# 5 DISCUSSION

## 5.1 Is FRB 20210202D a repeater?

The discovery of a faint post-cursor burst or emission component in FRB 20210202D is intriguing and makes it a good repeater candidate. Repeating FRBs often show the so-called 'sad trombone' effect, i.e. complex time-frequency structure with subbursts that drift down in frequency with increasing time (Hessels et al. 2019). They also generally appear to have significantly wider burst profiles and are more band-limited than the apparent non-repeaters, at least at CHIME frequencies (CHIME/FRB Collaboration 2021). FRB 20210202D exhibits none of those characteristics. However, repeaters also emit more broad-band spiky bursts, as seen for instance in FRB 20121102A ('R1'; Platts et al. 2021), FRB 20180916B ('R3'; Marthi et al. 2020), and FRB 20201124A (Marthi et al. 2022). Given its extremely narrow width, FRB 20210202D could be one of those spiky repeater bursts. Although narrower, it looks qualitatively similar to the broadband FRB 20221102A bursts with pre- or post-cursors presented in Platts et al. (2021).

## 5.2 Post-cursor burst separations

FRB 20210202D is already the second MeerTRAP FRB in which a post-cursor burst was detected, with the other being FRB 20210123A (Rajwade et al. 2022). Interestingly, the observed post-cursor separations are surprisingly similar, with values of around 200 ms in each case. The FRBs are at the lower end of the DM distribution of the current MeerTRAP sample with observed DMs of ~609 and 434 pc cm⁻³ and extragalactic DMs of ~486 and 109 pc cm⁻³ above the Galactic ISM and halo contributions. This could indicate that they are indeed reasonably nearby repeaters, especially FRB 20210123A. Aside from this, their parameters differ significantly. For instance, FRB 20210123A's pulse width is about double that of FRB 20210202D's. In comparison, the histogram of the sub-burst separations in the CHIME catalogue 1 sample peaks around ~10 ms with only two bursts above 30 ms and a maximum separation near 65 ms. Out of those, repeaters seems to show somewhat larger values (Pleunis et al. 2021). The ~200 ms separations in the two MeerTRAP FRBs is significantly larger than this, which supports the idea that they are faint repeat pulses.

Why are their post-cursor separations almost precisely the same? We are not aware of any obvious instrumental reasons for why that should be the case. This has neither been seen so far in other MeerTRAP FRBs nor any of thousands of pulsar or RRAT pulses. A shift in arrival time of one or multiple frequency sub-bands could happen in exceptional cases when the beam-former nodes get out of sync. However, the sky signal would get shifted in time and not copied. We are currently commissioning a real-time system to write out voltage data whenever FRBs are discovered. This will allow us in the future to test whether similar post-cursors are coherent copies of the primary bursts and determine their polarization properties. It is hard to imagine how a delayed mirror image of the primary burst could be introduced into the data stream, and we conclude that it must be astrophysical. We also caution that these are very small number statistics. None the less, perhaps the 200 ms separation corresponds to an oscillation frequency, activity or rotation period, or any of its harmonics in the FRB progenitor or its emission mechanism. Or maybe it is related to the quasi-periodic sub-components that have been reported in some FRBs (CHIME/FRB Collaboration 2022; Pastor-Marazuela et al. 2022). Most notable here is FRB 20191221A with a closely comparable and statistically significant

component periodicity of 216.8 ms (CHIME/FRB Collaboration 2022). Similarly, it could be a significantly scaled-up version of the quasi-periodic microstructure observed in several radio pulsars (Cordes, Weisberg & Hankins 1990). Alternatively, the post-cursor burst might be an attenuated echo of the primary, for which the separation would correspond to a light travel time difference. The same is true in the case of gravitational lensing of FRBs. For example, our data captures around the bursts are sensitive to FRB millilensing with delays of ~milliseconds and above by intermediate-mass black holes or dark matter halos (Connor & Ravi 2023). The lensed copies of the FRB signal will be fainter than the primary burst, as in our post-cursors. The phenomenon offers exciting prospects for studying cosmology and fundamental physics using FRBs (Zheng et al. 2014; Li et al. 2018a). Our new voltage buffer dump system will allow us to test if that is the case too.

The inter-burst arrival or waiting times between bursts from repeaters are of scientific interest and have been studied by several authors. Already early on it was realized that their bursts arrive often clustered in time (Wang & Yu 2017; Oppermann et al. 2018). For instance, both FRBs 20121102A and 20200120E show clustering seen as bimodality in their waiting time distributions. The short-duration clustering is most relevant for this discussion. In FRB 20121102A, the fast clustering peak occurs around 22–24 ms (Hewitt et al. 2022; Jahns et al. 2023) if sub-bursts are excluded and around 3.4 ms if they are not (Li et al. 2021b). In FRB 20200120E, the fast peak in the waiting time distribution appears around 1 s (Nimmo et al. 2023). Neither values are close to the ~200 ms separation seen here and they differ already significantly among the two repeaters. Hence, it is unclear whether the post-cursors are sub-bursts or repeat pulses. A larger sample of well-constrained repeater waiting time distributions is needed to inform the distinction.

## 5.3 A deficit of low-fluence FRBs

The FRB all-sky rate inferred from the MeerTRAP coherent survey is significantly below that expected from the best-fitting power-law scaling from surveys at higher limiting fluences $\ge 1$ Jy ms, see Fig. 9. Equivalently, the power-law scaling between the MeerTRAP coherent and incoherent surveys is appreciably flatter than that among the high-fluence surveys. The flatter power-law scaling from MeerTRAP extrapolates near the FAST rate at a limiting fluence ~45 times lower. The flattening of the scaling of the FRB all-sky rate with limiting fluence and the apparent deficit of low-fluence FRBs could have important implications for the FRB population and cosmology.

In the following, we discuss several possible explanations for the FRB deficit. (1) The MeerTRAP coherent rate is only based on six CB detections and therefore still in the small number statistics regime. Further detections might either strengthen the trend or reduce the tension with the high-fluence estimates. (2) The rate inferred from the MeerTRAP coherent survey could be slightly underestimated due to the more complex FoV than that of the incoherent survey, perhaps even by a factor of two. However, it is unlikely to be off by an order of magnitude. To illustrate the point, when extrapolating from the best-fitting high-fluence power law down to the MeerTRAP coherent survey fluence limit (shown as a slightly transparent yellow line in Fig. 9), we would expect to detect a rate of $25_{-12}^{+21} \times 10^3$ instead of $8.2_{-4.6}^{+8.0} \times 10^3$ sky$^{-1}$ d$^{-1}$ in the MeerTRAP coherent survey, which translates to about $34_{-16}^{+28}$ FRB CB detections above the completeness threshold instead of 6. Where are those missing FRBs? It seems unlikely that we missed such a large number of FRBs in our detection pipeline. (3) The MeerTRAP
# 5 DISCUSSION

## 5.1 Is FRB 20210202D a repeater?

The discovery of a faint post-cursor burst or emission component in FRB 20210202D is intriguing and makes it a good repeater candidate. Repeating FRBs often show the so-called 'sad trombone' effect, i.e. complex time-frequency structure with subbursts that drift down in frequency with increasing time (Hessels et al. 2019). They also generally appear to have significantly wider burst profiles and are more band-limited than the apparent non-repeaters, at least at CHIME frequencies (CHIME/FRB Collaboration 2021). FRB 20210202D exhibits none of those characteristics. However, repeaters also emit more broad-band spiky bursts, as seen for instance in FRB 20121102A ('R1'; Platts et al. 2021), FRB 20180916B ('R3'; Marthi et al. 2020), and FRB 20201124A (Marthi et al. 2022). Given its extremely narrow width, FRB 20210202D could be one of those spiky repeater bursts. Although narrower, it looks qualitatively similar to the broadband FRB 20221102A bursts with pre- or post-cursors presented in Platts et al. (2021).

## 5.2 Post-cursor burst separations

FRB 20210202D is already the second MeerTRAP FRB in which a post-cursor burst was detected, with the other being FRB 20210123A (Rajwade et al. 2022). Interestingly, the observed post-cursor separations are surprisingly similar, with values of around 200 ms in each case. The FRBs are at the lower end of the DM distribution of the current MeerTRAP sample with observed DMs of ~609 and 434 pc cm⁻³ and extragalactic DMs of ~486 and 109 pc cm⁻³ above the Galactic ISM and halo contributions. This could indicate that they are indeed reasonably nearby repeaters, especially FRB 20210123A. Aside from this, their parameters differ significantly. For instance, FRB 20210123A's pulse width is about double that of FRB 20210202D's. In comparison, the histogram of the sub-burst separations in the CHIME catalogue 1 sample peaks around ~10 ms with only two bursts above 30 ms and a maximum separation near 65 ms. Out of those, repeaters seems to show somewhat larger values (Pleunis et al. 2021). The ~200 ms separations in the two MeerTRAP FRBs is significantly larger than this, which supports the idea that they are faint repeat pulses.

Why are their post-cursor separations almost precisely the same? We are not aware of any obvious instrumental reasons for why that should be the case. This has neither been seen so far in other MeerTRAP FRBs nor any of thousands of pulsar or RRAT pulses. A shift in arrival time of one or multiple frequency sub-bands could happen in exceptional cases when the beam-former nodes get out of sync. However, the sky signal would get shifted in time and not copied. We are currently commissioning a real-time system to write out voltage data whenever FRBs are discovered. This will allow us in the future to test whether similar post-cursors are coherent copies of the primary bursts and determine their polarization properties. It is hard to imagine how a delayed mirror image of the primary burst could be introduced into the data stream, and we conclude that it must be astrophysical. We also caution that these are very small number statistics. None the less, perhaps the 200 ms separation corresponds to an oscillation frequency, activity or rotation period, or any of its harmonics in the FRB progenitor or its emission mechanism. Or maybe it is related to the quasi-periodic sub-components that have been reported in some FRBs (CHIME/FRB Collaboration 2022; Pastor-Marazuela et al. 2022). Most notable here is FRB 20191221A with a closely comparable and statistically significant

component periodicity of 216.8 ms (CHIME/FRB Collaboration 2022). Similarly, it could be a significantly scaled-up version of the quasi-periodic microstructure observed in several radio pulsars (Cordes, Weisberg & Hankins 1990). Alternatively, the post-cursor burst might be an attenuated echo of the primary, for which the separation would correspond to a light travel time difference. The same is true in the case of gravitational lensing of FRBs. For example, our data captures around the bursts are sensitive to FRB millilensing with delays of ~milliseconds and above by intermediate-mass black holes or dark matter halos (Connor & Ravi 2023). The lensed copies of the FRB signal will be fainter than the primary burst, as in our post-cursors. The phenomenon offers exciting prospects for studying cosmology and fundamental physics using FRBs (Zheng et al. 2014; Li et al. 2018a). Our new voltage buffer dump system will allow us to test if that is the case too.

The inter-burst arrival or waiting times between bursts from repeaters are of scientific interest and have been studied by several authors. Already early on it was realized that their bursts arrive often clustered in time (Wang & Yu 2017; Oppermann et al. 2018). For instance, both FRBs 20121102A and 20200120E show clustering seen as bimodality in their waiting time distributions. The short-duration clustering is most relevant for this discussion. In FRB 20121102A, the fast clustering peak occurs around 22–24 ms (Hewitt et al. 2022; Jahns et al. 2023) if sub-bursts are excluded and around 3.4 ms if they are not (Li et al. 2021b). In FRB 20200120E, the fast peak in the waiting time distribution appears around 1 s (Nimmo et al. 2023). Neither values are close to the ~200 ms separation seen here and they differ already significantly among the two repeaters. Hence, it is unclear whether the post-cursors are sub-bursts or repeat pulses. A larger sample of well-constrained repeater waiting time distributions is needed to inform the distinction.

## 5.3 A deficit of low-fluence FRBs

The FRB all-sky rate inferred from the MeerTRAP coherent survey is significantly below that expected from the best-fitting power-law scaling from surveys at higher limiting fluences $\ge 1$ Jy ms, see Fig. 9. Equivalently, the power-law scaling between the MeerTRAP coherent and incoherent surveys is appreciably flatter than that among the high-fluence surveys. The flatter power-law scaling from MeerTRAP extrapolates near the FAST rate at a limiting fluence ~45 times lower. The flattening of the scaling of the FRB all-sky rate with limiting fluence and the apparent deficit of low-fluence FRBs could have important implications for the FRB population and cosmology.

In the following, we discuss several possible explanations for the FRB deficit. (1) The MeerTRAP coherent rate is only based on six CB detections and therefore still in the small number statistics regime. Further detections might either strengthen the trend or reduce the tension with the high-fluence estimates. (2) The rate inferred from the MeerTRAP coherent survey could be slightly underestimated due to the more complex FoV than that of the incoherent survey, perhaps even by a factor of two. However, it is unlikely to be off by an order of magnitude. To illustrate the point, when extrapolating from the best-fitting high-fluence power law down to the MeerTRAP coherent survey fluence limit (shown as a slightly transparent yellow line in Fig. 9), we would expect to detect a rate of $25_{-12}^{+21} \times 10^3$ instead of $8.2_{-4.6}^{+8.0} \times 10^3$ sky$^{-1}$ d$^{-1}$ in the MeerTRAP coherent survey, which translates to about $34_{-16}^{+28}$ FRB CB detections above the completeness threshold instead of 6. Where are those missing FRBs? It seems unlikely that we missed such a large number of FRBs in our detection pipeline. (3) The MeerTRAP
fluence completeness threshold estimates could be systematically off. If both survey completeness limits were shifted by the same amount, the power-law exponent would be preserved. Shifting the CB survey fluence limit up would still mismatch the absolute rate expected from the high-fluence scaling. However, it would steepen the MeerTRAP intra-survey power-law exponent closer to the high-fluence value. (4) The lack of low-fluence FRBs could naturally be explained by a genuine break or turn-over in the rate–fluence threshold relation below 2 Jy ms. It could, for example, indicate that FRBs transition from the Euclidean scaling ($\propto F_c^{-1.5}$) to the constant scaling ($\propto F_c^{-1}$) in that fluence range. Astrophysically, this could be due to the FRB population’s cosmic evolution in redshift or luminosity space or the Universe’s expansion, which both flatten the FRB fluence distribution (Macquart & Ekers 2018). Those effects would only become important for higher-redshift FRBs. Our analysis is consistent with that of James et al. (2019) who hinted at the existence of a low-fluence downturn or equivalently a high-fluence steepening based on an early sample of ASKAP and Parkes FRBs. (5) More simplistically, the more sensitive surveys might detect more distant populations of FRBs, which have shallower fluence distributions. We can see that when comparing the median FRB DMs of various surveys arranged from shallow to deep. The ASKAP sample has a median DM of ~400 pc cm⁻³ (Shannon et al. 2018), the CHIME sample a median DM of ~500 pc cm⁻³ (CHIME/FRB Collaboration 2021), the Parkes sample a median DM of ~900 pc cm⁻³ (Shannon et al. 2018), and the entire MeerTRAP sample considered here has a median DM of ~740 pc cm⁻³.

Irrespective of the origin of the discrepancy, it will be interesting to see whether future MeerTRAP CB discoveries and improved beam or survey modelling reduce the tension to the high-fluence results.

## 6 CONCLUSIONS

In this paper, we presented a sample of three well-localized FRBs discovered with the newly commissioned MeerTRAP transient search instrument at the *MeerKAT* telescope array in South Africa. We analysed their burst properties and showed their localizations within a multiwavelength context. We conclude the following.

Each FRB was discovered in the data from a single coherent tied-array beam. Based on the non-detections in adjacent beams, we localized them to about 1 arcmin² or better. Therefore, they are more precisely localized than about 97 per cent of the currently published FRBs.

All the FRBs occurred in the southern hemisphere, at high absolute Galactic latitudes over ~30 deg.

They have substantial observed DMs between about 609 and 1196 pc cm⁻³, with extragalactic contributions between about 490 and 1100 pc cm⁻³, indicating expected host galaxy redshifts from as low as 0.2 up to about 1.2.

The FRBs have refined S/N values of at least ~15, meaning they are robust detections. On the other hand, their inferred fluences of >0.4 Jy ms place them at the low-fluence end of the known FRB population.

We tried to associate the FRBs to host galaxy candidates from the literature. Our analyses are mostly inconclusive, as several galaxies within the localization regions have non-negligible association probabilities. The exception is FRB 20210408H, for which there are only four host galaxy candidates. We derived a photometric redshift of $z_{phot} = 0.45 \pm 0.08$ for the favoured host ($p(O|x) \simeq 0.35 - 0.53$), galaxy 1 (PS1 ID 74052043311949899). While lower than expected, the redshift is compatible with the FRB’s DM of almost 1196 pc cm⁻³ at the $2\sigma$ level when assuming a moderate host

DM contribution $\ge 150$ pc cm⁻³ and taking into account the uncertainty in the DM–redshift relation. Alternatively, the galaxy might be an unrelated foreground galaxy, and the actual host is not visible in our current imaging data. The probability of an unseen host is 34 per cent.

The FRBs are mostly unresolved in our data due to the broad channelization and the effects of intra-channel dispersive smearing. FRB 20201211A exhibits hints of a marginally significant scattering contribution at the $1\sigma-2\sigma$ level.

FRB 20210202D appears to be followed by a faint post-cursor pulse about 200 ms after the main burst component. The FRB is a good repeater candidate, although it does not show any typical repeater-like characteristics. We speculated that it is a broad-band spiky repeater burst.

Additionally, we analysed the properties of the two simultaneous MeerTRAP transient surveys at L-band based on the entire sample of 11 FRBs discovered by the end of 2021.

We used conventional approaches to estimate fluence completeness thresholds of 0.66 and 3.44 Jy ms for the coherent and incoherent MeerTRAP surveys, respectively.

Between 2019 June and the end of 2021 December, the MeerTRAP instrument spent approximately 317.5 d surveying the sky. Excluding known periods of reduced pipeline performance, and based on the entire FRB sample discovered in that time, we inferred FRB all-sky rates of $8.2_{-4.6}^{+8.0}$, $2.1_{-1.1}^{+1.8}$, and $1.7_{-1.0}^{+1.8} \times 10^3 \text{ sky}^{-1} \text{ d}^{-1}$ at 1.28 GHz above 0.66, 3.44, and 3.44 Jy ms and assuming 50 per cent detection efficiency.

The power-law scaling between the MeerTRAP FRB all-sky rates is flatter than those in the literature obtained at higher limiting fluences $\ge 1$ Jy ms at the $1.4\sigma$ confidence level. There appears to be a shortage of low-fluence FRBs, suggesting a break or turn-over in the rate versus fluence relation below 2 Jy ms. We speculated that the deficit could be progenitor-intrinsic or due to cosmological effects. Perhaps we see signs of progenitor evolution. The MeerTRAP coherent survey is one of the first to systematically explore the FRB population’s low-fluence end. Although the numbers of our current FRB discoveries are limited, the CB cumulative source count distribution within the survey appear to follow the Euclidean $\propto F^{-3/2}$ scaling. The IB counts are significantly flatter, but become consistent with Euclidean if the brightest IB FRB is excluded.

We constrained the repetition rates of the three FRBs to less than 4.3, 23.4, and 5.2 bursts per day at the 95 per cent confidence level. If we include FRB 20210202D’s post-cursor as a genuine repeat pulse, its detection rate is $\sim 10_{-8}^{+21}$ bursts per day above our detection threshold at the 95 per cent Poisson confidence level.

No clear band-limited FRBs were discovered. This suggests that they are scarce for our observing setup compared with FRBs with more band-filling emission. Their inferred all-sky rate must be less than 890 sky⁻¹ d⁻¹, i.e. less than about 40 per cent of the incoherent survey rate above a limiting fluence of 3.44 Jy ms.

## ACKNOWLEDGEMENTS

We thank the reviewer for their insightful comments and questions that significantly improved the paper. The authors thank the *MeerKAT* Large Survey Project teams for allowing MeerTRAP to observe commensally. FJ, MCB, MC, LND, MM, VM, KMR, SS, BWS, and MPS acknowledge funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 694745). We also acknowledge the use of TRAPUM infrastructure funded and installed by the
fluence completeness threshold estimates could be systematically off. If both survey completeness limits were shifted by the same amount, the power-law exponent would be preserved. Shifting the CB survey fluence limit up would still mismatch the absolute rate expected from the high-fluence scaling. However, it would steepen the MeerTRAP intra-survey power-law exponent closer to the high-fluence value. (4) The lack of low-fluence FRBs could naturally be explained by a genuine break or turn-over in the rate–fluence threshold relation below 2 Jy ms. It could, for example, indicate that FRBs transition from the Euclidean scaling ($\propto F_c^{-1.5}$) to the constant scaling ($\propto F_c^{-1}$) in that fluence range. Astrophysically, this could be due to the FRB population’s cosmic evolution in redshift or luminosity space or the Universe’s expansion, which both flatten the FRB fluence distribution (Macquart & Ekers 2018). Those effects would only become important for higher-redshift FRBs. Our analysis is consistent with that of James et al. (2019) who hinted at the existence of a low-fluence downturn or equivalently a high-fluence steepening based on an early sample of ASKAP and Parkes FRBs. (5) More simplistically, the more sensitive surveys might detect more distant populations of FRBs, which have shallower fluence distributions. We can see that when comparing the median FRB DMs of various surveys arranged from shallow to deep. The ASKAP sample has a median DM of ~400 pc cm⁻³ (Shannon et al. 2018), the CHIME sample a median DM of ~500 pc cm⁻³ (CHIME/FRB Collaboration 2021), the Parkes sample a median DM of ~900 pc cm⁻³ (Shannon et al. 2018), and the entire MeerTRAP sample considered here has a median DM of ~740 pc cm⁻³.

Irrespective of the origin of the discrepancy, it will be interesting to see whether future MeerTRAP CB discoveries and improved beam or survey modelling reduce the tension to the high-fluence results.

# 6 CONCLUSIONS

In this paper, we presented a sample of three well-localized FRBs discovered with the newly commissioned MeerTRAP transient search instrument at the *MeerKAT* telescope array in South Africa. We analysed their burst properties and showed their localizations within a multiwavelength context. We conclude the following.

Each FRB was discovered in the data from a single coherent tied-array beam. Based on the non-detections in adjacent beams, we localized them to about 1 arcmin² or better. Therefore, they are more precisely localized than about 97 per cent of the currently published FRBs.

All the FRBs occurred in the southern hemisphere, at high absolute Galactic latitudes over ~30 deg.

They have substantial observed DMs between about 609 and 1196 pc cm⁻³, with extragalactic contributions between about 490 and 1100 pc cm⁻³, indicating expected host galaxy redshifts from as low as 0.2 up to about 1.2.

The FRBs have refined S/N values of at least ~15, meaning they are robust detections. On the other hand, their inferred fluences of >0.4 Jy ms place them at the low-fluence end of the known FRB population.

We tried to associate the FRBs to host galaxy candidates from the literature. Our analyses are mostly inconclusive, as several galaxies within the localization regions have non-negligible association probabilities. The exception is FRB 20210408H, for which there are only four host galaxy candidates. We derived a photometric redshift of $z_{phot} = 0.45 \pm 0.08$ for the favoured host ($p(O|x) \simeq 0.35 - 0.53$), galaxy 1 (PS1 ID 74052043311949899). While lower than expected, the redshift is compatible with the FRB’s DM of almost 1196 pc cm⁻³ at the $2\sigma$ level when assuming a moderate host

DM contribution $\ge 150$ pc cm⁻³ and taking into account the uncertainty in the DM–redshift relation. Alternatively, the galaxy might be an unrelated foreground galaxy, and the actual host is not visible in our current imaging data. The probability of an unseen host is 34 per cent.

The FRBs are mostly unresolved in our data due to the broad channelization and the effects of intra-channel dispersive smearing. FRB 20201211A exhibits hints of a marginally significant scattering contribution at the $1\sigma-2\sigma$ level.

FRB 20210202D appears to be followed by a faint post-cursor pulse about 200 ms after the main burst component. The FRB is a good repeater candidate, although it does not show any typical repeater-like characteristics. We speculated that it is a broad-band spiky repeater burst.

Additionally, we analysed the properties of the two simultaneous MeerTRAP transient surveys at L-band based on the entire sample of 11 FRBs discovered by the end of 2021.

We used conventional approaches to estimate fluence completeness thresholds of 0.66 and 3.44 Jy ms for the coherent and incoherent MeerTRAP surveys, respectively.

Between 2019 June and the end of 2021 December, the MeerTRAP instrument spent approximately 317.5 d surveying the sky. Excluding known periods of reduced pipeline performance, and based on the entire FRB sample discovered in that time, we inferred FRB all-sky rates of $8.2_{-4.6}^{+8.0}$, $2.1_{-1.1}^{+1.8}$, and $1.7_{-1.0}^{+1.8} \times 10^3 \text{ sky}^{-1} \text{ d}^{-1}$ at 1.28 GHz above 0.66, 3.44, and 3.44 Jy ms and assuming 50 per cent detection efficiency.

The power-law scaling between the MeerTRAP FRB all-sky rates is flatter than those in the literature obtained at higher limiting fluences $\ge 1$ Jy ms at the $1.4\sigma$ confidence level. There appears to be a shortage of low-fluence FRBs, suggesting a break or turn-over in the rate versus fluence relation below 2 Jy ms. We speculated that the deficit could be progenitor-intrinsic or due to cosmological effects. Perhaps we see signs of progenitor evolution. The MeerTRAP coherent survey is one of the first to systematically explore the FRB population’s low-fluence end. Although the numbers of our current FRB discoveries are limited, the CB cumulative source count distribution within the survey appear to follow the Euclidean $\propto F^{-3/2}$ scaling. The IB counts are significantly flatter, but become consistent with Euclidean if the brightest IB FRB is excluded.

We constrained the repetition rates of the three FRBs to less than 4.3, 23.4, and 5.2 bursts per day at the 95 per cent confidence level. If we include FRB 20210202D’s post-cursor as a genuine repeat pulse, its detection rate is $\sim 10_{-8}^{+21}$ bursts per day above our detection threshold at the 95 per cent Poisson confidence level.

No clear band-limited FRBs were discovered. This suggests that they are scarce for our observing setup compared with FRBs with more band-filling emission. Their inferred all-sky rate must be less than $890 \text{ sky}^{-1} \text{ d}^{-1}$, i.e. less than about 40 per cent of the incoherent survey rate above a limiting fluence of 3.44 Jy ms.

# ACKNOWLEDGEMENTS

We thank the reviewer for their insightful comments and questions that significantly improved the paper. The authors thank the *MeerKAT* Large Survey Project teams for allowing MeerTRAP to observe commensally. FJ, MCB, MC, LND, MM, VM, KMR, SS, BWS, and MPS acknowledge funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 694745). We also acknowledge the use of TRAPUM infrastructure funded and installed by the
Max-Planck-Institut für Radioastronomie and the Max-Planck-
Gesellschaft. We thank Chris Williams for his assistance in getting
the MeerTRAP pipeline up and running. MC acknowledges the
support of an Australian Research Council Discovery Early Career
Research Award (project number DE220100819) funded by the
Australian Government and the Australian Research Council Centre
of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D)
through project number CE170100013. KMR acknowledges support
from the Vici research programme 'ARGO' with project number
639.043.815, financed by the Dutch Research Council (NWO).
JXP as a member of the Fast and Fortunate for FRB Follow-up
team, acknowledges support from the National Science Foundation
(NSF) grants AST-1911140 and AST-1910471. Partially based on
observations made at the Cerro Tololo Inter-American Observatory at
NSF's NOIRLab as part of The Huntsman Survey (2018B-0904; PI:
L. Spitler), which is managed by the Association of Universities for
Research in Astronomy (AURA) under a cooperative agreement with
the NSF. The MeerKAT telescope is operated by the South African
Radio Astronomy Observatory, which is a facility of the National
Research Foundation, an agency of the Department of Science and
Innovation.

DATA AVAILABILITY

The localization region files, HEALPIX localization maps, and the
meta data for the three MeerTRAP FRB discoveries presented in this
paper are available from our Zenodo repository at https://doi.org/
10.5281/zenodo.6047539. Other data underlying this article will be
shared upon reasonable request to the corresponding author.

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Caleb M. et al., 2022, Nature Astron., 6, 828

Caleb M. et al., 2023, MNRAS in press

Carels C., Adámek K., Novotný J., Armour W., Ouannoughi N., Dimoudi
S., 2019, AstroAccelerate: Accelerated Software Package for Processing
Time-domain Radio Astronomy Data, ascl:1912.010

Chambers K. C. et al., 2016, preprint (arXiv:1612.05560)

Champion D. J. et al., 2016, MNRAS, 460, L30

Chatterjee S. et al., 2017, Nature, 541, 58

Chen W., Barr E., Karuppusamy R., Kramer M., Stappers B., 2021, J. Astron. Instrum., 10, 2150013

Connor L., Ravi V., 2023, MNRAS 521 4024

Cordes J. M., Chatterjee S., 2019, ARA&A, 57, 417

Cordes J. M., Lazio T. J. W., 2001, ApJ, 549, 997

Cordes J. M., Lazio T. J. W., 2002, preprint(arXiv:0207156)

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de Blok W. J. G. et al., 2018, in Proceedings of MeerKAT Science: On the Pathway to the SKA - PoS(MeerKAT2016). p. 007

de Villiers M. S., Cotton W. D., 2022, AJ, 163, 135

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Foreman-Mackey D., Hogg D. W., Lang D., Goodman J., 2013, PASP, 125, 306

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Gupta V. et al., 2021, MNRAS, 501, 2316

Haslam C. G. T., Salter C. J., Stoffel H., Wilson W. E., 1982, A&AS, 47, 1

Heintz K. E. et al., 2020, ApJ, 903, 152

Hessels J. et al., 2019, ApJ, 876, L23

Hewitt D. M. et al., 2022, MNRAS, 515, 3577

Hosenie Z., 2021, PhD thesis, The University of Manchester

Jahns J. N. et al., 2023, MNRAS, 519, 666

James C. W., Ekers R. D., Macquart J. P., Bannister K. W., Shannon R. M., 2019, MNRAS, 483, 1342

James C. W. et al., 2022, MNRAS, 516, 4862

Jankowski F., 2022, Scatfit: Scattering Fits of Time Domain Radio Signals (Fast Radio Bursts or pulsars), Astrophysics Source Code Library, ascl:2208.003

Jankowski F., van Straten W., Keane E. F., Bailes M., Barr E. D., Johnston
S., Kerr M., 2018, MNRAS, 473, 4436

Jankowski F. et al., 2022, in Ruiz J. E., Pierfedereci F., Teuben Peds, ASP Conf. Ser. Vol. 532, Astron. Soc. Pac., San Francisco. p. 273

Jonas J., MeerKATTeam, 2018, in Proceedings of MeerKAT Science: On the Pathway to the SKA - PoS(MeerKAT2016). p. 1

Keane E. F., Petroff E., 2015, MNRAS, 447, 2852

Keane E. F. et al., 2018, MNRAS, 473, 116

Kocz J. et al., 2019, MNRAS, 489, 919

Kouwenhoven M. L. A., Voûte J. L. L., 2001, A&A, 378, 700

Kron R. G., 1980, ApJS, 43, 305

Kumar P. et al., 2021, MNRAS, 500, 2525

Law C. J. et al., 2020, ApJ, 899, 161

Lawson K. D., Mayer C. J., Osborne J. L., Parkinson M. L., 1987, MNRAS,
225, 307

Levenberg K., 1943, Q. Appl. Math., 1, 536

Li Z.-X., Gao H., Ding X.-H., Wang G.-J., Zhang B., 2018a, Nature Commun., 9, 3833

Li D. et al., 2018b, IEEE Microw. Mag., 19, 112

Li C. K. et al., 2021a, Nature Astron., 5, 378

Li D. et al., 2021b, Nature, 598, 267

Löhmer O., Kramer M., Mitra D., Lorimer D. R., Lyne A. G., 2001, ApJ, 562, L157

Lorimer D. R., Bailes M., McLaughlin M. A., Narkevic D. J., Crawford F., 2007, Science, 318, 777
Max-Planck-Institut für Radioastronomie and the Max-Planck-
Gesellschaft. We thank Chris Williams for his assistance in getting
the MeerTRAP pipeline up and running. MC acknowledges the
support of an Australian Research Council Discovery Early Career
Research Award (project number DE220100819) funded by the
Australian Government and the Australian Research Council Centre
of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D)
through project number CE170100013. KMR acknowledges support
from the Vici research programme 'ARGO' with project number
639.043.815, financed by the Dutch Research Council (NWO).
JXP as a member of the Fast and Fortunate for FRB Follow-up
team, acknowledges support from the National Science Foundation
(NSF) grants AST-1911140 and AST-1910471. Partially based on
observations made at the Cerro Tololo Inter-American Observatory at
NSF's NOIRLab as part of The Huntsman Survey (2018B-0904; PI:
L. Spitler), which is managed by the Association of Universities for
Research in Astronomy (AURA) under a cooperative agreement with
the NSF. The MeerKAT telescope is operated by the South African
Radio Astronomy Observatory, which is a facility of the National
Research Foundation, an agency of the Department of Science and
Innovation.

DATA AVAILABILITY

The localization region files, HEALPIX localization maps, and the
meta data for the three MeerTRAP FRB discoveries presented in this
paper are available from our Zenodo repository at https://doi.org/
10.5281/zenodo.6047539. Other data underlying this article will be
shared upon reasonable request to the corresponding author.

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APPENDIX A: VERIFICATION OF OUR FRB SOURCE COUNT ESTIMATION METHOD

We tested how accurately our analysis method estimated the scaling
indices of the cumulative or integral source count distributions with
emphasis on the low number regime. We did that by randomly
drawing synthetic FRB fluence data from Pareto distributions with
parameters close to those of our data sets. Specifically, we used
values of (0.66, 1.9) and (3.44, 0.8) for the $x_m$ and $\alpha$ parameters in
equation (16) for the simulated CB and IB fluences. These Pareto
indices $\alpha$ correspond to power-law exponents $\delta$ of -1.9 and -0.8,
respectively. We successively drew 50, 10, and 6 random samples
from each distribution with an equal number for each simulated
survey and ran those synthetic data through our estimation software.
We repeated the process 240 to 330 times to check the spread in
returned measurements. For 50 FRBs, the recovered values are well
within the $1\sigma$ fit errors from the injected ones, with a sample spread
of 0.1 (IB) and 0.3 (CB) standard deviations. For 10 FRBs, the
sample variation becomes more significant as the probability of
missing the rare high-fluence events increases, especially for the
steeper CB distribution. The median recovered $\alpha$ values are steeper
(higher) than the injected parameters by 0.2 and 0.3. The sample
standard deviations are 0.4 and 0.8, i.e. there is a significant scatter
towards steeper indices in the CB sample. When bright bursts are
present, the recovered indices match the injected values within the fit
uncertainties. For six FRBs, the fit errors are appreciably larger than
before. In most cases, the estimated indices are compatible with the
injected values within the $1\sigma$ uncertainties. The sample medians are
steeper by 0.3 (IB) and 0.7 (CB), and the sample standard deviations
amount to 0.9 and 1.4, respectively. In summary, detecting the rare
bright FRBs is crucial to accurately characterize the population's
fluence distribution. Without them, the measurements are biased
to exponents that are too steep with respect to the true underlying
distribution.

This paper has been typeset from a TeX/LA TeX file prepared by the author.
Lupton R., Blanton M. R., Fekete G., Hogg D. W., O'Mullane W., Szalay A., Wherry N., 2004, PASP, 116, 133

McKinnon M. M., 2014, PASP, 126, 476

Macquart J. P., Ekers R., 2018, MNRAS, 480, 4211

Macquart J.-P. et al., 2010, PASA, 27, 272

Macquart J. P., Shannon R. M., Bannister K. W., James C. W., Ekers R. D., Bunton J. D., 2019, ApJ, 872, L19

Macquart J. P. et al., 2020, Nature, 581, 391

Malenta M. et al., 2020, in Pizzo R., Deul E. R., Mol J. D., de Plaa J., Verkouter H. eds, ASP Conf. Ser. Vol. 527, Astronomical Data Analysis Software and Systems XXIX, Astron. Soc. Pac., San Francisco. p. 457

Marcote B. et al., 2020, Nature, 577, 190

Marcote B., Kirsten F., Hessels J., Nimmo K., Paragi Z., Project P., 2022, in Proceedings of European VLBI Network Mini-Symposium and Users' Meeting 2021 - PoS(EVN2021). p. 35

Marquardt D. W., 1963, J. Soc. Ind. Appl. Math., 11, 431

Marthi V. R., Gautam T., Li D. Z., Lin H. H., Main R. A., Naidu A., Pen U. L., Wharton R. S., 2020, MNRAS, 499, L16

Marthi V. R. et al., 2022, MNRAS, 509, 2209

Mauch T. et al., 2020, ApJ, 888, 61

Morello V., Rajwade K. M., Stappers B. W., 2022, MNRAS, 510, 1393

Newville M., Stensitzki T., Allen D. B., Rawlik M., Ingargiola A., Nelson A., 2016, Lmfit: Non-Linear Least-Square Minimization and Curve-Fitting for Python, ascl:1606.014

Nimmo K. et al., 2023, MNRAS 520 2281

Niu C.-H. et al., 2021, ApJ, 909, L8

Niu C. H. et al., 2022, Nature, 606, 873

Onken C. A. et al., 2019, PASA, 36, e033

Oppermann N., Yu H.-R., Pen U.-L., 2018, MNRAS, 475, 5109

Oswald L. S. et al., 2021, MNRAS, 504, 1115

Parent E. et al., 2020, ApJ, 904, 92

Park R. S., Folkner W. M., Williams J. G., Boggs D. H., 2021, AJ, 161, 105

Pastor-Marazuela I. et al., 2022, preprint (arXiv:2202.08002)

Petroff E., Hessels J. W. T., Lorimer D. R., 2019, A&AR, 27, 4

Petroff E., Hessels J. W. T., Lorimer D. R., 2022, A&AR, 30, 2

Planck Collaboration, 2020, A&A, 641, A6

Platts E., Weltman A., Walters A., Tendulkar S. P., Gordin J. E. B., Kandhai S., 2019, Phys. Rep., 821, 1

Platts E. et al., 2021, MNRAS, 505, 3041

Pleunis Z. et al., 2021, ApJ, 923, 1

Rajwade K. et al., 2020, in Evans C. J., Bryant J. J., Moto-hara K., eds, Proc. SPIE Conf. Ser. Vol. 11447, Ground-based and Airborne Instrumentation for Astronomy VIII, SPIE, Bellingham, p. 114470J

Rajwade K. M. et al., 2022, MNRAS, 514, 1961

Rane A., Lorimer D. R., Bates S. D., McMann N., McLaughlin M. A., Rajwade K., 2016, MNRAS, 455, 2207

Remazeilles M., Dickinson C., Banday A. J., Bigot-Sazy M. A., Ghosh T., 2015, MNRAS, 451, 4311

Salvatier J., Wiecki T. V., Fonnesbeck C., 2016, PyMC3: Python Probabilistic Programming Framework, Astrophysics Source Code Library, ascl:1610.016

Sanidas S., Caleb M., Driessen L., Morello V., Rajwade K., Stappers B. W., 2018, in Weltevrede P., Perera B. B. P., Preston L. L., Sanidas S., eds, S337 Vol. 13, Proceedings of the International Astronomical Union. Symposium S337: Pulsar Astrophysics the Next Fifty Years, Cambridge University Press, Cambridge, p. 406

Schlegel D. J., Finkbeiner D. P., Davis M., 1998, ApJ, 500, 525

Shannon R. M. et al., 2018, Nature, 562, 386

Shin K. et al., 2023, ApJ, 944, 105

Spitler L. G. et al., 2014, ApJ, 790, 101

Spitler L. G. et al., 2016, Nature, 531, 202

Tarrío P., Zarattini S., 2020, A&A, 642, A102

Thornton D. et al., 2013, Science, 341, 53

Tonry J. L. et al., 2012, ApJ, 750, 99

van Leeuwen J., 2014, in Wozniak P. R., Graham M. J., Mahabal A. A., Seaman R., eds, Proceedings of The Third Hot-wiring the Transient Universe Workshop, eConf C000821 (2013), p. 79

van Leeuwen J. et al., 2023, A&A, 672, A117

van Straten W., Jameson A., Oslowski S., 2021, PSRDADA: Distributed Acquisition and Data Analysis for Radio Astronomy, ascl:2110.003

Vanderlinde K. et al., 2019, in Canadian Long Range Plan for Astronomy and Astrophysics White Papers. p. 28

Wang F. Y., Yu H., 2017, J. Cosmol. Astropart. Phys., 2017, 023

Yamasaki S., Totani T., 2020, ApJ, 888, 105

Yao J. M., Manchester R. N., Wang N., 2017, ApJ, 835, 29

Zhang B., 2018, ApJ, 867, L21

Zheng Z., Ofek E. O., Kulkarni S. R., Neill J. D., Juric M., 2014, ApJ, 797, 71

## APPENDIX A: VERIFICATION OF OUR FRB SOURCE COUNT ESTIMATION METHOD

We tested how accurately our analysis method estimated the scaling indices of the cumulative or integral source count distributions with emphasis on the low number regime. We did that by randomly drawing synthetic FRB fluence data from Pareto distributions with parameters close to those of our data sets. Specifically, we used values of (0.66, 1.9) and (3.44, 0.8) for the $x_m$ and $\alpha$ parameters in equation (16) for the simulated CB and IB fluences. These Pareto indices $\alpha$ correspond to power-law exponents $\delta$ of -1.9 and -0.8, respectively. We successively drew 50, 10, and 6 random samples from each distribution with an equal number for each simulated survey and ran those synthetic data through our estimation software. We repeated the process 240 to 330 times to check the spread in returned measurements. For 50 FRBs, the recovered values are well within the $1\sigma$ fit errors from the injected ones, with a sample spread of 0.1 (IB) and 0.3 (CB) standard deviations. For 10 FRBs, the sample variation becomes more significant as the probability of missing the rare high-fluence events increases, especially for the steeper CB distribution. The median recovered $\alpha$ values are steeper (higher) than the injected parameters by 0.2 and 0.3. The sample standard deviations are 0.4 and 0.8, i.e. there is a significant scatter towards steeper indices in the CB sample. When bright bursts are present, the recovered indices match the injected values within the fit uncertainties. For six FRBs, the fit errors are appreciably larger than before. In most cases, the estimated indices are compatible with the injected values within the $1\sigma$ uncertainties. The sample medians are steeper by 0.3 (IB) and 0.7 (CB), and the sample standard deviations amount to 0.9 and 1.4, respectively. In summary, detecting the rare bright FRBs is crucial to accurately characterize the population's fluence distribution. Without them, the measurements are biased to exponents that are too steep with respect to the true underlying distribution.

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