2.1. Implementation in the zDM code

The zDM code was originally developed for the analysis of James et al. (Reference James2022b), and has since been extended as per James et al. (Reference James2022c) and Baptista et al. (Reference Baptista2023). It calculates the expected number of FRBs from an FRB survey as a function of their redshift z, dispersion measure DM, and relative fluence F compared to threshold fluence $F_{\rm th}$ . The key parameters of the code, and their current best-fit values, are given in Table 1.

Table 1. FRB population parameter sets used in this work. Shown are the best-fit parameter sets from Shin et al. (Reference Shin2023) and James et al. (Reference James2022c), and a set of 12 parameter sets from James et al. (Reference James2022c) when each parameter takes the minimum/maximum value within its 90% confidence interval. The p-values from KS tests against the observed rate of CHIME single bursts is given as $p_{\rm KS}$ for different sets of FRB population parameters.

$^{\rm a}$ Downturn ( $\sim$ ‘maximum’) FRB energy, assuming a 1 GHz rest-frame emission bandwidth.

$^{\rm b}$ Frequency scaling of rate: $C \propto \nu^\alpha$ .

$^{\rm c}$ Cumulative power-law index of the FRB luminosity function.

$^{\rm d}$ Scaling of FRB density with star-formation: $C \propto {\rm SFR}^{{n_{\rm sfr}}}$ .

$^{\rm e}$ Log-mean of host galaxy DM contribution.

$^{\rm f}$ Log-standard deviation of host galaxy DM contribution.

$^{\rm g}$ Absolute total burst rate above $10^{39}$ erg at 1.3 GHz at $z=0$ .

$^{\rm h}$ p-value from the KS test assuming the distributed repeaters scenario.

$^{\rm i}$ p-value from the KS test assuming the strong repeaters scenario.

For this analysis, only the measured values of z and DM for an FRB are considered—unlike previous works, where signal-to-noise ratio (SNR) is also used. I add an additional observable: whether or not an FRB is observed as a repeater. This is treated as a boolean, that is, the number of observed repeats is not used. Furthermore, only repetition information obtained through initial blind surveys is considered, that is, this analysis is not suited to modelling FRBs determined to repeat or not through targeted follow-up observations. While CHIME remains the only FRB instrument to observe repeat bursts by this definition, the non-observation of repeaters in other blind surveys can still be included.

The total number of repeaters in any given z–DM bin is given by:

(7) \begin{eqnarray}N(z,{\rm DM}) & = & C_r(z) \frac{dV}{dz d\Omega} \Delta z f_z({\rm DM}) \Delta {\rm DM} \Delta \Omega,\end{eqnarray}

where $\Delta \Omega$ is the solid angle observed by the beam, $\Delta z$ and $\Delta$ DM are the bin sizes of the z–DM grid, $dV (dz d\Omega)^{-1}$ is the size of the cosmological volume element, and $f_z({\rm DM})$ is the distribution of DMs of FRBs at that redshift (itself a function of cosmological and host galaxy contributions, as given in James et al. Reference James2022b).

The distribution of intrinsic FRB rates within that volume element is given by (4), which for any given threshold energy ${E_{\rm th}}$ , produces a distribution in $R_{\rm obs}$ as per (2). Thus the expected number of once-off FRBs from that volume becomes

(8) \begin{align}& \langle N_1(z,{\rm DM}) \rangle =\frac{dV}{dz d\Omega} \Delta z f_z({\rm DM}) \Delta {\rm DM} \Delta \Omega \\ &\cdot \int_{{R_{\rm min}}}^{{R_{\rm max}}} R_{\rm obs} T_{\rm obs} e^{-R_{\rm obs} T_{\rm obs}} \frac{d \Phi_r(R)}{d R} \frac{dR}{dR_{\rm obs}} dR_{\rm obs}. \nonumber\end{align}

Here, I have left off the dependence of R on $R_{\rm obs}$ , expressed through (2). Similarly,

(9) \begin{align}& \langle N_0(z,{\rm DM}) \rangle =\frac{dV}{dz d\Omega}\Delta z f_z({\rm DM}) \Delta {\rm DM} \Delta \Omega \\&\cdot \int_{{R_{\rm min}}}^{{R_{\rm max}}} e^{-R_{\rm obs} T_{\rm obs}} \frac{d \Phi_r(R)}{d R}\frac{dR}{dR_{\rm obs}} dR_{\rm obs} \nonumber\end{align}

calculates the number of progenitors for which no bursts are detected. Thus, the expected number of repeating FRBs can be deduced as

(10) \begin{align} \langle N_{\rm reps} \rangle = N - \langle N_1\rangle - \langle N_0\rangle .\end{align}

This therefore allows a likelihood to be assigned to both once-off and repeating FRBs as a function of redshift z and dispersion measure DM.

4.1. Data sample

Data is taken from CHIME/FRB Collaboration et al. (2021) (hereafter, ‘ ${\rm Cat1}$ ’), consisting of 536 FRBs. Nominally, this is divided into 474 once-off bursts, and 62 bursts from 18 repeaters. However, two repeaters—FRB 20190417A, and FRB 20181119D from FRB 20121102A—are only identified as such from observations external to ${\rm Cat1}$ . Thus it is more proper to say that ${\rm Cat1}$ identifies 16 repeaters and 476 non-repeaters, which is the statistic used in this work.

CHIME have also recently published a search for repeating FRBs in an updated dataset, announcing the discovery of 25 new sources based on coincidences in DM–localisation space (CHIME/FRB Collaboration et al. 2023). I denote this the ‘Gold25’ sample, and discuss this in detail—including why it is not used for fitting—in Section 8.3.

The analysis in ${\rm Cat1}$ used three data quality cuts which are not implemented here, for the following reasons:

1. 1. Cut removing events with ${{\rm SNR}} < 12$ . This is implemented due to human inspection falsely rejecting low- ${{\rm SNR}}$ events. Since this work is primarily concerned with the ratio of repeaters to non-repeaters, rather than absolute FRB numbers, such a cut is not used. Repetitions were searched for at a slightly lower (but undocumented) threshold than initial bursts (CHIME/FRB Collaboration et al. 2021,; see Section 8.3 for further discussion of this effect) and using the ${{\rm SNR}} \ge 12$ cut would eliminate this effect. However, implementing it would leave only 7 of 16 repeating FRBs. I believe that the resulting loss of precision will be worse than the associated systematic error.

2. 2. Cut removing events with ${\rm DM} < 1.5\,{\rm max}({\rm DM}_{\rm NE2001},$ ${\rm DM}_{\rm YMW16})$ . Here, such events are retained—it is true that ${{\rm DM}_{\rm MW}}$ is poorly known, but this also means that the effect of such a cut is equally unknown.

3. 3. Far-sidelobe events. These will have poor localisation, and are identified using the FRB spectrum. However, this work is not concerned with localisation accuracy, but rather the number of FRBs detected, for which precise sidelobe details are not as important. Thus such a cut is not implemented.

In the following analysis, the declination $\delta$ of each FRB, its DM, and whether or not it has been observed to repeat is used.

4.2. Beamshape

The CHIME beamshape is both unique and complex, with a broad primary beam with full-width at half-power of approximately 60 $^\circ$ –120 $^\circ$ in the N–S direction and $1.3^{\circ}$ – $2.5^{\circ}$ in the E–W direction (CHIME/FRB Collaboration et al. 2018, 2021). Within this envelope are 1024 coherently formed beams used for FRB detection, with full-width, half-max (FWHM) of 20’–40’. I use the frequency-dependent beamshape given in CHIME/FRB Collaboration et al. (2021), averaged over XX and YY polarisations, and implemented in the GitHub library chime-frb-beam-model.Footnote a This is sampled at 16 frequencies in a $300 \times 1000$ grid in RA, DEC, taking points within $8^{\circ}$ of the meridian. The final beamshape is taken as the envelope over all $1024$ formed beams after averaging over frequency. I discuss an alternative method in Appendix A.

Figure 3. Top: declination-dependent exposure of the CHIME experiment—simulation from this work based on the beamshape described in CHIME/FRB Collaboration et al. (2021) and scaled using a total of 220 d’s observation time, ‘CHIME’ taken directly from CHIME/FRB Collaboration et al. (2021). Bottom: $T(\overline{B})$ , calculated from the simulated beam pattern, and averaged over the indicated declination ranges.

In James et al. (Reference James2022b), a telescope’s beam pattern on sky is described via the ‘inverse beamshape’, $\Omega(B)$ , being the solid angle of sky viewed at any given sensitivity. The total time observing that solid angle is then a simple scaling constant between rate and the number of observes FRBs. When considering the response of a transit instrument such as CHIME to repeating FRBs, the relevant metric is T(B), being the time spent observing a given sky position at beam sensitivity B (relative to the nominal sensitivity at beam centre, where $B=1$ ). For a source at a given declination, T(B) can be calculated from the RA-dependence of the beam pattern—this is similar to the approach used by Gardenier et al. (Reference Gardenier, Connor, van Leeuwen, Oostrum and Petroff2021).

4.2.1. Declination dependence

I calculate the declination dependence of CHIME’s beamshape by calculating T(B) for 1000 declinations between $-11^{\circ}$ and $+90^{\circ}$ . The effective exposure, $T_{\rm eff}$ , is defined as the sum of times spent observing the source $T_i$ , weighted by beam sensitivity b, assuming a Euclidean distribution of event rates:

(11) \begin{eqnarray}T_{\rm eff} & = & \sum_i T_i B^{1.5}. \end{eqnarray}

It is calculated for each declination as above, using discrete samples of source position (hence a sum, rather than an integral, in (11)), and is shown in Fig. 3. The oscillatory behaviour is due to the 256 rows of tied beams, with sources passing over beam centre having significantly more exposure than those passing between beams. In the region $\delta \gtrsim 70^{\circ}$ , the exposure is additionally increased by CHIME viewing sources transiting twice daily, both sides of the North Celestial Pole (NCP). Also plotted is the CHIME exposure, taken from CHIME/FRB Collaboration et al. (2021), defined as the time in which a source is within the full-width half-max (FWHM) of a tied beam at 600 MHz. This is used to normalise the total effective observation time to 311 d. by using CHIME’s simulation of the 600 MHz tied beam, measuring the time a source spends within the FWHM, and fitting to the published CHIME exposure. This total effective time is less than the 342 d duration of the catalogue, which is expected when allowing for equipment down-time. Since CHIME’s total exposure is a function only of the tied beamshape, it is therefore significantly greater than $T_{\rm eff}$ at low declinations where the primary beam is less sensitive.

For calculation purposes, declination is divided into six regions, spanning the full $[{-}11^{\circ},90^{\circ}]$ range. The bounds of each region are chosen so that the change in exposure between regions is not too large (at most a factor of three), while preserving reasonable statistics in each region. The mean exposures in each region are also shown as histograms in Fig. 3.

4.3. Sensitivity

The zDM code by default uses a two-dimensional function of FRB width w and DM to calculate the effective threshold $F_{\rm th}$ to an FRB with those properties. An important input into this is the telescope’s time- and frequency resolution used for FRB searches: in the case of CHIME, 0.983 ms, and 0.0244 MHz respectively (CHIME/FRB Collaboration et al. 2021). Integrating over a modelled distribution of intrinsic FRB widths w (James et al. Reference James2022b) and intrinsic scattering measures $\tau$ (CHIME/FRB Collaboration et al. 2021; James et al. Reference James2022c), this produces an efficiency function $\epsilon$ that acts to decrease measured ${{\rm SNR}}$ or equivalently, increase the effective detection threshold $F_{\rm th}$ above a nominal threshold $F_0$ in a DM-dependent manner. Including the effects of the beam B, $F_{\rm th}$ is given by

(12) \begin{eqnarray}F_{\rm th} & = & \frac{F_0}{B \epsilon(w,\tau,{\rm DM})}. \end{eqnarray}

CHIME/FRB Collaboration et al. (2021) extend this to a more complicated selection function, $P({{\rm SNR}}|F, DM, \tau, w, \gamma, r)$ , giving ${{\rm SNR}}$ as a function of DM, $\tau$ , w, F, and also including spectral shape parameters $\gamma$ and r. This is based on a sophisticated pulse injection system (Merryfield et al. 2022), and thus accounts for not only the dedispersion code, bonsai, but also the full detection pipeline, including RFI rejection. The full selection function P is not published, although it can presumably be inferred from the published library of injected pulses. Rather, one-dimensional selection functions, integrated over all other variables and modelled population probability distributions in those variables, are given.

The DM selection function, $s({\rm DM})$ , gives the relative fraction of FRBs passing selection cuts as a function of DM—which is precisely what is required to model CHIME’s DM distribution. To use CHIME’s bias selection function, it is fit with a $4{\rm th}$ -order polynomial as shown in Fig. 4. It is then corrected to a peak value of unity, such that $N_{\rm obs} \le N_{\rm true}$ , and converted to a modifier to the measured burst ${{\rm SNR}}$ by assuming a Euclidean relationship between event number and ${{\rm SNR}}$ , that is, ${{\rm SNR}}_{\rm bias} \sim s^{2/3}({\rm DM})$ . This is compared to the efficiency function $\epsilon$ produced by the zDM code, normalised such that the two efficiencies are equal at a DM of 1500 ${{\rm pc \, cm^{-3}}}$ .

Figure 4. DM bias correction for CHIME data. Shown are values of s(DM) from CHIME/FRB Collaboration et al. (2021) (red points), a cubic spline fit (blue solid line), the $4{\rm th}$ order polynomial fit from this work (orange solid line), the renormalised fit (orange dashed line), and implied SNR bias (green dotted line).

There are two clear regimes present in Fig. 4. Above 1000 ${{\rm pc \, cm^{-3}}}$ , the CHIME ${{\rm SNR}}$ bias agrees well with that estimated by zDM, which is indicative of sensitivity being fundamentally limited by the time–frequency resolution of the instrument. Below 1000 ${{\rm pc \, cm^{-3}}}$ , zDM flattens to represent efficient detection, while the efficiency of CHIME decreases. This is likely due to the effects of CHIME’s system for mitigating RFI (CHIME/FRB Collaboration et al. 2021; Merryfield et al. 2022).

To estimate the fluence threshold at beam centre and peak DM efficiency, $F_0$ , CHIME quotes a 95% completeness threshold of 5 Jy ms, compared to a theoretical minimum detection threshold of $\sim$ 1 Jy ms (CHIME/FRB Collaboration et al. 2021). The factor of five between these thresholds is comparable to the factor of 6.6 by which the zDM efficiency had to be increased to match the CHIME efficiency in Fig. 4. Thus the fitted ${{\rm SNR}}$ bias of CHIME’s selection function has been implemented in the zDM code, such that it acts in (12) as an efficiency factor; and $F_0=5$ Jy ms to a 1 ms burst is used. The value of the beam B is parameterised according to Sections 4.2 and 4.2.1, so that repeating FRBs in each declination bin are modelled as being exposed to sensitivity thresholds $F_{\rm th}$ for times $T_{\rm obs}$ .

6.1. Dispersion measure distribution

Within the range allowed by Fig. 7, the z–DM distribution predicted for each will in general be different. To illustrate, I take four scenarios, a–d, from the corners of the allowed region. In each case, the predicted DM distribution of repeating FRBs is plotted in Fig. 8.

Figure 8. Predicted DM distribution of repeating FRBs compared to that from CHIME Catalogue 1 (CHIME/FRB Collaboration et al. 2021), calculated using cases a–d from Fig. 7, and the golden sample of repeaters from CHIME/FRB Collaboration et al. (2023) (renormalised to 16). Note that b and c overlap.

From Fig. 8, it can immediately be seen that strong repeaters are more likely to be found at large distances (higher DM $_{\rm EG}$ ). Case d has a significantly higher distribution of DMs compared to the other three, and it is the only case with a significant number of strong repeaters (cases a and b have low ${R_{\rm max}}$ , while case c has such a steep ${{\gamma_r}}$ that the number of strong repeaters is negligible).

I also compare these DM distributions with those found from the ${\rm Cat1}$ and Gold25 samples. While formally the DM distribution of the Gold25 sample is statistically consistent with that of ${\rm Cat1}$ (CHIME/FRB Collaboration et al. 2023), the distribution is biased (see Section 8.3). That case d well reproduces the Gold25 sample DM distribution therefore should not be taken as evidence for it.

This comparison highlights a prediction of all repeating FRB models, which is the stronger upward skew of the DM distribution compared to single FRBs. This effect is seen in CHIME data, with two (one) high-DM repeaters in the ${\rm Cat1}$ (Gold25) samples.

To quantify agreement in DM space, a KS test using the ${\rm Cat1}$ FRBs and predicted DM distributions over ${R_{\rm max}}$ , ${{\gamma_r}}$ space is performed. The results are shown in Fig. 9.

Figure 9. P-values from a KS test of the DM distribution of ${\rm Cat1}$ repeating FRBs, $p_{\rm ks}({\rm DM}_r)$ , against predictions from models with different values of ${R_{\rm max}}$ and ${{\gamma_r}}$ . Other features are identical to Fig. 7, including cases b and c overlapping.

An implicit assumption of the above analysis is that the intrinsic distribution of ${{\rm DM}_{\rm host}}$ is identical regardless of the repetition rate of repeaters. The observation of PRSs at the locations of at least two bright repeaters (Marcote et al. Reference Marcote2017; Niu et al. Reference Niu2022) suggests that these presumably young objects would be more likely to have a larger ${{\rm DM}_{\rm host}}$ , bearing in mind that this term includes material in the vicinity of the progenitor, as well as the host galaxy’s ISM and halo contributions. This could then be responsible for observing repeating FRBs (which are on-average intrinsically stronger repeaters) to have slightly more DM than expected, and would not constitute hard evidence against the model. This might be the case for the ${{\gamma_r}} \lesssim -2.4$ region of Fig. 9, which is slightly disfavoured because it over-predicts the number of low-DM repeating FRBs. However, should observed repeaters have less DM than expected, this is clear evidence to reject the model. This is the case for the already ruled-out lower region of Fig. 9, which predicts more high-DM repeating FRBs than observed.

6.2. Declination distribution

The declination distribution of CHIME FRBs also holds information that allows us to discriminate between scenarios. For repeating FRBs, the difference between observing a small patch of the sky around the NCP almost continuously, and surveying several steradians near the equator for only a few minutes each day, is very important. Near the equator, only the strongest repeaters will be detected as such, while near the Pole, the small probed volume makes observations subject to cosmic variance.

In Fig. 10, I plot the declination distribution of CHIME once-off and repeating FRBs, and compare this against model predictions. For this plot, the number of declination bins into which CHIME was divided was increased to 30, whereas six declination bins was found to be sufficient to model the total number of repeaters, and their DM distribution.

Figure 10. Cumulative histogram of the CHIME repeating FRB declination ( $\delta$ ) distribution, for both the ${\rm Cat1}$ and Gold25 samples, compared to Monte Carlo predictions from four example cases.

Since the x-axis of Fig. 10 is increasing linearly with $\delta$ , most of the solid angle is concentrated on the left-hand-side of the figure. Despite this, the number of repeaters—both observed and predicted—increases as fast as, or faster, than linearly with $\delta$ . That the Gold25 sample shows the least steep rise with $\delta$ is likely because of the previously discussed bias against high declinations due to the increased background rate. This is evidence that the repeating population is dominated by progenitors with low apparent repetition rates that are best probed with deep observations (i.e. at high declinations), rather than sources with high apparent rates that are best detected in broad shallow surveys (i.e. at low declinations).

Of the four cases analysed, a–c show good agreement with ${\rm Cat1}$ in the $\delta \lesssim 60^{\circ}$ range, while not even d can match the rapid rise in repeater rates above this range. This suggests a simple fluctuation in the data, either a deficit at low declinations, or an excess at high declinatons—though an alternative explanation is the influence of non-Poissonian repetition (see Section 8.1).

I characterise the agreement in declination distributions via a KS test, with associated p-values given in Fig. 11. The greatest discrepancy with data is the aforementioned excess of high- $\delta$ repeaters, and the upper region of the figure is disfavoured because it reproduces this particularly poorly. The lower right region is disfavoured because this predicts mostly bright repeaters that should be found in the greater region of sky viewed at low declinations.

Figure 11. Results of the KS test against the declination distribution of identified repeating FRBs. Shown is the p-value as a function of ${R_{\rm max}}$ and ${{\gamma_r}}$ .

6.3. Repetition rate distribution

Most repeating CHIME FRBs are not localised, so that scaling between intrinsic and apparent repetition rates, which requires the luminosity distance to be known, is not possible. This precludes a direct fit to the rate distribution. Nonetheless, different combinations of ${R_{\rm min}}$ , ${R_{\rm max}}$ , and ${{\gamma_r}}$ lead to more/less repeating FRBs being observed with different apparent repetition rates.

Directly computing the number of FRBs with any given repetition rate is highly inefficient however—the algorithm currently estimates the number of repeaters by explicitly calculating $N_0$ and $N_1$ only. Extending this to a large number of $N_{\rm burst}$ values is computationally prohibitive. A Monte Carlo sampling algorithm was therefore implemented that generates repeating FRBs according to their underlying modelled distribution in z–DM–R space, and simulates the number of observed bursts assuming a Poissonian distribution.

To overcome Monte Carlo fluctuations, at least 1000 times as many repeating FRBs as expected are simulated, and a histogram produced in terms of the observed number of bursts by CHIME. This is then fit with a power-law distribution, and for histogram bins with less than 10 simulated repeaters, the observed number is replaced with the fitted number for purposes of evaluating likelihoods. An example of this procedure is shown in Fig. 12.

Figure 12. Top: histogram of observed number of repetitions in CHIME repeating FRBs from ${\rm Cat1}$ , compared to Monte Carlo predictions from four example cases, a–d (points). A power-law fit (lines) is given for each. Bottom: the same data, but shown as a cumulative distribution.

Since the data are discrete (integer numbers of bursts only), a KS test to assign a goodness of fit is inapplicable. Instead, the likelihood of the observed histogram of $N_{\rm burst}$ values for the 16 CHIME repeating FRBs from the ${\rm Cat1}$ sample is calculated, given predictions from the Monte Carlo histogram. This is then repeated for at least 1000 sets of 16 Monte Carlo FRBs, and the fraction of likelihoods that are lower than that observed is determined. This produces a p-value, $p_{\rm bursts}$ , under the null hypothesis that the Monte Carlo sample is the truth. Results are plotted in Fig. 13.

Figure 13. Maximum-likelihood estimates of FRB repeat parameters based on the distribution of the number of observed bursts from each repeater in the ${\rm Cat1}$ sample.

The repeat-rate distribution is best-reproduced by models with a large number of bright FRBs, since the two CHIME FRBs with high repetition rates in ${\rm Cat1}$ —FRB20180814A (11 bursts), and FRB20180916B (19 bursts)—are difficult to reproduce with models of low ${R_{\rm max}}$ and/or steep ${{\gamma_r}}$ .

6.4. Combined likelihood

Combining the evidence from the DM, $\delta$ , and $N_{\rm burst}$ probabilities derived above, the combined probability $p_{\rm tot}$ is constructed as

(13) \begin{eqnarray}p_{\rm tot} & = & p_N \, p_\delta \, p_{\rm DM} \, p_{\rm bursts},\end{eqnarray}

where $p_N$ is a Poissonian probability of observing 16 repeaters, which suppresses the region of the parameter space that over-produces repeaters. The probabilities are renormalised to sum to unity over the investigated range, excluding ${R_{\rm max}} < 0.5$ d $^{-1}$ , and confidence intervals assuming flat priors in ${{\gamma_r}}$ and $\log {R_{\rm max}}$ are constructed. This results in the probability distribution, and confidence intervals (CIs), shown in Fig. 14.

Figure 14. Posterior probability of repeating FRB parameters assuming that all FRBs repeat. Shown are 68% (red dotted lines) and 95% (white dot-dash lines) confidence intervals.

The 95% CI encompasses almost the entire allowed region from Fig. 7, showing that DM, $\delta$ , and $N_{\rm burst}$ are not strong discriminators between different models of the repeating FRB population. However, a preference for ${{\gamma_r}}=-2.2_{-0.8}^{+0.6}$ (68% CI), and ${R_{\rm max}} \ge 0.75$ , is found.

The above formulation for $p_{\rm tot}$ ignores correlations between variables: closer (low DM) FRBs, and those viewed closer to the zenith, will be more likely to have more bursts detected. A better analysis would use the full 3D distribution of $p({\rm DM}, \delta, N_{\rm bursts})$ , similarly to the use of redshift, DM, and burst energy in standard z–DM analyses. However, generating the distribution of $N_{\rm bursts}$ is computationally intensive, and thus it is only performed in one dimension. This method should be revisited once other systematic effects, as discussed in Section 8, are treated.

6.5. What if not all FRBs are repeaters?

It is of course possible that repeating FRBs do not constitute the total FRB population. Evidence for this comes from the different spectro-temporal properties of repeaters compared to non-repeaters (Pleunis et al. Reference Pleunis2021), and a tentative association of FRB 20190425A with binary neutron star merger GW190425 (Moroianu et al. Reference Moroianu2023). If such a population exists, it is likely subdominant—most cataclysmic events, which would produce intrinsically once-off FRBs, have a rate which is much too low to explain the total FRB rate (Ravi Reference Ravi2019). Therefore, observations of the total FRB population still serve as good constraints on the total repeating population, and the predictions made here remain valid. Nonetheless, in this section, the case where repeating FRBs are responsible for a sub-dominant fraction of the total number of bursts observed by CHIME is investigated.

The above analysis is repeated by first optimising ${R_{\rm min}}$ to produce 16 CHIME repeaters and some fraction ${F_{\rm single}}$ of the total singles burst rate, and calculating the joint probability $P_{\rm tot}({{\gamma_r}}, {R_{\rm max}}, {F_{\rm single}})$ . The peak likelihood over ${{\gamma_r}}$ and ${R_{\rm max}}$ for each ${F_{\rm single}}$ , Max $[P_{\rm tot}({{\gamma_r}}, {R_{\rm max}})]({F_{\rm single}})$ , is then plotted in Fig. 15.

Figure 15. Maximum value of the joint probability $P_{\rm tot}$ over all analysed ${{\gamma_r}}$ , ${R_{\rm max}}$ , as a function of the fraction of all CHIME single bursts explained by repeating FRBs, ${F_{\rm single}}$ .

In the range $0.5 \le {F_{\rm single}} \le 1$ , the peak probability is essentially identical, with fluctuations likely due to the coarse gridding in ${{\gamma_r}}$ – ${R_{\rm max}}$ space. The likelihood decreases for lower values of ${F_{\rm single}}$ —this is driven almost entirely by $p_{\rm DM}$ , since decreasing ${F_{\rm single}}$ increases the fraction of true repeaters detected as such, which requires on-average stronger repeaters that are invariably detectable at greater distances. This pushes the predicted DM distribution to higher values, inconsistent with CHIME data.

A note of caution is warranted however: if a small fraction of all single bursts are produced by repeaters, then the assumption that repeating FRB population parameters are the same as that of the total population is a bad one. Therefore, while it can be concluded that these results are consistent with a best-fit of all FRBs being from repeaters, it cannot be concluded that this excludes a large fraction of FRBs being from intrinsically once-off events.

7.1. Rate of new repeater discoveries with CHIME

As time spent observing a particular field increases, the number of repeating FRBs should eventually saturate, as essentially all such objects in the field are detected. Seeing the rate of detected repeaters plateau at a level where a large number of once-off bursts have no associated repeater would be a clear indication of two populations. This raises the question: how long might CHIME have to wait until the rate of new repeating FRB detections decreases?

Figure 17. Predicted z–DM distribution of left: repeating and right: single FRBs for cases b (top) and d (bottom). Contours enclose 50% (dotted), 90% (dot-dash), and 99% (dashed) of the probability space. Repeating CHIME FRBs identified in ${\rm Cat1}$ with host galaxies are shown as red circles; repeater hosts not from ${\rm Cat1}$ are shown as blue stars; and repeating FRBs with no firm host association from ${\rm Cat1}$ have their 0–99% probable redshift range indicated with red lines at their known ${{\rm DM}_{\rm EG}}$ , calculated assuming the Cordes & McLaughlin (Reference Cordes and McLaughlin2003) model for ${{\rm DM}_{\rm ISM}}$ , and a value of 50 ${{\rm pc}\,{\rm cm}^{-3}}$ for ${{\rm DM}_{\rm halo}}$ .

The answer is a very long time. Regardless of the scenario under consideration, the number of repeating FRB progenitors at high redshifts will vastly outnumber those at low redshifts due to the increased volume of the Universe. As observation time increases, the number of repeating FRBs in the nearby Universe will saturate, but the rate of repeater discoveries—both as single and repeat bursts—in the distant Universe increases. This effect is seen in Fig. 1—in Fig. 16, this is simulated for CHIME, by simply increasing the observation time in units of ${\rm Cat1}$ , $T_{{\rm Cat1}}$ (which is approximately a year’s worth of exposure). In case b, there are relatively few strong repeaters, and saturation is expected to be seen after $\sim$ 300 yr. In case d, with many strong repeaters, saturation will not occur in the next thousand years, and a steadily increasing repeat rate is expected.

7.2. z–DM distribution

Only six repeating CHIME FRBs have been localised to their host galaxies (Marcote et al. Reference Marcote2020; Bhardwaj et al. Reference Bhardwaj2021b,Reference Bhardwaja; Fong et al. Reference Fong2021; Ibik et al. Reference Ibik2023), though three more associations are highly likely (Michilli et al. 2022; Ibik et al. Reference Ibik2023); and only one once-off CHIME FRB has a tentative host association (Panther et al. 2022). Of these repeaters, four were identified as such in ${\rm Cat1}$ . Furthermore, these have only been localised either because they are nearby, and hence CHIME’s angular resolution—effectively enhanced when using multiple bursts (Michilli et al. 2022)—is sufficient to identify the host; or because they repeat rapidly, allowing follow-up observations with arrays with a better angular resolution to identify the host. Thus these represent a highly biased sample, and are unsuited to fitting to data. Nonetheless, the full z–DM distribution of repeating FRBs observed by CHIME can be predicted by the models. These distributions are given in Fig. 17 for cases b and d, and are compared to the distribution of singly detected FRBs.

Fig. 17 illustrates how the large tail of the DM distribution arises: it is almost entirely from objects lying well above the Macquart relation. Since repeating FRBs tend to only be detected as such in the nearby Universe, those repeating FRBs lying on the Macquart relation have a smaller range of DMs—using the 90% contours, up to $\sim$ 400 ${{\rm pc}\,{\rm cm}^{-3}}$ in case b, and $\sim$ 800 ${{\rm pc}\,{\rm cm}^{-3}}$ in case d. Thus the high-DM tail of low-z FRBs doesn’t become over-ridden by the larger number of FRBs lying on the Macquart relation in the more-distant Universe, as it is for singly detected FRBs.

This closer proximity of repeating FRBs means that the reduced ${{\rm DM}_{\rm EG}}$ of the CHIME repeater sample ( $436 \pm 49$ ${{\rm pc}\,{\rm cm}^{-3}}$ for repeaters, $597 \pm 24$ ${{\rm pc}\,{\rm cm}^{-3}}$ for apparently once-off bursts; CHIME/FRB Collaboration et al. 2023), which has been suggested to be evidence for two populations (Woods Reference Woods2023), is entirely consistent with expectations from the models. These predict mean repeater ${{\rm DM}_{\rm EG}}$ values in the range 460–540 ${{\rm pc}\,{\rm cm}^{-3}}$ , and mean single DMs in the range 640–660 ${{\rm pc}\,{\rm cm}^{-3}}$ . While the mean DMs of both samples are slightly over-predicted, the difference is a very good match with expectations.

Even more importantly for future observations, models b and d predict different z distributions. Case b has more low-rate repeaters, from which repeat bursts are only likely at low z; while case d has a significant population of high-rate repeaters, which can be detected as repeaters from the more-distant Universe. The z–DM values of repeaters with likely or confirmed host galaxies matches observations for both cases, but this has only been probed in the low-z, low-DM region. However, case d predicts redshifts will continue to increase with ${{\rm DM}_{\rm EG}}$ , while case b predicts $z \lesssim 0.5$ for all repeaters. Thus, if a large fraction of repeating CHIME FRBs could be localised, this would enable much more powerful tests of the repeating FRB population.

I illustrate this using a toy example, using simulated true values ${{\gamma_r}}=-2.2$ , ${R_{\rm max}}=30$ , and 100 Monte Carlo instances of repeating FRBs from ${\rm Cat1}$ . All FRBs detected as repeaters are assumed to be localised to their host galaxies, yielding their correct z and DM values. For each Monte Carlo sample, the likelihood $p(z,{\rm DM})$ is calculated for all values on the ${{\gamma_r}}$ , ${R_{\rm max}}$ grid. I do not calculate $p(N_{\rm reps})$ , that is, only the position in z, DM is accounted for, not the number of repeating FRBs or bursts per repeater. Bayesian 1-, 2-, and 3- $\sigma$ confidence intervals are then constructed for each sample, and the number of MC iterations in which any given value lies in each interval is counted. The result is shown in Fig. 18, which shows the expected confidence intervals at each of the three levels. This shows the power of being able to localise repeating FRBs: if all ${\rm Cat1}$ repeating FRBs could be localised, the expected $1 \sigma$ accuracy on ${{\gamma_r}}$ would be $\pm 0.2$ , and ${R_{\rm max}}$ would be determined to within a factor of $\sim 10$ .

Figure 18. Fraction of Monte Carlo iterations in which trial values of ${{\gamma_r}}$ , ${R_{\rm max}}$ fall within the $3\sigma$ confidence interval (CI), using a sample of simulated repeating FRBs with truth values ${{\gamma_r}}=-2., {R_{\rm max}}=31.62$ . The contours correspond to regions that fall within the 1 (red, dotted) 2 (white, dash-dot) and 3 (black, dashed) confidence intervals 50% of the time.

7.3. Predicted effects on other instruments

I now use cases b and d to estimate the relative rates of single and repeat observations for a sample of other FRB-hunting instruments. Four systems are considered: ASKAP, in Fly’s Eye (FE), ICS, and coherent (CRACO) mode at 1.3 GHz; and the Five-hundred-metre Aperture Spherical Telescope (FAST). ASKAP/FE and ASKAP/ICS are modelled as per James et al. (2022b), while the model of the CRAFT Coherent Upgrade (CRACO) system is described in James et al. (Reference James2022c). The parameters for FAST FRB searches are taken from Niu et al. (Reference Niu2021), namely a detection threshold of 0.0146 Jy ms for a 1 ms pulse width at a central frequency of 1.25 GHz, and time and frequency resolutions of $196.608\,\mu s$ and 0.122 MHz respectively. The FAST receiver is a 19-beam multibeam (Li et al. Reference Li2018), similarly designed to the 13-beam Parkes multibeam (Staveley-Smith et al. Reference Staveley-Smith1996). I therefore take the inverse beamshape $\Omega_b$ used for Parkes, scale up by the ratio of the number of beams ( $19/13 \approx 1.46$ ), and down by the ratio of effective collecting areas ( $64^2/300^2 \approx 0.0456$ ).

All these instruments have searched for FRBs with different dwell times. Here, I consider the longest time spent on any given field for each instrument, which will prove most sensitive to the repeating FRB population: 1338.9 h for ASKAP/FE (James et al. Reference James2020b), 879.1 h for ASKAP/ICS to the end of 2022 (Shannon et al. in preparation), and for ASKAP/CRACO predictions, the expected on-source time of 800 h for each of the Deep Investigation of Neutral Gas Origins (DINGO; Rhee et al. Reference Rhee2023, see also https://dingo-survey.org/) fields is used. For FAST, it is 59.5 h when performing follow-up observations on FRB 20121102A (Li et al. Reference Li2021). The normalised estimates are given in Fig. 19, as a function of DM for surveys with poor localisations, and as a function of z for those that typically identify host galaxies.

Figure 19. Predictions of the z or DM distributions of repeating FRBs for a selection of past and future FRB surveys for their longest pointing times (see Section 7.3), for cases (b) and (d). Shown are the distributions of those repeating FRBs detected as single bursts, as repeaters, the total progenitor distributions, and the total burst distributions, as per Fig. 1.

Qualitatively, all predictions are very similar. The total number of single bursts ranges from 27% of the total burst distribution (FAST, case b) to 70% (ASKAP/FE, case d). The difference between case b, which models a repeating FRB population spread over a narrow repetition rate, and case d, with a very broad distribution of rates, is marked, predicting 16% (case b) and 6% (case d) of FRB progenitors to repeat for all ASKAP models, and 31% and 10% for cases b and d for FAST.

The deficit between total burst number and total progenitors in the low-DM range for ASKAP/FE is not sufficient however to explain the observed deficit that has been previously noted by James et al. (2022b), especially when accounting for the average pointing time for that survey being less than that modelled here. Thus I conclude this effect—which originally motivated this work—is most likely a statistical fluctuation.

Jankowski et al. (Reference Jankowski2023) have noted that the FAST FRB rate is much lower than predicted. Here, the progenitor rate is predicted to be 40–60% of the burst rate, which certainly accounts for some, but not all, of the deficit. However, this would have no influence on the observations in drift-scan mode reported by Niu et al. (Reference Niu2021) due to the very short dwell times ( $\sim$ 13 s). Thus this deficit must have some other explanation.

A single FRB (20220531A; Shannon et al. in preparation) has been discovered in the ASKAP field with 879.1 h of observations, against a mean ASKAP detection rate of 350 h FRB $^{-1}$ . This could simply be a Poissonian under-fluctuation (p-value of 0.285 on a one-sided test), but repetition offers a partial explanation, which would reduce the expected number of progenitors from 2.5 to 1.75–1.86.

Overall, I expect that correct modelling of repeating FRBs will be important for these observations to account for repeater bias in the observed z–DM distribution.

8.1. Non-Poissonian repetition

All FRBs with sufficiently many detected bursts to allow studies of their repetition rates show non-Poissonian behaviour. On timescales of order seconds to hours, bursts from repeaters tend to be clustered (e.g. Gajjar et al. Reference Gajjar2018; Zhang et al. Reference Zhang2021; Nimmo et al. Reference Nimmo2023), in a process which is often modelled as a Weibull distribution (Oppermann, Yu, & Pen Reference Oppermann, Yu and Pen2018). On longer timescales ( $\sim$ 16–160 d), two repeating FRBs appear to have activity cycles (CHIME/FRB Collaboration et al. 2020; Rajwade et al. Reference Rajwade2020), with evidence for frequency dependence in the timing of the windows (Pastor-Marazuela et al. Reference Pastor-Marazuela2021). Other behaviours include a rapidly increasing/decreasing event rate (Zhang et al. Reference Zhang2022), or ‘turning on’ despite several years of monitoring (see the time dependence of bursts in CHIME/FRB Collaboration et al. 2023). What the true underlying nature of the time distribution of repeat rates of FRBs is is still under debate—what is sure is that they are most certainly not Poissonian.

Thus it should be asked: what effect does this have on the modelling? On sufficiently long timescales, FRBs will become inactive, and new repeating FRBs will be born. Therefore, the repeating FRB population studied here can only refer to those FRBs which have been active during the approximate year (three years) corresponding to the CHIME ${\rm Cat1}$ (Gold25) samples. However, FRB 20121102A has now been studied for over a decade since its first detection (Spitler et al. Reference Spitler2014), and while its properties (DM, RM etc.) do vary (Michilli et al. Reference Michilli2018), no evidence of a systematically reducing rate has been published. Therefore, these considerations likely won’t be relevant to current or near-future studies.

Of more relevance are FRBs with inactive windows comparable to, or longer than, the current survey. CHIME/FRB Collaboration et al. (2023) shows that at least three FRBs—20201130A, 20200929C, and 20201124A—have numerous bursts in the latter six months studied, but none in the first two years. While some of this may be reflective of a changing search sensitivity and analysis methods (it would be useful for CHIME to release a time-dependent sensitivity to account for this), it is also suggestive that these objects have very long inactive phases. By generating either many bursts or none, a larger population of such objects would mimic a flatter value of ${{\gamma_r}}$ than the true long-term rate, with repeaters being found at larger distances/DM values. Conversely, for a fixed observation, the fitted value of ${{\gamma_r}}$ will be flatter than the true value. Since the observed DM distribution of repeaters already favours ${F_{\rm single}} \ge 0.5$ and ${{\gamma_r}} \le -1.4$ , allowing for such behaviour would constrain ${F_{\rm single}}$ to higher, and ${{\gamma_r}}$ to lower, values. That is, the limits from this work are sensitive to activity windows on yearly timescales. Activity windows significantly shorter than a year however will have no consequence, since CHIME’s coverage is spread uniformly in time, unless the period of these windows lies extremely close to a sidereal day.

Finally, the effect of bursty behaviour in general is to reduce the number of singly detected FRBs, and increase the likelihood of viewing zero or many bursts. The effect of time correlation in bursts will be most pronounced when observations occur all in one block—when observations are individually very short and spaced far apart in time, any intrinsically bursty distribution will exhibit a Poissonian distribution of burst numbers.

To gauge the impact of this effect, I simulate a Weibull distribution of arrival times, using a shape parameter $k=0.34$ , as found for FRB 20121102A by Oppermann et al. (Reference Oppermann, Yu and Pen2018). CHIME detections are simulated at three declinations: near the NCP, with sources observed continuously ( $f_{\rm obs}=1$ ); approximately $7^{\circ}$ away from the NCP, where sources will be observed a fraction $f_{\rm obs}=10\%$ of the time; and $\sim$ $30^{\circ}$ from the NCP, which is modelled as a source being observed $f_{\rm obs}=1\%$ of the time. The expected repetition rates are modelled relative to the total time on-source in a calendar year, and this rate $R_{\rm obs}$ is varied from 0.1 to 10. I simulate 1000 Weibull sequences over 365 sidereal days, beginning each sequence well before the start of the year to ensure the sequence start time does not influence results. If a burst occurs in the first $f_{\rm obs}$ fraction of a day, it is counted as detected. The number of simulations resulting in none, one, or multiple detections are recorded.

The results are given in Fig. 20. As predicted, the fraction of FRBs detected to repeat twice or more is greater than Poissonian for low expected rates, and less for high expected rates. However, only very near the NCP is this effect large, where a deficit of $\sim$ 30% of repeating FRBs relative to Poisson rates are found.

Figure 20. Probability of an FRB being detected as a repeater given its true expected rate R, as a function of the fractional observation time $f_{\rm obs}=T_{\rm obs}/T$ , for a Weibull burst time distribution with shape index $k=0.34$ ; and for a Poissonian distribution.

A more accurate estimate of the effect of burstiness on these results however is the ratio of single to repeat bursts. Weighting the results by $R^{-2}$ , to represent there being less rapidly repeating FRBs than rarely repeating ones, $f_{\rm obs} = 0.01$ , $0.1$ , and 1 respectively produces 30%, 120%, and 310% more repeating FRBs relative to single bursts than expected for a Poissonian distribution. This effect would also alter the z–DM distribution of repeating FRBs, with bursty distributions favouring discoveries in the more-distant Universe; and flatten the distribution of observed burst rates, undermining the fitting of Section 6.3. It is quite possible that the excess of repeating FRBs in ${\rm Cat1}$ at high declinations, while not statistically significant, is due to this effect.

Nonetheless, I do not wish to over-emphasise this effect. For the majority of the sky seen by CHIME, the increase in observed repeaters due to burstiness with $k=0.34$ is tens of percent. Furthermore, this choice of $k=0.34$ is an extreme example—analysis of FRB 20121102A has shown that at timescales of seconds to hours, the wait time distribution is close the Poissonian, with a best-fit Weibull index of $k=0.79$ (Nimmo et al. Reference Nimmo2023).

Given the complexity of the issue, I defer an analysis of repetition that includes bursty behaviour to a future work—which should also include a fit of such behaviour to CHIME arrival time data.

8.2. Time-frequency structure of bursts

CHIME has shown that FRBs identified to repeat tend to have bursts which are broader in time by a factor of approximately two, with complex time-frequency structure (CHIME/FRB Collaboration et al. 2021; Pleunis et al. Reference Pleunis2021). Whether this is due to two intrinsically different FRB populations, or a smooth transition in the properties of one population (e.g. an ageing effect whereby less active—and presumably older—objects have modified emission physics, perhaps due to a change in beaming angle, as suggested by Connor, Miller, & Gardenier Reference Connor, Miller and Gardenier2020) is still up for debate. What is clearly true is that the increased time width of FRBs more likely to be detected as repeating will make them harder to detect for a given fluence. This effect is not included in the current work.

There are two methods of analysing this effect. The first is to modify the simulated burst width to increase with FRB repetition rate. This will act to suppress the number of repeating FRBs in the $z \lesssim 1$ range where intrinsic FRB width, rather than dispersion smearing, dominates the apparant FRB width, and hence sensitivity. The result will be that more active repeaters—which are in any case only observed as such at low z—become less detectable. Including this effect would require direct use of the CHIME pulse injection sample, since the published efficiency function analysed in Section 4.3 has already been averaged over the burst width distribution.

The second method to tackle this problem is to note that repetition rate R can be thought of as an effective repetition rate. After all, R can only ever be defined as the rate above a given energy threshold (here, $10^{39}$ erg), which must also be coupled to some assumption about the time–frequency properties of those bursts which affects their detectability. If more-strongly repeating FRBs tend to produce wider bursts, this reduces their detectability, and hence their apparent rate will decrease. In other words, this will cause ${{\gamma_r}}$ to steepen slightly from its true value. In the regime where detectability is limited by intrinsic width w, ${{\rm SNR}}\sim w^{0.5}$ , and hence for a burst luminosity function with cumulative slope $\gamma = -1$ , the detection rate will reduce as $w^{-0.5}$ . If the width w scales linearly with the intrinsic rate R (and the real effect is unlikely to be this strong), this would then steepen the apparent value of ${{\gamma_r}}$ by an extra factor of $-0.5$ .

Since either method requires an accurate estimate of the relationship between intrinsic FRB rate and measured width w, and this is not currently possible, I consider the second method appropriate, which means that a little care must be taken in interpreting the estimate of ${{\gamma_r}}$ found in this work.

8.3. Systematic effects in CHIME data

I conclude the discussion of systematics with an analysis of the data being used. As noted in Section 4.1, the threshold for identifying a repeat burst in ${\rm Cat1}$ was lower than for an initial burst. The best way of removing this effect would be for CHIME to publish an FRB catalogue with those repeat bursts that passed threshold only because they were repeaters removed or otherwise identified as such. The dependence on ${{\rm SNR}}$ should be overcome by use of the CHIME pulse injection data (Merryfield et al. 2022), or a publication of the parameterised multi-dimensional selection function used in CHIME/FRB Collaboration et al. (2021). It has been checked that excluding FRBs with high Galactic DMs does not significantly change the DM fits, but future analyses with zDM should in any case include an uncertainty term for this contribution. Overall, I do not think that this analysis is currently constrained by such systematic effects.

I have been reluctant to make quantitative comparisons in this work with CHIME’s new Gold25 sample of 25 repeating FRBs however. There are several reasons why interpretation is difficult. Firstly, note that this work is in fact a discovery of 34 new repeating FRBs, given that the ‘gold’ sample of 25 sources has an estimated contamination of 0.5 from coincidences between two or more unrelated FRBs, and the authors also publish a ‘silver’ sample of 14 repeaters, with a total contamination rate in the combined gold and silver samples of 5. It is unknown whether or not CHIME have detected two or more bursts from a true repeater which has a higher contamination fraction and is thus not included in any sample. This measurement therefore has a $5^{0.5} = 2.2$ systematic uncertainty reflecting the uncertainty in the expected contamination, and a $34^{0.5} = 5.8$ statistical deviation from the expected mean number of repeater discoveries, for a total uncertainty of $(5+34)^{0.5} = 6.2$ , or 18%.

The time period used for the search, from 2019 September 30 to 2021 May 1 (579 d), is approximately 70% longer than and does not overlap with, the period of 2018 July 25 to 2019 July 1 used for ${\rm Cat1}$ (342 d) where 16 repeating FRBs were identified. Hence, the detection rate has increased by a factor of $1.26 \pm 0.23$ , consistent both with the predictions of an increasing discovery rate in Section 7.1 and with a constant rate. Aside from Poisson error, this increase could also be due to a higher efficiency of CHIME data-taking, or a lower threshold for including bursts in the analysis. This question should be revisited once it becomes possible to normalise the two samples, for an accurate rate comparison.

Secondly, the identification of repeating FRBs in Gold25 placed a cut on the chance coincidence probability. This cut is strictest where the rate of FRBs in DM– $\delta$ space is highest: at high declinations and intermediate DMs. Therefore, the gold sample of 25 repeating FRBs is biased towards very low or high DMs at low declinations. Conversely, the silver sample may have the opposite bias, both because repeaters at high declinations and intermediate DMs will preferentially be placed there and because this region has more chance coincidences. CHIME/FRB Collaboration et al. (2023) do not publish estimates of the contamination probability in $\delta$ -DM space, which would be required to account for this effect.

9.1. ASKAP follow-up observations

The only other result of which I am aware that has limited ${R_{\rm min}}$ , ${R_{\rm max}}$ , ${{\gamma_r}}$ James et al. (Reference James2020a). Those authors used the observation of repetition in only one of 27 FRBs detected by ASKAP (Kumar et al. Reference Kumar2019; James et al. Reference James2020b) to constrain ${{\gamma_r}} < -1.94$ (those authors use $\zeta$ for ${{\gamma_r}}$ ) and ${R_{\rm min}} < 10^{-2.9}$ d $^{-1}$ . Limits on ${R_{\rm max}}$ are not published, though values of ${R_{\rm max}} \le 100$ are investigated.

The values of ${R_{\rm min}}$ and ${R_{\rm max}}$ used in that work are applicable to $1.3$ GHz observations and are measured relative to an energy threshold of $10^{38}$ erg. I therefore scale to a threshold of $10^{39}$ erg by reducing those rates by a factor of $10^\gamma \approx 0.13$ , and to the mean CHIME frequency of 600 MHz by increasing the rates by a factor of $(600/1300)^\alpha \approx 4.4$ , for a total adjustment factor of $0.55$ .

I extract the likelihoods from that work at the values of ${R_{\rm min}}$ , ${R_{\rm max}}$ , and ${{\gamma_r}}$ used here and plot the Bayesian posterior in Fig. 21. The best-fit region agrees remarkably with results from CHIME FRB data, but has a much tighter constraint on ${{\gamma_r}}$ . That two very different measurements with very different instruments converge to the same properties of the repeating FRB population is strong evidence that this model is a reasonable approximation to the underlying truth. This has implications for FRB progenitor models, as discussed in James et al. (Reference James2020a).

Figure 21. Bayesian posterior likelihoods, $p_{\rm ASKAP}$ , from follow-up observations of ASKAP repeaters (James et al. Reference James2020a) for the parameter values investigated here.

The one major discrepancy between the results which is hidden by Fig. 19 however is that the best-fit values of ${R_{\rm min}}$ found by James et al. (Reference James2020a) are a factor of $\sim$ 100 lower than that found for CHIME. Equivalently, the results of James et al. (Reference James2020a) would under-predict the number of repeating CHIME FRBs. There may be several causes for this.

Firstly: the results for James et al. (Reference James2020a) assumed a Weibull distribution with $k=0.34$ . As discussed in Section 8.1, a bursty distribution requires less-rapid repeaters to produce the same number of repeating FRBs and could therefore allow the values of ${R_{\rm min}}$ found here to be lower and more consistent with the ASKAP results. Secondly: the results here are only weakly constraining on ${R_{\rm min}}$ , and only the probabilities at the best-fit values have been used, rather than marginalising over ${R_{\rm min}}$ . Thirdly: repetition behaviour at 1.3 GHz and 600 MHz might be more different than the simple scaling above would suggest. A fuller investigation will require an improved model of FRB time–frequency structure. Fourthly: the limits from James et al. (Reference James2020a) used non-localised repeaters, with conservatively large distance estimates from the Macquart relation assuming no host contribution, thus introducing a bias in those results (albeit one which would push ${R_{\rm min}}$ and ${R_{\rm max}}$ to lower values). Fifthly and finally, not all FRBs may repeat. The upper limit on ${R_{\rm min}}$ found by James et al. (Reference James2020a) is driven primarily by the lack of observed repetition from FRB 20171020A. If this FRB is intrinsically a once-off event, then those results weaken significantly, allowing higher values of ${R_{\rm min}}$ .

The current implementation of repetition in the zDM code does not allow for easy estimation of FRB repetition parameters from follow-up observations. It would be useful to develop such a method to allow the results of follow-up observations to also be fit in a self-consistent manner.

9.2. Absolute number of repeaters and persistent radio sources

The total number of repeating FRBs, $C_r$ , is a function of their rate of birth and active lifetime. Estimates of their birth rate vary greatly according to their progenitor model and range from $10^5$ Gpc $^{-3}$ yr $^{-1}$ for core-collapse supernova (Taylor et al. Reference Taylor2014), to 0.02 Gpc $^{-3}$ yr $^{-1}$ for NS-NS binary mergers in globular clusters (Ye et al. Reference Ye2020). Their lifetime is also unknown: while magnetar fields are expected to decay on timescales of order $10^4$ yr (Colpi, Geppert, & Page Reference Colpi, Geppert and Page2000), precisely how this relates to their time as an active emitter of FRBs will depend on the microphysics of emission.

Fig. 22 plots the implied value of $C_r$ as a function of ${R_{\rm max}}$ and ${{\gamma_r}}$ , compared to limits on those parameters from Fig. 14. By far the largest number of FRB progenitors are found near ${{\gamma_r}}=-2$ . At flatter (less negative) values of ${{\gamma_r}}$ , the repeater distribution is dominated by so many rapidly repeating FRBs that very few are required, while at steep (more negative) values, ${R_{\rm min}}$ must be relatively high (and hence $C_r$ not too large) to prevent singly detected objects overwhelming the total FRB rate. That this region is just allowed within the current limits means that the total uncertainty on $C_r$ is huge, ranging approximately between $10^{-5}$ and 10 Mpc $^{-3}$ at $z=0$ . Assuming a lifetime of $10^{4}$ yr, this corresponds to birth rates of 1– $10^6$ Gpc $^{-3}$ yr $^{-1}$ , which is compatible with most FRB progenitor models.

Figure 22. FRB progenitor population density $C_r$ , as a function of ${R_{\rm max}}$ and ${{\gamma_r}}$ with 68% and 95% contours from Fig. 14 overplotted.

The total number of strong repeating FRBs is also of interest. Aside from their obvious identification via detection of their bursts, these objects might be identified via their association with PRS, which can be identified in radio surveys. There is some evidence that PRS are more likely to be associated with strong repeaters (Chatterjee et al. Reference Chatterjee2017; Niu et al. Reference Niu2022), though this evidence is not conclusive (Law et al. Reference Law, Connor and Aggarwal2022).

In this context, Law et al. (Reference Law, Connor and Aggarwal2022) estimate the total number density, and differential power-law slope, of the observed repeating FRB population using CHIME data, as in Section 6.3. Assuming all CHIME FRBs come from intrinsic repeaters, they use the observed distribution of the number of bursts to have a power-law slope of $=-1.5$ , calculated via the maximum-likelihood estimator of Crawford, Jauncey, &Murdoch (Reference Crawford, Jauncey and Murdoch1970). This method thus determines the slope of the ‘source counts’ distribution of repeaters, which is largely insensitive to the intrinsic value of ${{\gamma_r}}$ (see Fig. 12). They also estimate the mean number of observed repeat bursts per source, $\langle R_{\rm obs} \rangle =1.2$ d $^{-1}$ , assuming a minimum repetition rate equal to the inverse of $T_{\rm {\rm Cat1}}^{-1}$ . Again, this estimate is based on the observed repetition rates, rather than the intrinsic rates, which explains why the derived mean rate is somewhat higher than the values of ${R^*}$ found here. Their derived number density of repeaters, being between 22 and $5.2 \cdot 10^3$ Gpc $^{-3}$ in the no-beaming case, are both more constraining and on-average lower than derived here. Law et al. (Reference Law, Connor and Aggarwal2022) estimate that the fraction of FRB sources with associated PRS is between 0.06 and 0.36: the much higher uncertainty on the intrinsic source density found in this work suggests that surveys identifying PRS independently of FRBs might be a better tracer of the intrinsic number of FRB sources.

It should be noted that the estimates above assume isotropic emission. However, to first order, beaming should have no effect. Including beaming means that the intrinsic number of bursts per repeater is greater by $4 \pi / \Omega_{\rm frb}$ , where $\Omega_{\rm frb}$ is the solid angle subtended by each FRB. However, so is the intrinsic single-burst emission rate. Hence, the implied number of repeaters to reproduce the detected rate is identical, and beaming angle has no effect. Only in the case that $\Omega_{\rm frb}$ varies with repetition rate R, as suggested by Connor et al. (Reference Connor, Miller and Gardenier2020), is this scaling broken.