2.2.1 Swift triggers

MWA rapid-response observations were triggered on Swift-BAT events GRB 190627A (Sonbas et al. Reference Sonbas2019a) and GRB 191004A (Cenko et al. 2019). The advantage of Swift– BAT detected GRBs over Fermi detected events is that they are often localised by the Swift X-ray Telescope (XRT; Burrows et al. Reference Burrows2005), which provides arcsec position localisations and makes afterglow detections/redshift determinations much more likely. The XRT also performs these observations rapidly following the BAT trigger (109.8 and 81.5 s for GRB 190627A and GRB 191004A, respectively; Sonbas et al. Reference Sonbas2019a; Cenko et al. 2019), with the resulting X-ray light curves allowing us to constrain coherent radio emission models (see Section 4.1).

GRB 190627A has a brief duration of $1.6\,\mathrm{s}$ (Barthelmy et al. Reference Barthelmy2019), placing it in the short GRB category based on the usual criterion ${T}_{90}\leq2\,\mathrm{s}$ (Kouveliotou et al. Reference Kouveliotou, Meegan, Fishman, Bhat, Briggs, Koshut, Paciesas and Pendleton1993). Nonetheless, there are a few caveats about this identification. Despite the short duration of GRB 190627A, the relative softness of its spectrum makes it intermediate between most short and long bursts detected by Swift– BAT (Barthelmy et al. Reference Barthelmy2019). GRB 190627A is the only event with a spectroscopic redshift of $z=1.942$ (Japelj et al. Reference Japelj2019) in our sample, which helps us to constrain its DM when searching for associated dispersed signals as described in Section 2.4.3. However, this redshift is unusually high for a short GRB, requiring a very high efficiency to produce observable emission given the limited energy reservoir of binary mergers (Berger Reference Berger2014). Additionally, GRB 190627A has a bright optical afterglow (Japelj et al. Reference Japelj2019), making it unusual in the population of short GRBs (Kann Reference Kann2013). Therefore, there is ambiguity in classifying GRB 190627A as being short.

GRB 191004A has been included in our sample as one of our rare triggers on Swift short GRBs, though it had a duration of 2.44 s (Sakamoto et al. Reference Sakamoto2019). We calculated its hardness ratio (i.e., fluence in 50–100 keV over fluence in 25–50 keV) to be $\approx1.6$ , intermediate between the short and long population of Swift GRBs (see Figure 8 in Lien et al. Reference Lien2016). There is no other information available for determining whether it is short or long, such as rest-frame duration (Zhang et al. Reference Zhang2009; Belczynski et al. Reference Belczynski, Holz, Fryer, Berger, Hartmann and O’Shea2010) or the isotropic gamma-ray energy and spectral peak (LÜ et al. Reference LÜ, Liang, Zhang and Zhang2010). A few high-redshift long GRBs with rest-frame durations shorter than 2 s have been found to possess the properties of short GRBs, such as a hard spectrum and a large offset from the host galaxy centre (e.g. Ahumada et al. Reference Ahumada2021). Given there is no clear distinction in the durations of long and short GRBs (Berger Reference Berger2014), and the rest-frame duration of GRB 191004A would be $<2\,\mathrm{s}$ should it occur at $z>0.2$ , we treated GRB 191004A as if it were short. Note that the first three 2 min observations following the Swift trigger were corrupted so our first MWA observation of this source was delayed 6.35 min with respect to the GRB detection, as shown in Table 1.

2.2.2 Fermi triggers

Seven GRBs were triggered by the Fermi-GBM and detected only in the gamma-ray band. For five out of the seven events, we were able to obtain 15 continuous MWA snapshot observations, covering 30 min post-burst. During our MWA observation of GRB 190804A, the source coordinates were updated by Fermi by more than 20 deg, resulting in the first few MWA pointing centres being well separated from the final GRB position, and subsequently discarded. Note that in Table 1, the quoted start time is the earliest time that the MWA was actually pointed at the GRB, with the time post-burst corresponding to this delay with respect to the GRB detection by the Fermi-GBM. We also discarded the last few observations of GRB 190420.98, of which the pointing centres were driven away from the GRB location (beyond 50% of the primary beam) by the Sun suppression algorithm. Among the seven Fermi GRBs, three were further localised by the Interplanetary Network (IPN; Hurley et al. Reference Hurley2013), one was later localised by Swift– BAT, whereas for the other four events, the only position was provided by the Fermi-GBM. The positional information contained in the GBM Final Position Notice usually have uncertainties of about a 1–10 deg radius, which are less accurate than the IPN and BAT localisations.

2.3.1 Calibration

We used two calibration methods depending on the location of the GRB relative to bright sources in the field. Our preferred method is in-field calibration, which uses the GaLactic and Extragalactic All-sky MWA (GLEAM) survey (Hurley-Walker et al. Reference Hurley-Walker2017) as a source model within the field of view and avoids the bulk refractive offset resulting from transferring solutions from dedicated calibration observations. However, for the GRBs with bright sources such as PicA or CenA within the primary beams, the in-field calibration was not possible as these sources were not included in the GLEAM catalogue. Instead, we derived our calibration solution from a nearby calibrator that had been observed within 12 h of the GRB observation.

The calibration methods adopted for each of the nine GRBs are listed in the last column of Table 1. We applied the in-field calibration to five GRBs. Since there was a significant amount of extended emission in the region of GRB 190712.02, we discarded the baselines shorter than 500 m before applying the in-field calibration. Self-calibration was applied on GRB 190420.98 to improve the image quality. For the other four GRBs, we derived calibration solutions using calibrator observations of a single bright source including PicA, HerA, and 3C353.

2.3.2 Imaging

The MWA-fast-image-transients pipeline, which incorporates the WSClean algorithm (Offringa et al. Reference Offringa2014; Offringa & Smirnov Reference Offringa and Smirnov2017), was used to image and deconvolve the 2 min observations of each GRB. We adopted a pixel scale of 32 arcsec for the observations taken in the MWA phase I configuration, 16 arcsec for the phase II extended configuration, and 1.6 arcmin for the phase II compact configuration. The default image size was $4\,096\times 4\,096$ pixels. For the Fermi GRBs in our sample with poor localisations, we increased the image size to as large as $8\,000\times8\,000$ pixels. For the GRBs observed with the lower angular resolution compact configuration, we made smaller images of $1\,000\times1\,000$ pixels. See Figures A.1 and A.2 in Appendix A for the first 2 min snapshots of the nine short GRBs in our sample.

Next we made images on 30 and 5 s timescales, which accommodate the expected dispersive smearing of a prompt radio signal across the MWA observing bandwidth for a redshift of $z\sim0.1$ (DM $\sim\!120\,\text{pc}\,\text{cm}^{-3}$ ) and $z\sim0.7$ (DM $\sim\!840\,\text{pc}\,\text{cm}^{-3}$ ), the lowest and average redshift known for short GRBs (Rowlinson et al. Reference Rowlinson, O’Brien, Metzger, Tanvir and Levan2013). This was done by splitting each 2 min observation into 4 intervals of 30 s, which were cleaned and imaged separately. We then split the 2 min observation into 22 intervals of 5 s starting from the first timestamp with real data (i.e. abandoning the first 4 s and last 6 s of data, see Anderson et al. Reference Anderson2021a) and created the 5 s images without cleaning. Instrumental XX and YY polarisation images were converted into primary beam corrected Stokes I images using the fully embedded element MWA beam model (Sokolowski et al. Reference Sokolowski2017).

To allow for a de-dispersion search for prompt signals, we created sub-band images by splitting the 2 min observation into 0.5 s intervals and the $30.72\,\text{MHz}$ bandwidth into 24 1.28 MHz channels, which is the native coarse channel resolution of the MWA correlator (for the MWA system design see Tingay et al. Reference Tingay2013). Note that we estimate our largest sensitivity loss due to intrachannel smearing within a 1.28 MHz channel at the maximum DM search value of 2 500 $\text{pc}\,\text{cm}^{-3}$ (see Section 2.4.3) to be $\sim\!60$ %, which we consider acceptable when weighed against the massive disk space and computational resources required to process images with a higher frequency resolution. We also created 0.5 s full band images, which could be used to correct source positions in the 0.5 s sub-band images as discussed below. We performed no cleaning on the 0.5 s timescale images.

Source position offsets caused by ionospheric effects and the errors in the absolute flux density calibration were corrected using fits_warp ^{Footnote b} (Hurley-Walker & Hancock Reference Hurley-Walker and Hancock2018) and flux_warp ^{Footnote c} (Duchesne et al. Reference Duchesne, Johnston-Hollitt, Zhu, Wayth and Line2020), which apply corrections to MWA images via comparisons to the GLEAM catalogue. Ionospheric corrections using fits_warp were applied to each individual 2 min, 30 s, and 5 s image. There were too few sources found in the 0.5 s/1.28 MHz images to perform a reliable position correction, so instead we generated a solution from each 0.5 s/30.72 MHz image and applied it to the sub-band images in the same time bin. We did not apply a chromatic correction to the frequency dependent ionospheric position offsets ( $\propto1/\nu^2$ ). At our central observing frequency of 185 MHz, the relative difference in the position offset within the 30.72 MHz bandwidth is expected to be $\sim\!18\%$ . Given the ionospheric position offset has a typical value of $\sim\!$ arcmin (Hurley-Walker & Hancock Reference Hurley-Walker and Hancock2018), this difference would be smaller than the MWA synthesized beam size (see Section 2.1) and thus negligible. However, the flux density calibration was computed by running flux_warp on the ionospherically corrected high signal-to-noise 2 min images, with the resulting solutions then being transferred to the 30, 5, and 0.5 s images.

We did not correct the source positions and flux scales for the observations of GRB 190903A and GRB 191004A taken in the compact configuration as their low angular resolution meant that few sources could be matched to the GLEAM catalogue. We expected the ionospheric correction, typically a few tens of arcsec, to be much smaller than the $\sim\!10\,\mathrm{arcmin}$ resolution of the compact configuration and thus should not distort our images or analysis. We also expected a consistent flux calibration for the different timescale images across the 30 min compact configuration observations of GRB 190903A and GRB 191004A as all 15 snapshots were calibrated using a single solution derived from an external calibrator.

2.4.1 Transient and variable search

We adopted the Robbie (Hancock, Hurley-Walker, & White Reference Hancock, Hurley-Walker and White2019a) work flow, which was further updated in Anderson et al. (Reference Anderson2021a) to process the MWA images and search for variable and transient events within the positional error regions of each GRB. For each GRB data set, Robbie first runs fits_warp to correct for ionospheric positional shifts in the individual images and then creates a mean image, which is then used to extract a persistent source catalogue and corresponding light curves. Comparisons between the mean and individual images, and a statistical analysis of the catalogue, are then used to identify variable and transient candidates.

Provided the GRB position is known to within the synthesised beam of the images, we can add a monitoring position into the catalogue of persistent sources, which forces Robbie to perform priorized fitting and extract a light curve at the best known position of the GRB to search for associated radio emission (Hancock, Trott, & Hurley-Walker Reference Hancock, Trott and Hurley-Walker2018). We performed this analysis on the two Swift GRBs in our sample as they were localised by XRT, which resulted in smaller position errors than the angular resolution of MWA (Wayth et al. Reference Wayth2018).

We characterised the variability of all light curves output by Robbie through the derivation of three parameters: the modulation index (m); the de-biased modulation index ( $\text{m}_d$ ), which takes into account the errors on the flux densities (see Equation (4) in Hancock et al. Reference Hancock, Hurley-Walker and White2019a); and the probability of being a non-variable source (p_val). The value p_val is calculated from the light curves after being normalised by the uncertainty on each data point to account for the effect of varying uncertainties caused by the telescope changing its pointing centre throughout the observation (see Equation (3) in Anderson et al. Reference Anderson2021a).

For the less well-localised Fermi GRBs, we assumed that any associated variable or transient candidates were located within the Fermi-GBM $1\sigma$ error region (Narayana Bhat et al. Reference Narayana Bhat2016) if they were not further localised by Swift– BAT or the IPN. In addition, since the noise level increases towards the edge of the MWA primary beam that can create spurious signals, we restricted our source search to the inner part of the MWA primary beam, within 50% of the maximum sensitivity. Among the different MWA pointings in the $15\times2\,\mathrm{min}$ snapshot observations of a GRB, we picked the first pointing to determine the MWA primary beam as the prompt radio emission we are targeting is most likely to appear in the first few snapshots (see Section 2.1). We therefore only searched for candidates within the overlap between these two regions. An example of the overlap region between the IPN position of GRB 170827B and 50% of the MWA primary beam within which we searched for transients and variables is shown in Figure 1 (for other Fermi GRBs see Figures A.1 and A.2 in Appendix A). The final list of variable and transient candidates within the region of interest (ROI) of all Fermi GRBs were retained for further inspection.

2.4.2 Transient and variable selection

For the Swift GRBs, we looked for any transients or variables detected by Robbie at their known positions. We also inspected the light curves and corresponding variability statistics as output by Robbie that were generated via priorized fitting at the GRB positions for evidence of transient or variable behaviour.

For the Fermi GRBs, we inspected the Robbie transient and variable candidates found within the ROI. In order to make a first cut on transient and variable candidate selection, we devised a set of tests to filter out false positives (such as noise fluctuations or imaging artefacts) as described in the following.

All transient candidates had to have a signal-to-noise ratio (SNR) $\geq6$ , which corresponds to a false positive rate of $\sim\!10^{-9}$ under the assumption of Gaussian noise that is independent in both the space and time dimensions. The number of trials was estimated by the number of synthesised beams in the ROI. For example, for GRB 190420.98 there were $\sim\!2\times10^5$ synthesised beams in the ROI, and there were ten 2 min snapshots, resulting in a final trial number of $\sim\!2\times10^6$ . Therefore, we expected a false positive transient rate of $2\times10^{-3}$ at a $6\,\sigma$ level in the ROI for GRB 190420.98 for the full 20 min observation. See Table B.2 in Appendix B, which lists the number of synthesised beams, the transient false positive rate $\geq6\sigma$ within each ROI for each transient timescale (2 min, 30 s, and 5 s) assuming Gaussian statistics, and the total number of transient candidates detected by Robbie for each Fermi GRB.

Table 2. Upper limits on the radio flux density of transient and variable emission associated with the nine short GRBs in our sample. We quote $6\,\sigma$ upper limits (6 times the average RMS within the ROI) for Fermi GRBs and $3\,\sigma$ for Swift GRBs. Given the noise evolves with time for each GRB, we quote the maximum value for the upper limit. We also include the $3\sigma$ deep limit derived from the 30 min full observation for each GRB.

As a sanity check, we used another method to estimate the expected transient false positive rate in each GRB ROI. Again assuming the image noise conforms to Gaussian statistics, we estimated the expected number of false positive events in the ROI by taking the number of candidates found in a larger image region defined by the 50% MWA primary beam and then multiplying by the ratio of the region areas. In the case where more transient candidates were detected by Robbie in the ROI than predicted via this method, we inspected the individual images and removed any candidates that were consistent with sidelobes from bright sources or other imaging artefacts. For example, while we detected a transient candidate in the ROI for GRB 190420.98, there were significant sidelobe artefacts from the nearby radio galaxy PKS 2153-69, indicating the candidate was unlikely to be real. The expected transient false positive rate within the ROI of each of the seven Fermi GRBs based on comparisons to the number of false events in the MWA 50% primary beam can be found in Table B.2.

For variable candidates, we followed the threshold set in Hancock et al. (Reference Hancock, Hurley-Walker and White2019a), i.e. p_val $< 10^{-3}$ and $m_d$ > 0.05, to distinguish variables from non-variables. As a sanity check, we again computed the number of variables in the MWA 50% primary beam to estimate the expected variable false positive rate in the ROI as we did for transient candidates. In the case of an excess of variables in the ROI, we compared their light curves with the light curves of nearby sources. If a variable candidate showed the same trend of variation as the nearby sources, the variation was probably caused by short-timescale calibration, measurement, or instrumental errors rather than being intrinsic to the source (Bell et al. Reference Bell2019). All variable candidates found by Robbie and the expected variable false positive rate within each Fermi GRB ROI are listed in Table B.2.

Following this analysis, no transients or variables were identified in the ROIs of the Fermi GRBs. Based on minimising the false positive rate assumed from Gaussian statistics, we quote $6\,\sigma$ flux density upper limits (6 times the average RMS within the ROI) for each of the Fermi GRBs on timescales of 2 min, 30 s, and 5 s, as shown in Table 2. As the light curves generated at the position of the Swift GRBs on different timescales are consistent with the noise, we quote $3\,\sigma$ flux density upper limits. We also include a $3\,\sigma$ deep limit derived from the 30 min full observation for each GRB in Table 2, which can be used to constrain persistent emission models (see Section 4.2.3).

2.4.3 Search for dispersed signals

Our observations are specifically targeting prompt, coherent radio signals predicted to be associated with short GRBs, which will be dispersed in time by the intervening medium between their origin and the Earth. We therefore perform a search for dispersed signals across a wide range of DMs over the entire $\sim\!$ 30 min triggered observation. A technique has been developed to search for dispersed signals in short timescale, sub-band radio images of a transient event, which generates a de-dispersed time series in units of SNR at a single sky position (for details see Anderson et al. Reference Anderson2021a). The de-dispersion code has four main steps for creating de-dispersed time series, which was run on all relevant sky positions within the ROI of each short GRB.

1. 1. Create a dynamic spectrum at each relevant sky position using the 0.5 s sub-band images. For Swift GRBs, we created a dynamic spectrum at the pixel coincident with the GRB position. For Fermi GRBs with large positional errors, we only processed the independent sky/field positions within the ROI, which were essentially one position per synthesised beam as illustrated in Figure 1. The search areas of GRB 190712.02 and GRB 190804A were prohibitively large (see Figure A.2 in Appendix A) so we did not perform this analysis on these two Fermi GRBs.

2. 2. Create a de-dispersed time series from each dynamic spectrum. For every 0.5 s time step and $12\,\text{pc}\,\text{cm}^{-3}$ DM trial across the whole 30 min observation (see details in the next paragraph), we calculated the average de-dispersed flux density over the dynamic spectrum pixels crossed by the dispersive sweep (see Anderson et al. Reference Anderson2021a for a visualisation of the dynamic spectrum and de-dispersed time series).

3. 3. Estimate the noise levels of the de-dispersed time series. For Swift GRBs, this was calculated by running the de-dispersion code on 100 nearby position pixels (in addition to the signal pixel) to create 100 de-dispersed time series. To remove any persistent emission that may be at that position, we averaged each of the 101 de-dispersed time series in time and subtracted this mean from their corresponding parent time series. We then calculated the standard deviation of the 100 mean subtracted time series (not including the signal pixel), which we defined as the de-dispersed time series of the noise. For Fermi GRBs, the noise was estimated in the same way but by averaging the de-dispersed time series created at each independent pixel in the ROI.

4. 4. Create the final de-dispersed time series in SNR units. This was done by dividing the de-dispersed time series at the signal pixel (Swift) or independent pixels in the ROI (Fermi) by the de-dispersed time series of the noise derived in the previous step.

The DM resolution of our search for dispersed signals ( $12\,\text{pc}\,\text{cm}^{-3}$ ) was chosen by equating the expected dispersion smearing across the full 30.72 MHz bandwidth to the temporal resolution of 0.5 s (see Equation (1) in Anderson et al. Reference Anderson2018a). All but one of the GRBs in our sample have no redshift measurement so we searched for dispersed signals across the DM space that covers the known redshift range of short GRBs ( $0.1<z<2.5$ ; Rowlinson et al. Reference Rowlinson, O’Brien, Metzger, Tanvir and Levan2013). The contribution of the intergalactic medium to the DM of a short GRB can be estimated from the redshift for the cosmological paradigm of a flat universe. We adopted the method described by Macquart et al. (Reference Macquart2020) to calculate $\text{DM}_{\text{IGM}}$ , taking into account the redshift evolution of the fraction of cosmic baryons in diffuse ionized gas. The redshift range corresponded to a range of $90 < \text{DM}_{\text{IGM}} < 2\,400$ $\text{pc}\,\text{cm}^{-3}$ . Considering the typically large offset of short GRBs from the centers of their host galaxies (Fong & Berger Reference Fong and Berger2013), we assumed the DM contribution from their host galaxies to be small ( $\text{DM}_{\text{host}}\sim30\,\text{pc}\,\text{cm}^{-3}$ ; Cordes & Lazio Reference Cordes and Lazio2002). Assuming a similarly small contribution from the Milky Way based on the YMW16 DM model ( $\text{DM}_{\text{MW}}\sim30\,\text{pc}\,\text{cm}^{-3}$ ; Yao, Manchester, & Wang Reference Yao, Manchester and Wang2017), we adopted a DM range of 150–2 500 $\text{pc}\,\text{cm}^{-3}$ for our search for dispersed signals associated with GRBs without a known redshift.

The redshift of GRB 190627A corresponded to a $\text{DM}_{\text{IGM}}$ of $\sim\!1\,800\,\text{pc}\,\text{cm}^{-3}$ . However, this value can vary depending on the number of galactic halos intersected by the line of sight, corresponding to a possible DM range between 1 400 and 2 400 $\text{pc}\,\text{cm}^{-3}$ , which encompasses 90% of the expected values (Macquart et al. Reference Macquart2020). The DM contribution from the Milky Way in the direction of GRB 190627A ( $l=8.17^{\circ}$ , $b=30.26^{\circ}$ ) is estimated to be $\text{DM}_{\text{MW}}\sim50\,\text{pc}\,\text{cm}^{-3}$ based on the YMW16 electron-density model (Yao et al. Reference Yao, Manchester and Wang2017). We therefore estimate a DM of $\text{DM}_{\text{host}}+\text{DM}_{\text{MW}}+\text{DM}_{\text{IGM}} = 1\,900^{+600}_{-400}\,\text{pc}\,\text{cm}^{-3}$ for GRB 190627A.

In order to set a threshold for selecting dispersed signal candidates for further investigation in the resulting de-dispersed time series, we first considered the Swift GRBs and determined the number of trials based on the time and DM steps used in our analysis. Given that there are $\sim\!3\times10^6$ trials for GRB 191004A and $\sim\!10^{6}$ for GRB 190627A, we set a threshold of $5\sigma$ , corresponding to less than one false positive for each Swift GRB. In the case of Fermi GRBs, which are not localised to a single pixel, we assessed the noise statistics of the de-dispersed time series in the ROI for each event by creating a set of time series from the same data set using (nonphysical) negative DM values. If the GRB error region contains only noise, a de-dispersion analysis with positive and negative DMs (same range of absolute values) should give similar SNR distributions. Table 3 shows a comparison of the maximum SNR and number of high SNR events above $5\sigma$ for the set of positive and negative DMs for each Fermi GRB. We also include the expected number of false positive events $>5\,\sigma$ , along with the maximum SNR for which we expect there to be only one false positive event assuming a Gaussian distribution. From Table 3, one can see that the high SNR events observed in our data set are consistent with noise. There are fewer detected events above $5\sigma$ than the expected number of false positive events, which may be caused by an overestimation of the noise. The noise calculated in our data using the standard deviation of a population of pixels was affected by the sensitivity changing across the image, which is higher than the value expected in the case of an unchanged sensitivity. If any signals are detected with a higher SNR in the positive de-dispersed time series than the maximum measured in the negative de-dispersed time series, then it is possible that they are real signals.

Table 3. Results following the search for dispersed signals associated with Fermi GRBs. This includes a comparison of high SNR events arising from the positive and negative DM time-series analysis described in Section 2.4.3. We also list the $7\,\sigma$ upper limits on the fluence for each GRB in the last column. Given the noise varies with DM, time and the GRB position within the ROI due to the MWA primary beam, we present a range for the fluence upper limits (see details in Section 2.4.5). GRB 190712.02 and GRB 190804A are not included in this analysis due to their poor localisations (see Section 2.4.3), and GRB 200325A is analysed along with the Swift GRBs (see Section 2.4.4).

1 :The positive (P) or negative (N) DMs used to create the de-dispersed time series;

2 : The trial number estimated from the number of time steps, DM trials, and synthesised beams (for which a de-dispersed time series was generated) within the ROI;

3 : The maximum SNR event detected;

4 : The critical SNR beyond which we expect there to be just one event;

5 : The number of events detected with SNRs above $5\sigma$ in the positive and negative DM de-dispersed time series;

6 : The expected false positive event rate above $5\sigma$ assuming a Gaussian distribution;

7 : The $7\sigma$ fluence limits on dispersed signals associated with each GRB.

In the case of no dispersed signal detections in the de-dispersed time series, we derived an upper limit for each Swift GRB using signal simulations (see Section 2.4.4) and adopted a $7\sigma$ upper limit for each Fermi GRB given less than one event above $7\sigma$ is expected from Gaussian statistics (see Table 3). We present our dispersed signal search results in units of fluence (Jy ms, the integrated flux density over the pulse width), which is common in the fields of FRB and pulsar astrophysics (see Tables 3 and 4).

Table 4. The fluence limit on dispersed signals for the two Swift GRBs and one Fermi GRB. These limits correspond to the 90% detection efficiency of our detection algorithm to simulated signals injected into the dedispersed time series of these three events (see Section 2.4.4).

2.4.4 Dispersed signal simulations for well-localised GRBs

As GRBs 190627A, 191004A, and GRB 200325A were well localised by Swift (Fermi GRB 200325A was localised by Swift– BAT to within 50 synthesised beams), we were able to inject simulated pulses into the de-dispersed time series to determine our sensitivity to such signals in each MWA observation, thereby allowing us to derive fluence limits as a function of DM (as done by Anderson et al. Reference Anderson2021a). As GRB 191004A and GRB 200325A have no known redshift, we simulated pulses over a large DM range, including 150, 500, 1 000, 1 500, 2 000, and 2 500 $\text{pc}\,\text{cm}^{-3}$ . As the redshift is known for GRB 190627A, we simulated pulses over a smaller DM range between 1 500 and 2 500 $\text{pc}\,\text{cm}^{-3}$ . By injecting signals over a wide range of fluence values, we were able to test the efficiency of our detection algorithm in the three GRB data sets, which are plotted in Figure 2. The fluence limits quoted in Table 4 are the signal fluence corresponding to the 90% detection efficiency of our algorithm. The performance of our algorithm was different for each MWA observation depending on many factors such as the presence of bright sources in the field, the GRB location within the primary beam, and the elevation of the observation.

2.4.5 Fluence limits for Fermi GRBs

Signal injection was not a viable method for calculating the fluence limits of the Fermi GRBs as their poorer localisations means that the sensitivity changes significantly across the ROI, and the performance of our algorithm is dependent on the sky position we choose to inject the signals. Although we could provide a fluence limit as a function of DM and signal position in the ROI using the signal injection technique for a given Fermi GRB, it would be hugely computationally expensive. Instead we created a de-dispersed time series for each of the independent positions in the ROI and derived a $7\,\sigma$ fluence upper limit for each of the Fermi GRBs using the noise calculated from these de-dispersed time series, corresponding to a false positive rate of $\sim\!10^{-12}$ under the assumption of Gaussian noise (see Table 3). Given the noise varies with DM, time, and the position in the MWA primary beam, we present a range for the fluence upper limits for the Fermi GRBs.

4.2.1 Interactions of NS magnetic fields

The earliest coherent radio emission may come from the inspiral phase, where interactions between the NS magnetospheres just preceding the merger may spin them up to millisecond spin periods and lead to a revival of the pulsar emission mechanism (Lipunov & Panchenko Reference Lipunov and Panchenko1996; Metzger & Zivancev Reference Metzger and Zivancev2016). The emission starts from a few seconds prior to the merger and peaks at the first contact of the NS surfaces, giving rise to a short duration radio flash. Its flux density is predicted to be

(1) \begin{equation}F_{\nu} \sim 2\times10^8\,(1 + z)\,\frac{B_{15}^{2}M_{1.4}^{3}}{R_6\nu_{9,\text{obs}}D^2}\,\epsilon_{\text{r}}\,\text{Jy},\end{equation}

where $\nu_{9,\text{obs}}$ is the observing frequency in units of $10^9\,\mathrm{Hz}$ , D is the distance to the binary system in Gpc, the radio emission efficiency ( $\epsilon_{\text{r}}$ ) is the fraction of the wind luminosity being converted into coherent radio emission (typically $10^{-4}$ for known pulsars; Taylor, Manchester, & Lyne Reference Taylor, Manchester and Lyne1993), and we assume typical NS parameters for the magnetic field of $B_{15}=10^{-3}$ in units of $10^{15}\,\mathrm{G}$ , a mass of $M_{1.4}=1$ in units of $1.4\,\text{M}_\odot$ , and a radius of $R_6=1$ in units of $10^6\,\mathrm{cm}$ (Rowlinson & Anderson Reference Rowlinson and Anderson2019). Note that the parameters $B_{15}$ , $M_{1.4}$ , and $R_6$ are assumed properties of the merging NSs, not of the magnetar remnant. The distance can be estimated from the redshift z using the predefined cosmology parameters (Wright Reference Wright2006).

4.2.2 Interaction of relativistic jets with the ISM

If a Poynting flux dominated wind is produced by the magnetar remnant formed following the binary merger (Usov Reference Usov1994; Thompson Reference Thompson1994), its interaction with the ISM would generate a low frequency, coherent radio pulse at the highly magnetised shock front (Usov & Katz Reference Usov and Katz2000). As this emission is linked to the relativistic gamma-ray jet and only propagates through the pre-existing low density surrounding medium, it should be detectable within the first few minutes of a GRB trigger. The radio fluence is given by

(2) \begin{equation} \Phi_{\nu}\sim\frac{0.1\epsilon_B(\beta-1)}{\nu_{\text{max}}}\left(\frac{\nu}{\nu_{\text{max}}}\right)^{-\beta}\Phi_{\gamma}\,\text{erg}\,\text{cm}^{-2}\,\text{Hz}^{-1}\end{equation}

where $\epsilon_B$ is the fraction of magnetic energy in the relativistic jet, and $\beta\simeq1.6$ is the spectral index (Usov & Katz Reference Usov and Katz2000). The peak frequency of the coherent radio emission ( $\nu_{\text{max}}$ in GHz) is determined by the magnetic field at the shock front (see Equations (11)–(13) in Section 2.4 of Rowlinson & Anderson Reference Rowlinson and Anderson2019), and $\Phi_{\gamma}$ is the observed gamma-ray fluence in $\text{erg}\,\text{cm}^{-2}$ .

4.2.3 Persistent emission following the formation of a magnetar

If the merger remnant is a magnetar, the coherent radio emission powered by dipole magnetic braking may be detectable during the lifetime of the magnetar (Totani Reference Totani2013; Metzger, Berger, & Margalit Reference Metzger, Berger and Margalit2017). If the beam of the magnetar aligns with the GRB jet and points towards the Earth, the coherent emission will be detected as persistent emission. The flux density of this emission is given by

(3) \begin{equation} F_{\nu}\sim8\times10^{7}\,\nu^{-1}_{\text{obs}}\,\epsilon_{\text{r}}\,D^{-2}\,B^{2}_{15}\,R^6_6\,P^{-4}_{-3}\,\text{Jy},\end{equation}

where $P_{-3}$ is the magnetar spin period in units of $10^{-3}\,\mathrm{s}$ (Totani Reference Totani2013; Rowlinson & Anderson Reference Rowlinson and Anderson2019). This persistent radio emission may only be detectable for up to a few hours following the merger as its rapid spin-down rate would cause the dipole radiation to weaken very quickly or (if unstable) it collapses into a BH (which is $\lesssim2\,\mathrm{h}$ , depending on the magnetar parameters B and P derived in Section 4.1).

4.3.1 GRB 190627A

GRB 190627A is the only event amid our nine short GRBs that has a known redshift ( $z=1.942$ ). Thanks to the redshift measurement, we can calculate a luminosity distance of 15.2 Gpc and directly constrain the fundamental parameters of the emission models, such as the efficiency of radio emission $\epsilon_{\text{r}}$ , which directly scales with the predicted flux densities from the interaction of the merging NS magnetic fields, and that of the persistent pulsar emission from the magnetar remnant (Equtions (1) and (3)). We can also constrain the fraction of magnetic energy $\epsilon_B$ that directly scales with the predicted fluence from the interaction between the relativistic jet and ISM (Equation (2)). This has not been possible in previous works as low frequency radio upper limits of coherent radio emission have only poorly constrained these emission parameters under a presumed redshift (e.g. Rowlinson et al. Reference Rowlinson2019; Rowlinson et al. Reference Rowlinson2020; Anderson et al. Reference Anderson2021a). In addition, as this Swift GRB has also been localised to within an MWA synthesised beam, we are able to derive a constraining dispersed signal (fluence) limit.

In Figure 4, we plot the predicted flux density of a signal produced by the alignment of the merging NS magnetic fields in GRB 190627A, as a function of the radio emission efficiency $\epsilon_r$ (see Equation (1) in Section 4.2.1). We compare this prediction to the upper limit derived from the 2 min snapshots of GRB 190627A as this timescale is comparable to the dispersive smearing of a prompt, coherent signal across the MWA observing band for a redshift of 1.942 ( $\sim\!90\,\mathrm{s}$ ). We can see from Figure 4 that this coherent emission would only be detectable on a 2 min timescale if it has an efficiency $\gtrsim10^{-2}$ . We also plot the deeper limit derived from the dispersed signal simulations (see Section 2.4.4) in Figure 4. In this case, the emission would be marginally detectable for a typical pulsar efficiency.

Figure 4. The predicted 185 MHz flux density (blue line) of the prompt signal emitted by the alignment of the merging NS magnetic fields (Section 4.2.1) in GRB 190627A as a function of the radio emission efficiency ( $\epsilon_r$ ). The horizontal dotted line shows the least constraining flux density upper limit derived from the 2 min snapshots of GRB 190627A (Table 2). The horizontal dashed line shows the flux density upper limit converted from the least constraining fluence limit derived from the image de-dispersion analysis (Table 4), and the vertical line shows the typical efficiency observed for known pulsars ( $\epsilon_r\sim10^{-4}$ ).

In Figure 5, we plot the predicted fluence produced by the relativistic jet and ISM interaction as a function of the fraction of magnetic energy $\epsilon_B$ (see Equation (2) in Section 4.2.2). The gamma-ray fluence of GRB 190627A was measured to be $(9.9\pm2.2)\times10^{-8}\,\text{erg}\,\text{cm}^{-2}$ by Swift– BAT in the 15–150 keV energy band (Barthelmy et al. Reference Barthelmy2019). In this case, the jetted outflow is assumed to be a magnetised wind that is powered by a magnetar central engine (Usov & Katz Reference Usov and Katz2000) so is dependent on the magnetar parameters derived for GRB 190627A (see Table 5). By also adopting a typical value of $10^{49}\,\mathrm{erg}$ for the kinetic energy from short GRBs (Fong et al. Reference Fong, Berger, Margutti and Zauderer2015), $\Gamma=1\,000$ for the Lorentz factor of the relativistic wind (Ackermann et al. Reference Ackermann2010), and $n=10^{-2}$ $\text{cm}^{-3}$ for the poorly known electron density of the surrounding medium (Fong et al. Reference Fong, Berger, Margutti and Zauderer2015) we were able to derive $\nu_{\max}$ (see Section 4.2.2 and Rowlinson & Anderson Reference Rowlinson and Anderson2019). We then compared the fluence upper limit we derived through our de-dispersion analysis (Table 4; horizontal dotted line in Figure 5) to the fluence prediction (solid line), constraining the fraction of magnetic energy in the relativistic jet launched by GRB 190627A to $\epsilon_B<2\times10^{-4}$ , which is consistent with the limit $\epsilon_B\lesssim10^{-3}$ given by the requirement that the magnetic stress at the shock front should not disrupt the thin colliding shells in internal shock models (vertical dashed line in Figure 5; Katz Reference Katz1997).

Figure 5. The predicted fluence (blue line) of a prompt signal produced by the relativistic jet and ISM interaction (Section 4.2.2) for GRB 190627A as a function of the fraction of magnetic energy in the GRB jet ( $\epsilon_B$ ). The horizontal dotted line shows the least constraining fluence upper limit derived from our image de-dispersion analysis (Table 4), and the vertical dashed line shows a typical value for the magnetic energy fraction of $\epsilon_B=10^{-3}$ (Katz Reference Katz1997).

In Figure 6, we plot the predicted flux density of persistent pulsar emission from the magnetar remnant as a function of the radio emission efficiency $\epsilon_{r}$ using the magnetar parameters listed in Table 5 and Equation (3) in Section 4.2.3. We use the upper limit obtained from the 30 min integration of GRB 190627A to constrain our detection of the persistent emission (Table 2). Figure 6 shows that for our limit, the magnetar remnant likely has a radio emission efficiency of $\epsilon_r\lesssim10^{-3}$ .

Figure 6. The predicted flux density of the persistent emission from a magnetar remnant (Section 4.2.3) resulting from GRB 190627A as a function of the radio emission efficiency ( $\epsilon_r$ ). The shaded region corresponds to the $1\,\sigma$ uncertainty on the fitted magnetar remnant parameters listed in Table 5 (see Section 4.1). The horizontal line shows the flux density upper limit obtained from the 30 min integration of GRB 190627A (Table 2), and the vertical line shows the typical efficiency observed for known pulsars.

From the above comparison between the upper limits and the theoretical predictions, we obtained the following constraints on the model parameters for GRB 190627A: the radio emission efficiency of the nearly merged NSs $\epsilon_r\lesssim10^{-4}$ ; the fraction of magnetic energy in the GRB jet $\epsilon_B\lesssim2\times10^{-4}$ ; and the radio emission efficiency of the magnetar remnant $\epsilon_r\lesssim10^{-3}$ . While the merging NSs are predicted to have a lower radio emission efficiency than typical pulsars, the magnetar remnant may have a higher efficiency. The constraint on the fraction of magnetic energy in the GRB jet is consistent with the range $10^{-6} \lesssim \epsilon_B \lesssim 10^{-3}$ resulting from a systematic study of GRB magnetic fields (Santana, Barniol Duran, & Kumar Reference Santana, Barniol Duran and Kumar2014). GRB afterglow analyses also show that $\epsilon_B$ downstream of the shock is much larger than $10^{-9}$ , which is the typical value in the surrounding medium of short GRBs, assuming a density of $1\,\text{cm}^{-3}$ and a magnetic field of $\sim\! \mu G$ similar to the Milky Way (e.g., Panaitescu & Kumar Reference Panaitescu and Kumar2002; Yost et al. Reference Yost, Harrison, Sari and Frail2003 and Panaitescu Reference Panaitescu2005). If the magnetic field in the surrounding medium of GRB 190627A has a similar value to that of the Milky Way, the amplification factor of the magnetic energy fraction for this GRB would be $\lesssim2\times10^5$ . Several magnetic field amplification mechanisms have been proposed, including the Weibel instability, the cosmic-ray streaming instability, and the dynamo effect (e.g., Lucek & Bell Reference Lucek and Bell2000; Medvedev et al. Reference Medvedev, Fiore, Fonseca, Silva and Mori2005; MilosavljeviĆ & Nakar Reference MilosavljeviĆ and Nakar2006; Inoue, Asano, & Ioka Reference Inoue, Asano and Ioka2011; Mizuno et al. Reference Mizuno, Pohl, Niemiec, Zhang, Nishikawa and Hardee2011).

4.3.2 GRB 191004A

GRB 191004A is one of the two events detected by Swift. Given that the first three 2 min snapshots were corrupted, the delay between the MWA first being on-target with respect to the GRB detection (see Section 2.2.1) has meant we cannot test coherent emission models that predict the production of prompt radio signals either just prior to, or concurrent with, the merger (see Sections 4.2.1 and 4.2.2) for redshifts $z\lesssim2.7$ . We are therefore only able to constrain the persistent emission from a magnetar remnant for this GRB, which is the model presented in Section 4.2.3. Given we do not know the redshift of this event, we need to explore how the predicted flux density changes between redshifts of $0.1<z<2$ . The magnetar parameters were derived by fitting the magnetar model to the rest frame X-ray light curve (assuming $z=0.7$ ; see Section 4.1). These parameters therefore vary with redshift (Rowlinson & Anderson Reference Rowlinson and Anderson2019), with the magnetic field and spin period scaling according to

(4) \begin{equation} B_{15}\propto D^{-1}\,(1+z),\end{equation}

(5) \begin{equation} P_{-3}\propto D^{-1}\,(1+z)^{1/2}.\end{equation}

Figure 7 shows the predicted persistent pulsar emission from a magnetar remnant produced by GRB 191004A as a function of redshift. We used the fitted magnetar parameters derived for GRB 191004A (listed in Table 5) and the above scaling relations (Equations (4) and (5)) to calculate the predicted flux density. Otherwise, we assumed the same parameters as adopted for GRB 190627A in Section 4.3.1. Since the observations of GRB 191004A were conducted with the MWA compact configuration, the confusion noise caused the upper limit on the 30 min integration to be less constraining compared to other GRB observations taken in the extended configuration. Our MWA flux density limit for GRB 191004A is therefore insufficient for constraining this model.

Figure 7. The flux density of persistent emission (solid red line) predicted to be produced by a remnant magnetar resulting from GRB 191004A as a function of redshift (Section 4.2.3). The shaded region corresponds to the $1\,\sigma$ uncertainties on the fitted magnetar parameters (see Sections 4.1 and Table 5). The radio emission efficiency is assumed to be $\epsilon_r=10^{-4}$ , which is the typical value for pulsars. The horizontal dashed line indicates the flux density upper limit of 1.104 Jy derived from the 30 min integration of GRB 191004A.

4.3.3 Fermi GRBs

The other GRBs in our sample were detected by Fermi-GBM so no follow-up X-ray data are available to derive their magnetar remnant parameters. Assuming a typical magnetar remnant for all the Fermi GRBs (see Section 4.1), we now compare the radio emission upper limits derived from our MWA observations to theoretically predicted values described by models presented in Sections 4.2.2 and 4.2.3. Note that since none of the Fermi GRB fluence upper limits place constraints on the NS magnetic field interactions described in Section 4.2.1, we do not consider this model further (see discussion in Section 5.1.1).

Figure 8 shows the predicted prompt radio emission produced by the GRB jet– ISM interaction as a function of redshift assuming a typical magnetar remnant was formed. The large uncertainties (the shaded region) come from the scatter in the distribution of the spin period and magnetic field strength of magnetar remnants from previously studied short GRBs (Rowlinson et al. Reference Rowlinson, O’Brien, Metzger, Tanvir and Levan2013; Rowlinson & Anderson Reference Rowlinson and Anderson2019). The predicted radio fluence directly scales with the gamma-ray fluence (see Equation (2)), which we list in Table 6 for each GRB as measured by Fermi-GBM in the 10–1 000 keV energy band. For this comparison, we adopt the median gamma-ray fluence value, i.e. $7.8\times10^{-7}\,\text{erg}\,\text{cm}^{-2}$ , for predicting the prompt emission fluence (thick black curve in Figure 8). We also plot the model radio fluence predictions corresponding to the minimum and maximum gamma-ray fluence in our sample (Table 6) in Figure 8 (thin black curves). Note that the uncertainty in the predicted emission due to the magnetar parameters encompasses the range in predictions caused by different gamma-ray fluence measurements. The other model parameters are assumed to be the same as for GRB 190627A.

Table 6. The gamma-ray fluences (10–1 000 keV) measured by Fermi-GBM for those Fermi events for which we derived radio fluence limits.

References include:

a : Fermi-GBM burst catalog at HEASARC: https://heasarc.gsfc.nasa.gov/W3Browse/fermi/fermigbrst.html;

b : Mailyan & Meegan (Reference Mailyan and Meegan2019);

c : Veres et al. (Reference Veres, Hui, Meegan and Wood2020a);

d : Veres, Meegan, & Fermi GBM Team (2020b).

We overplot all the fluence limits derived from our GRB sample (Tables 3 and 4) on Figure 8 to constrain the GRB jet-ISM interaction model (Section 4.2.2). Only those GRBs for which MWA was on-target $\lesssim1\,\mathrm{min}$ post-burst were included in this Figure as any prompt signals emitted at cosmological distances at the time of burst would have been dispersion delayed by up to $\sim\!2\,\mathrm{min}$ at MWA frequencies. We incorporated the dependence of the fluence upper limits on the DM value and redshift, as was done in Anderson et al. (Reference Anderson2021a). In each case, we plot the maximum fluence limit in the range quoted in Tables 3 and 4. It can be seen that the majority of our fluence limits in this paper are constraining for GRBs at low redshifts $z \lesssim 0.5$ for a subset of magnetar parameters. In Section 5.1.2, we further explore the implications of our upper limits on this emission model.

Figure 8. The fluence of the prompt radio signal predicted to be produced by the relativistic jet and ISM interaction using the mean values of the magnetic field and spin period of known magnetar remnants (see Figure 8 in Rowlinson & Anderson Reference Rowlinson and Anderson2019) and assuming the median value of the gamma-ray fluences measured for different Fermi GRBs in Table 6 (thick black curve). The two thin black curves show the radio fluence predictions corresponding to the minimum and maximum gamma-ray fluence measured for the Fermi GRBs, and the shaded region corresponds to the $1\,\sigma$ scattering in the distribution of the parameters of typical magnetars. Different from Figure 7, there is no rescaling of magnetic field and spin period with redshift. The fluence limit for GRB 190627A is plotted as a black triangle. The solid coloured curves represent the fluence upper limits as a function of DM (redshift) derived from the de-dispersion image analysis performed on the Fermi GRBs. We also include the fluence upper limits published for individual short GRBs (dashed coloured curves), including GRB 150424A (132 MHz; Kaplan et al. Reference Kaplan2015), GRB 170112A (56 MHz; Anderson et al. Reference Anderson2018b), GRB 180805A (185 MHz; Anderson et al. Reference Anderson2021a), and GRB 181123B (144 MHz; Rowlinson et al. Reference Rowlinson2020). The dotted black line indicates a potential fluence limit we could achieve if we instead trigger observations using the MWA Voltage Capture System (VCS; see further details in Section 5.1.2.)

Figure 9 shows the predicted persistent dipole radiation from a typical magnetar as a function of redshift (thick black curve). Again, the shaded region illustrates the uncertainty in the prediction due to the scatter in the distribution of known magnetar parameters (Rowlinson & Anderson Reference Rowlinson and Anderson2019), which spans six orders of magnitude. We plot the flux density upper limits derived from the 30 min integrations of the GRBs as listed in Table 2. For a typical magnetar remnant, the majority of our MWA observations could have detected persistent dipole radiation up to a redshift of $z\sim0.6$ . We explore this further in Section 5.1.3.

Figure 9. Similar to Figure 8, here we plot the predicted flux density for the persistent radio emission from the dipole radiation of a magnetar remnant (see Section 4.2.3). The solid black curve represents the predicted emission from a typical magnetar with the shaded region corresponding to the $1\sigma$ scatter in the distribution of magnetar parameters. The solid coloured curves represent the flux density upper limits derived from the 30 min integration of our sample of short GRBs. We also plot the flux density upper limits from observations of other individual GRBs (dashed coloured curves), including GRB 150424A (Kaplan et al. Reference Kaplan2015), GRB 180706A (a long GRB; Rowlinson et al. Reference Rowlinson2019), GRB 180805A (Anderson et al. Reference Anderson2021a), and GRB 181123B (Rowlinson et al. Reference Rowlinson2020).