An FRB Sent Me a DM: Constraining the Electron Column of the Milky Way Halo with Fast Radio Burst Dispersion Measures from CHIME/FRB

Our Galactic halo connects the baryon-rich intergalactic medium (IGM) to the disk of the Milky Way (MW). Gas from the halo is a combination of new and recycled material, is a consequence of galactic feedback processes, and represents a galaxy's future star formation fuel. The MW halo contains both neutral and ionized gas, although it is dominated in mass by the latter component, which extends to hundreds of kiloparsecs (Reynolds 1991; Putman et al. 2012).

Meaningful theoretical predictions of the composition and size of the MW halo come from our knowledge of cosmology and galaxy formation theory (for a review, see Putman et al. 2012). In this sense, the total mass and extent of the halo can be used to check our understanding of these topics. Unfortunately, due to the diffuse and hot nature of the ionized halo gas and our position within the MW, the MW halo gas cannot be imaged directly.

Existing indirect constraints for the total amount of hot halo gas have been placed using observations of the ∼0.1–1 keV diffuse soft X-ray background, which find typical emission measures (EMs) of (1.4–3.0) × 10⁻³ cm⁻⁶ pc (Gupta et al. 2009; Yoshino et al. 2009; Henley et al. 2010; Henley & Shelton 2013). Indirect constraints for the total amount of plasma have also been placed using absorption lines of oxygen ions in X-ray and far-ultraviolet spectroscopy of active galactic nuclei; however, there are considerably fewer useful sight lines (Gupta et al. 2012; Sakai et al. 2012; Fang et al. 2015). Most of the hot gas detected in X-ray emission is thought to be within a few kiloparsecs of the MW disk (Fang et al. 2006; Yao & Wang 2007), although evidence exists for extended hot halo gas with density on the order of 10⁻⁵–10⁻⁴ cm⁻² at distances of 50–100 kpc (Sembach et al. 2003; Stanimirović et al. 2006; Grcevich & Putman 2009). More accurate estimates of the total mass of the ionized medium in the halo require a more precise knowledge of the physical properties of this extended ionized gas. Evidence is emerging suggesting more structure within the MW halo gas, although most models previously had assumed spherical symmetry. Yamasaki & Totani (2020) and Ueda et al. (2022) found evidence for a disklike component to the MW halo gas. The gas closest to us (within 50 Mpc) suggests that the hot halo gas cannot be the host of all of the missing baryons (Bregman et al. 2015, 2018). However, Faerman et al. (2017) showed that if the gas density beyond 50 kpc were to flatten, the hot halo gas could account for the missing baryons.

Another constraint on the mass and extent of the plasma in the MW halo comes from radio observations of pulsars in the Large and Small Magellanic Clouds (LMC and SMC; e.g., Ridley et al. 2013). The LMC and SMC are located ∼50 and 60 kpc away, respectively (Pietrzyński et al. 2019; Graczyk et al. 2020), but this distance is only a small fraction of the virial radius of the MW (using the definition of Bryan & Norman 1998, current estimates of the latter are typically between 180 and 250 kpc; Bovy 2015; Cautun et al. 2020; Shen et al. 2022).

The key to the LMC and SMC pulsar-based halo constraint is the significant dispersive effect of ionized gas on radio waves. Precise measurements of arrival times at the top (high frequencies, ν₁) versus bottom (low frequencies, ν₂) of a radio survey's observing band or the detectable emission for short bursts allow for a quantification of the dispersive delay known as the dispersion measure (DM). This effect is mainly due to the electrons along the line of sight and thus approximately (i.e., good to within one part per thousand; see Kulkarni 2020 for an in-depth discussion) proportional to the column density of free electrons,

$$\begin{eqnarray}&&\mathrm{DM}={\int }_{0}^{L}{n}_{e}\ {d}l,\end{eqnarray} \tag{ 1 }$$

where L is the distance to the source, in this case the pulsars in the LMC or SMC, and n_e is the free electron number density. The DM is determined from observations via the relationship

$$\begin{eqnarray}&&{\rm{\Delta }}t=\displaystyle \frac{{\left({e}^{-}\right)}^{2}}{2\pi {m}_{e}c}\left(\displaystyle \frac{1}{{\nu }_{2}^{2}}-\displaystyle \frac{1}{{\nu }_{1}^{2}}\right)\mathrm{DM},\end{eqnarray} \tag{ 2 }$$

where Δt is the wave arrival time delay between ν₁ and ν₂. ¹⁸ The DM is a direct probe of the intervening plasma between observers and radio transients. The radio pulsars within the SMC and LMC set a lower bound on the Galactic DM contribution at their respective distances of 70 ± 3 and 45 ± 1 pc cm⁻³ (Manchester et al. 2006).

The DM is also useful for constraining the plasma within the Galactic disk. One can characterize this medium using pulsars with independent distance measurements, typically through annual parallax, which enables the modeling of the scale height and midplane density of the warm ionized medium (WIM) disk (Cordes & Lazio 2002; Gaensler et al. 2008; Savage & Wakker 2009; Schnitzeler 2012; Yao et al. 2017; Ocker et al. 2020).

Galactic plasma models NE2001 (Cordes & Lazio 2002) and the more recent YMW16 (Yao et al. 2017) include more components than scale height, filling factor, and vertical electron column, in contrast to the other models listed above, but NE2001 and YMW16 are also based on DM measurements of pulsars with independent distance measurements. Both models include components for the thin and thick disk, spiral arms, and local structures like the Local Bubble and the Gum Nebula. The NE2001 model parameters were fit using data from 112 pulsar distances and 269 scattering measurements. The YMW16 used 189 independent pulsar distances and an updated estimate of the WIM disk scale height. ¹⁹ Both models have been shown to fail in predicting the DMs of certain populations, like high-latitude pulsars, pulsars in H ii regions, and several relatively local pulsars (Chatterjee et al. 2009). Price et al. (2021) gave a comprehensive review and comparison of these two models.

Unfortunately, there are very few known pulsars available to probe significant fractions of the MW halo. One expects to find the highest density of canonical pulsars, remnants of short-lived massive stars, within the disk; hence, historical pulsar surveys most commonly target this area. Typical pulsar emission is also too faint to readily observe at great distances like the edge of the Galaxy or beyond.

Fast radio burst (FRB) DMs are a new way to constrain the total mass and extent of the halo. The class-defining observation of an FRB (Lorimer et al. 2007), FRB 20010724A, caught the attention of astronomers in part because the burst had a DM higher than could be contributed by our Galaxy along that line of sight according to Galactic electron density models like NE2001.

Most observed FRBs have DMs many times what these models estimate can be expected from our Galaxy based on the above models or the measured scale height and average vertical electron column of the MW. In all published instances of precise FRB localizations to date, the FRB is spatially coincident with a galaxy (for a review of host galaxy associations, see Heintz et al. 2020). These associations confirm their extragalactic nature, as the chance coincidence of finding a galaxy that is physically unrelated to the source in their small localization regions is negligible. The FRBs with DMs substantially larger than that maximally predicted by Galactic density models in their line of sight can be assumed to be extragalactic. ²⁰

We can define the measured DM of an extragalactic FRB as the sum of the following four components:

$$\begin{eqnarray}&&\mathrm{DM}={\mathrm{DM}}_{\mathrm{disk}}+{\mathrm{DM}}_{\mathrm{halo}}+{\mathrm{DM}}_{\mathrm{cosmic}}+\displaystyle \frac{{\mathrm{DM}}_{\mathrm{host}}}{1+z},\end{eqnarray} \tag{ 3 }$$

where the terms refer to the DM contributions from electrons in the MW disk, MW halo, cosmic web, and FRB host galaxy. The first two terms, DM_disk and DM_halo, when summed, are denoted DM_Gal, as they comprise the contribution from the MW in a given line of sight, i.e.,

$$\begin{eqnarray}&&{\mathrm{DM}}_{\mathrm{Gal}}={\mathrm{DM}}_{\mathrm{disk}}+{\mathrm{DM}}_{\mathrm{halo}}.\end{eqnarray} \tag{ 4 }$$

The DM_host likely includes contributions from the halo and disk of the host galaxy and potentially includes a local component around the source of the burst. The DM_cosmic includes contributions from the IGM and ionized gas in the Local Group (Prochaska & Zheng 2019) and could include intervening galaxies or galaxy halos along the line of sight to the FRB (Prochaska et al. 2019).

The MW halo contribution was first constrained using a population of FRBs by Platts et al. (2020). After subtracting the DM_disk estimated by NE2001 from the total measured DM of each FRB, the authors modeled the excess DM distributions using asymmetric kernel density estimation and set conservative limits of −2 pc cm⁻³ < DM_halo <123 pc cm⁻³. The authors concluded by emphasizing that they expected that a larger sample of FRBs would tighten these constraints.

In this paper, we derive observation-based upper limits of DM_halo as a function of Galactic latitude from the most extensive sample of FRBs to date. In Section 2, we outline the extragalactic source sample from the FRB back end of the Canadian Hydrogen Intensity Mapping Experiment (CHIME), which allows us to make direct upper limits of the column density of ionized halo gas without relying on models for DM_halo, DM_cosmic, and DM_host, each of which remain loosely constrained on a population scale. In Section 3, we compare this extragalactic sample with information from Galactic pulsar DMs and show that there is a distinct gap between the extragalactic and Galactic populations. Then, in Section 4, we derive estimates of DM_Gal as a function of Galactic latitude, which, when combined with estimates of DM_disk, also describe the structure of DM_halo as a function of Galactic latitude. We discuss the biases in our data collection in Section 5.1, as well as how these biases could produce the lack of radio pulse detections between our Galactic and extragalactic populations. The uncertainties of our derived models are discussed in Section 5.2.

Our extragalactic FRB sample comes from CHIME/FRB. CHIME is a radio telescope operating over 400–800 MHz (CHIME Collaboration et al. 2022). It is a transit telescope with no moving parts; it observes the sky above it as the Earth rotates. CHIME is located at the Dominion Radio Astrophysical Observatory near Penticton, British Columbia, Canada. The CHIME telescope is comprised of four 20 m × 100 m, north/south-oriented, semicylindrical parabolic reflectors, each of which has 256 dual-polarization feeds, giving the entire instrument a more than 200 deg² field of view. CHIME's FX correlator forms 1024 beams over this large field of view, and the FRB back end searches the beams for radio pulses with durations of ∼1 to hundreds of milliseconds, such as pulsars and FRBs (CHIME/FRB Collaboration 2018). For this study, we selected all 93 sources detected by CHIME/FRB through 2021 February that satisfied our selection criteria, namely, having a low measured DM (<250 pc cm⁻³) and high Galactic latitude (∣b∣ > 30°). Thirty-four of these FRBs are reported in the first CHIME/FRB catalog (CHIME/FRB Collaboration 2021). We inspected the events for any evidence that they were detected away from the meridian of the telescope (i.e., in a side lobe), as this can result in a given burst's reported position being inaccurate due to imperfect modeling of the inherent side-lobe-beam structure. None of the bursts in our sample show evidence of being side lobe events, but, especially with the lower signal-to-noise ratio bursts, we cannot completely rule out this possibility with intensity data alone.

We define high latitude as those FRBs with a measured absolute Galactic latitude (∣b∣) greater than 30°. We made this selection to avoid contamination in the measured DM by H ii regions and other small-scale local structures. At these latitudes, the maximal DM_Gal predicted by Galactic free electron density models YMW16 and NE2001 show significantly less scatter in Galactic longitude, a dimension we collapse over in this study. CHIME/FRB's pipeline imposes another selection criterion on our sample. The pipeline only saves intensity data from bursts with measured DMs greater than at least one of YMW16's or NE2001's maximal DM_disk estimate in the burst's apparent line of sight. The impact of this pipeline-imposed selection criterion is discussed further in Section 5.1. Our low-DM samples at these latitudes are FRBs with DM < 250 pc cm⁻³ and are chosen because they are the most constraining on DM_halo. Most MW halo models typically translate into DM_halo predictions of less than 100 pc cm⁻³, and at Galactic latitudes greater than 30°, DM_disk is predicted to be less than 70 pc cm⁻³ according to NE2001, YMW16, and Ocker et al. (2020). Thus, we choose to consider only FRBs with a measured DM of less than 250 pc cm⁻³ to conservatively explore the range within which models predict DM_Gal.

Our selected sample includes four repeating sources, one of which, FRB 20200120E, is associated with spiral galaxy M81 and, at 3.6 Mpc away, is the closest known extragalactic FRB source (Bhardwaj et al. 2021; Kirsten et al. 2022). Repeating FRB 20200120E, which we will denote M81R for brevity, is a particularly interesting source for this study, not only because it likely has a low DM_cosmic contribution (Kirsten et al. 2022 estimated this contribution to be on the order of 1 pc cm⁻³) but also because it is located in a globular cluster on the outskirts of M81 (the globular cluster's offset from the center of M81, measured in projection, is approximately 20 kpc). This circumstance means that we expect a negligible DM contribution from the disk of M81. Additionally, we do not expect that a globular cluster would contribute significant amounts of internal dispersion (Freire et al. 2001).

The extragalactic FRBs with the lowest DMs provide the most constraining upper limits on DM_Gal. Figure 1 shows the DMs of all high-latitude and low-DM FRB candidates from CHIME/FRB (triangles) as a function of $\sin | b|$. Repeating FRB sources are shown as red triangles, indicating the best-measured latitude and DM considering all published bursts. The FRB with the smallest DM in our current sample is M81R, with DM = 87.8 pc cm⁻³.

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We plot all Galactic pulsars in DM versus Galactic latitude from the Australia Telescope National Facility (ATNF) pulsar catalog ²¹ (Manchester et al. 2005; light gray stars) and indicate the sources from this sample that have been detected by the real-time CHIME/FRB pipeline (dark gray stars). Additionally, if prepublication pulsars or rotating radio transients (RRATs) from the pulsar survey scraper ²² have been detected by CHIME/FRB's real-time pipeline through 2021 February, we also include them in this plot (dark gray stars). This addition includes new Galactic sources seen by CHIME ²³ (Good et al. 2021; Dong et al. 2022). We exclude ATNF and pulsar survey scraper sources that were detected in lines of sight with EMs above the 95th percentile of the sky as measured by the Planck 2015 astrophysical component separation analysis (Planck Collaboration et al. 2016a). This exclusion is enacted to avoid higher-than-representative DMs due to contamination by H ii regions ²⁴ and other small-scale local structure. This cut affects about 30% of pulsars across the sky but does not remove any pulsars from the Galactic latitudes and declinations we consider. The pulsar decl. criterion removes sources that are outside of CHIME's sky coverage (i.e., those with declinations <−11°). The pulsars and RRATs all sit below a DM of ≈50 pc cm⁻³, and the highest pulsar or RRAT DMs are largely found at lower $\sin | b|$.

There is a distinct gap in DMs at around 50–87.8 pc cm⁻³ (the exact values depend on the latitude considered, but the gap is not narrower than this in any direction). As discussed more in Section 5.1, for ∣b∣ > 30°, this gap separates known or suspected Galactic sources and extragalactic FRBs.

4.1. Basic Methodology

We seek to describe the constraints placed on DM_Gal using FRBs as a function of Galactic latitude. Reasonable possibilities for models of DM_Gal include those that assume a purely spherical ionized MW halo and those that imply more latitudinal variation in the density of the ionized MW halo. We will fit a model that assumes that DM_Gal is constant, a model that assumes that DM_halo is constant, and models for DM_Gal that take the form of third-order polynomials but still bound the DMs of the FRBs from below. Since measured extragalactic FRB DMs must include the contribution from DM_Gal along their line of sight, and each of these models bound the FRB DMs from below, the models represent the upper limits of DM_Gal derived when DM_cosmic = DM_host = 0. We then turn the upper limit models of DM_Gal at each latitude into upper limits of DM_halo by subtracting the DM_disk component as a function of Galactic latitude (b) found by Ocker et al. (2020),

$$\begin{eqnarray}&&{\mathrm{DM}}_{\mathrm{disk}}=\displaystyle \frac{23.5\pm 2.5}{\sin | b| }\ \mathrm{pc}\,{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 5 }$$

4.2. Galactic DM Estimates

In Figure 2, we fit and plot four structure estimates that either strictly or roughly bound the DMs of FRBs from below, since the lowest extragalactic FRB DMs are the most constraining upper limits of the MW halo. The first estimate (red dotted–dashed line in Figure 2) assumes a constant value for the total Galactic contribution DM_Gal across the sky. That is, DM_Gal = 87.8 pc cm⁻³ for all ∣b∣ ∈ (30°, 90°). This is the measured DM of FRB 20200120E, associated with spiral galaxy M81 (Bhardwaj et al. 2021; Kirsten et al. 2022). Below absolute latitudes of 30°, this model is not supported, as many pulsars have been detected at DMs higher than 87.8 pc cm⁻³.

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The next model (solid yellow line in Figure 2) assumes that the halo has a constant contribution at a given latitude. If DM_disk is assumed to be the central value predicted by Ocker et al. (2020) at the latitude of our lowest-DM FRB, likely the most constraining single estimate of DM_halo, our observations support a DM_halo of no more than 87.8 pc cm⁻³ $-(23.5/\sin (0.64))$ pc cm⁻³ = 52 pc cm⁻³.

Written explicitly,

$$\begin{eqnarray}&&{\mathrm{DM}}_{\mathrm{Gal}}(b)=\left(\displaystyle \frac{23.5}{\sin | b| }+52\right)\,\mathrm{pc}\,{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 6 }$$

We also fit a model for DM_Gal using locally weighted scatterplot smoothing (LOWESS; Cleveland 1979) to the local minimum of the measured FRB DMs (blue dashed line in Figure 2). LOWESS is a method for smoothing a scatterplot in which the fitted value at a given point is the value of a polynomial fit to the data using weighted least squares. The weight is determined by how close the original value is to a local regression so that the weight is large if the proposed value is close to the data and small if not. At each point in $\sin | b|$, we bin all FRB DMs within 5° and select the minimum as the value for that latitude. Using these minima, we fit a LOWESS line with a polynomial degree of 3 and bandwidth of 0.55. A bandwidth of 0.55 means that 55% of the data are considered when smoothing each point. The polynomial degree was chosen because both quadratic and quartic functions predicted behavior near the ∣b∣ boundaries that was unphysical, and higher polynomial degrees did not offer a better fit. The DM_Gal predictions from this model fall between 88 pc cm⁻³ at $\sin | b| =0.65$ (b = 408) and 111 pc cm⁻³ at $\sin | b| =0.77$ (501). The LOWESS line demonstrates a lot of variation with Galactic latitude. This model is not intended to suggest a physical representation of structure in the halo. Rather, we wanted to demonstrate what conservative upper limits on DM_Gal might be reasonable in lines of sight that do not have a particularly constraining FRB DM using more constrained lines of sight nearby in Galactic latitude. These more constrained lines of sight are still relatively sparse at our sample size. This model is likely most appropriate if a conservative estimate is desired.

The final method we apply to model DM_Gal as a function of Galactic latitude is polynomial boundary regression. The polynomial boundary regression assumes that the boundary of a given scatterplot can be described by a polynomial and optimizes that polynomial such that it envelopes the data and minimizes the area under its graph (Hall et al. 1998). We computed this estimate assuming a third-degree polynomial using the CRAN package npbr ²⁵ (Daouia et al. 2017). For a cubic polynomial with coefficients defined as

$$\begin{eqnarray}&&{\mathrm{DM}}_{\mathrm{Gal}}(b)=\displaystyle \sum _{i=0}^{3}{x}_{i}{\sin }^{i}| b| ,\end{eqnarray} \tag{ 7 }$$

the best-fit coefficients for our model can be found in Table 1. The cubic boundary regression predicts values for DM_Gal between 87.6 and 130.1 pc cm⁻³ at $\sin | b| =0.67$ and 0.50, respectively. We plot this model as the green dotted line in Figure 2. When considering the error associated with this estimate of the structure of DM_Gal across Galactic latitude, it is important to consider not only the error in estimation of the fit parameters but also the error introduced by the scatter in DM_halo over Galactic longitudes. We provide pointwise bootstrap errors implied for the fit parameters at doi:10.11570/22.0079. These boundary estimates are summarized for their comparison in Section 5.3 to existing models of DM_halo in Table 2.

Table 1. Best-fit Parameters Derived for the FRB DM Cubic Boundary Estimate Described in Equation (7)

Coefficient	Value (pc cm−3)
x0	1304
x1	−4747
x2	6044
x3	−2490

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Table 2. Summary of Our Boundary (Upper Limit) Estimates of DM_halo and DM_Gal as Presented in Section 4

Model	DMGal Upper Limit Range	DMhalo Upper Limit Range
	(pc cm−3)	(pc cm−3)
Constant DMhalo	76–99	52
Cubic boundary estimate	87.8−130	52–83
LOWESS estimate	88–141	52–111

Note. The DM_halo upper limits are derived from the DM_Gal estimates by subtracting an estimate of DM_disk from Ocker et al. (2020), shown in Equation (5).

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We plot DM_disk in Figure 2 as estimated by the two popular free electron density models, NE2001 (Cordes & Lazio 2002) and YMW16 (Yao et al. 2017). We again removed from the data lines of sight with EMs above the 95th percentile of the sky. We remove lines of sight outside of CHIME's sky coverage as in Figure 1 to ensure an appropriate comparison. In Figure 2, NE2001 and YMW16 are shown by blue and pink shaded regions representing the range of maximum Galactic contribution over Galactic longitudes. The dashed lines of the same color located within the shaded regions of both models represent the median value over the relevant Galactic longitude. The implied DM of the WIM disk component as a function of Galactic latitude b found by Ocker et al. (2020), shown in Equation (5), is shown in yellow in Figure 2, where the solid line represents the best-fit model, and the surrounding region represents the model fit uncertainty. The estimated DM_disk from YMW16, NE2001, and Ocker et al. (2020) mostly bound the DMs of the pulsars and RRATs from above. There are a few exceptions where an observed Galactic source is only a few pc cm⁻³ above the largest expected DM_disk from a given model at the relevant Galactic latitude. The FRB DMs are all >30 pc cm⁻³ larger than the largest expected DM_disk from any model.

In summary, our constant Galactic contribution model, constant DM_halo model, LOWESS boundary estimate, and cubic boundary estimate result in high-latitude upper limits on DM_Gal from 87.8 to 141 pc cm⁻³ depending on the model and line of sight considered. By subtracting the DM_disk estimate from Ocker et al. (2020) assuming slab geometry of the disk, we can place upper limits on the DM_halo alone. These constraints range from 52 to 111 pc cm⁻³ depending on the Galactic latitude and model considered.

5.1. Potential Biases in Sample Collection

5.2. Model Uncertainties and Unmodeled Contributions

There are three main sources of error when describing the boundary of the halo as in Section 4. There is random error in parameter estimation, error due to unmodeled longitudinal variation, and the contribution of DM_host and DM_cosmic.

As discussed in Section 4 when introducing the cubic boundary estimate of DM_Gal, the first source of uncertainty is due to parameter estimation. We resample our original data set of pairs of FRB DMs and latitudes, which were used to fit our DM_halo models, 1000 times with replacement (bootstrapping) to estimate pointwise 90% confidence intervals. We show this 90% confidence interval of the cubic boundary estimate of DM_Gal as the region bound by solid green lines in Figure 3.

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The second source of uncertainty, also discussed in Section 4, is longitudinal variation in DM_halo. This variation introduces error in the models, which is unaccounted for (due to the small sample size) in this analysis.

A scatter of 0.3–0.4 dex is seen in both Suzaku (Nakashima et al. 2018) and HaloSat X-ray EM data (Kaaret et al. 2020) of the MW halo. If we knew exactly what fraction of this scatter can be attributed to the fluctuation of the MW halo gas density, it would tell us the approximate scatter of DM_halo, as EM is proportional to the path integral of n_e ², while DM is proportional to the path integral of n_e . It is worth noting that the instrumental limitations of X-ray telescopes are such that these observations are sensitive to only the densest hot gas, whereas the integrated DM_halo will include more distant, diffuse gas (Fang et al. 2006; Yao & Wang 2007). As such, one may expect DM_halo to have much less scatter. We assume that the upper limit on the fluctuation of DM_halo is approximately ≈0.2 dex. This also constrains the total amount of longitudinal variation we expect. For each $\sin | b|$, we illustrate in Figure 3 the extent of a 0.2 dex variation around our cubic boundary estimate of DM_Gal (the region bounded by solid orange lines; see Section 3 for more information on the cubic boundary estimation).

To investigate the third source of error, that due to each FRB's nonzero and unaccounted-for DM_cosmic and DM_host, one can study the most constraining FRB sight line, that of M81R. It is exceptional both in being the lowest-DM source in our sample and in being precisely localized within a globular cluster on the outskirts of the halo of M81 (Bhardwaj et al. 2021; Kirsten et al. 2022). In Bhardwaj et al. (2021), the authors discussed the uncertainties in estimating this source's exact DM_host and DM_cosmic but ultimately concluded a minimal, conservative, expected DM_host + DM_cosmic = 15 pc cm⁻³.

Even given the conservative nature of the quadrature sum of these three sources of uncertainty, the region does not encompass any of the Galactic sources in our sample. This is indicative of the conservative nature of the upper limits presented in the paper and a nice consistency check for our models.

Given that the source is relatively nearby (so we expect very little DM_cosmic) and has essentially no local or disk DM contributing to DM_host, it could be argued that this is an edge case of the FRB population, and it would be appropriate to subtract this same lower bound on DM_cosmic + DM_host = 15 pc cm⁻³ from every line of sight. We refrain from making this generalization in our DM_halo estimates given that our sample is not universally localized to the precision required to estimate the DM_host contribution, nor is there sufficient knowledge about the population of FRB hosts to make a meaningful distribution-based argument. We still demonstrate, however, the magnitude of this minimal expected contamination of DM_host + DM_cosmic in Figure 3 (the region bounded by a solid blue line).

5.3. Constraints on Existing Halo Models

Our study was motivated by wanting to observationally constrain the gas content of the MW halo to distinguish between different galaxy formation models. We obtain this constraint by comparing the upper limits implied by FRBs to the estimates made by models with different physical assumptions. First, we review these models and compare their estimates to our DM_halo boundary estimates. Table 2 summarizes our DM_halo boundary (upper limit) estimates from Section 4.

Keating & Pen (2020) computed most of these estimates of DM_halo using gas profiles of halo models. Two total masses of the MW halo are considered in each of their estimates, M_halo = 1.5 × 10¹² and 3.5 × 10¹² M_⊙. We compare our FRB constraints with the range bounded by the halo model estimates from the lower- and higher-mass scenario. ²⁶ In addition, for some of the models, multiple physical scenarios are considered. In this case, which is noted in the brief descriptions of the models that follow, we additionally consider the range in values between these multiple scenarios. We briefly summarize the models included for their comparison to our upper limits.

Navarro et al. (1996) and Mathews & Prochaska (2017). The Navarro–Frenk–White (NFW; Navarro et al. 1996) profile describes well the density profile of virialized dark matter halos in cosmological simulations. A simple model for the baryonic matter is to assume that it traces dark matter near the cosmic ratio (Ω_b /Ω_m ∼ 0.2; Planck Collaboration et al. 2016b) down to 10% of the MW virial radius, in which case the gas density profile (ρ) as a function of distance from the center of the Galaxy (r) can be described using the NFW profile,

$$\begin{eqnarray}&&\rho (r)=\displaystyle \frac{{\rho }_{0}}{y(1+{y}^{2})},\end{eqnarray} \tag{ 8 }$$

where y = c(r/r_V ) with concentration c and r_V is the virial radius, and ρ₀ is a characteristic density. This model predicts DM_halo ≈ 300–500 pc cm⁻³ and hence was inconsistent with previous observations (Keating & Pen 2020) and remains inconsistent given our observations. This simple model does not account for nonlinear effects facilitated by, e.g., feedback, accretion, and shocks. Mathews & Prochaska (2017) modify the NFW profile with an additional two parameters (y₀, a) based on measurements of O vi absorption in quasar spectra caused by intervening galactic halos:

$$\begin{eqnarray}&&\rho (r)=\displaystyle \frac{{\rho }_{0}}{y{\left({y}_{0}+y\right)}^{2+\alpha }}.\end{eqnarray} \tag{ 9 }$$

This extension to baryonic matter accounts for feedback. We consider (as in Prochaska & Zheng 2019 and Keating & Pen 2020) profiles with y₀ = 2 and 4 in the span of this modified NFW (mNFW) profile predicted DM_halo in Figure 4 while keeping α = 2 fixed for both cases. In the y₀ = 2 case, the profile is disfavored for both of the halo masses considered (i.e., between 1.5 and 3.0 × 10¹² M_⊙), as it predicts DM_halo = 66–86 pc cm⁻³, which is higher than the upper limits in most lines of sight of each of our four models. The profile with y₀ = 4 remains more plausible for lower masses, as it predicts DM_halo = 41–51 pc cm⁻³.

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Maller & Bullock (2004) (MB04). MB04 created their gas density profile by assuming that the halo gas is adiabatic and in hydrostatic equilibrium, taking into account the expectation that the hot gas in halos is prone to fragmentation during cooling due to its thermal instability. The resulting density profile is defined as

$$\begin{eqnarray}&&\rho (r)={\rho }_{c}{\left(1+\displaystyle \frac{3.7}{y}\mathrm{ln}(1+y)-\displaystyle \frac{3.7}{{C}_{c}}\mathrm{ln}(1+{C}_{c})\right)}^{3/2},\end{eqnarray} \tag{ 10 }$$

where again, y = c(r/r_V ). Here ρ_c is a normalization constant set by the assumed gas mass of the halo, and ${C}_{c}=c\tfrac{{r}_{c}}{{r}_{V}}$ with r_c = 147 kpc as assumed by Prochaska & Zheng (2019) and Keating & Pen (2020). All masses considered are compatible, as they are slightly lower than the upper limits given by our observations, as DM_halo is estimated to be 42 and 56 pc cm⁻³ in the low and high halo mass scenarios, respectively. If this model is correct and the mass of the MW halo is within (1.5, 3.5) × 10¹² M_⊙, when this estimate is used in the line of sight of M81R, it would suggest that DM_host (including that from the likely significant fraction of M81's halo that the burst encounters) would need to contribute considerably less DM than the MW halo.

Miller & Bregman (2013) (MB13). MB13 used archival soft X-ray data from XMM-Newton's Reflection Grating Spectrometer to measure O vii Kα. This is used to find the best-fit parameters (n₀ = 0.46 cm⁻³, r_c = 0.35 kpc, and β = 0.71) for an underlying spherical density of the hot Galactic halo (n) model of the form

$$\begin{eqnarray}&&n(r)={n}_{0}{\left(1+\displaystyle \frac{r}{{r}_{c}}\right)}^{-3\beta /2}\end{eqnarray} \tag{ 11 }$$

with the addition of an ambient density component due to ram pressure stripping of n = 1 × 10⁻⁵ cm⁻³ out to 200 kpc. As the density profile with the lowest estimated DM_halo, this model is not ruled out at these masses. Keating & Pen (2020) estimated the contribution from these as ≈6 pc cm⁻³ in the low-mass halo scenario and ≈7 pc cm⁻³ in the higher-mass scenario.

Pen (1999) (P99). P99 used an entropy-floor singular isothermal sphere model motivated by observations of the soft X-ray background. In this model, the halo gas is assumed to have two phases, an outer region in which gas traces mass isothermally and an inner region in which the gas has been heated to constant entropy, invoking baryonic feedback. Keating & Pen (2020) considered two cases of the model, one with a heated core radius (r_c ) that produces X-ray emission at the limit of the observational constraints of Moretti et al. (2003) and one that maximizes the effect of feedback by choosing r_c equal to the virial radius of the MW. We consider each of these profiles in Figure 4. When M_halo = 1.5 × 10¹² M_⊙ is assumed, and r_c = 0.34r_V in order to match the X-ray emission, Keating & Pen (2020) estimated DM_halo ≈ 79 pc cm⁻³, which is larger than some of our upper limits and hence largely inconsistent with our observations. When M_halo = 3.5 × 10¹² M_⊙ and r_c = 0.86r_V (predicted DM_halo = 34 pc cm⁻³), the measurements are consistent with our observations. Similarly, in either the high- (M_halo = 3.5 × 10¹² M_⊙) or low- (M_halo = 1.5 × 10¹² M_⊙) mass scenario, when the heated core radius r_c is set equal to r_V (DM_halo = 28 and 21 pc cm⁻³ for the high- and low-mass scenario, respectively), the results are consistent with our observations.

Voit (2019) (V19). V19 constructed a model for the halo, called the pNFW model, that assumes a confining gravitational potential with a constant circular velocity at small radii. At larger radii, the circular velocity profile is assumed to decline like that of an NFW halo with scale radius r_{s, NFW}. These two profiles are joined continuously at a radius of 2.163r_{s, NFW}. The author provided a table of formula coefficients as a function of input halo mass for the resultant density profile. In this model, only the lower halo mass scenarios are consistent with all of our lines of sight, since the model estimates DM_halo= 24–84 pc cm⁻³ for MW halo masses between 1.5 and 3.0 × 10¹² M_⊙.

In addition to the estimates derived by Keating & Pen (2020) from the above density profiles, we compare our observations to the following estimates for DM_halo.

Yamasaki & Totani (2020) (YT20). YT20 modeled the MW halo with a spherical component of isothermal gas in hydrostatic equilibrium and a disklike hot gas component to reproduce the directional dependence of X-ray EM observed by Nakashima et al. (2018). They presented an analytic formula for DM_halo that we plot as a function of Galactic latitude while representing the longitudinal variation as a span in the left panel of Figure 4. At ∣b∣ > 30°, this model predicts a DM_halo of between 29 and 68 pc cm⁻³ depending on the line of sight considered. As can be seen in Figure 4, at each b, the minimum and median values lie below our cubic boundary estimate of DM_halo, and in most cases, the maximum DM_halo prediction also lies below our model. At the sky position of M81R, YT20 predicted DM_halo ≈ 30.5 pc cm⁻³. Our constraint is DM_halo < 52 pc cm⁻³ for all boundary models; hence, we find that the YT20 model is consistent with our FRB observations.

Dolag et al. (2015) (D15). D15 performed cosmological simulations of an MW-like galactic halo including hot thermal electrons in order to estimate DM_halo. Their probable values for DM_halo, depending on which inner radius one expects from the edge of the Galactic disk, range over ≈24–67 pc cm⁻³. This range, particularly for the larger radii of the edge of the Galactic disk, remains highly relevant and agrees well with our observations, as does the commonly cited representative halo electron column estimate of DM_halo = 30 pc cm⁻³ selected by the authors. This representative value assumes integration radii beginning 17 kpc from the Galactic center, the maximal extent of NE2001 that was used by Dolag et al. (2015) to model DM_disk.

Prochaska & Zheng (2019) (PZ19). PZ19 looked at tracers of the "hot" (T ∼ 10⁶ K) and "cool" (T ∼ 10⁴ K) components of the halo gas. These tracers, namely, observations of O vi and O vii absorption (Fang et al. 2015), Si II and Si III (Richter et al. 2017), and high-velocity clouds (HI4PI Collaboration et al. 2016), are combined with hydrostatic models of the halo to estimate DM_halo = 50–80 pc cm⁻³ integrated to 200 kpc. This is within the upper limit range of our various models, but most of the range is above the excess DM of M81R (see also Bhardwaj et al. 2021).

We compare our DM_halo boundary estimates and upper limits to estimates of DM_halo implied by various models in Figure 4. The red region in Figure 4 shows the DM range that cannot be supported by our observations regardless of model chosen or line of sight. Within the yellow region, we show the DM range that encompasses all upper limits from our models, ranging between 52 and 111 pc cm⁻³. We highlight the excess DM of M81R (FRB 20200120E; Bhardwaj et al. 2021), which is the lowest extragalactic DM in our sample, with the black dotted line.

To summarize, the NFW profile can be unambiguously ruled out (as was previously known by, e.g., Fang et al. 2013 and Pen 1999), and MB13, YT20, and D15 are consistent with our observations. In the case of mNFW, MB04, and V19, the models are mildly in tension with our observations for the high MW halo mass considered (3.5 × 10¹² M_⊙) but not for the low-mass (1.5 × 10¹² M_⊙) scenario. Similarly, for P99, the model is supported only when the heated core radius is set at the virial radius or the MW halo is assumed to be of lower mass. The majority of the DM_halo range proposed by PZ19 is higher than our estimates but remains possible in the scenario where there is significant DM_halo scatter in the sky, as acknowledged for the M81R sight line (Bhardwaj et al. 2021).

Baryonic feedback processes and their overall effect are still relatively uncertain in galaxy formation; however, it is interesting to note that both the NFW/mNFW model and P99's models flip from inconsistent with our observations to consistent when consideration for the effect of feedback is increased. The cosmological simulation of D15 that results in a DM_halo estimate that is in great agreement with our observations also accounts for the energy released in explosions of massive stars as supernovae, a type of baryonic feedback, and for feedback from active galactic nuclei.

We explore the constraints on the total Milky Way (MW) dispersion measure (DM), as well as the MW halo DM using CHIME/FRB's large, extragalactic, fast radio burst (FRB) source population. This sample of DM measurements offers a unique opportunity to constrain the distribution of the Galactic plasma and estimate the MW halo DM contribution upper limits as a function of Galactic latitude. The observation-based high-latitude upper limits on the Galactic DM contribution range over 87.8–141 pc cm⁻³ depending on the chosen model and the Galactic latitude of interest. Subtracting estimates of the disk contribution from Ocker et al. (2020), we derive upper limits on the MW halo DM contribution ranging over 52–111 pc cm⁻³. These results agree with the recently reported constraint of DM_halo ≤ 47.3 pc cm⁻³ along the line of sight toward FRB 20220319D, located at a comparatively low Galactic latitude of b ∼ +91 (Ravi et al. 2023).

Although there is a DM gap between Galactic and extragalactic radio pulses, assuming the rate at which FRB sources are detected can be described using Poisson statistics, and, using measured population statistics from the first CHIME/FRB catalog, we find that this lack of intermediate DM radio sources is compatible with having arisen from volume effects and pipeline bias alone. The presence of the gap is therefore not evidence of a DM_halo contribution that is nonzero.

Our constraints on the MW halo DM contribution seem in tension with most popular estimates of DM_halo (e.g., Maller & Bullock 2004; Mathews & Prochaska 2017; Voit 2019) when assuming an MW halo mass of 3.5 × 10¹² M_⊙, with the exception of Miller & Bregman (2013) and Pen (1999). In part, this tension arises because our estimates are necessarily an overestimate of the true value, as we do not estimate and remove DM contributions from the intergalactic medium or host galaxy of each FRB. If we assume a lower MW halo mass estimate of of 1.5 × 10¹² M_⊙, our constraints agree with more models, including those proposed by Maller & Bullock (2004), Mathews & Prochaska (2017), and Voit (2019). The estimates of the MW halo DM contribution produced by Dolag et al. (2015) using cosmological simulations of an MW-like galactic halo are supported by our observations. So too is the MW halo model of Yamasaki & Totani (2020), which combines a spherical isothermal gas component and a disklike hot gas component. The majority of the DM_halo range proposed by PZ19 is higher than our estimates but remains possible in a scenario where there is significant DM_halo scatter in the sky, as acknowledged for the M81R sight line (Bhardwaj et al. 2021). For some models, these results seem to emphasize the importance of the role of baryonic feedback in Galaxy formation.

While many models of the density of the halo gas invoke strict or quasi-spherical symmetry, one expects the ionized gas in the Local Group to be ellipsoidal, extended from our Galaxy toward M31 due to the inflows, outflows, and tidal interactions between our Galaxy and M31 (Bregman & Lloyd-Davies 2007). Simulation work (Nuza et al. 2014) also finds evidence for this gas excess between a pair of galaxies resembling M31 and the MW compared to another random line of sight. In searches for an excess in DM from FRBs that intersect the dark matter halos of other galaxies, Connor & Ravi (2022) found a higher excess DM in these lines of sight than expected from diffuse gas surrounding isolated galaxies. The authors suggested that this DM excess is potentially due to ionized media in galaxy groups, including the Local Group. Wu & McQuinn (2022) presented a similar analysis but introduced a weighted-stacking scheme that minimizes the effect of the variance of the observed DM distribution and derives a significance for the result that is lower than that found by Connor & Ravi (2022; probability >0.99 versus >0.68 to >0.95). We plan to repeat our study observed in two dimensions (i.e., producing a sky map) once the known FRB population has roughly doubled. This 2D map will allow us to search for evidence of asymmetries in the Galaxy or such an ellipsoidal halo gas distribution that is extended by interactions with our Galaxy group, expected to be dominated by interactions with M31.

We thank Jo Bovy, Stanislav Volgushev, and Jeremy Webb for discussions vital to preparing this work. We are also grateful to the referee for the very thoughtful and constructive comments.

A.M.C. is funded by an NSERC Doctoral Postgraduate Scholarship. M.B. is a McWilliams Fellow. The Dunlap Institute is funded through an endowment established by the David Dunlap family and the University of Toronto. B.M.G. acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) through grant RGPIN-2022-03163 and of the Canada Research Chairs program. G.M.E. acknowledges funding from NSERC through Discovery Grant RGPIN-2020-04554 and U of T through the Connaught New Researcher Award. V.M.K. holds the Lorne Trottier Chair in Astrophysics & Cosmology, a Distinguished James McGill Professorship, and receives support from an NSERC Discovery grant (RGPIN 228738-13), an R. Howard Webster Foundation Fellowship from CIFAR, and the FRQNT CRAQ. K.W.M. holds the Adam J. Burgasser Chair in Astrophysics and is supported by an NSF grant (2008031). A.P.C. is a Vanier Canada Graduate Scholar. F.A.D. is supported by the UBC Four Year Fellowship. C.L. was supported by the U.S. Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program. A.P. is funded by an Ontario Graduate Scholarship. A.B.P. is a McGill Space Institute (MSI) Fellow and a Fonds de Recherche du Quebec—Nature et Technologies (FRQNT) postdoctoral fellow. Z.P. is a Dunlap Fellow. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. S.M.R. is a CIFAR Fellow and supported by NSF Physics Frontiers Center awards 1430284 and 2020265. K.S. is supported by the NSF Graduate Research Fellowship Program. The FRB research at UBC is supported by an NSERC Discovery Grant and the Canadian Institute for Advanced Research. D.C.S. acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), RGPIN-2021-03985. We acknowledge that CHIME is located on the traditional, ancestral, and unceded territory of the Syilx/Okanagan people. We are grateful to the staff of the Dominion Radio Astrophysical Observatory, which is operated by the National Research Council of Canada. CHIME is funded by a grant from the Canada Foundation for Innovation (CFI) 2012 Leading Edge Fund (project 31170) and contributions from the provinces of British Columbia, Québec, and Ontario. The CHIME/FRB Project is funded by a grant from the CFI 2015 Innovation Fund (project 33213), contributions from the provinces of British Columbia and Québec, and the Dunlap Institute for Astronomy and Astrophysics at the University of Toronto. Additional support was provided by the Canadian Institute for Advanced Research (CIFAR), McGill University, and the McGill Space Institute thanks to the Trottier Family Foundation and the University of British Columbia.

Given the biases discussed in Section 5.1, in this Appendix we will answer the question "Is this gap astrophysical?" or, equivalently, "Does one need more than volume and selection effects to explain this gap?" We do not assert what fraction of the gap can be attributed to DM_halo and DM_host accordingly. To derive the astrophysical significance, we will quantify the likelihood of observing zero events within the "gap," given the observation of 93 FRB sources in the remaining DM range of our sample and considering only the volume and pipeline biases. In order to estimate this likelihood, we make some simplifying assumptions, and we highlight these assumptions as they appear in the derivation. We show at the end of the section that ultimately, these assumptions result in a conservative estimate.

First, we define DM−DM_disk as the excess DM and assume that it is contributed solely by the IGM. That is, we are assuming that there is no contribution from the MW halo (DM_halo = 0 pc cm⁻³) or the host galaxy (DM_host = 0 pc cm⁻³). If we assume that both are zero, and we know that in a given line of sight, the DM_cosmic is proportional to distance (d) and distance is proportional to redshift (z), ²⁷ we are essentially extending the volume in which FRB sources can exist right to the edge of our MW WIM disk. We can estimate the relative rate of FRBs between two volumes using the fluence distribution (commonly referred to as $\mathrm{log}(N)/\mathrm{log}(F)$) of FRBs. We will compare the relative rate of FRBs in the DM gap (excess DMs [0, 52) pc cm⁻³) and in the rest of the sample, which spans excess DMs from [52, 215] pc cm⁻³.

To simplify, assume that FRBs are standard candles; that is, each burst has equal intrinsic energy. The number of FRBs (N) contained in a given spherical volume of radius d is

$$\begin{eqnarray}&&N\propto {F}^{\alpha }\propto {\left({d}^{-2}\right)}^{\alpha },\end{eqnarray} \tag{ 12 }$$

where F is the FRB fluence, d is distance, and α is a power-law index for the cumulative fluence distribution (α < 0). For a nonevolving population in Euclidean space, one expects α = −3/2, and this is in agreement with the $\alpha \,=-1.40\pm 0.11(\mathrm{stat}.{)}_{-0.09}^{+0.06}(\mathrm{sys}.)$ measured by CHIME/FRB when including bursts at all DMs/distances in the first FRB catalog (CHIME/FRB Collaboration 2021). At small d, where space is approximately Euclidean, we can assume d ∝ z and hence

$$\begin{eqnarray}&&N(\lt z)\propto {z}^{-2\alpha }.\end{eqnarray} \tag{ 13 }$$

Now compare the ratio of the volume in which we detect no FRBs (the gap, v₁) and the volume containing our FRB sample (v₂). We can estimate the redshifts at the DM excesses that define our volumes of interest (52 and 215 pc cm⁻³) as in Macquart et al. (2020), who assumed cosmological parameters as measured by Planck Collaboration et al. (2016b). The expected relation between DM_cosmic and redshift results in redshift estimates of 0.06 at 52 pc cm⁻³ and 0.23 at 215 pc cm⁻³. These redshift estimates have uncertainty due to scatter within the IGM on the order of the estimates (0.04 and 0.11 for the first and second volume boundary, respectively) according to the 90% confidence interval on the fit of the DM_cosmic–z. However, we simply select the central value of the redshift estimate to define our boundaries and discuss the effect of the scatter in the IGM on this calculation at the end of this section.

We expect the ratio of the number of sources detected in the first and second volumes to be

$$\begin{eqnarray}&&\frac{{N}_{\mathrm{FRBs},{v}_{1}}}{{N}_{\mathrm{FRBs},{v}_{{2}}}}=\frac{{N}_{\mathrm{FRBs}}\left(z\in [0,0.06)\right)}{{N}_{\mathrm{FRBs}}\left(z\in [0,0.23]-N\in [0,0.06)\right)}\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&=\,\displaystyle \frac{{z}_{1}^{-2\alpha }}{{z}_{2}^{-2\alpha }-{z}_{1}^{-2\alpha }}\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&=\,\displaystyle \frac{1}{{\left(\tfrac{{z}_{2}}{{z}_{1}}\right)}^{-2\alpha }-1},\end{eqnarray} \tag{ 16 }$$

where N_FRBs describes the number of detectable FRBs in a given volume or redshift range, and z₁ = 0.06 and z₂ = 0.23 are the redshifts that define the boundaries of the two volumes of interest (v₁ and v₂, respectively).

Finally, we must account for the sensitivity of CHIME/FRB to radio pulses in the DM range considered. We correct for this effect using information from CHIME/FRB's synthetic signal injection system. For injected FRB signals with excess DM ∈[0, 52) and ∈[52, 250], the fraction of detected events (μ) is ${\mu }_{{{v}}_{1}}=0.346$ and ${\mu }_{{{v}}_{2}}=0.366$, respectively. Hence, we adjust Equation (16) to account for this bias,

$$\begin{eqnarray}&&\displaystyle \frac{{N}_{\det ,{{v}}_{1}}}{{N}_{\det ,{{v}}_{2}}}=\displaystyle \frac{{\mu }_{{{v}}_{1}}{N}_{\mathrm{FRBs}\ {{v}}_{1}}}{{\mu }_{{{v}}_{2}}{N}_{\mathrm{FRBs}\ {{v}}_{2}}},\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&=\,\displaystyle \frac{{\mu }_{{{v}}_{1}}}{{\mu }_{{{v}}_{2}}}\displaystyle \frac{1}{{\left(\tfrac{{z}_{2}}{{z}_{1}}\right)}^{-2\alpha }-1},\end{eqnarray} \tag{ 18 }$$

where ${N}_{\det }$ is the number of FRBs we expect CHIME/FRB's real-time pipeline to detect. We only use the FRBs that were detected while the CHIME/FRB pipeline was in the configuration these injections are intended to gauge, leaving us with 83 of our sample 93 FRBs. The other 10 FRBs were detected after 2020 November, when a significant change was implemented in our real-time dedispersion algorithm.

Since we have measured a nonzero ${N}_{\det ,{{v}}_{2}}$, we estimate the expected ${N}_{\det ,{{v}}_{1}}$ in the same time frame. Treated as a rate, we can then use Poisson statistics to describe the likelihood of our observation of zero events in volume 1 under our initial assumptions,

$$\begin{eqnarray}&&{N}_{\det ,{{v}}_{1}}=\displaystyle \frac{{\mu }_{{{v}}_{1}}}{{\mu }_{{{v}}_{2}}}\displaystyle \frac{{N}_{\det ,{{v}}_{2}}}{{\left(\tfrac{{z}_{2}}{{z}_{1}}\right)}^{-2\alpha }-1}.\end{eqnarray} \tag{ 19 }$$

The probability of detecting no events given a rate of ${N}_{\det ,{{v}}_{1}}$, assuming that FRB source detection can be modeled as a Poisson process, is

$$\begin{eqnarray*}&&P(\mathrm{no}\ \mathrm{events}\,| \ \mathrm{rate}={N}_{\det ,{{v}}_{1}})={e}^{-{N}_{\det ,{{v}}_{1}}},\end{eqnarray*}$$

where P is the likelihood of the observation.

The probability of obtaining zero events in volume 1 under the null hypothesis of there being no astrophysical DM gap is therefore

$$\begin{eqnarray}&&P=\exp \left[-\displaystyle \frac{{\mu }_{{{v}}_{1}}}{{\mu }_{{{v}}_{2}}}\displaystyle \frac{{N}_{\det ,{{v}}_{2}}}{{\left(\tfrac{{z}_{2}}{{z}_{1}}\right)}^{-2\alpha }-1}\right]\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&=\,0.24.\end{eqnarray} \tag{ 21 }$$

Hence, under these assumptions, the resulting likelihood suggests that the lack of FRB detections in the first volume, considering the number of FRBs detected in the second volume, is consistent with being due to pipeline biases and volume effects alone.

If we look at the assumptions that we have made, we find that this likelihood is quite conservative (i.e., likely an overestimate; hence, we expect to see zero FRBs in this region due to volume and selection effects alone less frequently than 24% of the time if we could repeat the experiment again and again). The assumption that FRBs are standard candles predicts fewer low-fluence bursts compared to the true underlying luminosity distribution as seen in the first CHIME/FRB catalog (CHIME/FRB Collaboration 2021), as well as in observations of repeaters (e.g., Li et al. 2021; Lanman et al. 2022). As volume 1 is closer than volume 2, the omission of these low-fluence bursts would decrease the number of detectable bursts per unit volume more in volume 1 than in volume 2. That is, if there were a population of intermediate-fluence FRBs, there should be bursts that are detectable in volume 1 and not volume 2. Hence, we can safely assume that the fraction of detected bursts in volume 1 compared to volume 2, and the true rate, would be larger than what we estimate. This underestimation of the rate will overestimate the likelihood of our observation and hence is a conservative estimate.

In CHIME/FRB Collaboration (2021), the authors investigated the DM–distance relation by splitting the FRB sample into "low DM" (100–500 pc cm⁻³) and "high DM" (above 500 pc cm⁻³) and measuring the α for the subsets. One might believe, then, a more appropriate α to be used would be that measured by CHIME/FRB Collaboration (2021), who inferred $\alpha =-0.95\pm 0.15(\mathrm{stat}{)}_{-0.19}^{+0.06}(\mathrm{sys})$ for events with DMs of 100–500 pc cm⁻³. However, since we have assumed a standard candle luminosity function, it is not appropriate to use this α value.

Additionally, when we estimate the redshifts that correspond to DM_cosmic = 52, 215 pc cm⁻³, hence defining our volume boundaries, we do not consider the uncertainty presented by Macquart et al. (2020). These uncertainties represent the 90% confidence intervals on the fit of DM_cosmic versus redshift (DM−z), accounting for scatter within the IGM (cosmic structure). However, as above, this variance is more likely to result in additional FRBs being detected in the first volume, so our estimate remains conservative.