The fast radio burst population energy distribution

Abstract

We examine the energy distribution of the fast radio burst (FRB) population using a well-defined sample of 63 FRBs from the Australian Square Kilometre Array Pathfinder (ASKAP) radio telescope, 28 of which are localised to a host galaxy. We apply the luminosity-volume ($V/V_{\mathrm{max}}$) test to examine the distribution of these transient sources, accounting for cosmological and instrumental effects, and determine the energy distribution for the sampled population over the redshift range $0.01 \lesssim z \lesssim 1.02$. We find the distribution between $10^{23}$ and $10^{26}$ J Hz$^{-1}$ to be consistent with both a pure power-law with differential slope $\gamma=-1.96 \pm 0.15$, and a Schechter function with $\gamma = -1.82 \pm 0.12$ and downturn energy $E_\mathrm{max} \sim 6.3 \, \times 10^{25}$ J Hz$^{-1}$. We identify systematic effects which currently limit our ability to probe the luminosity function outside this range and give a prescription for their treatment. Finally, we find that with the current dataset, we are unable to distinguish between the evolutionary and spectral models considered in this work.

1. Introduction

Fast radio bursts (FRBs) are short duration (millisecond timescale), dispersed, transient events in the radio spectrum known to originate from cosmological distances (Lorimer et al. 2007; Thornton et al. 2013; Chatterjee et al. 2017; Bannister et al. 2019). Current research has two major directions: to determine their progenitor source(s) and to use them as cosmological probes (Macquart et al. 2020). Accordingly, the FRB population statistics continues to be a topic of considerable conjecture (see e.g. Petroff, Hessels, & Lorimer 2022, and references therein).

Determining the intrinsic energy distribution (i.e. luminosity function) of FRBs has, hitherto, proven to be problematic. The first impediment stems from radio telescopes with optics that make an accurate determination of the FRB location within the telescope beam difficult, such as Parkes/Murriyang (e.g. Thornton et al. 2013; Keane et al. 2018), UTMOST (Farah et al. 2019), FAST (Niu et al. 2021), and the first FRB searches with CHIME (CHIME/FRB Collaboration: Amiri et al. 2021). This complicates the construction of a fluence-complete sample and determining the effective survey area (see Keane & Petroff 2015; Macquart & Ekers 2018a). This issue is effectively mitigated when using telescope arrays to search for FRBs, as pioneered by the Australian Square Kilometre Array Pathfinder (ASKAP) telescope – which uses phased array feeds (PAFs) to provide a wide field of view (FoV) with dense coverage of the focal plane – permitting reliable estimates of the survey area and FRB fluences to be made (Bannister et al. 2017; Shannon et al. 2018). Even then, however, the relationship between a detected FRB’s signal-to-noise (S/N) and its fluence is distorted by dispersion measure smearing, scattering, and the nuances of the particular detection algorithm. A significant literature is now dedicated to modelling these effects (Cordes & McLaughlin 2003; Keane & Petroff 2015; Qiu et al. 2023; Merryfield et al. 2023; Hoffmann et al. 2024a), allowing them to be accounted for in FRB energy determination.

The second impediment is the difficulty in obtaining an FRB distance estimate, which yields the FRB energy and survey volume. This requires either arcsecond-precision FRB localisations, thereby permitting the identification of the host galaxy, or the existence of a relation between the FRB dispersion measure, DM, and redshift, z. ASKAP has helped provide both, with a large sample of FRBs localised to their host galaxies (Shannon et al. 2024), and the establishment of a z-DM relationship, known as the Macquart Relation (Macquart et al. 2020). Other instruments with similar capabilities include DSA 110 (Law et al. 2024), MeerKAT (Rajwade et al. 2022), the VLA (Law et al. 2018), and CHIME’s outriggers (Leung et al. 2021).

A great deal of literature has attempted to draw conclusions on the FRB luminosity function. Relatively few, however, have made a proper account for the above-mentioned effects, as emphasised by Connor (2019) – examples include Luo et al. (2020), James et al. (2021a), Shin et al. (2023), Hoffmann et al. (2024b). However, these fits rely on assumptions about the functional form of the FRB energy distribution and source evolution, which may differ from that of other classes of transients. A non-parametric way to determine both – the $V/V_{\mathrm{max}}$ method – was described by Schmidt (1968), in the context of studies of the quasar population. The simplest application of this method is to test for a spatially uniform distribution of FRB sources, which has been applied to FRB data by several authors (Oppermann, Connor, & Pen 2016; Shannon et al. 2018; Locatelli et al. 2019). Others have applied the analysis determining the FRB energy distribution from non-localised FRBs (Lu & Piro 2019; Hashimoto et al. 2022; Zhang et al. 2024), which has the aforementioned uncertainties of fluctuations in the Macquart relation.

In this work, we update these analyses using FRBs detected by the Commensal Real-time ASKAP Fast Transients (CRAFT; Macquart et al. 2010) survey with ASKAP (Hotan et al. 2021). In particular, we use a large sample of FRBs with known redshift, allowing for the first time an accurate measurement of both V and $V_{\mathrm{max}}$ for FRBs. This allows unbiased estimates of their energy and spatial distributions to be used.

In Section 2 we review the volumetrics and formation of the energy distribution for a sample being analysed. We then apply the approach to the ASKAP sample and outline our results and observations in Section 3. In Section 4 we discuss the implications of the energy distributions and present our conclusions in Section 5.

2. The energy function

2.1 The $\textbf{\textit{V/V}}_{\mathbf{max}}$ test

The discovery of FRBs in 2007 (Lorimer et al. 2007) has many similarities to the discovery of quasars (Schmidt 1963); both are new classes of extragalactic objects catalogued in surveys with well-defined but complex detection limits.

To estimate the spatial distribution and luminosity function of quasars (then referred to as QSOs), Schmidt (1968) introduced the $V/V_{\mathrm{max}}$ parameter which, for each source, provides a measure of its position within the maximum volume over which it would have been observed in the complete sample. Due to the uncertainty in cosmological models at the time, Schmidt (1968) calculated volumes in co-moving coordinates using two cosmological models: luminosity distance $D_L \propto z$ , and $D_L \propto z (1+0.5 z)$ . Schmidt notes that $V/V_{\mathrm{max}}$ provides a very simple test of uniformity for the spatial distribution in a sensitivity-limited sample, with an expectation value $ \lt +V/V_{\mathrm{max}}$ $ \gt = 0.5$ . In the case of quasars, $V/V_{\mathrm{max}}$ was found to be significantly larger than 0.5, and Schmidt concluded that the sample was strongly evolving. The expected uniformity in $V/V_{\mathrm{max}}$ was achieved by weighting the Cartesian volume $V \sim D_L^3$ by an assumed source evolution of $(1+D_L)^2$ . Schmidt then estimated the local luminosity function by using $1/V_{\mathrm{max}}$ to weight the contribution to the spatial density from each source separately, and then grouped the sources in luminosity bins, wherein these luminosities were converted to the rest frequency.

Like quasars, FRBs are also cosmologically distributed, and the problems of analysing their redshift evolution and luminosity function are very similar. The $V/V_{\mathrm{max}}$ method requires a complete sample of sources above a well-defined flux (or fluence) limit. Even if the redshifts of these sources are unknown or poorly defined, the mean value of $V/V_{\mathrm{max}}$ can indicate whether the sources are distributed uniformly through the sample volume. A uniform distribution (with $ \lt +V/V_{\mathrm{max}}$ $ \gt \approx 0.5$ ) implies a population that is non-evolving (i.e. not changing with distance) within the sample volume, while a larger or smaller value implies either an incompleteness in selection, or a population that undergoes some form of redshift evolution within the sample volume.

For a source population where redshift measurements are available for individual objects, and where there is also little or no evolution within the sample volume, a luminosity function can be calculated by summing the values of 1/ $V_{\mathrm{max}}$ within different luminosity bins. Local radio luminosity functions (RLFs) for large, complete samples have been calculated by several authors (e.g. Condon, Cotton, & Broderick 2002; Sadler et al. 2002; Best et al. 2005; Mauch & Sadler 2007), and Pracy et al. (2016) calculated the RLF for high- and low-excitation radio galaxies in several redshift bins out to $z\sim0.75$ . Avni & Bahcall (1980) extended Schmidt (1968)’s method to samples with different completeness limits in two (or more) different parameters; this technique may be used to measure a bivariate luminosity function, for example, a set of RLFs for different bins in optical luminosity (Mauch & Sadler 2007) or black hole mass determination (Best et al. 2005).

If the redshift range covered by a survey is large enough that redshift evolution occurs within the sample-volume (i.e. $V/V_{\mathrm{max}}$ has a value significantly different from 0.5), then this evolution must be taken into account. Examples from the literature include studies of the luminosity function of gamma-ray bursts (Schmidt 2009) and the redshift evolution of powerful radio galaxies (Dunlop & Peacock 1990).

Schmidt’s methods may be applied directly to the FRB population, with a few significant differences. Since FRBs are transient rather than static sources of emission, the observing time should be included in the analysis as well as the survey area. Transients are typically characterised by their fluence and energy distribution rather than their flux and luminosity function. To keep notation consistent with Schmidt (1968), hereinafter we refer to FRB luminosities and their RLF when describing the distribution of their spectral energy density, $E_\nu$ . If the positions are not determined well enough during the outburst, the location in the FoV cannot be determined. Thus the sensitivity of the telescope beam at the detection point, hence the correction for that sensitivity, cannot be made. For the population of FRBs that have not been observed to repeat, only surveys which determine the position in the FoV can therefore be used – significantly reducing the applicable sample size.

First, we use the $V/V_{\mathrm{max}}$ test to check whether the FRBs in our sample are uniformly distributed in space. Then, following Schmidt, we use $1/V_{\mathrm{max}}$ for each FRB to estimate its contribution to the density of FRBs of that luminosity. The estimation of $V_{\mathrm{max}}$ is the critical aspect introduced by this analysis: for FRB surveys it can be applied on a per source basis, provided the survey detection limit, the detected S/N and the position in the FoV are known for each FRB. This requirement significantly limits the sample of FRBs that can be used, and we therefore confine our analysis to suitable FRBs from the CRAFT survey, which satisfy these criteria. We do this for FRBs with known host galaxies for which V and $V_{\mathrm{max}}$ can be calculated. We also investigate the effect of using DM as a distance proxy by comparing this result to that obtained when estimating FRB distances from their DMs using the cosmological DM-z (‘Macquart’) relation (Macquart et al. 2020). For simplicity, in the main body of this work, we ignore FRB spectral dependence and source evolution; however, we consider both in Appendix A, and show that neither have a strong influence given current data. We do not explicitly calculate the time-dependence of the survey volume, thus we cannot calculate the FRB rate. Moreover, since we use data from both ASKAP’s Fly’s Eye and Incoherent Sum (ICS) modes in different proportions for the two samples, the relative normalisation is arbitrary. We discuss this further in Appendix B.

The ratio of volumes from which the FRB has been detected, V, to that in which it could have been detected, $V_{\mathrm{max}}$ , is a measure of the position of the detected event within the probed volume. The statistic $\langle V/V_{\mathrm{max}}\rangle$ is the algebraic mean of events in a sample and is expressed as:

(1) \begin{align} \langle V/V_{\mathrm{max}}\rangle = \frac{1}{N} \sum_{i=1}^{N} \frac{V_{i}}{V_{\mathrm{max},i}} \text{,}\end{align}

where i represents the $i^{\mathrm{th}}$ event in a sample of N events. A spatially uniform sample would be uniformly distributed over the range $\left[0, 1\right]$ with $\langle V/V_{\mathrm{max}}\rangle=0.5$ (Schmidt 1968). The luminosity function may be determined from a contribution of each event by taking the reciprocal of the volume in which each event could have been observed (i.e. $1/V_{\mathrm{max},i}$ ), and binning in terms of energy.

In the case of an evolving population (e.g. source density evolving with redshift, or source luminosity variations; Schmidt 1968; Macquart & Ekers 2018b) or incorrect assumptions regarding the nature of the volume, the distribution given through equation (1) will not, in general, be uniform. Re-weighting V by the correct source density, $\psi$ , within that volume, that is, $V \rightarrow V^\prime = V \cdot \psi\left(V\right)$ , would, however, restore the distribution to uniformity.

Measurement of the FRB luminosity distribution presents a number of complications not typically encountered with static sources, since it is not possible to find all objects by scanning an area of sky with uniform sensitivity. For a sample of static sources, one may clearly define the volume over which a source would have been detectable, viz., the volume of a spherical sector whose radius is governed by the luminosity distance out to which an object could have been detected, given the telescope sensitivity. For radio transients such as FRBs, however, this is not the case: the instantaneous sensitivity across the FoV, when the FRB is detected, is non-uniform and the volume probed is therefore not a section of a sphere. When one is interested in the event rate rather than the source density per comoving volume, the additional effect of the observing time and time dilation as a function of distance needs to be taken into account.

The spectral energy density, $E_{\nu, 0}$ , of a given FRB, its observed fluence, $F_{\nu,0}$ , and its luminosity distance, $D_{L}$ , are related via equation (2)

(2) \begin{align} E_{\nu,0} = \frac{4 \pi F_{\nu,0} D_L^2}{ (1+z_b)^{2+\alpha}} \text{,}\end{align}

where $z_b$ is the redshift of the FRB and $\alpha$ the fluence spectral index. We define $\alpha:F_{\nu}\propto\nu^{\alpha}$ – this is now common usage, however it is the opposite sign convention to that used in Macquart & Ekers (2018b) and subsequently in Arcus et al. (2020) and (2022).

2.2 The survey volume

Here we derive measures of both V and $V_{\mathrm{max}}$ that account for the beamshape of the FRB discoveryp antenna. We note that, for a non-evolving population in a Cartesian Universe, the antenna beamshape will not affect calculations of $V/V_{\mathrm{max}}$ , since both V and $V_{\mathrm{max}}$ will scale identically for all beam positions. Assuming a constant value of beam (hence telescope) sensitivity will, however, smear-out the luminosity function due to uncorrected-for differences between true fluence (which requires the beamshape at point of detection to be known) and the fluence calculated when ignoring beamshape.

In the case of an evolving population, objects requiring spectral corrections, and/or a non-Euclidean Universe, the proportionality between V and $V_{\mathrm{max}}$ is lost, and beamshape corrections can become important in calculating $V/V_{\mathrm{max}}$ as well as the luminosity function. Given that the FRB population may experience both source evolution and spectral dependence, and FRB observations now probe the $z \gt 1$ regime at which cosmological distance measures are significantly different from their Euclidean equivalents, we consider a proper treatment crucial when applying the $V/V_{\mathrm{max}}$ method.

Figure 1. The geometry of the $V_{\mathrm{max}}$ region defining the total comoving volume out to which a given FRB may be detected with an S/N being a factor of X above the threshold detection S/N for a generic beam. Note that $V_{\mathrm{max}}$ must be computed separately for each FRB since the S/N of a given FRB depends upon both the FRB fluence and duration: the $D_{L,{\mathrm{max}}}$ surface cannot be specified solely in terms of a threshold fluence.

2.2.1 The maximum volume probed by a generic beam

Consider a FRB event occurring at a given offset, $\theta_{d}$ , from the beam centre of a generic telescope beam at an S/N value that is a factor of X above the cut-off S/N flux threshold $S_{\mathrm{cutoff}}$ (see Fig. 1). We would like to know over what volume this particular event, with FRB spectral energy density, $E_\nu$ , and burst-width, $\Delta t$ , could have been detected. If the telescope beam is circularly symmetric, the comoving volume of space probed out to a redshift $z_{\mathrm{max}}$ is given by

(3) \begin{align} \begin{split} V_{\mathrm{max}} &= \int \int \frac{D_H}{E(z)} \frac{D_L^2(z)}{(1+z)^2} dz d\Omega \\ &=2 \pi \int_0^{\theta_{\mathrm{outer}}} \int_0^{z_{\mathrm{max}}(\theta)} \frac{D_H}{E(z)} \frac{D_L^2(z)}{(1+z)^2} \sin \theta d\theta dz \text{,} \end{split}\end{align}

where $\Omega$ is the solid angle of the telescope beam on the sky; $\theta$ , the bore-sight angle of the telescope beam; and $\theta_{\mathrm{outer}}$ the outermost detectable angle of the beam. Moreover, $z_{\mathrm{max}}(\theta)$ is the redshift of the maximum luminosity distance that an event could be detected in the telescope beam and $D_H$ and $D_L(z)$ are the Hubble distance and luminosity distance for a given redshift, z, respectively.

We write $z_{\mathrm{max}}(\theta)$ as an explicit function of $\theta$ to emphasise that the telescope probes to a larger redshift at the beam centre relative to its periphery. We take the integral over the angular distance to extend out to an effective beam cut-off point; the objective here being to find $z_{\mathrm{max}}(\theta)$ for a given FRB so that the effective survey volume may be determined.

We may compute the maximum detectable luminosity distance for each FRB at its particular location within the telescope beam via equation (2), to determine $E_{\nu, 0}$ , then find the luminosity distance at which the FRB of this energy density would be detectable at the threshold $S_{\mathrm{cutoff}}$ .

An additional complication is that the detection S/N is not determined just by the FRB flux density; rather, S/N is proportional to a product involving the FRB flux density and its duration. Thus the threshold fluence is obtained by solving

(4) \begin{align} \begin{split} \frac{S_{\mathrm{cutoff}}}{S} \equiv X(\theta_d) &= \frac{S_{\nu,0} \Delta t_0^{1/2}}{S_{\nu,{\mathrm{cutoff}}} \Delta t_{\mathrm{cutoff}}^{1/2} } \\ &= \frac{F_{\nu,0} }{F_{\nu,{\mathrm{cutoff}}}} \frac{(1+z)^{-1/2}}{(1+z_{\mathrm{max}})^{-1/2}} . \end{split}\end{align}

The solution of equation (4) yields the following transcendental equation for the limiting detectable fluence for a given FRB:

(5) \begin{align} F_{\nu,{\mathrm{cutoff}}} = \frac{F_{\nu,0}}{X(\theta_d)} \left( \frac{1+z_b}{1+z_{\mathrm{max}}} \right)^{-1/2} \text{,}\end{align}

and we solve this equation to determine $z_{\mathrm{max}}(\theta)$ .

Yet a further complication is that the telescope detection efficiency decreases with increasing DM, which is nearly linearly proportional to redshift at $z \lesssim 1$ (see e.g. Arcus et al. 2022). If the telescope efficiency, $\eta$ , is written in terms of DM, the maximum luminosity distance out to which the FRB is detectable ^a is given by:

(6) \begin{align} \begin{split} D_{L,{\mathrm{cutoff}}} = D_{L,b} \, X^{1/2}(\theta_d) & \left( \frac{1+z_{\mathrm{max}}}{1+z_b} \right)^{(3/2+\alpha)/2} \\ & \left( \frac{\eta(\mathrm{DM}(z_{\mathrm{max}}))}{\eta(\mathrm{DM}(z_b))} \right)^{1/2} \text{,} \end{split}\end{align}

where we note explicitly that $\mathrm{DM}=\mathrm{DM}(z)$ .

$z_{\mathrm{max}}(\theta)$ may therefore be determined by noting that $X(\theta)$ changes according to the position in the beam. For a telescope beam whose sensitivity falls off as $B(\theta)$ , telescope sensitivity changes as

(7) \begin{align} X(\theta) = X(\theta_d) \frac{B(\theta)}{B(\theta_d)}.\end{align}

Thus the limiting redshift at an angular distance, $\theta$ , from the beam centre may be found by solving the following equation for $z_{\mathrm{max}}(\theta)$ via

(8) \begin{align} D_{L,{\mathrm{max}}} = D_{L,b} X^{1/2}(\theta_d) & \left( \frac{1+z_{\mathrm{max}}}{1+z_b} \right)^{(3/2+\alpha)/2} \nonumber \\ & \left(\frac{B(\theta)}{B(\theta_d)} \frac{\eta(\mathrm{DM}(z_{\mathrm{max}}))}{\eta(\mathrm{DM}(z_b))} \right)^{1/2} . \end{align}

Determination of $z_{\mathrm{max}}(\theta)$ via equation (8) is fully prescribed in terms of: (i) the FRB detection angle from beam centre, $\theta_{d}$ ; (ii) the factor above the cut-off S/N threshold, $X(\theta_d)$ ; (iii) the FRB redshift, $z_b$ ; (iv) the beam pattern, $B(\theta)$ ; and (v) the telescope efficiency, $\eta(\mathrm{DM}; w)$ . The maximum volume in which the FRB could have been detected, $V_{\mathrm{max}}$ , may then be determined using equation (3).

2.2.2 The detection volume of a FRB within a generic beam

The volume in which a FRB is detected, V, for a generic beam may be determined via

(9) \begin{align} \begin{split} V = 2 \pi & \left(\int_0^{\theta_{\mathrm{max}}} \int_0^{z_{\mathrm{b}}} \frac{D_H}{E(z)} \frac{D_L^2(z)}{(1+z)^2} \sin \theta d\theta dz \right. \\ &\left. +\,\int_{\theta_{\mathrm{max}}}^{\theta_{\mathrm{outer}}} \int_0^{z_{\mathrm{max}}(\theta)} \frac{D_H}{E(z)} \frac{D_L^2(z)}{(1+z)^2} \sin \theta d\theta dz\right) \text{,} \end{split}\end{align}

where we integrate the constant luminosity distance of the detected FRB out to maximum angle, $\theta_{\mathrm{max}}$ , at the detection threshold (i.e. at the beam cut-off fluence, see equation (10)), then add the residual volume out to the limit of integration, $\theta_{\mathrm{outer}}$ . For an overview of determining $V/V_{\mathrm{max}}$ in relation to a non-uniform sensitivity, see Appendix B.

By rearranging and relabelling equation (8), and making the substitution $X(\theta_{\mathrm{max}}) B(\theta_{\mathrm{max}}) \rightarrow B^{2}(\theta_{\mathrm{max}}) X(\theta_{\mathrm{d}}) / B(\theta_{\mathrm{d}})$ via equation (7), $\theta_{\mathrm{max}}$ may be determined by solving

(10) \begin{align} \begin{split} B^{2}(\theta_{d}) \, & D^{2}_{L}(z_\mathrm{b}) \, \eta(\mathrm{DM}(z_{\mathrm{b}})) \, (1+z_{\mathrm{max}})^{2+\alpha} = \\ & B^{2}(\theta_{\mathrm{max}}) \, D^{2}_{L}(z_{\mathrm{max}}) \, \eta(\mathrm{DM}(z_{\mathrm{max}})) \, (1+z_\mathrm{b})^{2+\alpha} . \end{split}\end{align}

2.3 The volume probed by a specific beam

We further adapt the treatment of Section 2.2 to admit telescopes with arbitrary beamshapes (later in Section 3 we specifically admit the ASKAP telescope beamshape).

For a beam viewing a solid angle of sky, the inverse beamshape, $\Omega(B)$ (James et al. 2021a) with beam function $B(\theta)$ , the maximum volume in which a FRB may have been detected, may be recast as

(11) \begin{align} \begin{split} V_{\mathrm{max}} &= \int \int \frac{D_H}{E(z)} \frac{D_L^2(z)}{(1+z)^2} dz d\Omega \\ &= \int_{0}^{1} \int_0^{z_{\mathrm{max}}(B)} \frac{D_H}{E(z)} \frac{D_L^2(z)}{(1+z)^2} \, \Omega(B) \, dz \, dB . \end{split}\end{align}

Likewise the volume in which the FRB was detected, for a specific beam-shape, is recast as

(12) \begin{align} \begin{split} V = & \int_{0}^{B_{0}(\theta_{d})} \int_0^{z_{\mathrm{max}}(B)} \frac{D_H}{E(z)} \frac{D_L^2(z)}{(1+z)^2} \, \Omega(B) \, dz \, dB \\ &+ \int_{B_{0}(\theta_{d})}^{1} \int_0^{z_{\mathrm{b}}} \frac{D_H}{E(z)} \frac{D_L^2(z)}{(1+z)^2} \, \Omega(B) \, dz \, dB \text{,} \end{split}\end{align}

where $B_{0}(\theta_{d}) = B(\theta_{d}) / X(\theta_{d})$ .

In order to determine the limits of integration in equation (12) and to solve equation (8) for $z_{\mathrm{max}}(B)$ , we utilise an Airy beam function as the underlying beam model where necessary (Arcus et al. 2022).

Furthermore, consistent with (James et al. 2021a, see Section 4.3 Numerical implementation), equations (11) & (12) are implemented as histogram approximations (i.e. Riemann sums), such that $\int B(\Omega) \, dB \approx \sum_{i=1}^{N_B} \Omega(B) \Delta B$ where we choose $N_{B} = 10$ (James et al. 2021a).

As the source evolution function for FRBs is hitherto unknown, considered hypotheses generally take on the form of some function of the star formation rate (SFR; e.g. Macquart & Ekers 2018b), or delayed with respect to star formation (e.g. Cao, Yu, & Zhou 2018). Current fitting methods favour source evolution consistent with the cosmic star formation rate (James et al. 2022; Shin et al. 2023), although this is equally consistent with a generic $(1+z)^n$ model. We show in Appendix A that with current data, the $V/V_{\mathrm{max}}$ method cannot currently discriminate between source evolution models. Accordingly, we choose the simpler case of no source evolution and set $V^\prime = V$ as discussed in Appendix A.

3. Application to ASKAP

We consider two discrete samples from the ASKAP telescope: the full set of 63 FRBs and a subset of 28 FRBs for which an identified host galaxy with measured redshift z has been obtained. We treat these two samples separately in order to determine whether the use of FRBs from the DM-only inferred redshift sample yields an energy distribution consistent with those from which redshift has been independently determined. We examine the FRB population and apply the luminosity-volume- or $V/V_{\mathrm{max}}$ -test to examine the source distribution of these transient sources, accounting for cosmological and instrumental effects, in order to determine the RLF for the sampled population. In Appendix A, we consider both $\alpha=0$ and $\alpha=-1.5$ (Macquart et al. 2019), and also a cosmological evolution of the source population. However, we find little discriminating power between the two, hence we choose, hereinafter, $\alpha=0$ and no source evolution for simplicity.

We use the formalism outlined in Section 2.3 to determine the volumetrics necessary to conduct the $V/V_{\mathrm{max}}$ -test and apply the beamshape for the ASKAP telescope, as given by James et al. (2021a, see Figure 3 of Section 4.1 therein), via the inverse beamshape, $\Omega(B)$ . We choose this approach to represent a realistic beamshape for ASKAP and to avoid complications in cases where a FRB detection occurs either in multiple beams or in an outer beam. Due to ASKAP’s beams overlapping approximately at the half-power points, the effect of beamshape in this analysis is not strong, and a short investigation shows that the effect of ignoring it for both and the luminosity function is minor.

Table 1 lists the candidate localised sample of FRBs along with their relevant observational parameters applicable to our analysis. Since there is some suggestion that ASKAP ICS observations are incomplete in the range $\text{S/N} \lt 14$ (Shannon et al. 2024), and we wish to ensure the localised sample has minimum bias, only those FRBs for which the S/N exceeds the threshold of $\mathrm{S/N}_{\mathrm{cutoff}}=14$ were subsequently admitted for further analysis. These are listed in Table 2 and are hereinafter identified as the Localised High S/N Sample, comprising 19 FRBs.

Table 1. Properties of the 28 localised ASKAP candidate FRBs for which a host galaxy redshift has been determined. FRBs identified with an asterisk below (*) are excluded from subsequent analysis since their detected $\text{S/N} \lt \text{S/N}_{\mathrm{cutoff}} (= 14)$ . Variables listed in this table are: (i) DM – Observed DM; (ii) $\nu_{c}$ – Centre Frequency; (iii) S/N – Primary S/N; (iv) S/N $_{\mathrm{cuttoff}}$ – S/N threshold; (v) DM $_{\mathrm{Gal}}$ – DM of Milky Way disc using the NE2001 model; (vi) $\Delta t$ – Sample Interval; (vii) w – Fitted Pulse-width (FWHM); (viii) $\theta_{d}$ – Detection Angle; (ix) $F_{\nu}$ – Corrected Fluence; & (x) $z_{\mathrm{loc}}$ – Localized host redshift. References are: a: Bannister et al. (2017), b: Shannon et al. (2018), c: Mahony et al. (2018), d: Macquart et al. (2019), e: Agarwal et al. (2019), f: Qiu et al. (2019), g: Bhandari et al. (2019), h: Bannister et al. (2019), i: Prochaska et al. (2019b), j: Macquart et al. (2020), k: Heintz et al. (2020), l: Bhandari et al. (2020), m: Bhandari et al. (2022), n: Bhandari et al. (2023) o: (Shannon et al. 2024), p: James et al. (2022), q: Baptista et al. (2024), r: Ryder et al. (2023), s: Gordon et al. (2024).

Table 2. Derived properties of the 19 Localised High S/N Sample of ASKAP FRBs for which the S/N exceeds the threshold $\text{S/N} \ge \text{S/N}_{\mathrm{cutoff}} (= 14)$ , for fluence spectral indices of $\alpha=0.0$ , and no source evolution. Note that this sample is a subset of the FRBs listed in Table 1.

Table 3. Properties of the Full Sample. Columns and references are the same as in Table 1, excepting $z_{\mathrm{DM}}$ – Redshift inferred from the $z-\text{DM}$ relation.

$^*$ These FRBs have $z_{\mathrm{DM}} \lt 0$ , and are excluded from initial analysis.

$^{\#}$ This FRB has S/N $_{\mathrm{cutoff}}=8$ ; all others are 9.5.

$^*$ These FRBs have $z_{\mathrm{DM}} \lt 0$ , and are excluded from initial analysis.

Table 3 lists the candidate full sample comprising 63 ASKAP FRBs along with their relevant observational parameters applicable to our analysis. In this sample, we include the 28 FRBs localised to their host galaxies. This constitutes the Full Sample (see Table 3), where the detection threshold of $\text{S/N}_{\mathrm{cutoff}} = 9.5$ as used in the CRAFT detection pipeline, is used for all FRBs irrespective of considerations of potential bias. The derived parameters of the Full Sample are provided in Table 4, whereby redshifts, even for FRBs with measured redshift, have been estimated from their DM budget via

(13) \begin{align} \mathrm{DM}_{\mathrm{Obs}} = \mathrm{DM}_{\mathrm{MW}} + \mathrm{DM}_{\mathrm{Halo}} + \mathrm{DM}_{\mathrm{cosmic}} + \mathrm{DM}_{\mathrm{Host}}/(1+z) \text{,}\end{align}

where $\mathrm{DM}_{\mathrm{Obs}}$ is the observed DM of the FRB, while DM_MW, $\mathrm{DM}_{\mathrm{Halo}}$ and $\mathrm{DM}_{\mathrm{Host}}$ are the DM contributions due to the Milky Way disc, its halo, and the FRB host environment, respectively. We set the cosmological contribution $\mathrm{DM}_{\mathrm{cosmic}}$ to its mean, $\overline{\mathrm{DM}}(z)$ , using equation (14) and assume a constant host contribution of $\mathrm{DM}_{\mathrm{Host}} = 50$ pc cm $^{-3}$ and halo contribution of $\mathrm{DM}_{\mathrm{Halo}} = 50$ pc cm $^{-3}$ consistent with Arcus et al. (2020). $\mathrm{DM}_{\mathrm{MW}}$ is determined via the NE2001 model of Cordes & Lazio (2003). We note that uncertainties in these quantities can be large – of order a factor of two for $\mathrm{DM}_{\mathrm{MW}}$ (Schnitzeler 2012), and perhaps a similar uncertainty for $\mathrm{DM}_{\mathrm{Halo}}$ (Prochaska & Zheng 2019; Keating & Pen 2020). Fluctuations in $\mathrm{DM}_{\mathrm{Host}}$ are not directly measured, but are estimated to be large (James et al. 2022). This results in potentially large fluctuations about the Macquart relation, as evinced by FRBs with exceptionally low or high DMs for their redshifts, for example, FRB 20200120E with DM 87.82 pc cm $^{-3}$ at 3.6 Mpc (Bhardwaj et al. 2021), and FRB 20190520B with DM $1204.7$ pc cm $^{-3}$ at $z=0.241$ (Niu et al. 2022a).

Table 4. Derived properties of the Full Sample for a fluence spectral index of $\alpha = 0.0$ and no source evolution. For those with $z_{\mathrm{DM}} \lt 0$ (marked with a $^*$ : FRB 20171020A, FRB 20180430A, and FRB 20230718A), an assumed distance of 2 Mpc is used. Columns are identical to those of Table 2.

Consistent with Macquart et al. (2020) and Arcus et al. (2022), we determine the mean DM of a homogeneously distributed intergalactic medium (IGM) as given by Ioka (2003), Inoue (2004), updated to include the fraction $f_d$ of baryons in diffuse ionised gas as per Deng & Zhang (2014)

(14) \begin{align} \begin{split} \overline{\mathrm{DM}}(z)=&\dfrac{3H_0c\Omega_b}{8 \pi G m_p}\\ &\int_{0}^{z} f_d(z') \dfrac{(1+z') \left[\frac{3}{4} X_{\mathrm{e,H}}(z') + \frac{1}{8}X_{\mathrm{e,He}}(z')\right] }{\sqrt{(1+z')^3 \Omega_m + \Omega_{\Lambda}}} dz' \text{,} \end{split}\end{align}

where the ionised fractions of Hydrogen and Helium are taken to be $X_{\mathrm{e,H}}=1$ for $z \lt 8$ and $X_{\mathrm{e,He}}=1$ for $z \lt 2.5$ , respectively, or zero otherwise. Throughout this work we adopt a $\Lambda$ CDM universe with the cosmological parameters $(h, H_{0}, \Omega_{b}, \Omega_{m}, \Omega_{\Lambda}, \Omega_{k}) = (0.7, 100 h \mathrm{km s}^{-1} \mathrm{Mpc}^{-1}, 0.0486, 0.308, 0.691, 0)$ , that is, an intermediate value of $H_0$ (Abdalla et al. 2022), but otherwise in accordance with the (Planck Collaboration et al. 2016). We use the estimate of $f_d(z)$ from the FRB code base (Prochaska et al. 2019a), which ranges from 0.844 at $z=0$ , and increases slowly with redshift. This relation between FRB redshift and expected DM was first verified by Macquart et al. (2020), and is now known as the Macquart relation. Fig. 2 illustrates the scatter about the Macquart relation for the Localised High S/N Sample of FRBs. The scatter is large – up to a factor of two in redshift – and dominates over the uncertainties in the mean of the Macquart relation. We therefore estimate the error from using the Full Sample by comparing it to that of the Localised High S/N Sample, rather than marginalising over uncertainties in estimating $z_\mathrm{DM}$ . Three FRBs have an implied negative $z_\mathrm{DM}$ , and hence are omitted from our initial analysis of the Full Sample. The effects of this are discussed in Section 4.2.

Figure 2. Scatter plot of spectroscopically measured host galaxy redshifts, $z_\mathrm{loc}$ , and those derived from the Macquart relation, $z_\mathrm{DM}$ , for the Localised High S/N Sample.

Figure 3. Histograms of $V/V_{\mathrm{max}}$ for both the Localised High S/N Sample (top) and Full Sample (bottom), under the assumption of no spectral dependence ( $\alpha=0$ ) or cosmological evolution ( $n_\mathrm{SFR}=0$ ). Three FRBs with negative $z_\mathrm{ DM}$ values have been omitted from the Full Sample.

4. Discussion

4.1 The FRB radio luminosity function

The distributions of $V/V_{\mathrm{max}}$ for both samples are shown in Fig. 3. As discussed in Appendix A, the major deviation from uniformity is the deficit of FRBs with low $V/V_{\mathrm{max}}$ , which cannot be rectified for any reasonable source evolution function. Hence, we proceed to calculate the RLF from these samples, under the assumption of no spectral dependence (i.e. $\alpha=0$ ) and no cosmological evolution of the source population.

Fig. 4 depicts the derived RLF from the Localised High S/N Sample and Full Sample. Also shown are their best-fit functions (fitted parameters given in Table 5) and comparisons to values from the literature. A flatter RLF is preferred by the Full Sample ( $\gamma=-1.82 \pm 0.12$ ) compared to the Localised High S/N Sample ( $\gamma=-2.11 \pm 0.18$ ). At high luminosities, the Full Sample shows some evidence for a high-energy down-turn near $\log_{10} E_\mathrm{max}\,(\mathrm {J\,Hz}^{-1})=25.8 \pm 0.39$ — likely due to the smaller Localised High S/N Sample containing no data in the $10^{26}{-}10^{27}\,{{\mathrm {J\,Hz}^{-1}}}$ bin. Conversely, the RLF data at $E_\nu \lt 10^{23}{{\mathrm {J\,Hz}^{-1}}}$ from the Localised High S/N Sample shows an excess which is inconsistent with both a power-law or Schechter function, and the Full Sample contains no data in that luminosity bin. Such a low-energy excess has been observed in several repeating FRBs, with low-energy peaks becoming dominant in the $\sim+10^{22}{-}10^{23}{{\mathrm {J\,Hz}^{-1}}}$ range (Niu et al. 2022b; Li et al. 2021). Furthermore, Kirsten et al. (2024) have found evidence for a flatter power-law index at energies above $10^{24}{{\mathrm {J\,Hz}^{-1}}}$ for FRB 20201124A. This suggests that apparently once-off FRBs localised with ASKAP exhibit a qualitatively similar hardening of the RLF above $10^{23}\,{{\mathrm {J\,Hz}^{-1}}}$ , though this is an ensemble average over the behaviour of many objects, and there are quantitative differences both within and between the RLFs measured for repeating FRBs; these samples may be subject to systematic biases, as discussed below.

Figure 4. Radio luminosity functions (RLFs) calculated from the Localised High S/N Sample (using $z_\mathrm{loc}$ ) and Full Sample (using $z_\mathrm{DM}$ ). The (arbitrary) normalisation is fixed to unity at the $10^{23}$ – $10^{24}$ bin. The best-fit Schechter functions for each sample are depicted for reference purposes. Also shown are luminosity functions derived from ASKAP and Parkes data by Ryder et al. (2023), CHIME data by Shin et al. (2023), and a mixed sample by Luo et al. (2020). The data are binned in log-space, so that the ordinate (y-axis) is effectively the RLF multiplied by the spectral fluence, $E_\nu$ .

4.2 Systematic biases – full sample

The Full Sample includes three low-DM FRBs with implied negative redshifts, which cannot therefore be trivially included in calculations. This results in the RLF that uses $z_\mathrm{DM}$ missing these events, which invariably occur in the nearby Universe, where under-fluctuations in $\mathrm{DM}_\mathrm{Host}$ , $\mathrm{DM}_\mathrm{Halo}$ , and/or $\mathrm{ DM}_\mathrm{MW}$ could result in low measured values of $\mathrm{DM}_\mathrm{Obs}$ , such that only a negative value of z will satisfy equation (13). This effect can be seen most clearly in the missing data point for the Full Sample in the $10^{22}$ – $10^{23}$ J Hz $^{-1}$ bin in Fig. 4, which in the Localised High S/N Sample, is entirely due to FRB 20171020A. One method of avoiding such a bias is to marginalise over distributions of Milky Way and host galaxy DM contributions, as performed by Locatelli et al. (2019) – see Section 4.5 for further discussion of this approach.

The effect of this bias can be estimated by placing robust bounds on the true distance to these $z_\mathrm{DM} \lt 0$ FRBs. A lower bound assumes they are not located in Local Group galaxies, limiting the luminosity distance $D_L \gtrsim 2$ Mpc (which equates to $z_\mathrm{min} = 0.00024$ , ignoring peculiar velocities). An upper bound assumes that the entire DM contribution is cosmological in nature, that is, $z_\mathrm{max} = z_\mathrm{DM}(\mathrm{DM}_\mathrm{cosmic} = \mathrm{ DM}_\mathrm{Obs})$ . We vary between these extremes, using $z = z_\mathrm{min} + k (z_\mathrm{max}-z_\mathrm{min})$ , for $k=0,0.1,1.0$ . We find that for $k\ge0.2$ , the effect on the RLF is negligible. However, for very low values of k, the RLF extends to very low luminosities, with a dependence $\propto E_\nu^{-1.5}$ , since these FRBs invariably occupy the local Universe with approximately Euclidean geometry. The case of $k=0$ only is shown in Fig. 5.

When assuming very nearby FRBs, the low-luminosity form of the RLF is significantly changed, and we are unable to obtain consistent fits. Excluding data below $10^{22}\,{{\mathrm {J\,Hz}^{-1}}}$ produces almost identical values for $\gamma$ and $E_\mathrm{max}$ as those previously found for the Full Sample. We therefore conclude that this bias limits our ability to probe the low-luminosity end of the RLF.

4.3 Systematic biases – localised high S/N sample

The inclusion of FRB 20171020A in the Localised High S/N Sample highlights our second source of systematic bias. FRB 20171020A only has a confident redshift precisely because it is nearby, thus its host galaxy can be identified despite the relatively large localisation errors of the CRAFT Fly’s Eye observations. The analysis presented here has no means of accounting for the likely more-distant, higher-DM FRBs of the Fly’s Eye sample (those from FRB 20170107A to FRB 20180525A) which cannot be included in the Localised High S/N Sample. A similar effect also occurs for high-redshift – and necessarily high-luminosity – FRBs, the host galaxies of which may be unidentifiable due to their large distance. An example of this is FRB 20210912A, where optical limits on the as-yet unseen host galaxy suggests $z \gt 0.7$ , with $z=1$ implying $E_\nu = 9.7 \cdot 10^{25}\,\mathrm{J}\,\mathrm{Hz}^{-1}$ in the case of $\alpha=0$ (Marnoch et al. 2023). However, without this firm localisation, this undoubtedly energetic FRB cannot be included in the Localised High S/N Sample.

The biases mentioned above can be overcome in the case of the Localised High S/N Sample by using a limiting redshift $z_\mathrm{ lim}$ such that all FRBs with $z \lt z_\mathrm{lim}$ are guaranteed to have their host galaxies identified. To do this, we first remove FRB 20171020A from the sample, since $z_\mathrm{lim}$ for the CRAFT Fly’s Eye observations are poorly defined and set $z_\mathrm{lim}=0.7$ for the remaining FRBs localised with ICS observations. All integrals over z in the calculations for V and $V_{\mathrm{max}}$ in Section 3 are then terminated at $z_\mathrm{lim}$ , while FRBs located outside this volume are excluded. Thus, the definition of $V_{\mathrm{max}}$ becomes ‘the volume within which this FRB would have been included in the analysis’. A limiting case of this method is to use only FRBs in a thin slice of redshift, between z and $z+dz$ . In such a case, $V=V_\mathrm{max}$ , and is constant for each and every FRB, such that every FRB has equal weight in the calculation of the luminosity function, consistent with expectation.

Table 5. Mean $V/V_{\mathrm{max}}$ and best fit parameters ( $\gamma$ , $E_\mathrm{max}$ ) of the pure power-law and Schechter function fits to the FRB luminosity function for different data-sets, assuming no spectral dependence (i.e. $\alpha=0$ ) or evolution of the source population. The p-values for the power-law fits are the probability of observing a $\chi^2$ that value or higher should the power-law be the true model (high values indicate a good fit); for the Schechter function, the p-value is the probability of observing such a significant improvement in $\chi^2$ should the power-law be the true model (low values are evidence for a Schechter function).

$^*$ Assumes FRBs with a negative $z_\mathrm{DM}$ are located at a distance of $z_\mathrm{min}=0.00024$ (2 Mpc); fit excludes data below $10^{22}$ J Hz $^{-1}$ .

${}^{\#}$ No errors can be estimated for these parameters, due to the number of degrees of freedom (ndf.) of the fit being zero.

$^{\#\#}$ The best-fit value of $E_\mathrm{max}$ is effectively infinite, rendering error calculations for the parameters meaningless.

Figure 5. Data on radio luminosity functions (RLFs) calculated so as to account for observational biases, showing the full range allowing for a minimum distance at $z_\mathrm{min}$ . The line indicating the $E_\nu$ RLF $ \propto E_\nu^{-1.5}$ is to guide the eye only.

Fig. 5 shows the RLF for this updated sample of FRBs. It is only measured in the range $10^{23}$ – $10^{25}$ J Hz $^{-1}$ , and in this range, is consistent with a pure power-law with slope $\gamma = -1.96 \pm 0.15$ (p-value of linear fit 0.09); a Schechter function produces a much flatter differential slope $\gamma = -1.58$ and turn-over energy of $E_\mathrm{max} = 1.2\,\times 10^{25}$ J Hz $^{-1}$ (errors cannot be estimated since the number of variables equals the number of points, i.e. it is statistically ill-posed).

4.4 Comparison with other results

Fits of the FRB RLF have been undertaken by several authors. While many assume a 1:1 z–DM relation, we concentrate on those which have fully modelled uncertainties in FRB redshift given DM, and/or used a sample of localised FRBs, while accounting for selection effects as per Connor (2019). Luo et al. (2020) uses a mixed sample of mostly unlocalised FRBs from several instruments – including Parkes and ASKAP – to fit a Schechter function with differential index $\gamma = -1.79^{+0.31}_{-0.35}$ and $E_\mathrm{max} = 2.9_{-1.7}^{+11.9} \cdot 10^{25}{{\mathrm {J\,Hz}^{-1}}}$ (assuming a 1 ms burst width and 1 GHz bandwidth). James et al. (2022) uses a sample of 16 FRBs with host redshifts, and approximately 60 without, to find an index of $-1.95^{+0.18}_{-0.15}$ , with Ryder et al. (2023) updating $E_\mathrm{max}$ to $5_{-2}^{+3} \cdot 10^{25}{{\mathrm {J\,Hz}^{-1}}}$ . Shin et al. (2023) fits the dispersion measure of 536 FRBs observed by CHIME to find an index of $-1.3^{+0.7}_{-0.4}$ and $E_\mathrm{max} = 2.38^{+5.65}_{-1.64} \cdot 10^{25}{{\mathrm {J\,Hz}^{-1}}}$ . These results are broadly consistent with the range of RLFs derived in this work, although they would have difficulty fitting the possible low-energy excess observed in the potentially biased Localised High S/N Sample, and the fit of Shin et al. (2023) is flatter, and has a downturn which is stronger, than allowed by our data. We note that James et al. (2022) accounts for the biases discussed in Section 4.2 by not using the localisation of FRB 20171020A or FRBs above DM $_\mathrm{EG}$ of 1 000 pc cm $^{-3}$ , while Shin et al. (2023) fits FRBs at lower frequency which may have a different underlying RLF. The possible minimum energy (or downturn) at $23.47_{-1.28}^{0.54}$ J Hz $^{-1}$ suggested by Hoffmann et al. (2024a) is not observed, but lies at the lower end of the RLF which we can probe. We therefore conclude that, given uncertainties in FRB spectral behaviour and source evolution, and possible biased effects in our own analysis, we cannot discriminate between these previous fits.

4.5 Comparison with Locatelli et al. (2019)

Locatelli et al. (2019) also apply the $V/V_{\mathrm{max}}$ test to FRBs by comparing 23 FRBs discovered by ASKAP with 20 of the FRBs found by Parkes up to 2019. Their paper focuses on the use of the $V/V_{\mathrm{max}}$ distribution to explore evolution, while our paper uses the $V_{\mathrm{max}}$ method to estimate the luminosity distribution. These are different uses, and analysis of $V/V_{\mathrm{max}}$ needs complete unbiased samples which is quite problematic for FRBs as discussed in Section 4.3. They find evidence for cosmological source evolution in the ASKAP sample, with $\langle V/V_{\mathrm{max}}\rangle$ $= 0.68 \pm 0.05$ , but less so for the Parkes data, with $\langle V/V_{\mathrm{max}}\rangle$ $ = 0.54 \pm 0.04$ , assuming a spectral evolution of $\alpha=1.6$ .

For ASKAP, Locatelli et al. (2019) only analyse the ICS sample, which was all that was available at the time. They also include the Parkes FRB sample; however, for these FRBs the location in the beam is not known, so neither $V_{\mathrm{max}}$ nor the actual beam-corrected fluences are known; we therefore excluded the Parkes FRB sample in our analysis. This difficulty will also apply to the much larger CHIME sample. The authors also omit discussion of beam areas for Parkes and ASKAP.

As discussed in Section 4.2, Locatelli et al. (2019) builds appropriate probability functions to estimate the redshift PDF(z) instead of using a unique value. This treatment is an elegant way to avoid the bias due to negative apparent redshifts encountered when only the mean DM correction is used. We note that the zDM code is able to produce such PDFs, for example, as per Lee-Waddell et al. (2023).

Our results for $\langle V/V_{\mathrm{max}}\rangle$ for the Localised High S/N Sample and Full Sample are 0.58 and 0.63, respectively (0.61 and 0.62 when debiased); when we include a spectral dependence of $\alpha=-1.5$ (see Appendix A), we find $\langle V/V_{\mathrm{max}}\rangle$ $=0.67$ for the Full Sample, consistent with the result of Locatelli et al. (2019).

5. Conclusions

We have shown how to apply the $V/V_{\mathrm{max}}$ method of Schmidt (1968) to a population of transient sources and applied this to FRBs. We find that the current sample of FRBs detected by ASKAP/CRAFT is insufficient to distinguish between competing evolutionary and spectral models, with the greatest departure from uniformity in the $V/V_{\mathrm{max}}$ distribution being due to a dearth of very high S/N FRBs.

Using both FRBs with known redshift, $z_\mathrm{loc}$ , and a larger sample of FRBs with $z_\mathrm{DM}$ estimated from the Macquart relation, we plot the FRB energy distribution in the range $10^{23}$ – $10^{26}$ J Hz $^{-1}$ . We find it to be fairly consistent ( $p=0.09$ ) with a power-law with differential slope $\gamma=-1.96 \pm 0.15$ using $z_\mathrm{loc}$ . Above this energy, we find some evidence of a downturn consistent with a Schechter function with $E_\mathrm{max} = 6.3 \, \times 10^{25}$ when using $z_\mathrm{DM}$ . We have also identified several systematic effects in the analysis and shown how to take these into account. In particular, the difficulty of identifying high-z host galaxies limits our knowledge of the tip of the FRB energy distribution, as it is unclear if the downturn in the energy distribution seen in the $z_\mathrm{DM}$ result is physical, or an artefact of smearing in the Macquart relation.

In the near future, FRB surveys will detect too many bursts to follow up their host galaxies individually with 8 m-class telescope time (e.g. CHORD; Vanderlinde et al. 2019). Low-DM, near-Universe host galaxies can likely be identified in existing or impending (e.g. LSST) optical surveys without further follow-up, allowing an unbiased sample of the $E_\nu \lt 10^{23}$ J Hz $^{-1}$ region to be formed. Moreover, we find that the use of $z_\mathrm{DM}$ compared to $z_\mathrm{loc}$ does not significantly affect the luminosity function in the range $10^{23}$ – $10^{26}$ J Hz $^{-1}$ . We therefore recommend that optical follow-up time be focused on identifying high-DM/ $z_\mathrm{loc}$ FRBs, to allow the high-end of the FRB luminosity function to be studied.