The Statistical Similarity of Repeating and Non-Repeating Fast Radio Bursts

Abstract

In this paper, we present a sample of 21 repeating fast radio bursts (FRBs) detected by different radio instruments before September 2021. Using the Anderson–Darling test, we compared the distributions of extra-Galactic dispersion measure ($D{M}_{\mathrm{E}}$) of non-repeating FRBs, repeating FRBs and all FRBs. It was found that the $D{M}_{\mathrm{E}}$ values of three sub-samples are log-normally distributed. The $D{M}_{\mathrm{E}}$ of repeaters and non-repeaters were drawn from a different distribution on basis of the Mann–Whitney–Wilcoxon test. In addition, assuming that the non-repeating FRBs identified currently may be potentially repeators, i.e., the repeating FRBs to be universal and representative, one can utilize the averaged fluence of repeating FRBs as an indication from which to derive an apparent intensity distribution function (IDF) with a power-law index of $a_1=$$1.10\pm 0.14$ ($a_2=$$1.01\pm 0.16$, the observed fluence as a statistical variant), which is in good agreement with the previous IDF of 16 non-repeating FRBs found by Li et al. Based on the above statistics of repeating and non-repeating FRBs, we propose that both types of FRBs may have different cosmological origins, spatial distributions and circum-burst environments. Interestingly, the differential luminosity distributions of repeating and non-repeating FRBs can also be well described by a broken power-law function with the same power-law index of −1.4.

1. Introduction

Fast radio bursts (FRBs) are mysterious millisecond-duration radio pulses which occur randomly on the sky (e.g., [1,2,3]). FRBs were first found in archived pulsar survey data more than a decade ago [1], and more than 600 FRBs were reported as of April 2022 (e.g., [4,5,6]). However, the detection rate of FRBs is considered to be $\sim {10}^3$–${10}^4{\mathrm{sky}}^{-1}{\mathrm{day}}^{-1}$ [2,6,7,8,9,10,11,12,13,14,15,16,17], which means that FRBs are not uncommon in the Universe; e.g., 1652 repeating events from FRB 20121102A [18] and 1863 bursts from FRB 20201124A [19] were recently detected by the Five-hundred-meter Aperture Spherical radio Telescope (FAST, Li et al. [20]). The observed dispersion measures ($DM$) are larger than the $DM$ contributions from the Milk Way ($D{M}_{MW}$) (except FRB 200428, which was confirmed to be from the Galactic magnetar SGR 1935+2154 [21,22,23,24,25,26,27]), which suggests an extragalactic, even cosmological origin in most cases.

It should be mentioned that there is no unambiguous physical origin for FRBs now. In general, FRBs’ millisecond duration and high $DM$ indicate the high brightness temperature (${10}^{32}$K – $3.5\times {10}^{35}$ K) [28] and the corresponding isotropic energy (E) released (${10}^{35}\mathrm{erg}$–${10}^{44}$ erg) [5,22,29,30]. Many progenitor models have been proposed to figure out what FRBs are (for a review, see, e.g., [31]), such as mergers of compact objects [32], flaring magnetars [33], young magnetars in supernova remnants [34], collisions between neutron star/magnetar and asteroids [35,36,37,38,39], collisions between episodic magnetic blobs [40], and massive black hole model [41]. However, none of the current models can explain all the observational properties of FRBs. Fortunately, great progress in observations can help to constrain the progenitor model of FRBs; there has been a great leap forward in the research of FRBs. For example, the periodic activity of repeating FRB 20180916B suggests that the source is modulated by the orbital motion of a binary system [42]. Additionally, the discovery of Galactic FRB 200428 indicates the origin of a magnetar (e.g., [43]). In addition, some other models, e.g., the precession like a gyroscope model [43,44,45,46,47] and the spin period of isolated neutron star/magnetars model [48,49], can also work.

With the increase in FRBs monitoring by many radio telescopes, such as Very Large Array (VLA, Thompson et al. [50]), Australian Square Kilometer Array Pathfinder (ASKAP) [51], Canadian Hydrogen Intensity Mapping Experiment (CHIME, CHIME/FRB Collaboration et al. [52]) and FAST, it was found that some events are apparently “non-repeating,” i.e., they did not repeat within a monitoring period. On the other hand, some sources have been repeating (e.g., [53,54,55,56,57]), which led to the successful identification of host galaxies and the precise mensuration of redshifts. Li et al. [5] analyzed 133 FRBs, including 110 non-repeating and 23 repeating ones, and proposed to classify FRBs into short and long groups according to pulse duration less than 100 ms or not. Interestingly, they found long FRBs are on average more energetic than short ones about two orders of magnitude. Moreover, they pointed out that FRBs could be used as a standard candle because peak luminosity becomes weakly dependent of the cosmological distance at higher redshift. Some observational properties of FRBs are similar to those of short and long gamma-ray bursts [58,59,60]. However, it is still uncertain whether this classification of FRBs is derived from the intrinsic physical characteristics, and whether repeating or “non-repeating” is due to observational selection bias—e.g., the monitor is not in the most active window for the “non-repeating” FRBs. Although the astrophysical origins of repeating and non-repeating FRBs are considered to be different (e.g., [61,62,63,64]), another viewpoint that most FRBs may be repeating sources has also been proposed [5,65,66,67]. It is now accepted that there are usually two types of FRBs, i.e., repeating FRBs and non-repeating FRBs. It is worth noting that the luminostiy function can place important constraints on the physical origins of FRBs and possible progenitors (e.g., [31,61,67,68,69,70,71,72]). Niino [69] has hypothesized three luminosity distribution function models (i.e., standard candle, power-law, and power-law + exponential cutoff) to investigate how differences in luminosity functions (LF) affect the observation properties of FRBs, and used the LF model and the cosmic FRB rate density to examine the distribution of $D{M}_{\mathrm{E}}$, the $logN-logS$ distribution, and the $D{M}_{\mathrm{E}}-S_{\mathrm{peak}}$ correlation. Luo et al. [61] constructed a Schechter luminosity function of 33 FRBs samples with a power-law index ranging from −1.8 to −1.2. Unfortunately, the LF of repeating FRBs has not been constructed yet due to the limit of numbers in the past. Hence, it is important and necessary to investigate their statistical characteristics and build the apparent intensity distribution function (IDF) and the LF of these repeating FRBs in order to disclose more natural differences from those non-repeating ones.

Our article is organized as follows. In Section 2, we introduce the FRB sample and present the statistical analyses of their parameters. The intensity distribution function of our repeating FRB sample is derived in Section 3. In Section 4, we construct the differential broken power-law LF of repeating FRBs and non-repeating FRBs based on the k-corrected isotropic luminosity (L). In Section 5, our conclusion and discussion are presented. The flat $\mathrm{\Lambda }$CDM cosmological parameters $H_0$ = 67.74 km s${}^{-1}$ Mpc${}^{-1}$, ${\mathrm{\Omega }}_b$ = 0.0486, ${\mathrm{\Omega }}_m$ = 0.3089, ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$ = 0.6911 have been adopted throughout the paper [73].

2. The Statistical Properties of Repeating FRBs

2.1. Distributions of Extra-Galactic Dispersion Measure and Total Energy

Currently, about 600 FRBs have been reported (including repeating bursts and apparent non-repeating bursts). Here, we collect the observation data of 21 repeating FRBs and 571 non-repeating FRBs. The sample of repeating FRBs was extracted from FRBCAT and several reported observational datasets (e.g., [53,54,55,56,74]). We list the key physical parameters of 21 repeating FRBs in Table 1. It is worth mentioning that for the observed peak flux density ($S_{\mathrm{peak}}$) listed in Table 1, which is used as a representative of the brightness of an FRB, we chose the brightest one in every monitoring period for one repeating event. Additionally, the last column of Table 1 is the average fluence $\overline{F}$, which is calculated based on all archived observed fluence ($F_{\mathrm{obs}}$) for one repeating FRB. The reason is that multiple observations at different duty-circles inevitably suffer from the observational biases. Another reason is that number of sub-bursts within a repeater varies among distinct FRBs even for a comparable observation time, and thus is very difficult to count accurately. For instance, 1652 sub-bursts from FRB 20121102A [18] and 1863 sub-bursts from FRB 20201124A [19] were recently detected by FAST. In addition, the fluence fluctuations between different sub-bursts for one repeating FRB could be very large. For example, the fluences of FRB 20171019A were found to range from 0.37 to 388 Jy ms [56]. Although it is uncertain whether repeating or “non-repeating” is due to the observational select bias (e.g., [13,56,62] for a discussion of repeating FRBs), here we ignore the probability that the present non-repeating events may be found to be repeating in future, and treat them as real non-repeating events, whose corresponding parameters are not presented in the text because of their large number. Note that in Table 1, we consider the observed pulse width instead of the intrinsic one. The reason is that the intrinsic pulses of the narrow FRBs would be broadened due to scattering, and the scattering time scale is affected by the local environment of the FRB and is model-dependent, which results in a larger uncertainty. An appendix of non-repeating FRBs samples has been added separately at the end of the paper (for more details, see Table A1). Moreover, we did not consider FRB 200428 in our non-repeating burst sample, which is the only event detected in the Milky Way [21,22,23,25,75].

Table 1. Key physical parameters of 21 repeating FRBs before September 2021.

TNS Name	${\mathit{\nu }}_{\mathbf{c}}{}^{\mathbf{a}}$ MHz	$\mathit{DM}$ (pc cm−3)	${\mathit{DM}}_{\mathit{MW}}$ (pc cm−3)	${\mathit{DM}}_{\mathit{E}}{}^{\mathbf{b}}$ (pc cm−3)	${\mathit{W}}_{\mathbf{obs}}$ (ms)	${\mathit{S}}_{\mathbf{peak}}$ (Jy)	${\mathit{F}}_{\mathbf{obs}}$ (Jy ms)	z	E (1039 erg)	$\overline{\mathit{F}}{}^{\mathbf{d}}$ (Jy ms)
FRB 20121102A	1375	$557.00\pm 2.00$	188.00	369.00	$3.00\pm 0.50$	$0.{40}_{-0.10}^{+0.40}$	$1.{20}_{-0.55}^{+1.60}$	0.31	0.13	0.36
FRB 20171019A	1297	$460.80\pm 1.10$	37.00	423.80	$5.40\pm 0.30$	40.50	219.00	0.35	34.01	101.62
FRB 20180814A	600	$189.38\pm 0.09$	87.00	102.38	$2.60\pm 0.20$	8.08 ${}^{\mathrm{c}}$	21.00	0.09	0.13	22.57
FRB 20180908B	600	$195.70\pm 0.90$	38.00	157.70	$1.91\pm 0.10$	$0.60\pm 0.40$	2.70	0.13	0.04	2.03
FRB 20180916B	600	$349.70\pm 0.70$	200.00	149.70	$1.06\pm 0.05$	7.64 ${}^{\mathrm{c}}$	8.10	0.12	0.12	10.26
FRB 20181017A	600	$1281.00\pm 0.60$	43.00	1238.00	$20.20\pm 1.70$	0.79 ${}^{\mathrm{c}}$	16.00	1.03	60.07	8.50
FRB 20181030A	600	$103.50\pm 0.70$	40.00	63.50	$0.59\pm 0.08$	12.37 ${}^{\mathrm{c}}$	7.30	0.05	0.02	4.75
FRB 20181119A	600	$364.00\pm 0.30$	34.00	330.00	$2.66\pm 0.10$	0.94 ${}^{\mathrm{c}}$	2.50	0.28	0.24	1.77
FRB 20181128A	600	$450.20\pm 0.30$	112.00	338.20	$2.43\pm 0.16$	1.81 ${}^{\mathrm{c}}$	4.40	0.28	0.46	3.45
FRB 20190116B	600	$443.60\pm 0.80$	20.00	423.60	$1.50\pm 0.30$	1.87 ${}^{\mathrm{c}}$	2.80	0.35	0.52	1.80
FRB20190117A	600	$393.30\pm 0.10$	48.00	345.30	$1.44\pm 0.03$	$1.70\pm 0.60$	5.90	0.29	0.64	6.36
FRB 20190208A	600	$580.20\pm 0.20$	72.00	508.20	$1.31\pm 0.14$	$0.60\pm 0.30$	2.00	0.42	0.60	1.70
FRB 20190209A	600	$424.60\pm 0.60$	46.00	378.60	$3.70\pm 0.50$	0.54 ${}^{\mathrm{c}}$	2.00	0.32	0.28	1.25
FRB 20190213A	600	$651.50\pm 0.40$	43.00	608.50	4.00	$0.50\pm 0.30$	3.00	0.51	1.46	1.80
FRB 20190212A	600	$301.40\pm 0.20$	49.00	252.40	$2.10\pm 0.30$	$1.10\pm 0.60$	2.50	0.21	0.13	2.67
FRB 20190222A	600	$460.60\pm 0.10$	87.00	373.60	$2.97\pm 0.90$	2.53 ${}^{\mathrm{c}}$	7.50	0.31	1.00	5.45
FRB 20190303A	600	$221.80\pm 0.50$	29.00	192.80	$2.00\pm 0.30$	$0.50\pm 0.30$	2.30	0.16	0.06	2.47
FRB 20190417A	600	$1378.50\pm 0.30$	78.00	1300.50	$1.19\pm 0.02$	$0.70\pm 0.20$	1.70	1.08	7.40	3.10
FRB 20190604A	600	$552.60\pm 0.20$	32.00	520.60	$3.00\pm 0.40$	$0.90\pm 0.40$	8.30	0.43	2.64	5.00
FRB 20190711A	23.8	$593.10\pm 0.40$	56.40	536.70	$6.50\pm 0.50$	5.23 ${}^{\mathrm{c}}$	34.00	0.45	9.88	17.70
FRB 20190907A	600	$309.50\pm 0.30$	53.00	256.50	$0.54\pm 0.14$	$0.40\pm 0.20$	0.90	0.21	0.05	1.10

Note: The all data were taken from http://www.frbcat.org [4] (accessed on 3 May 2022), and we report the properties of the brightest bursts for each repeater. ^a ν_c is the central frequency within the observational bandwith for an FRB. ^b DM_E is the extra-Galactic dispersion measure, which is defined as DM_E = DM – DM_MW. ^c For S_peak which is not given in FRBCAT, the corresponding value is estimated by S_peak = F_obs/W_obs. ^d $\overline{F}$ is the average fluence, which is calculated based on all archived F_obs of one repeating FRB.

The $DM$ is a key quantity in the study of FRBs. CHIME/FRB Collaboration et al. [55] argued that FRBs can be divided into two subclasses (repeating and non-repeating FRBs) according to the current observations. If repeating FRBs indeed differ from the apparent non-repeating FRBs—for example, the two subclasses have different host or local environments, or one population is intrinsically more bright—the $DM$ distributions of the sub-samples could be obviously different. Meanwhile, their extra-Galactic dispersion measures ($D{M}_{\mathrm{E}}=DM-D{M}_{MW}$) could be distinctly distributed. We applied the Anderson–Darling (A–D) test to different kinds of FRBs, see Figure 1a,b, and list the statistical results in Table 2, where it is shown that the $D{M}_{\mathrm{E}}$ distributions of the non-repeating, repeating and all FRBs can be well described by a log-normal function. Furthermore, we found the mean values of $D{M}_{\mathrm{E}}$ are $496.9\pm 16.9\mathrm{pc}{\mathrm{cm}}^{-3}$ (0.70 dex) for non-repeating FRBs and $349.1\pm 8.4\mathrm{pc}{\mathrm{cm}}^{-3}$ (0.39 dex) for repeating FRBs via a Gauss fit. Simultaneously, we found that the mean values of $D{M}_{\mathrm{E}}$ of all FRBs are $485.1\pm 15.2\mathrm{pc}{\mathrm{cm}}^{-3}$ (0.67 dex). It is notable that the mean $D{M}_{\mathrm{E}}$ value of non-repeating FRBs is evidently larger than that of repeating FRBs. To check whether the $D{M}_{\mathrm{E}}$ distributions of non-repeaters and repeaters are same or not, we used a Mann–Whitney–Wilcoxon (M–W–W) test [76,77] and obtained the statistic $W=4328.5$ with a p-value of 0.031, less than the significance threshold of 0.05, which demonstrates that the $D{M}_{\mathrm{E}}$ distributions of the two kinds of FRBs may have different progenitors or different physical mechanisms [31,64,78]. Notably, the $D{M}_{\mathrm{E}}$ is mainly contributed by the intergalactic medium, the host galaxies or the local environments. Unfortunately, what we have learned about the host galaxies is so little that the $D{M}_{\mathrm{E}}$ distributions are still uncertain and need to be verified by more observations in future.

Figure 1. The logarithmic distributions of $D{M}_{\mathrm{E}}$ and radio energy of FRBs. Panel (a) presents the $D{M}_{\mathrm{E}}$ distributions of non-repeating and repeating FRBs and panel (b) displays the $D{M}_{\mathrm{E}}$ distributions of all FRBs. Panel (c) shows the radio energy distribution of repeating events. The dash lines are the best fits to data with a Gauss function.

Table 2. The statistical results of $D{M}_{\mathrm{E}}$ for non-repeating FRBs and repeating FRBs.

$\mathbf{DM}$	FRB Sample	Statistic Value	The Critical Value	p-Value	Methods
$D{M}_E$	Non-repeating	0.814	1.084	0.035 ${}^{\mathrm{b}}$	A–D test ${}^{\mathrm{a}}$
$D{M}_E$	Repeating	0.427	0.963	0.285 ${}^{\mathrm{b}}$	A–D test ${}^{\mathrm{a}}$
$D{M}_E$	All	0.727	1.085	0.058 ${}^{\mathrm{b}}$	A–D test ${}^{\mathrm{a}}$
$D{M}_{\mathrm{E}}$	Repeating and Non-repeating FRBs	4328.5	...	0.031 ${}^{\mathrm{d}}$	M–W–W test ${}^{\mathrm{c}}$

^a The A–D test was executed in logarithmic scale with a significance level of α = 0.01. ^b A p-value larger than α indicates a log-normal distribution is favored [79]. ^c The M–W–W test was used to check different distributions with a significance Level of α = 0.05. ^d A p-value greater than α means that the two distributions are the same [76].

In addition, we found that the radio energy of repeating FRBs ranges from $2.00\times {10}^{37}$ to $6.01\times {10}^{40}$ erg. We also used the Gauss function to fit the energy statistical distribution of repeating FRBs. As shown in panel Figure 1c, the total energy of repeating FRBs roughly follows a log-normal distribution with a mean value of $2.{51}_{-0.59}^{+0.81}\times {10}^{38}$ erg and a scatter of 1.03 dex. Interesting, our result is roughly consistent with the early estimates of Li et al. [5] (${10}^{39}$–${10}^{42}$ erg) for the repeating FRBs. Meanwhile, it is worth mentioning that this result is different from the bimodal energy distribution (a log-normal function and a generalized Cauchy function, with the peak of the energy distribution of $4.8\times {10}^{37}$ erg) of a sub-sample of all the bursts from FRB 20121102A found by Li et al. [18] with FAST. Alternatively, it is possible that the bimodal energy distribution could be existent for a single burst but disappear when many bimodal distributions of diverse repeating FRBs are randomly mixed. However, Li et al. [5] noticed that the total energy of non-repeaters is on average larger than that of repeaters about one order of magnitude, which may demonstrate that at least some of repeating and non-repeating FRBs have different physical origins.

2.2. Correlations between Some Characteristic Parameters

In Figure 2, the relationships of the $D{M}_{\mathrm{E}}-S_{\mathrm{peak}},D{M}_{\mathrm{E}}-F_{\mathrm{obs}},D{M}_{\mathrm{E}}-E,$$W_{\mathrm{obs}}-S_{\mathrm{peak}},W_{\mathrm{obs}}-D{M}_{\mathrm{E}},W_{\mathrm{obs}}-E$ of repeating FRBs are illustrated. Figure 2a shows that $S_{\mathrm{peak}}$ is not correlated with $D{M}_{\mathrm{E}}$. Note that the intrinsic width of the narrow FRBs is difficult to discern due to dispersion smearing and scattering broadening. Scattering is model dependent, and the assumptions of the model introduce high uncertainty. Therefore, we used the observed pulse width instead of the intrinsic pulse width in our work. Figure 2b shows that $F_{\mathrm{obs}}$ does not correlate with $D{M}_{\mathrm{E}}$. As shown in Figure 2c, the radio energy and the $D{M}_{\mathrm{E}}$ are positively correlated with a Pearson correlation coefficient of 0.81 and can be well-fitted. The power-law relation is $E\propto D{M}_{\mathrm{E}}^{2.60\pm 0.04}$, which is roughly consistent with the tight correlation of non-repeating FRBs found by Li et al. [80], who argued that the positive correlation may be attributed to an observation selection effect—i.e., a fainter event is easier to be observed at a nearer distance. We found from Panels (d) and (e) that $W_{\mathrm{obs}}$ is not correlated with $S_{\mathrm{peak}}$ but obviously correlated with the $D{M}_{\mathrm{E}}$. Note that our $W_{\mathrm{obs}}-S_{\mathrm{peak}}$ relation is inconsistent with that of non-repeaters in [80]. The $S_{\mathrm{peak}}$ of the repeated FRBs spans three magnitudes and is more dispersive than the $S_{\mathrm{peak}}$ of the non-repeating FRBs sample of Li et al. [80], so that $W_{\mathrm{obs}}$ and $S_{\mathrm{peak}}$ are not correlated. Meanwhile, we found that there is a positive correlation between $W_{\mathrm{obs}}$ and the radio energy, as shown in Figure 2f. Li et al. [80] analyzed the correlations of key parameters of 16 non-repeating FRBs and found no clear correlation between energy and pulse width. For our repeating FRBs, the radio energy E and the pulse width $W_{\mathrm{obs}}$, respectively, span four and three orders of magnitude. The best-fitted power-law relation is $E\propto W_{\mathrm{obs}}^{0.43\pm 0.06}$, with a correlation coefficient of 0.71 and a chance probability of $1.49\times {10}^{-6}$. This means that the repeating FRB pulse with a longer duration has a greater energy release generally, which is similar to those non-repeaters reported by Li et al. [80], and both of which are consistent with the findings in [5], where the averaged isotropic energies of long FRBs were found to be larger than those of short FRBs by at least two orders of magnitude, not only for non-repeating FRBs but also for repeating FRBs.

Figure 2. The correlations of $D{M}_{\mathrm{E}}$ with $S_{\mathrm{peak}}$, $F_{\mathrm{obs}}$ and E are shown in Panels (a–c) respectively. The $W_{\mathrm{obs}}$ is plotted against $S_{\mathrm{peak}}$ in Panel (d), $D{M}_{\mathrm{E}}$ in Panel (e) and E in Panel (f). The observed data are symbolized with the filled squares. The solid lines in Panels (c,f) stand for the best fits to data with a power-law function.

Figure 2e,f seems to show that $D{M}_{\mathrm{E}}$, $W_{\mathrm{obs}}$ and E may be related. This motivated us to perform multiple linear regression fitting for these three parameters in Figure 3. The three-parameter relation can be well described by a binary linear regression function as

$$logE=(-1.31\pm 0.97)+(1.85\pm 0.41)logD{M}_{\mathrm{E}}+(1.04\pm 0.36)log{W}_{\mathrm{obs}}$$

(1)

with a Pearson correlation coefficient of 0.88, which implies that wider FRBs usually hold larger energy outputs and higher $D{M}_E$ values, and vice versa.

Figure 3. Left panel: Correlation among E, $D{M}_{\mathrm{E}}$ and $W_{\mathrm{obs}}$ of repeating FRBsis illustrated by the 3D scatter plot. Right panel: The isotropic energy E is plotted against the energy estimated by Equation (1). The solid line is the best fit to the data. The light and the heavy shaded regions are the $95\%$ confidence and the $95\%$ prediction ranges, respectively.

3. Apparent Intensity Distribution Function of Repeating FRBs

Li et al. [80] applied the $F_{\mathrm{obs}}$ of 16 non-repeating FRBs to derive an intensity distribution function (IDF). We considered the IDF of repeating FRBs to be determined by the average fluence ($\overline{F}$) and observed fluence ($F_{\mathrm{obs}}$) of the brightest burst for each repeater. We then utilized the average fluence $\overline{F}$ presented in Table 1 to investigate the IDF of 21 repeating FRBs. This was done for several reasons: (1) For one repeating burst, multiple monitor observations in different time periods may lead to the observational bias. (2) The repeaters contain multiple bursts, and the number of monitored subbursts between different repeaters varies—e.g., FAST has recently detected 1652 repeating events from FRB 20121102A [18] and 1863 bursts from FRB 20201124A [19]. (3) There is a certain degree of fluctuation between the fluences of repetitions for one repeating FRB; for example, the fluence of the repetitions from faint FRB 20171019A have a large range of 0.37 to ∼ 388 $\mathrm{Jy}\mathrm{ms}$ [56]. Therefore, we took the mean fluence as a statistical variable for each repeater in order to build an apparent IDF of the repeating FRBs. In addition, we got rid of the faint FRB 20171019A from our repeating FRB sample because of its large fluctuations of fluence in the following calculations.

We assumed that FRBs can be considered as standard candles, and defined that the apparent IDF of repeating FRBs is in the same form as the function given by Li et al. [80]: $dN/d{F}_{\mathrm{obs}}=A{F}_{\mathrm{obs}}^{-a}$ (or $N(>F_{\mathrm{obs}})\propto F_{\mathrm{obs}}^{-a+1}$), where $\alpha$ is the power-law index, $a=2.5$ is expected for a uniform distribution and A is a constant which depends on the observations and the event rates of FRBs. We group the repeating FRB sample into several fluence intervals with a bin width of $\Delta \overline{F}$, and created the exemplary distributions of $\overline{F}$, as presented in Figure 4. Then we acquired a best-fit power-law curve for $dN/d{F}_{\mathrm{obs}}$. In Figure 4a, best fit results are shown for a bin width of $\Delta \overline{F}=3.6\mathrm{Jy}\mathrm{ms}$; the corresponding power-law index and the coefficient of determination (R-Square) are $a_1=1.07\pm 0.05$ and 0.99, respectively. Note that the error bars along the x-axis and y-axis are, respectively, $\Delta \overline{F}/2$ and the square root of FRB count in each interval. Meanwhile, we also investigated the influence of bin width selection on the fitted results. In our work, the range of the bin width $\Delta \overline{F}$ was selected to be from 0.2 to 6.0 $\mathrm{Jy}\mathrm{ms}$, and the best-fit power-law indices corresponding to each $\Delta \overline{F}$ are shown in Figure 4b, where one can see that when $\Delta \overline{F}$ is in a range of $3.2$–$4.2$ Jy ms, the fitted $a_1$ has smaller fluctuations and the corresponding error bar is also small. Hence, the $a_1$ values of $\Delta \overline{F}=3.2-4.2\mathrm{Jy}\mathrm{ms}$ were selected to calculate the final average value, which is $a_1=1.10\pm 0.14$. To constrain the unknown constant A, we adopted similar calculation processes to those described in detail in Li et al. [80] and compare them with the published event rates of FRBs in the literature in Table 3. The A value in our IDF was obtained to be $(2.37\pm 0.76)\times {10}^3{\mathrm{sky}}^{-1}{\mathrm{day}}^{-1}$. As a result, the final IDF of 20 repeating FRBs can be written as

$$\frac{dN}{dF}=(2.37\pm 0.76)\times {10}^3{F}_{\mathrm{obs}}^{-1.10\pm 0.14}{\mathrm{sky}}^{-1}{\mathrm{day}}^{-1}.$$

(2)

Figure 4. Panel (a) shows an exemplary distribution of $\overline{F}$ for a bin width of $\Delta \overline{F}=3.6$ Jy ms. The solid line is the best-fit curve. Panel (b) illustrates the best-fit power-law values $a_1$ for each bin width; the solid short horizontal line shows the relatively stable range of $\Delta \overline{F}=3.2$–4.2 Jy ms for $a_1$. Panel (c) shows an example distribution of $F_{\mathrm{obs}}$ for a bin width of $F_{\mathrm{obs}}=3.8$ Jy ms. The solid line is the fitted curve. Panel (d) presents the best-fit power-law values $a_2$ for each bin width; the solid short horizontal line shows the relatively stable range of $\Delta F_{\mathrm{obs}}=3.3$–4.4 Jy ms for $a_2$.

Table 3. Comparison of the deduced event rates of different FRB samples.

${\mathit{F}}_{\mathbf{Limit}}$ (Jy ms)	$\mathit{R}\left(>{\mathit{F}}_{\mathbf{Limit}}\right)$ (sky −1 day −1)	Reference	Coefficient A (103 sky −1 day −1)	FEB Class
3	$1.0_{-0.5}^{+0.6}\times {10}^4$	Thornton et al. [2]	$5.02\pm 1.62$	non-repeaters
0.35	$3.1_{-3.1}^{+12}\times {10}^4$	Spitler et al. [84]	$7.49\pm 1.15$	non-repeaters
2	$2.5\times {10}^3$	Keane and Petroff [7]	$1.06\pm 0.32$	non-repeaters
1.8	$1.2\times {10}^4$	Law et al. [8]	$4.88\pm 1.38$	non-repeaters
4	$4.4_{-3.1}^{+5.2}\times {10}^3$	Rane et al. [11]	$2.53\pm 0.87$	non-repeaters
0.13–1.5	$7_{-3}^{+5}\times {10}^3$	Champion et al. [9]	$2.87\pm 0.82$	non-repeaters
3.8	$3.3_{-2.2}^{+3.7}\times {10}^3$	Crawford et al. [85]	$1.85\pm 0.63$	non-repeaters
0.03	$(3.03\pm 1.56)\times {10}^4$	Li et al. [80]	$4.19\pm 0.22$	non-repeaters
6	587	Lawrence et al. [86]	$0.42\pm 0.16$	non-repeaters
2	$1.7_{-0.9}^{+1.5}\times {10}^3$	Bhandari et al. [12]	$0.72\pm 0.21$	non-repeaters
26	$37\pm 8$	Shannon et al. [15]	$0.15\pm 0.01$	non-repeaters
8	${98}_{-39}^{+59}$	Farah et al. [16]	$0.01\pm 0.003$	non-repeaters
2	$3.4_{-3.3}^{+15.4}\times {10}^3$	Parent et al. [17]	$1.44\pm 0.42$	non-repeaters
5	$818\pm 64$	CHIME/FRB Collaboration et al. [6]	$0.53\pm 0.19$	non-repeaters
0.36	$(9.64\pm 3.09)\times {10}^3$	this work	$2.37\pm 0.76$	repeaters

Similarly, we derived, while adopting $F_{\mathrm{obs}}$, the power-law $a_2=1.01\pm 0.16$ of IDF for repeating FRBs (see Figure 4d). Furthermore, we got $A=(1.96\pm 0.41)\times {10}^3{\mathrm{sky}}^{-1}{\mathrm{day}}^{-1}$ for our IDF. The IDF of repeating FRBs is given by

$$\frac{dN}{dF}=(1.96\pm 0.41)\times {10}^3{F}_{\mathrm{obs}}^{-1.01\pm 0.16}{\mathrm{sky}}^{-1}{\mathrm{day}}^{-1}.$$

(3)

In particular, we found that power-law indices $a_1$ and $a_2$ have small differences, which indicates that our results are stable and reliable. If assuming the minimum fluence of 0.36 Jy ms as the threshold, one can estimate that the detection rate of repeating FRBs is at least $(9.64\pm 3.09)\times {10}^3$ sky${}^{-1}$ day${}^{-1}$ for $a_1$ and $(1.38\pm 0.19)\times {10}^4$ sky${}^{-1}$ day${}^{-1}$ for $a_2$. Combined with all parameters of FAST [81,82,83], we utilized the method of Li et al. [80], and estimate the detection rate for the 1000 h observation time is about $3\pm 1$; that is slightly smaller than that for non-repeating FRBs predicted by Li et al. [80]. It is reasonable, since the current detection rate of non-repeating FRBs is obviously larger than that of repeating FRBs observationally.

4. Differential Bolometric Luminosity Distributions

The luminosity functions of FRBs can be applied to reveal the origins of FRBs, design the optimal searching plan, guide future observations [61,67,80,87], etc. The detection rate of the telescope can be calculated by the luminosity function, and the event rate density at different luminosity ranges sheds light on the origins of FRBs [31,78]. Building the LF requires not only the flux and distance, but also a k-correction of luminosity because of the different observational central frequencies (${\nu }_c$) in the mixed sample of FRBs. According to Zhang [29], the isotropic peak luminosity of a FRB can be calculated with $L_p\simeq 4\pi D_{\mathrm{L}}^2\left(z\right)S_{\mathrm{peak}}{\nu }_c$, where $D_{\mathrm{L}}\left(z\right)$ is the lumonisity distance calculated with the redshift given by FRBCAT (https://www.frbcat.org/). (accessed on 3 May 2022). We utilized ${\nu }_c$ to substitute the observational bandwidth of radio telescopes, which was proposed by Petroff et al. [4] and Aggarwal [88]. Additionally, Zhang [29] also suggested to use the central frequency instead of the receiver bandwidth since the FRB spectrum must be unknown and emissions may extend beyond the receiver bandwidth. In addition, when the observed flux densities of cosmological objects at different ${\nu }_c$ are considered, k-correction is also an important effect. After the k-correction, the bolometric luminosity can be given by $L\equiv L_p\simeq 4\pi D_{\mathrm{L}}^2\left(z\right)S_{\mathrm{peak}}{\nu }_{c}k$, in which the k-correction factor is taken as $k={(1+z)}^{{\alpha }_t-\beta }$ with a temporal index of ${\alpha }_t\sim 0$ and a spectral index $\beta \sim 1/3$ [5,89], which are similar to those parameters of normal pulsar spectra [28,90,91].

The differential distributions of the k-corrected luminosity for 21 repeaters and 571 non-repeaters are shown in Figure 5, where one can notice that the repeaters are normally less luminous than the non-repeaters by about two orders of magnitude. Subsequently, we adopted a broken power-law from [92,93,94] to build the LF as

$$\varphi \left(L\right)=\left\{\begin{array}{cc}A{\left(\frac{L}{L_{\mathrm{b}}}\right)}^{{\alpha }_1},\hfill & L\le L_{\mathrm{b}}\hfill \\ A{\left(\frac{L}{L_{\mathrm{b}}}\right)}^{{\alpha }_2},\hfill & L>L_{\mathrm{b}}\hfill \end{array}\right.,$$

(4)

where A is the normalized factor; $L_b$ is the break luminosity; ${\alpha }_1$ and ${\alpha }_2$ are two power-law indices corresponding to lower and higher luminosity portions. In Figure 5, we can see that the luminosity distributions of both repeating and non-repeating FRBs can be well described by the broken power-law function, and the best-fit parameters are provided in Table 4. We excitingly found that the power-law indices of repeating and non-repeating bursts in the luminous end are the same as $\approx -1.4$ within the range from −1.8 to −1.2, constrained by the Schechter luminosity function by Luo et al. [61]. However, we noticed that non-repeaters decline slower that repeaters in the less luminous end instead. In addition, after k-correction for observational frequency, the distributions of luminosity for repeating and non-repeating FRBs are similar at their brightest ends. The characteristic luminosity values $L_b$ are not significantly different.

Figure 5. The luminosity functions of the repeating (Panel a) and non-repeating (Panel b) FRBs. The solid lines stand for the best fits with the power-law function of Equation (3). In each panel, the error bars along the x-axis are simply the standard errors in the corresponding luminosity bins, and the y-axis error bars are the 1-$\sigma$ Poisson errors [95].

Table 4. The best-fit parameters of a broken power-law LF for repeating and non-repeating FRBs.

Sample	A	${\mathit{\alpha }}_1$	${\mathit{\alpha }}_2$	${\mathit{L}}_{\mathit{b}}\left(\mathbf{erg}{\mathbf{s}}^{-1}\right)$	${\mathit{\chi }}_{\mathit{\nu }}^2$
Repeating FRBs	$(4.80\pm 1.41)\times {10}^{-42}$	$-0.45\pm 0.20$	$-1.41\pm 0.08$	$(1.78\pm 0.44)\times {10}^{42}$	0.17
Non-repeating FRBs	$(8.59\pm 2.01)\times {10}^{-41}$	$-0.09\pm 0.10$	$-1.49\pm 0.05$	$(1.74\pm 0.29)\times {10}^{42}$	3.17

5. Conclusions and Discussion

In this study, we analyzed the observed data of 21 repeating and 571 non-repeating FRBs published and drew the following conclusions:

• The extra-Galactic dispersion measures $D{M}_{\mathrm{E}}$ of non-repeating and repeating FRBs were found to be log-normally distributed with mean values of $496.9\mathrm{pc}{\mathrm{cm}}^{-3}$ and 349.1 $\mathrm{pc}{\mathrm{cm}}^{-3}$, respectively. The M–W–W test showed that the $D{M}_{\mathrm{E}}$ is drawn from a different distribution.
• It was found that the total radio energies of repeating FRBs are log-normally distributed with a mean value of $2.51\times {10}^{38}$ ergs (1.03 dex), which is smaller than to that of non-repeating FRBs, which is consistent with the conclusion in [5]. Surprisingly, the bimodal energy distribution discovered in FRB 20121102A by Li et al. [18] with FAST was not recovered in our repeating FRB sample any more.
• We statistically analyzed the relationships among $D{M}_{\mathrm{E}}$, $S_{\mathrm{peak}}$, $F_{\mathrm{obs}}$, E and $W_{\mathrm{obs}}$, and found that most correlations between them are similar to those of non-repeaters given by Li et al. [80], except that the $E-W_{\mathrm{obs}}$ relation of repeaters is tighter. The statistical results hint that the spatial distribution and the local environments of two samples of FRBs may be different, although more samples are needed to verify our argument, as noted by CHIME/FRB Collaboration et al. [55] and Fonseca et al. [74].
• We constructed a three-parameter relation to be $logE\simeq -1.31+1.85logD{M}_{\mathrm{E}}+1.04$$log{W}_{\mathrm{obs}}$, indicating that longer FRBs usually have larger extra-galactic dispersion measures and more energy releases, which supports the early findings by Li et al. [5], no matter whether a FRB is repeating or not.
• We assume that FRBs can be considered as standard candles homogeneously in a flat Euclidean space, with an IDF of $dN/d{F}_{\mathrm{obs}}=A{F}_{\mathrm{obs}}^{-a}$ (or $N(>F_{\mathrm{obs}})\propto F_{\mathrm{obs}}^{-a+1}$), where a should be theoretically 2.5. Using the averaged fluence as a characteristic quantity, we built the IDF of repeating FRBs as $dN/dF=(2.37\pm 0.76)\times {10}^3{F}_{\mathrm{obs}}^{-1.10\pm 0.14}{\mathrm{sky}}^{-1}{\mathrm{day}}^{-1}$. Likewise, using the observed fluence as a statistical indicator, we obtained the IDF as $dN/dF=(1.96\pm 0.41)\times {10}^3{F}_{\mathrm{obs}}^{-1.01\pm 0.16}{\mathrm{sky}}^{-1}{\mathrm{day}}^{-1}$. The power-law index ($a_1,a_2$) of IDF deviates from the theoretical value of 2.5 in a flat Euclidean space, and shows that the repeating FRBs may be not uniformly distributed. Assuming the averaged minimum fluence as the threshold, we predicted the detection rate of repeaters to be about $(9.64\pm 3.09)\times {10}^3$ sky${}^{-1}$ day${}^{-1}$ or $(1.38\pm 0.19)\times {10}^4$ sky${}^{-1}$ day${}^{-1}$; that is slightly lower in contrast with those non-repeating ones.
• Finally, we constructed and compared the luminosity functions of repeating and non-repeating FRBs. Interestingly, we found that the luminosity functions for both kinds of FRBs can be well characterized by a broken power-law relation; their power-law indices at their luminous ends are equal to $\approx -1.4$, despite the discrepancy at their less-luminous ends.

Observationally, the power-law relation of $N\propto F^{\alpha }$, especially its power-law index a, indeed changes largely for different telescopes, owing to many factors, such as the selection effects/biases, the instrumental sensitivity, the sample incompleteness, the poor localizations, the small sample size or the cosmological effect (e.g., [12,90,96]), which may cause the relation to deviate from the theoretical form of $N\propto F^{-3/2}$ in a Euclidean space. Caleb et al. [68] pointed out by simulations that the slope of the logarithmic relation is mainly determined by cosmological effect and got $a=-0.9\pm 0.3$, which matches a uniform distribution of FRBs roughly. Even for the same telescope, the deduced power-law indices could be inconsistent. For instance, Macquart and Ekers [90] found $a=-2.6_{-1.3}^{+0.7}$ and James et al. [96] obtained $a=-1.18\pm 0.24$ for Parkes FEBs. However, these power-law indices cannot show that the FRBs are not distributed in a Euclidean space. It is also why we took the average fluence as a characteristic quantity. Considering all above complex situations, we have assumed a universal slope in order to reduce the influences of uncertain factors on the $logN-logF$ relation.

In addition, we have constructed the luminosity functions for our samples of repeating and non-repeating FRBs on the basis of the k-corrected isotropic luminosities. It was found that the k-corrected luminosity of repeaters and non-repeaters span three and six orders of magnitude, respectively. On average, the repeaters are less energetic than the non-repeaters, which is coincident with [5]. The luminosity distributions of the two samples of FRBs can be well described by a broken power-law function with a break luminosity of $L_{\mathrm{b}}=1.74\times {10}^{42}\mathrm{erg}{\mathrm{s}}^{-1}$ for non-repeaters and $L_{\mathrm{b}}=1.78\times {10}^{42}\mathrm{erg}{\mathrm{s}}^{-1}$ for repeaters. This may imply that either repeaters or non-repeaters can be redivided into low- and high-luminosity types originating from different progenitors.

As addressed above, the $D{M}_{\mathrm{E}}$ distributions of the two samples have different distributions, which suggests that the majority of FRBs might have different astrophysical origins. Recent observations indicate that both kinds of FRBs may have distinct environments, such as dense, offset and star formation distribution (e.g., [97,98]). Additionally, there exist some special observed properties (such as polarization, rotation measures and the complex frequency–time structure) and a difference between FRB host galaxies [99,100,101]. Thus, these observational differences imply the emergence of subpopulations of FRBs. However, there are several recent works claiming no significant differences in the host galaxy properties (e.g., [102,103]). One possible reason is that the number of FRBs with known host galaxies is still limited currently. There is a debate about repeating and non-repeating (or more subclasses) or having a single population [62,104,105]. This means that the current classification results from the observations may be due to observational selection bias, e.g., if the observational period is not in the most active window for the individual FRBs [63,105], and one non-repeating burst identified currently may be a potential repeating FRB, as was the case for FRB 20171019A and FRB 20180301A. It may be just a few lucky cases, but we have probably missed many faint bursts of other FRBs. Nonetheless, the large sample of known FRBs is starting to show trends that suggest subclasses based on burst morphology and frequency–time structure (the downward frequency drift of sub-bursts [55,74,106], $\sim 100\%$ linear polarization and negligible circular polarization [107]; and some sources show periodicity [108,109,110]). The observations in future should be focused on the FRB events which have relatively high probabilities of becoming repeating FRBs (see also [5]). When more FRBs are accurately localized, more useful information (such as $DM$, redshift, hosts and local environments of repeaters and apparently one-off FRBs) can be obtained from the direct observations, and the true LF will be accurately built as a cosmological probe to reveal the nature of FRBs.