The Fast Radio Burst Dispersion Measure Distribution

Ever since their discovery by Lorimer et al. (2007), it has been conjectured that the large dispersion measures (DMs) of fast radio bursts (FRBs) encode information about their distances and evolutionary history (e.g., see the discussion in Macquart & Ekers, 2018b). Early suppositions that DMs contain a sizeable contribution due to their passage through the intergalactic medium (IGM), thereby placing the population at cosmological distances (Lorimer et al., 2007; Thornton et al., 2013), have been vindicated by recent localisations of two repeaters (Chatterjee et al., 2017; Marcote et al., 2020) and at least seven single events (Bannister et al., 2019; Prochaska et al., 2019; Ravi et al., 2019; Macquart et al., 2020).

The cosmological nature of the FRB population has been further substantiated by the discovery of a relation between mean DM and fluence, $F$, in the bright non-repeating FRB population observed by the Australian SKA Pathfinder (ASKAP) and Parkes radio telescopes (Shannon et al., 2018). Therein, the authors note the average DM of the ASKAP sample, 440 pc cm^-3, is half that of the fainter bursts detected by Parkes at 881 pc cm^-3 . This result is akin to the redshift-flux density relations observed in other cosmological populations – for active galactic nuclei see the discussion in von Hoerner (1973). Further, von Hoerner (1973) draws attention to the critical value of the luminosity function (in our case energy distribution) of $\gamma\approx 2.5$ when sources of a given fluence are equally probable in distance (in Euclidean space). The relevance of this will be discussed in Section 4.

The DM distribution is an observationally underexploited means of probing both the nature of FRB emission and the media through which they propagate. The redshift distribution of FRBs detected in a survey of finite sensitivity is shaped by the underlying burst luminosity, the evolution with redshift and by the spectral index of the emission. The DM distribution provides an additional means of accessing information on the mapping between redshift and DM, which is not bijective (i.e., one-to-one) except when averaged over many lines of sight (Ioka, 2003; Inoue, 2004; McQuinn, 2014).

An especially powerful approach is to compare the DM distribution of two sample sets obtained from telescopes with significantly different detection thresholds. The DM-fluence relation noted by Shannon et al. (2018) exploits the first moment of the DM distribution. However, comparison of the shapes of the DM distributions obtained by surveys of differing sensitivities permits greater leverage to isolate key variables, since both distributions must be drawn from the same underlying luminosity and redshift distribution and with the same spectral index and host DM distributions.

It is known that the nature of the DM-fluence relation is a particularly useful probe of the average burst luminosity distribution¹¹1The nature of the relation also depends upon the parameters of the cosmological model (von Hoerner, 1973), however these are not regarded as free parameters in the present treatment. (Macquart & Ekers, 2018b). Whether the relation manifests as a correlation or anti-correlation depends upon the slope of the burst luminosity function, while the scatter of bursts about the relation contains information on the intrinsic spread of burst luminosities (Lorimer et al., 2007; Shannon et al., 2018).

It is the purpose of this paper to examine the DM distributions of the FRB populations detected by the ASKAP and Parkes radio telescopes. In Section 2, we present the samples used in our analysis, the DM histograms of those sample sets, a summary of the formalism relating the observed distributions to survey parameters and the underlying properties of the FRB distribution. In Section 3, we employ this formalism to infer the properties of the FRB distribution, the mapping between DM and distance and the effect of instrument performance on the detectability of bursts as a function of DM. The implications of our results and conclusions are discussed in Sections 4 and 5 respectively.

The treatment herein is based on the analysis of FRB event data detected by the Commensal Real-time ASKAP Fast Transients (CRAFT) survey and by various surveys with the Parkes radio telescope. The ASKAP-CRAFT data are drawn from Shannon et al. (2018) and Macquart et al. (2019), whilst Parkes data are drawn from FRBCAT (Petroff et al., 2016, http://frbcat.org) and are summarised in Table 1. In our analysis, we exclude the Lorimer Burst (FRB 010724) to avoid potential discovery bias, as discussed in Macquart & Ekers (2018a), although it is unclear as to the extent to which such a bias may affect the DM. (We include FRBs that are below the nominal fluence limits for their respective telescopes, since these limits are characteristic values averaged over telescope parameters – in particular, beam-shape.) We utilise the DM of the IGM, $DM_{\text{IGM}}$, determined via eq.(1), by estimating and removing the DM contributions due to the Milky Way disc, $DM_{\text{MW}}$, its halo, $DM_{\text{Halo}}$, and the FRB Host environment, $DM_{\text{Host}}$, thus:

For the ASKAP-CRAFT (lat50 survey) data, we assume $DM_{\text{MW}}\approx 30$  pc cm^-3 , due to the high galactic latitudes of the observations (NE2001, Cordes & Lazio, 2003), $DM_{\text{Halo}}\approx 30$  pc cm^-3 and $DM_{\text{Host}}\approx 50$  pc cm^-3 throughout (Dolag et al., 2015; Xu & Han, 2015; Tendulkar et al., 2017; Mahony et al., 2018; Macquart et al., 2020). We note the assumption that $DM_{\text{Host}}\approx 50$  pc cm^-3 , instead of using a distribution of possible $DM_{\text{Host}}$ values, will only have a small effect given the much larger observed DM values utilised in this analysis. Values of $DM_{\text{MW}}$ for individual Parkes events are drawn from FRBCAT.

The survey fluence limit at $DM=0$, $F_{0}$, of the Parkes and ASKAP telescopes are in the approximate ranges of 1-5  Jy ms and 21-31 Jy ms respectively and are dependent upon the slope of the source counts distribution at their limits (see Table 2 of James et al., 2018). While the DMs of the bursts are well determined for both FRB event datasets, the fluences of the Parkes events are lower limits due to the inherent inability to localise each burst within individual beams (Keane & Petroff, 2015; Macquart & Ekers, 2018a). The Parkes fluences are therefore referenced to the beam centre. In practice, this limitation is not expected to significantly affect the present analysis since the modelled DM distributions are referenced only to a limiting survey depth and do not require information pertaining to each burst.

Figure 1 depicts the DM histograms of the Parkes and ASKAP FRB event samples listed in Table 1. An interesting feature of the histograms is that they have similar shapes. The means of these distributions differ by 594 pc cm^-3, an update to the value of Shannon et al. (2018), due to the increased sample size. Whilst these overall shapes are expected²²2The initial increase at low DM is due to the volume sampled increasing as distance cubed. The counts then decrease at higher DMs (distances) as the fluences of the less luminous bursts drop below the survey sensitivity limit – i.e., they become incomplete., a quantitative analysis of the data necessitates we account for the finite instrumental spectral and temporal resolution of both telescope back-ends.

Whilst the DM-redshift relation is generally not expected to be bijective, except when averaged over many lines of sight (see e.g. McQuinn, 2014), we nonetheless make this assumption advisedly on the basis that the DM dataset utilised exhibits high DMs with low DM dispersion and on the basis of recent work establishing a DM-redshift relation for localised FRBs (Macquart et al., 2020). We further assume an homogeneous IGM and determine the best-fit population parameters of fluence spectral index, $\hat{\alpha}$, and energy power-law index, $\hat{\gamma}$, for an assumed energy curve. We determine these parameters simultaneously using the joint p-value via the K-S test and for different redshift evolutionary models, by comparing the  M&E 2018 modelled DM distribution shapes with the observed histograms.

The approach adopted circumvents a number of key unknowns regarding the FRB population. First, the model allows for the ready generation of the DM distribution using few population and instrument parameters (viz., $\alpha$, $\gamma$ & $F_{0}$) with a relatively simple assumption for the energy curve cut-offs, $E_{\text{min}}$ and $E_{\text{max}}$, even though their values are not well established. Second, it permits direct comparison between datasets from telescopes of different survey sensitivities, obviating the need to address difficulties around calibration: the absolute FRB event rates are not required. By using the relative FRB event rates, and given the demonstrated robustness to pulse-widths smaller than the instrument resolution, we find the overall sensitivity curve shapes do not change significantly (i.e., shape changes are higher-order effects) and they are subsequently normalised out during the fitting process – a process insensitive to the specific (log normal) distribution chosen. We find simultaneously fitting the FRB data for both telescopes, using the K-S method, to be robust. Third, even though the  M&E 2018 model derives $\alpha$ principally on cosmological k-correction grounds (i.e., it is measured via the correction $(1+z)L_{(1+z)_{\nu}}/L_{\nu}$ made because the radiation is observed in a different band from that emitted by the source), even with an assumed energy curve, it compares favourably to values determined from independent means – e.g., $\alpha=1.5_{+0.3}^{-0.2}$ (Macquart et al., 2019) and $\alpha=1.8\pm 0.3$ (Shannon et al., 2018). Furthermore, the fits suggest that the distribution of burst energies for the population is relatively flat (viz., $\gamma<2$) in this DM regime.

Other authors also fit the DM distribution for Parkes and ASKAP data. Lu & Piro (2019) examines the DM distribution of ASKAP samples (only), finding $\gamma=1.6\pm 0.3$ with a fitted $\log_{10}E_{\rm max}=27.1_{0.7}^{+1.1}$ J Hz^-1 (68% confidence) – two orders of magnitude less than that used herein. Their model is broadly similar to the  M&E 2018 model, with the following key differences: (a) the width distribution of the FRBs, hence its effect on sensitivity, is not included; (b) an exponential tail to the luminosity function beyond $E_{\rm max}$ is used, rather than a sharp cut-off as used herein; (c) the authors further assume, but do not fit, $\alpha=1.5$; and (d) the authors study source evolution via $\psi(z)\sim(1+z)^{\beta}$, finding $\beta=0.8^{+2.6}_{-2.9}$ (approximately corresponding to $n=0.3^{+1.0}_{-1.1}$). Nonetheless, those results, together with the large error of their fits, are comparable to ours. In Luo et al. (2020), the authors consider a wider sample of FRBs, including ASKAP and Parkes observations, as well as those from several other instruments. A key difference of that treatment is the inclusion of DM scatter about the expectation for a given redshift, however they assume a flat spectrum ($\alpha=0$) and do not consider source evolution. These authors find $E_{\rm max}=2.9^{+11.9}_{-1.7}\times 10^{28}$ J Hz^-1 (converted assuming a 1 GHz bandwidth), and $\gamma=1.79^{+0.35}_{-0.31}$ – results also comparable with those found here.

Given the results attained, our assumption that the telescopes observe the same FRB population is consistent with the observations. As discussed at length in Macquart & Ekers (2018b), the behaviour of the DM distribution depends upon the slope of the FRB energy (or luminosity) function. Between $\gamma\approx 2-3$ there is expected to be a dramatic change in character of the DM distribution, with flatter distributions probing to higher redshifts, where they contain a large fraction of observed events at large distances. At a critical value of $\gamma=2.5$ there is no distance dependence on fluence, hence no information on evolution. For the high redshift evolution fits (viz., $n=2$), the solutions push $\gamma$ closer to this critical value, and whilst this may be the correct interpretation, it is more likely to be finding a solution that is independent of the imposed evolution.

From Figure 7, it can be seen that this effect predicts an excess of FRBs in the nearby Universe ($DM\sim 0$), particularly for the ASKAP sample. This effect does not appear to be observed: recent localisations of FRBs by ASKAP (Bannister et al., 2019; Prochaska et al., 2019; Macquart et al., 2020) do not show this excess. Whilst we therefore cannot definitively exclude a strongly evolving population (viz., $n=2$), it does seem unlikely. Future surveys with a greater fluence range will reduce this degeneracy. Conversely, for steeper distributions, observations with higher sensitivity will be dominated by nearby events. In our analysis, we determine the FRB energy function to be relatively flat (see Table 3), suggesting FRBs should be readily detectable to higher redshifts. Accordingly, Parkes and other more sensitive telescopes such as CHIME and FAST may be better able to discriminate the effects of population evolution, thereby aiding in the selection of progenitor model classes as the FRB event dataset grows.

We favour the case of no redshift evolution (i.e., $n=0$), based predominantly on constraining $\alpha=1.5$ – see Table 3, Figure 7 and the likelihood curves for $p(\gamma|\alpha=1.5)$ of Figure 6. Despite the relative p-values between redshift evolutionary models indicated in Table 3, we disfavour the model of quadratic redshift evolution ($n=2$) with respect to the CSFR due to the cuspiness exhibited in the modelled DM distribution (see Figure 7, green dot-dashed curves). The cuspiness being a direct consequence of $\gamma\geq 2$, representing an aggregation of FRBs at low DM, which is not present in the observed histograms. We estimate the mean redshift probed by the Parkes and ASKAP telescopes to be approximately $0.62$ and $0.32$, respectively.

Motivated by the results of the fits, we further determine the DM expectation, $\langle DM\rangle$, that a survey-limited telescope is expected to probe using eq.(9). We compute $\langle DM\rangle$ for fluence survey limits extending down to the anticipated regime of the Square Kilometre Array (SKA), viz., $F_{0}\sim 0.01$ Jy ms, as shown in Figure 8. Here, we use the best-fit energy power-law index, $\hat{\gamma}$, for the constrained fluence spectral index of $\alpha=1.5$, for the redshift evolutionary models of Table 3, and use the Parkes response curve (the ASKAP response-based curves, yielding ostensibly the same result, are omitted).

We compare the observed DM histograms of two FRB sample sets detected by the ASKAP and Parkes radio telescopes with distributions of the  M&E 2018 model and exploit the fact that the telescopes have different survey fluence limits.

After accounting for temporal and spectral resolution of the data, we show that the modelled distributions fit the observed histograms well, and that by comparing the distribution shapes, the absolute FRB event rate is not required – providing a significant advantage in not having to address calibration complexities or unknown survey rate corrections. In this DM regime, DM does seem to be a reasonable proxy for redshift thereby providing additional evidence over direct measurements for a handful of localised FRBs (cf. Macquart et al., 2020), that the IGM does indeed dominate the DM budget for the FRB population as a whole.

After fitting the modelled distributions to the observed datasets simultaneously, we determine the best-fit population parameters of fluence spectral index and energy power-law index, for an assumed energy curve, and for different redshift evolutionary models.

The fluence spectral index, manifest as a k-correction in our analysis, models the value obtained independently by direct fits to burst spectra. This seems remarkable given the irregular burst spectra often measured and that it approximates values determined by observationally independent means (see e.g., Macquart et al., 2019; Shannon et al., 2018). Based on these results, we find that the two telescopes likely do observe the same FRB population and that the energy curve may indeed be relatively flat in this DM regime.

Fits for an FRB population evolving faster than the star-formation rate predict $\gamma\approx 2.2$, leading to an expectation of many FRBs occurring in the nearby Universe, contrary to observations. After constraining the fluence spectral index to $\alpha=1.5$, we find no evidence that the FRB population evolves faster than linearly with respect to the star formation rate, which places further constraints on progenitor classes.

Motivated by the performance of the  M&E 2018 model, and the prospect of much larger FRB sample-sets in future, it seems worthwhile for future studies to investigate more realistic FRB evolutionary scenarios, such as those in which FRBs may exhibit a substantial finite time to evolve from the epoch at which their progenitors form. A further analysis of the FRB population, using telescopes of significantly different sensitivities, such as with FAST and CHIME would also be worthwhile: more sensitive telescopes will be more effective in discriminating the effect of changing sensitivity, $F_{0}$, as can be seen in Figure 4 (bottom right). Other areas of further investigation would include: (i) exploring the effect of $E_{\text{max}}$ of the burst energy distribution; (ii) examine the effects of the degeneracy in $\gamma$ and its mitigation; and (iii) extending the analysis to incorporate the effect that IGM inhomogeneities may have on the DM distribution (see §4 of Macquart & Ekers, 2018b).

This research was partly supported by the Australian Research Council through grant DP180100857. WA acknowledges the contribution of an Australian Government Research Training Program Scholarship in support of this research. In Memoriam: Vale J-PM, our dear friend and esteemed co-author. Like the FRBs you studied, you leave a blazing signal for others to follow; your enthusiasm and insight profound.

Data underlying this article are available in the article.