one of the first 5 exposures of the night, we take the quiescent magnitude to be the median of the final five detections.After two consecutive detections after  peak to below the quiescent magnitude, we consider this the end of the event, denoted by  max .For light curves without quiescent emission, the event is defined as the detections between the peak and the second non-detection.
We measure the variability of the detections that comprise the event with  −1 event .Only 0.02 % of all light curves in the DWF data and 30.6 % of injected afterglows brighter than 23 AB mag satisfy a cut of  −1 event > 4. We therefore separate all light curves satisfying this cut for human inspection.
For light curves with  −1 event < 4 and  −1 lc > 0.6, we use a XGBoost binary classifier which has been shown to be robust in light-curve classification tasks (Möller et al. 2016).The training data for our classifier comprises 848 each of a curated set of injected afterglow light curves (described in Section 3.3) and a curated set of randomly selected light curves from the DWF data that do not exhibit afterglowlike variability.While afterglow light curves vary in morphology, their fade is modelled well as a power-law decay.We fit a power-law decay to all DWF light curves that satisfy  −1 lc > 0.6.We do not model the rise phase of the light curve for two reasons.First, it allows us to treat events that peak before our observing window and those that peak during in the same manner.Second, our training data, generated using afterglowpy does not accurately model the rise (see Section 3.2).
A power-law of the form shown in Equation 6 is then fit to the detections between the  peak and  max .
is the time in hours after the peak of the light curve,  is the apparent magnitude of a given detection, m and   is the median and standard deviation of the magnitudes that comprise the light curve.,, and  are the parameters we fit for.The following features comprise those used in our classifier which achieves the performance metrics in table 3: (i) , , , , best fit parameters from equation 6 (ii) ∇ event , average gradient of the event, scaled by the error.
(iii) ∇ q , average gradient of the quiescent measurements, scaled by the error.
(iv) med 5 , the median time of the five brightest detections subtracted by  peak .
(v)  event , number of detections between the  peak and  max .
(vi)  peak −  max , difference between the  peak and  max .
(vii)  event / lc , the median magnitude error of the event scaled by the median error of the light curve.
(viii)  −1 event , the variability between  peak and  max .(ix)  −1 resid , the variability of the residuals of the power-law fit.The left-hand axis corresponds to the histogram of injected afterglows, the blue denotes the total afterglows that were injected, green denotes those that had at least five detections, orange denotes those that had  −1 event > 0.6 and purple denotes those that are recovered by our classifier.The right-hand axis and black line and points show the recovery efficiency of our classifier at each magnitude bin.The total classification efficiency for events with peak magnitudes brighter than where our classification efficiency drops below 50 %, 21 AB mag, is 82.86 ± 1.32 %.Light curves that are not recovered are primarily afterglows with slower fade rates in our observations and are outside the scope of this work.The efficiency curve from this plot has been propagated into our rate calculations and jet constraints shown in Figures 6 and 10.

Expected Rates of Orphan Afterglows in DWF
We use the sample of afterglows generated with the formalism described in Section 3.2 to evaluate the expected rate of OAs extracted from the DWF data 4.1.We simulate DWF's observing strategy when generating light curves, using a distribution of minute-cadence observing windows based on the distribution of those in the data itself.The values of time length of each observing window,  window , were binned for the data used in this search to generate typical  window values for the synthetic light curves.The simulated afterglow start times, defined by the time the prompt GRB is emitted, are uniformly distributed between 24 hr before the beginning of the observing window and the end of the observing window.
We inject afterglowpy light curves into DWF images (as discussed in Section 3.3) to determine classification efficiencies as a function of the magnitude of the light curve's peak magnitude, shown in Figure 5.These values are propagated into our rate estimates.For each simulated afterglow, we predict an efficiency based on its peak magnitude by interpolating the efficiency bins shown in Figure 5.Each afterglow with a detection with  < 23 AB mag is weighted based on this predicted efficiency.
The OA detection rate varies between different values of  and   −   .For each of combination of  and   −   we can calculate a ratio of DWF-detected afterglows to Swift-detected GRBs and an absolute rate with R Swift .Figure 6 shows the resultant distribution of expected values for OA detections in the DWF data analysed in this search, according to the assumptions made in Section 3.2.Higher rates of OA detection are expected for shallower (small values of ), wider (large values of   −   ) structures outside the core of the jet.

ANALYSING ORPHAN AFTERGLOW CANDIDATES
We present the candidates found in the DWF data using our classifier (Section 4.1) in Section 5.1 and analyse the nature of their coincident sources in Section 5.2.

Extracting Candidates
The total number of light curves available in the processed data amounts to 2.6 × 10 7 .Requiring  −1 lc > 0.6 reduces the sample to 3.6 × 10 6 light curves.Of these, 45961 were extracted from the data to be inspected based on a classification score > 0.7 or  −1 event > 4. We crossmatch our sample to Gaia DR3 (Gaia Collaboration et al. 2023), the ASAS-SN catalogue of variable stars (Jayasinghe et al. 2018) and the catalog for RR Lyrae variable stars in DES Y6 (Stringer et al. 2021) to determine which sources in our sample are known variable stars.We also crossmatched to the Gaia DR3 to search for sources associated with known stars.We require a match within one arcsecond and a parallax measurement with >3 statistical significance.Table 4 shows the numbers of candidates removed from these cuts, resulting in 33,123 candidates left for human inspection.
The largest contaminant of these candidates were artefacts including edge detections, cosmic rays, CCD pixel faults, crosstalk and saturated sources (Webb et al. 2020;Goode et al. 2022).The criteria for a candidate passing visual inspection is that no artefacts were evident in the candidate thumbnails during the event.Once a candidate has passed the visual inspection, a candidate would then be rejected if there was rebrightening on subsequent nights of observation or had brightened and faded on previous nights.It would also be rejected if a source was not detected with difference imaging of the candidate with stacked science and template images using hotpants (Becker 2015).

Analysing Candidates' Coincident Sources
After this process, 51 candidates remained.All of these candidates possess coincident sources detected in DELVE's second data release (Drlica-Wagner et al. 2021).We use sextractor SPREAD_MODEL parameter, a star/galaxy classifier based on PSF models, for these candidates and plot the results in Figure 7.We find that all but four of our candidates are consistent with a single point-source.The four sources which have large SPREAD_MODEL values, uncharacteristic of point sources, are resolved as two distinct sources in Gaia DR3.Therefore, we cannot conclude that any of our 51 candidates have extended, galaxy-like hosts.
In the absence of a conclusive detection of an extragalactic host for any of the candidates, we analyse the coincident sources' colours to determine whether they are consistent with M-stars or other main sequence stars, as OA and M-dwarf flare light curves can look very similar.To achieve this, we used DELVE PSF photometry, supplemented with our measurements in the case of highly blended sources.In Figure 8, we plot the coincident sources' colours along with the observed 3- distribution of spectroscopically confirmed M-stars reported in (West et al. 2011).We find that 42 of the coincident sources have colours that are placed within the M-star regions, shown as orange points in Figure 8. Their colours are a strong indication that the candidates may be stellar flares and we classify them here as such.The light curves of the remaining nine candidates are plotted in Figure 9 and are denoted by green or blue points in Figure 8.
The three green points in Figure 8 mark coincident sources where non-detection in two or more of the filters have prevented an M-star classification test.0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 The six blue points in Figure 8 mark coincident sources that are inconsistent with a single M-star.Two of these coincident sources lie on the main-sequence region in colour-colour space.While flares from stars bluer than M-stars are comparatively rare, it is likely that some subset of the flares we observe would originate from higher mass stars (Balona 2015).The other four candidates (10, 19, 47, and 48) have conicident sources that possess colours inconsistent with a main-sequence star.Candidate 10 is resolved as two distinct sources with an angular separation of less than one arcsecond in Gaia DR3.This explanation could apply to candidates 19 and 48, with smaller angular separations.This hypothesis is supported by the fact that they have a shallower colour evolution between  and -bands but otherwise have photometry in , and -band that are consistent with the M-star population.
We also note that candidates 8, 10, 19, 21 and 26 were identified in NGC 6101 field, which has Galactic latitude close to zero compared to other fields (see Table 1).At these galactic latitudes, the stellar density is higher which favours these candidates to be Galactic events.Candidate 47's coincident source is unusual in its colour evolution compared to the rest of the candidates.It falls within the M-star region of colour-colour space in Figure 8 but varies in colour between an M0 and M5 class star.Despite being a point source, the nature of candidate 47 is unknown.
We fit the light curves in Figure 9 to afterglowpy models with varying results in Figure A1.However, we note in Appendix A and Figure A2 that these fits are not sufficient to rule out a stellar flare explanation.
In Section 3.3 we note that the host galaxies of the theoretical population of OAs explored in this work are expected to predominantly exhibit apparent magnitudes fainter than 23 AB mag.In Figure 7 we see that none of our candidates satisfy this criteria.Thus, despite a coincident source having colours that are unexpected for a stellar flare, they are inconsistent with the expected properties of LGRB host galaxies.

Constraints on Jet Structure
For all of the candidates found in this search, we find that all have coincident sources that are consistent with a point source (Figure 7).A subset do not possess colours that are consistent with the M-star population and cannot be confidently associated with a stellar flare.However, there is a lack of evidence for any of them being associated with an extragalactic host.We therefore conclude that we have not found any OAs using the procedure in this work.
We calculate the rate at which we would expect to detect a single OA in the data searched using: Where R OA is the OA rate, the average sky coverage of a single pointing is Ω = 2.14 deg 2 , the efficiency for OAs with a peak detection brighter than  = 22, drawn from a uniform distribution of 0 <  < 3, is  OA = 0.68.Since we distribute the the burst times of our synthetic afterglows up to one day before the start of each observing window, the effects of OA burst times on their detectability is absorbed into the efficiency.We therefore consider the length of time searched to be one day per field per night,  search = 100 d.As a result, with no convincing OA detected in this work, we place an upper limit on the rate of OAs to  < 22 AB mag of R OA < 7.46 deg −2 yr −1 at the 95 % confidence level and R OA < 2.49 deg −2 yr −1 at 63.2 % confidence.
This is a novel search, probing an unexplored parameter space.It is therefore difficult to make a direct comparison to previous work.The DELVE sample is coloured with their the mean of the CLASS_STAR parameter across , ,  and -bands.CLASS_STAR is the output to a star/galaxy classifier that is run with sextractor where a source that appears star-like is near a value of 1.0 and galaxies typically range from 0.0 to ≳ 0.9.The candidates' coincident sources are plotted with the same colours as in Figure 8.They are plotted as stars if they are detected as two distinct sources in, within one arcsecond, in Gaia DR3.Two blended point sources within one arcsecond of each other would be detected as a single extended source with typical atmospheric seeing conditions.We therefore conclude that all of the coincident sources are amongst the distribution of stars in the DELVE sample.
Andreoni et al. (2020) placed a similar upper limit for extragalactic fast transients of 1.63 deg −2 d −1 .In this work, however, the authors restricted their search to transients rising and fading within a single observing window.As our search is sensitive to OAs with burst times up to one day before each observing window, the rate constraints in this work and Andreoni et al. (2020) are not directly comparable.Previous rate constraints on OAs, such as Ho et al. (2022) place the OA rate from dirty fireballs to be not significantly larger than the LGRB population.However, the work here probes down to minutetimescales and assumes a luminosity function and light curves from OAs originating from misaligned structured jets rather than dirty fireballs.Thus, we consider our upper limit on R OA to be independent of previous work.We show the predicted number of OAs with respect to the jet parameters in Figure 6.We expect more than 1 OA in the DWF if the jet has low power-law index, , and the difference between the angular extent of the wings and the core is large.If GRB 221009Alike events occurred in the DWF data, with large values of   , we would have expected at least one detection as Gill & Granot (2023) measure  = 0.8 for GRB 221009A.Thus, our non-detection of OA constrains the possible angles and power-law indices.Assuming a constant,  = 0.8, we place upper limits on   −   of 58.3 • and 56.6 • for smooth power-law and power-law with core jet models respectively.For a steeper angular profile,  = 1.2, we find upper limits of   −  of 75.3 • and 76.8 • for smooth power-law and powerlaw with core jet models respectively.These values are calculated for an expected value of 1 OA in the data, corresponding to a confidence of ∼ 63.2 %.
Using the expected number of afterglows in the data, shown in Figure 6, we can use Poisson statistics to calculate the probability of our non-detection with a given values of  and   −   (see Figure 10).We find that the non-detection of an OA in this paper disfavours shallow angular jet profiles with a large angular extent outside the -ray emitting region.Our results favour a scenario where the wings of the jet are small or steeply drop-off in energy outside the -ray emitting region.This is consistent with the results of hydrodynamical simulations of LGRB jets which predict  > 1 for most of the LGRB population (Gottlieb et al. 2021).

Prospects for Detection with Other Current and Future Surveys
In Figure 11, we see the importance of cadence with searches for OAs.We assume the same sky coverage and depth for each cadence to enable a direct comparison.Generally, a lower cadence allows for deeper observations with more sky coverage, which maximises the likelihood of achieving a single detection.However, to understand and classify a light curve, more detections are required.This is highlighted in Figure 11; a high cadence can also significantly boost OA detection rate by probing the OA population deeper.DWF's ∼ 50 s cadence, therefore, has a high detection rate per night and square degree observed, making uniquely positioned amongst transient surveys to search for OAs.The Vera C. Rubin Observatory's (Rubin) Legacy Survey of Space and Time has an unprecedented combination of depth and a large field-of-view that make it an extremely powerful facility for discovering transients.However, with a typical cadence of three days, Rubin will be inefficient for a study similar to this one.At this cadence, the survey will be sensitive to only the brightest and slowest evolving afterglows, detecting a small fraction of those detectable at high cadence, as shown in Figure 11.Fink, a broker for Rubin/LSST, provides alerts and classification in real-time which promises to provide the capability for fast cadenced follow-up to alerts (Möller et al. 2021).Supplementing Rubin alerts with other facilities to achieve a faster cadence could provide a promising avenue for OA detection.
The Transiting Exoplanet Survey Satellite (TESS) possesses an observing strategy similar to DWF, well suited towards searching for OAs.Since 2022, TESS has adopted a 200 second cadence, observing a given sector continuously for 27 days at a time.While TESS's typical 5- limiting magnitude of 16 AB mag is substantially shallower than the depths that Rubin and DECam are capable of, its 2300 deg 2 field-of-view, cadence and temporal coverage make it a promising instrument for transient detection (Ricker et al. 2015).TESS imaging has a comparatively large pixel scale at 21 arcseconds compared to DECam's 0.27 arcseconds.This will present challenges in OA searches, particularly in identifying host galaxies and disentangling them from M-stars.
Difficulties with TESS background subtraction for conducting image subtraction analysis has prevented comprehensive searches for extragalactic fast transients like OAs.However, recently TESSreduce has made searches like this possible (Ridden-Harper et al. 2021) and a number of optical afterglows coincident with GRB triggers, serendipitously detected by TESS, have been identified (Roxburgh et al. 2024;Jayaraman et al. 2023).
Evryscope (Law et al. 2015) is a ground based facility which utilises a similar, high cadence, observing strategy.With a depth of  ∼ 16. it continuously observes an 18400 square degrees at a two minute cadence for six hours per night.While its sky and temporal coverage fall short of TESS, it is still a powerful facility for searching Figure 8. Colour-colour diagrams of the coincident sources, from the DELVE catalog, associated with the OA candidates found in this work.The orange shaded regions are the expected distributions of M-star classes 1-5 (West et al. 2011).The grey regions denote the main-sequence from DES colour transformations (Abbott et al. 2018) to the spectral flux library described in Pickles (1998).An OA candidate coincident source falling in this region indicates that the observed transient is likely a stellar flare.Orange points are candidates that have a coincident source consistent with a given M-star class to within 3-, Green points are OA candidates without enough coincident source colour information to make a determination of their nature and blue points are OA candidates with coincident source colours inconsistent with M-star with 3- confidence.
for OAs and its smaller comparable pixel scale of 13 arcseconds will allow for more effective identification of OA host galaxies.
In the structured jet regime, the OAs luminosity function shifts to fainter peak luminosities as  becomes large.Due to TESS and Evryscope's comparative shallowness, they are sensitive to probing an OA population with shallower angular jet structure.It is useful, therefore, to use a two-pronged approach when probing the OA population, with both large field-of-view, shallow surveys like TESS and Evryscope and deep surveys with smaller fields-of-view like Rubin.

CONCLUSIONS
Orphan afterglows (OAs) provide a powerful probe into the geometry of the relativistic outflows that give rise to long-duration gamma-ray bursts (LGRBs).Understanding LGRB jet geometry will help constrain the true, beaming corrected LGRB rate and energy release.Recent observations, such as the follow-up to GRB 221009A (O'Connor et al. 2023;Williams et al. 2023), have supported the possibility of a structured jet where -ray emission is restricted to the jet's core, with OAs detectable at wider viewing angles.
In this work, we conduct a search for OAs in 100 nights of observations from The Deeper, Wider, Faster programme (DWF).DWF's deep ( ∼ 23), minute-cadence observations provide a unique opportunity to probe this theoretical population of OAs.We use a machine learning classifier, trained on afterglow models generated with afterglowpy to extract OA candidates in the archival DWF data.
We found 51 OA candidates.Of these 42 were found to likely originate from M-stars, suggesting they are Galactic, stellar flares.While their nature is not obvious, the other nine candidates are likely Galactic transients.
We find no strong OA candidates with, or without, apparent host galaxies, in 100 nights of the DWF data, comprising 9033 images and 145 hours of observing time.We measure an upper limit on the rate of OAs  < 22 AB mag of 7.46 deg −2 yr −1 (95 % confidence).We also place constraints on GRB jet structure with a structured jet where prompt, -ray emission is restricted to the jet's core.Setting the power-law index of the structured jet,  = 0.8 we measure upper limits on the difference between half opening angle of the -ray emitting core and the half opening angle of the jet   −   .These values are 58.3 • and 56.6 • (75.3 • and 76.8 • for  = 1.2) for smooth power-law and power-law with core jet models respectively with ∼ 63.2 % confidence.We encourage further searches for OAs with other fast-cadenced, wide-field surveys such as TESS to better constrain this parameter space.
The unique deep, fast-cadenced data from DWF allows for a search for OAs at their fastest timescales.We present the first observational constraints on GRB jet structures with a search for OAs.This work highlights the importance of untargeted, multi-wavelength searches in understanding LGRB jet structure.These efforts not only reveal OA rates but can also provide insights into the jet launching mechanism and the intrinsic properties of LGRBs.
0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 This research made use of matplotlib, a Python library for publication quality graphics (Hunter 2007), SciPy (Virtanen et al. 2020) and Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration et al. 2013Collaboration et al. , 2018)).We also used XGBoost and scikit-learn (Pedregosa et al. 2011).We have simulated the same number of events for each cadence, distributed between a day before the first observation and the last observation.The OAs were simulated with a jet structure of  = 0.8 and   = 57 • .We also require three observations where an OA is  < 23 AB mag before an event is detected.We find that at a day cadence, 5.5 % of the OAs are found compared to minute cadence.At a day cadence, we find that the median intrinsic peak magnitude for detected OAs are required to be 4.7 magnitudes brighter.
1.7 +0.3 −0.3 models at early times.In addition, the reverse-shock emission may result in departures from a typical, forward shock afterglow light curve (Sari & Piran 1999;Racusin et al. 2008;Vestrand et al. 2014;Oganesyan et al. 2023).We therefore, cannot rule out any of our candidates purely based off a poor fit.We also fit six of the candidates with M-star colour evolution in Figure A2, which are likely to be stellar flares.We also find qualitative agreement in some of these fits, particularly candidates 13 and 31.We conclude from this that stellar flares may have an evolution consistent with a GRB afterglow in -band, minutes to hours post-burst.
When using fast-cadenced imaging from a single filter, fitting light curves with currently available models like afterglowpy, is not effective in rejecting contaminants.A poor fit does not necessarily rule out a GRB afterglow due to limitations of models minutes to hours post-burst.In addition, a good fit does not rule out a stellar flare as a stellar flare light curve may provide good fits to afterglow models in a single filter.
The parameter estimations from our fits are shown in Figures A1 and A2 as well as Table A1.We find that the values of  0 and  extracted from the candidates are within the expected range for LGRBs (Cenko et al. 2011).
Simultaneous coverage in multiple optical filters may provide an avenue to more effectively use models to disentangle OAs from contaminants like stellar flares.GROND utilizes a similar approach when following up GRB triggers with simultaneous imaging in seven filters (Greiner et al. 2008).A1 but with a selection of six candidates that had coincident sources possessing colours consistent with an M-star.These light curves.therefore, likely originate from stellar flares.Despite this, our fits yield good results in some of these light curves.

Figure 1 .
Figure 1.Diagram of the structured jet model considered in this work, adapted from O'Connor et al. (2023).In this model, -ray emission is beamed at an angle of   .The energy of the jet decays as a power-law out to an angle of   according to Eq. 3 and Eq. 4. OAs are therefore detectable with viewing angles, that satisfy   <   <   .

Figure 2 .Figure 3 .
Figure 2. Example of an afterglow model injected into DWF images.The left-hand side shows cutouts at the location of the injected afterglow (centred in the images).The right-hand side shows the resultant light curve from the pipeline described in Section 2.1.The blue points are the extracted photometry of the injected source and the green points show the injected magnitude of the afterglow.The parameters for this afterglow are   = 72.8• ,   = 79.1 • ,   = 12.0 • ,  = 0.30,  = 2.95,  = 1.8 × 10 2 cm −3 ,  iso = 2.73 × 10 51 .

Figure 4 .
Figure 4. Steps involved in extracting afterglow light curves in the DWF data.

Figure 5 .
Figure5.Classification efficiency of injected afterglows with magnitude and the distribution of a sample of injected afterglows.The left-hand axis corresponds to the histogram of injected afterglows, the blue denotes the total afterglows that were injected, green denotes those that had at least five detections, orange denotes those that had  −1 event > 0.6 and purple denotes those that are recovered by our classifier.The right-hand axis and black line and points show the recovery efficiency of our classifier at each magnitude bin.The total classification efficiency for events with peak magnitudes brighter than where our classification efficiency drops below 50 %, 21 AB mag, is 82.86 ± 1.32 %.Light curves that are not recovered are primarily afterglows with slower fade rates in our observations and are outside the scope of this work.The efficiency curve from this plot has been propagated into our rate calculations and jet constraints shown in Figures6 and 10.

Figure 6 .
Figure 6.Expected value for the number of detectable afterglows in the DWF data for a range of jet profiles.The left-hand plot shows the results for a smooth power-law model and the right-hand plot shows the results for a power-law with core model.A jet with  = 0 and   −   → 90 • has isotropic afterglow emission whilst keeping the prompt GRB emission restricted to the jet's core.Conversely, a jet with  → ∞ and   −   ∼ 0 • describes a top-hat model, where an afterglow is only detectable where the prompt GRB is also detectable.

Figure 7 .
Figure 7.The mean of sextractor's SPREAD_MODEL parameter across , ,  and -bands plotted against the -band AB magnitude for the coincident sources of the candidates found in this work and a sample of sources detected in DELVE.The DELVE sample is coloured with their the mean of the CLASS_STAR parameter across , ,  and -bands.CLASS_STAR is the output to a star/galaxy classifier that is run with sextractor where a source that appears star-like is near a value of 1.0 and galaxies typically range from 0.0 to ≳ 0.9.The candidates' coincident sources are plotted with the same colours as in Figure8.They are plotted as stars if they are detected as two distinct sources in, within one arcsecond, in Gaia DR3.Two blended point sources within one arcsecond of each other would be detected as a single extended source with typical atmospheric seeing conditions.We therefore conclude that all of the coincident sources are amongst the distribution of stars in the DELVE sample.

Figure 9 .
Figure9.The light curves of each of the nine candidates that do not fall within the M-star regions in Figure8.We plot the -band AB magnitude against MJD across the observing window in which the transient was detected.In the top right of each panel, we show the candidate number which is consistent across Table4, Figure8and Figure7.Candidate 6 was reported inAndreoni et al. (2020), DWF17x.

Figure 10 .
Figure 10.Probability of achieving a non-detection in the DWF data for a range of jet profiles.Axes are as Figure 6.

Figure 11 .
Figure11.The rate and peak magnitude of OAs, in the top and bottom panel respectively, detected with a theoretical survey possessing a limiting magnitude of  = 23 AB mag at different observing cadences.We have simulated the same number of events for each cadence, distributed between a day before the first observation and the last observation.The OAs were simulated with a jet structure of  = 0.8 and   = 57 • .We also require three observations where an OA is  < 23 AB mag before an event is detected.We find that at a day cadence, 5.5 % of the OAs are found compared to minute cadence.At a day cadence, we find that the median intrinsic peak magnitude for detected OAs are required to be 4.7 magnitudes brighter.

Figure A1 .
Figure A1.The light curves in Figure9with 100 afterglowpy models sampled from the posteriors, assuming  = 3.The best-fit parameters are shown in the top-left of each light curve.The units of log  0 , log  and   are log erg  −1 , log cm −3 and degrees respectively.Each light curve is labelled with its candidate number.

Figure A2 .
Figure A2.Same as FigureA1but with a selection of six candidates that had coincident sources possessing colours consistent with an M-star.These light curves.therefore, likely originate from stellar flares.Despite this, our fits yield good results in some of these light curves.

Table 1 .
Fields and night coverage for this search.
* is the central value of the log Γ 0 distribution.For each value of Γ 0 generated, a value of log  jet is generated with a central value, log  * , jet.The best-fit parameters calculated in G13 are denoted  = 2.5 and  = 1.45.The log-normal distributions have dispersions of  log Γ 0 = 0.65 dex and  log jet = 0.3 dex.

Table 3 .
Performance (Pedregosa et al. 2011)er on the test set.Uncertainties are given by Poisson statistics.AUC denotes the Area Under the Curve statistic for classifier performance which is given by the integral of the curve tracing the true positive rate versus the false positive rate.Efficiency, purity and accuracy are given by TP/(TP + FP), TP/P and accuracy is the balanced accuracy score metric in scikit-learn(Pedregosa et al. 2011).TP is the number of true positives, FP is the number of false positives and P is the number of positives in the test set.

Table 4 .
Evaluation of candidates extracted by our search methodology presented in Section 5.1.

Table 5 .
West et al. (2011) sources associated with OA candidates found in 100 nights of DWF data.Colours are calculated from sources detected in DELVE DR2.A 3- association with M-stars calculated from these colours using spectroscopically classified M-stars fromWest et al. (2011).