Constraining the Cosmic Baryon Distribution with Fast Radio Burst Foreground Mapping

The first mock observable we need to define is the DM that is measured as part of the initial FRB detection in the radio.

We will separately calculate DM_igm, DM_halos, and DM_host contributions that would be observed for a simulated FRB sight line within the simulation. First, we will define as the primary observable for our analysis the extragalactic DM component:

$\begin{eqnarray}&&{\mathrm{DM}}_{\mathrm{eg}}={\mathrm{DM}}_{\mathrm{FRB}}-{\mathrm{DM}}_{\mathrm{MW}},\end{eqnarray} \tag{ 2 }$

which represents all the DM contributions from beyond the Milky Way and its halo. DM_eg will therefore constitute one of the key observables in our subsequent forecast:

$\begin{eqnarray}&&{\mathrm{DM}}_{\mathrm{eg}}={\mathrm{DM}}_{\mathrm{igm}}+{\mathrm{DM}}_{\mathrm{halos}}+\displaystyle \frac{{\mathrm{DM}}_{\mathrm{host}}}{1+{z}_{\mathrm{frb}}}+{\delta }_{N},\end{eqnarray} \tag{ 3 }$

where a random deviate arising from noise and Milky Way subtraction, δ_N , has also been added.

To compute the DM_igm component arising from large-scale IGM, we use the gridded 256³ Millennium density field snapshots with a side length of L = 480.279 h⁻¹ Mpc, which we smoothed with a Gaussian kernel with size R_sm. First, we computed the path between the FRB coordinate and light-cone origin, ⁸ keeping track of the exact path length l_s traced by the sight line through each grid cell s and the redshift z_i at the simulation snapshot corresponding to the line-of-sight comoving distance from the origin for the traversed cell. We calculate a discretized version of ${\mathrm{DM}}_{\mathrm{igm}}=\int {n}_{e}^{\mathrm{igm}}/(1+z)ds$ as follows:

$\begin{eqnarray}&&{\mathrm{DM}}_{\mathrm{igm}}={\bar{n}}_{e}^{\mathrm{igm}}({z}_{i})\sum _{s}(1+{\delta }_{m,s}^{\mathrm{sm}}){l}_{s}{\left(1+{z}_{s}\right)}^{-1},\end{eqnarray} \tag{ 4 }$

where ${\delta }_{m,s}^{\mathrm{sm}}$ and l_s are the smoothed matter overdensity and path length through the grid cell s, respectively; z_s is the redshift of the grid cell calculated from the distance−redshift relation; ⁹ and ${\bar{n}}_{e}^{\mathrm{igm}}$ is the mean cosmic electron density at the redshift of the snapshot z_i , defined as

$\begin{eqnarray}&&{\bar{n}}_{e}^{\mathrm{igm}}({z}_{i})={f}_{\mathrm{igm}}{{\rm{\Omega }}}_{b}\bar{{\rho }_{c}}({z}_{i})\left[\frac{{m}_{\mathrm{He}}(1-Y)+2{{Ym}}_{{\rm{H}}}}{{m}_{{\rm{H}}}{m}_{\mathrm{He}}}\right],\end{eqnarray} \tag{ 5 }$

where m_H and m_He are the atomic masses of hydrogen and helium, respectively; Y = 0.243 is the cosmic mass fraction in helium (assumed to be double-ionized); Ω_b = 0.044 is the fraction of the cosmic critical density in baryons; ${\bar{\rho }}_{c}({z}_{i})$ is the critical density of the universe at the redshift z_i ; and f_igm is the fraction of cosmic baryons residing in the diffuse IGM. The summation carried out in Equation (4) traverses different density snapshots of the Millennium simulation according to the snapshot redshift z_s that is closest to the cell redshift z_i : for example, a sight line originating at z_frb = 0.5 would pass through 14 different redshift snapshots.

Next, we calculate the DM_halos contribution from intervening galaxy halos close to the FRB line of sight, assuming that the halo gas is described by a modified Navarro−Frank−White (mNFW) profile as elucidated in Prochaska & Zheng (2019). The baryon density as a function of radius is given by

$\begin{eqnarray}&&{\rho }_{b}=\frac{{\rho }_{b}^{0}}{{y}^{1-\alpha }{\left({y}_{0}+y\right)}^{2+\alpha }},\end{eqnarray} \tag{ 6 }$

where ${\rho }_{b}^{0}$ is the central density, y is a rescaled radius (see Mathews & Prochaska 2017 for the full definitions), and we fix the profile parameters α = 2 and y₀ = 2. This choice of parameters is designed to produce halo gas profiles similar to those derived by Maller & Bullock (2004) for multiphase CGMs. For a halo with mass M_halo, the total mass in baryons is

$\begin{eqnarray}&&{M}_{\mathrm{halo}}^{b}\equiv {f}_{\mathrm{hot}}({{\rm{\Omega }}}_{b}/{{\rm{\Omega }}}_{m}){M}_{\mathrm{halo}},\end{eqnarray} \tag{ 7 }$

where f_hot represents the fraction of the halo baryons that are in a hot gas phase tracing the mNFW profile out to 1 virial radius. We truncate the mNFW profile at a given radius ${r}_{\max }$, which is usually of comparable magnitude to the virial radius r₂₀₀, and so we will usually quote ${r}_{\max }$ in units of r₂₀₀. Simha et al. (2020) showed that varying the assumed ${r}_{\max }$ for the foreground galaxies of FRB 190608 from r₂₀₀ to 2 × r₂₀₀ more than doubled the DM_halos contribution, so this will be one of the free parameters that we will consider in our mock analysis.

To speed up the computation, we only calculate the DM_halos contribution for foreground galaxies within $10^{\prime}$ of each simulated FRB sight line. This corresponds approximately to the transverse separation at r₂₀₀ of a 10¹³ M_⊙ halo at z = 0.05—for lower-mass or higher-redshift halos, the transverse extent r₂₀₀ will be correspondingly smaller. Since we are defining our mock observables at this point, there is no possibility of halos outside of this separation biasing our forecast so long as we remain internally consistent. In a real observation, we would have to carefully consider the possible contributions of more massive (cluster-sized) halos that might contribute at larger transverse separations if they are low redshift. In this regime, DM_halos and DM_igm will begin to overlap, as group- and cluster-sized halos will also represent clear overdensities on ∼Mpc scales. When analyzing real observations, we will need to carefully keep track of these massive halos to avoid double-counting their DM contributions, but in our mock analysis the separation is clear. This is because both mock "observations" and fitting functions are derived from exactly the same model as described in this section.

In addition to computing the DM_halos for each sight line, we also output a catalog listing the redshift, separation, halo mass, and DM contribution of all galaxies at $\lt 10^{\prime}$ within each sight line. This will be used in the subsequent Fisher matrix parameter forecast.

Now that we have defined DM_igm and DM_halos, we will remark on the effect of the Gaussian smoothing length, R_sm, applied to the N-body density field before calculating DM_igm from Equation (4). In principle, at fixed cosmology and for a given cosmic baryon distribution, there is no effect from varying R_sm on 〈DM_igm + DM_halos〉(z) (i.e., the IGM and halo dispersions averaged over the universe at fixed FRB redshift). However, the scatter in DM_igm, which we refer to as σ_igm, will vary as a function of R_sm—with large R_sm, the variance in DM_igm between different sight lines will get washed out. As a comparison, we use the results of Jaroszynski (2019, hereafter J19), who studied the statistics of the DM_igm distribution in the Illustris cosmological hydrodynamical simulations (see also Takahashi et al. 2021; Zhu & Feng 2021; Batten et al. 2021b; Zhang et al. 2021). We calculated 〈DM_igm〉(z) at several redshifts using our model (Equation (4)), for a few different values of R_sm. In each case, we averaged over 750 sight lines drawn from the 24 separate H15 light cones and set f_igm = 0.85 to match the average z < 1 value found by J19. The 〈DM_igm〉(z) from our models are shown as cyan symbols in Figure 1(a), in comparison with the J19 DM_igm curve shown as a cyan line.

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Our model—derived from the density field of the non-hydrodynamical Millennium simulation—has a reasonable agreement with J19, and, as expected, varying R_sm does not significantly modify 〈DM_igm〉(z).

We also computed 〈DM_igm + DM_halos〉(z) (red symbols in Figure 1(a)) for a range of $[{r}_{\max },{R}_{\mathrm{sm}}]$, which does exhibit significant changes with respect to parameter choice. Since 〈DM_igm(z)〉 alone shows little variation respect to R_sm, this variation in 〈DM_igm + DM_halos〉(z) must come from the response of DM_halos with respect to ${r}_{\max }$. For some of the larger values of ${r}_{\max }$, the overall 〈DM_igm + DM_halos〉(z) becomes large enough that it exceeds the total baryon budget of the universe. However, in this paper we will treat ${r}_{\max }$ as a free parameter for our forecast; therefore, we will not seek to match ${r}_{\max }$ with J19 or any other hydrodynamical simulation. However, we note in passing that this comparison with J19 implies that the halo CGM in the Illustris simulation has characteristic scales comparable to the halo virial radius.

The scatter of the DM_igm distribution (Figure 1(b)), however, shows considerably more variation with respect to the choice of parameters in our model, especially with respect to R_sm. Given the cell size of 1.875 h⁻¹ Mpc in the Millennium binned density field, we find that Gaussian smoothing by R_sm = 0.7 h⁻¹ Mpc provides an adequate match for σ_igm in our model in comparison with J19. In Figure 2, we show the probability distribution of DM_igm only computed at z = 0.5 for both our model and that from the Illustris hydrodynamical simulation by J19. We have set R_sm = 0.7 h⁻¹ Mpc and f_igm = 0.85 to get the best match with J19. The two distributions are similar, with a long non-Gaussian tail toward high DM_igm, although the J19 distribution extends toward lower DM_igm compared with ours.

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Note that these σ_igm values, in both our model and J19, are higher at all redshifts than the σ_igm = 10 pc cm⁻³ adopted by Walters et al. (2019) and Ravi (2019). Considering the variance in both DM_igm and DM_halos, our model slightly overestimates the overall scatter of (DM_igm + DM_halos) relative to the hydrodynamical model. Since the parameters governing DM_halos will be adopted as free parameters in our framework, we do not consider it necessary for our model to perfectly reproduce the overall (DM_igm + DM_halos) scatter of J19, but we deem the statistical properties in our DM_igm model accurate enough for realistic parameter forecasts.

The final free parameter in our forecast framework is the FRB host contribution to the DM signal, DM_host, which in principle is the sum of dispersions caused by electrons surrounding the progenitor object and the host galaxy. This is likely the most uncertain component of our model given our lack of knowledge on FRB progenitors. In this paper, we assume that the DM_host of the FRBs is drawn from a Gaussian distribution with a mean of ${\overline{\mathrm{DM}}}_{\mathrm{host}}=200\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$ and rms scatter of σ_host = 50 pc cm⁻³ in the FRB rest frame. This is a distribution that is consistent with the emerging consensus for the general population of FRBs (Cordes et al. 2021; James et al. 2022), although for the purpose of our forecast it is not crucial that we come up with an accurate model for the DM_host distribution. When this paper was in an advanced stage of preparation, new results emerged of FRBs with extremely high DMs given their host redshift (Niu et al. 2021 and other as-yet-unpublished results), implying DM_host ∼ 500–1000 pc cm⁻³, which is much larger than the "normal" population of FRBs with DM_host ∼ 100–200 pc cm⁻³. However, these high-DM_host FRBs exhibit properties that would likely allow them to be excluded from the kind of cosmic baryon analysis that we develop in this paper—we will discuss this aspect more in Section 7.

Finally, we assume that the Milky Way contribution, DM_MW, can be modeled with reasonable accuracy and subtracted from the overall signal, leaving only the IGM, halo, and host contributions. We therefore do not explicitly model DM_MW but assume that its subtraction introduces an uncertainty of σ_MW = 15 pc cm⁻³ in the measurement of the extragalactic FRB DM, which we add as a random Gaussian deviate, δ_N , with standard deviation σ_MW to the DM_eg (Equation (3)). In reality, DM_MW varies across the sky, and σ_MW for each sight line would vary as a proportional uncertainty on DM_MW. Our framework can easily incorporate individual sight-line errors on σ_MW, but for convenience we simply pick the same value for σ_MW for all of the mock sight lines.

As seen in the previous section, the IGM component is typically the dominant component of the extragalactic FRB DM (see Equation (3) and Figure 1) in both the mean value and scatter. The scatter, σ_igm, is driven primarily by fluctuations in the underlying large-scale (≳Mpc) cosmic web traversed by the FRB sight lines. We therefore hypothesize that spectroscopic galaxy surveys in the FRB foreground will allow us to significantly reduce σ_igm and improve our ability to constrain the parameters governing the CGM and IGM. The H15 light-cone catalog includes synthetic source magnitudes calculated for a variety of photometric filters. Therefore, it is straightforward to select samples of galaxies that simulate various spectroscopic surveys and then study the foreground DM contribution in front of hypothetical FRBs.

We defined several mock spectroscopic catalogs up to z_frb ≤ 0.8, utilizing multiplexed wide-field spectrographs on a range of telescope sizes. These are described in detail in Appendix A and summarized in Table 1, while Figure 3 provides a visual impression of the various surveys' coverage for a single light cone. Depending on the z_frb, we will assume a combination of surveys to build up to the desired redshift: for example, for a simulated z_frb = 0.3 sight line we will assume the availability of the 2m-z01 and 4m-z03 data in the foreground. These mock surveys are coarse grained in terms of the redshift ranges they cover. Thus, for the mock analysis on a z_frb = 0.6 FRB there might be many superfluous background galaxies at z_gal > 0.6 because 8m-z10 observations will cover up to z = 0.8 (see below). In real observations, the target selection will be fine-tuned individually for the specific z_frb fields that are being targeted in order to maximize the telescope efficiency. Nevertheless, these mock catalogs have been invaluable in helping to design the observational strategy for the FLIMFLAM Survey, which we briefly describe in Section 6.

The area densities and comoving number densities of the different surveys are shown in Figure 4, in which we have averaged all 24 of the available H15 light cones from the Millennium database in order to obtain a smooth redshift distribution—the distributions in the individual 3.1 deg² light cones are much more uneven owing to variations in large-scale structure. While it may appear from Figure 4 as if the 2m-z01 catalog contributes only negligibly to the foreground sampling, the wide-field coverage over hundreds of square degrees is crucial for accurately recovering transverse structures at very low redshifts. Similar data sets to 2m-z01 already exist for much of the extragalactic sky thanks to all-sky spectroscopic surveys such as SDSS (Abazajian et al. 2009) and 6dF (Jones et al. 2009).

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For the density reconstructions in the rest of this paper, we used mock catalogs from only 6 of the 24 light cones released by H15, which were the subsets in which the all-sky catalogs were available—the other 18 light cones had no data beyond the central 3.1 deg² footprints.

In addition to the wide-field survey redshift data defined above, for each mock FRB sight line we also output an "observational" list of all galaxies within $\lt 10^{\prime}$ and M_halo ≥ 3 × 10¹¹ M_⊙, listing their redshift, angular separation from the sight line, and estimated halo mass. To mimic a realistic observational scenario, we include all galaxies with the aforementioned criteria whether or not they actually contribute to the sight-line DM_halos given our fiducial parameters. Also, since there are significant uncertainties in estimating galaxy halo masses, we introduce random errors of $\sigma ({\mathrm{log}}_{10}{M}_{\mathrm{halo}}/{M}_{\odot })=0.3$ to the listed halo masses. Separately from the "observational" intervening galaxy list, we also kept track of the true halo masses and DM_halos contributions for each sight line to calculate the "true" values.

We now comment on the observations required to account for these intervening galaxies, which would be expected to be fainter than the large-scale tracer galaxies described in the previous section and thus need to be observed with larger telescopes. We analyze an ensemble of 120 random sight lines with z_frb ≤ 0.5 in Figure 5 to characterize the typical separations and galaxy magnitudes at which intervening halos are likely to contribute to the sight-line DM budget. In the top panel of Figure 5, we see that 64% of the mock sight lines have DM_halos contributions from angular separations of $\theta \lt 0\buildrel{\,\prime}\over{.} 5$, of which on average ∼40% of the total DM_halos contribution is from these small separations. But these closely intervening galaxies can reach down to relatively faint magnitudes of r > 24, with ∼10% of the DM_halos contribution missing from these sight lines if r > 24 galaxies are unaccounted for.

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At slightly larger separations of $0\buildrel{\,\prime}\over{.} 5\lt \theta \lt 5^{\prime}$, we see in the bottom panel of Figure 5 that nearly all (99%) sight lines have contributions from larger angular distances from FRB sight lines. At these larger separations, however, the contributions come from galaxies with typical magnitudes of r < 23. Indeed, an observational limit of r < 22.5 would only miss out on a few percent of the overall DM_halos contribution on these larger separations. We find that, on average, there are ∼4 galaxies (up to a maximum of ∼12) with r < 22.5 that would contribute to each sight line. However, within a radius of $\theta \lt 5^{\prime}$ there are likely to be ∼100–200 galaxies at this magnitude limit. Photometric redshifts would therefore be invaluable to preselect ∼20–30 candidate foreground galaxies that could then be observed with a typical imaging spectrograph on an 8–10 m class telescope, to confirm their redshifts.

These results are consistent with the estimates of Ravi (2019), who found that observing galaxies down to r < 24 would detect >90% of the DM_halos contribution, but our findings motivate a tiered observing strategy to target intervening galaxies. First, one desires integral field unit (IFU) spectrograph observations with an 8–10 m class telescope (e.g., VLT/MUSE or Keck II/KCWI) within $\theta \lesssim 0\buildrel{\,\prime}\over{.} 5$ with several-hour integration to reach depths of r > 24. For larger radii ($0.5\lesssim \theta \lesssim 5^{\prime}$), multiple short pointings ( <1 hr) with spectrographs on 8–10 m class telescopes such as Keck I/LRIS, VLT/FORS2, or Gemini/GMOS would confirm any intervening galaxies. Note that it is possible for halos to contribute at larger separations of $\theta \gt 5^{\prime}$, but these tend to be galaxy-group-sized halos or larger (M ≳ 10¹³ M_⊙), for which the central galaxies are typically massive and bright enough to be identified through the wide-area surveys described in Section 3.

The goal of Bayesian inference using Markov Chain Monte Carlo (MCMC) is to draw samples of a posterior probability density ${ \mathcal P }({\boldsymbol{\Theta }}| {\boldsymbol{d}})$, representing a set of parameters Θ given a data vector d . The posterior consists of a prior π and a likelihood function ${ \mathcal L }$,

$\begin{eqnarray}&&{ \mathcal P }({\boldsymbol{\Theta }}| {\boldsymbol{d}})\propto \pi ({\boldsymbol{\Theta }})\times { \mathcal L }({\boldsymbol{d}}| {\boldsymbol{\Theta }}).\end{eqnarray} \tag{ 8 }$

In our framework the variable of interest is the linearized matter density Θ = δ _L , which we infer on a rectangular grid with N_C cells. The transformation between the physical density contrast ${\boldsymbol{\delta }}={\boldsymbol{\varrho }}/\bar{{\boldsymbol{\varrho }}}-1$ (matter density ϱ) and the linearized density δ _L (Coles & Jones 1991) is

$\begin{eqnarray}&&{{\boldsymbol{\delta }}}_{L}=\mathrm{log}(1+{\boldsymbol{\delta }})-{\boldsymbol{\mu }},\end{eqnarray} \tag{ 9 }$

where the mean field μ (Kitaura & Angulo 2012) ensures 〈 δ _L 〉 = 0. As prior distribution for δ _L we utilize a multivariate Gaussian

$\begin{eqnarray}&&\pi ({{\boldsymbol{\delta }}}_{L})=\displaystyle \frac{1}{\sqrt{{\left(2\pi \right)}^{{N}_{c}}\det ({{\bf{C}}}_{L})}}\exp \left(-\displaystyle \frac{1}{2}{{\boldsymbol{\delta }}}_{L}^{{\rm{T}}}{{\bf{C}}}_{L}^{-1}{{\boldsymbol{\delta }}}_{L}\right),\end{eqnarray} \tag{ 10 }$

where ${{\bf{C}}}_{L}=\langle {{\boldsymbol{\delta }}}_{L}^{{\rm{T}}}{{\boldsymbol{\delta }}}_{L}\rangle$ is the covariance matrix of the linearized density.

The information content of the data is encoded in the likelihood function that depends on the expected number of galaxies λ given the underlying matter field δ and the observed number of galaxies N _G. We model the expectation number per cell i as

$\begin{eqnarray}&&{\lambda }_{i}={f}_{N}{w}_{i}{\left(1+{G}_{i}{\delta }_{i}\right)}^{{b}_{i}},\end{eqnarray} \tag{ 11 }$

with the normalization f_N ensuring correct number density $\bar{N}$, w_i the completeness at cell i, and b_i the galaxy bias parameter, accounting for the different clustering of galaxies as compared to the total matter. G_i is the growth factor ratio between a reference redshift z_ref and an arbitrary redshift z_i ,

$\begin{eqnarray}&&G({z}_{\mathrm{ref}},{z}_{i})=D({z}_{i})/D({z}_{\mathrm{ref}}),\end{eqnarray} \tag{ 12 }$

where

$\begin{eqnarray}&&D({z}_{i})=\displaystyle \frac{H({z}_{i})}{{H}_{0}}{\int }_{{z}_{i}}^{\infty }{\rm{d}}z\displaystyle \frac{1+z}{{H}^{3}(z)}/{\int }_{0}^{\infty }{\rm{d}}z\displaystyle \frac{1+z}{{H}^{3}(z)}.\end{eqnarray} \tag{ 13 }$

The factor G_i is necessary to account for the redshift-dependent clustering evolution if the density field is reconstructed from galaxy positions on a light cone (Ata et al. 2017) when compared to a reference redshift. We will explain this important point in the next section.

We can now write the Poisson likelihood expression for a single survey as

$\begin{eqnarray}&&{ \mathcal L }({{\boldsymbol{N}}}_{{\rm{G}}}| {\boldsymbol{\lambda }})=\prod _{i}\displaystyle \frac{{\lambda }_{i}^{{N}_{{\rm{G}}i}}{{\rm{e}}}^{-{\lambda }_{i}}}{{N}_{{\rm{G}}i}!}.\end{eqnarray} \tag{ 14 }$

As stated above, in this paper we seek a reconstruction given multiple surveys. Thus, we formulate a multisurvey likelihood with k surveys as

$\begin{eqnarray}&&{{ \mathcal L }}^{\mathrm{multi}}=\prod _{k}{{ \mathcal L }}^{(k)}({{\boldsymbol{N}}}_{{\rm{G}}}^{(k)}| {{\boldsymbol{\lambda }}}^{(k)}).\end{eqnarray} \tag{ 15 }$

This likelihood is then sampled using a Hybrid Monte Carlo technique (HMC; Duane et al. 1987; Neal 2011), in which artificial momentum variables are introduced to avoid inefficient random walks in our high-dimensional space of matter density grid cells. Our method is based on that first introduced in Ata et al. (2015), but with a generalization for the multisurvey likelihood. Due to its technical nature, we will describe it in Appendix B.

Another important aspect of the density reconstructions is the iterative correction of so-called redshift space distortions (Kaiser 1987; Hamilton 1997), where galaxy positions are displaced along line of sight owing to their peculiar motions.

Let us denote the apparent line-of-sight position of a galaxy with s , which is the real-space position r biased by the radial component of the galaxies' peculiar motion, ¹⁰

$\begin{eqnarray}&&{\boldsymbol{s}}={\boldsymbol{r}}+\frac{{{\boldsymbol{v}}}_{r}}{H(a)a},\end{eqnarray} \tag{ 16 }$

with

$\begin{eqnarray}&&{{\boldsymbol{v}}}_{r}=({\boldsymbol{v}}\cdot {\boldsymbol{r}}){{\boldsymbol{e}}}_{r},\end{eqnarray} \tag{ 17 }$

where e _r is the radial unit vector.

Solving Equation (16) for r demands the knowledge of v , which we can compute from the density contrast using linear perturbation theory as

$\begin{eqnarray}&&{\boldsymbol{v}}=-{fH}(a)a{\boldsymbol{\nabla }}{{\rm{\nabla }}}^{-2}\delta ,\end{eqnarray} \tag{ 18 }$

with $f=d(\mathrm{log}D)/d(\mathrm{log}a)$ being the logarithmic derivative of the growth factor, called the growth rate. We solve Equation (16) iteratively in a Gibbs sampling scheme for the real-space positions given the inferred density contrast of the previous iteration (see Kitaura et al. 2016).

The mock galaxy positions are given on the sky with R.A. α, decl. δ, and observed redshift z^obs. We map them to comoving Cartesian coordinates as

$\begin{eqnarray}\begin{array}{rcl}x & = & {d}_{\mathrm{com}}\cos \alpha \cos \delta ,\\ y & = & {d}_{\mathrm{com}}\sin \alpha \cos \delta ,\\ z & = & {d}_{\mathrm{com}}\sin \delta ,\end{array}\end{eqnarray} \tag{ 19 }$

where the comoving distance d_com is given by

$\begin{eqnarray}&&{d}_{\mathrm{com}}=\frac{c}{{H}_{0}}{\int }_{0}^{{z}^{\mathrm{obs}}}\frac{dz}{\sqrt{{{\rm{\Omega }}}_{{\rm{M}}}{\left(1+z\right)}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}}}.\end{eqnarray} \tag{ 20 }$

We organize the reconstructions with two consecutive rectangular volumes (dubbed as Volume 1 and Volume 2), each with N_C = 600 × 100 × 100 cells. With a cell resolution of d_Cell = 1.875 h⁻¹ Mpc (to match the H15 density field; see Section 2), this leads to a line-of-sight comoving length of d_LOS = 1125 h⁻¹ Mpc, while in transverse dimension it is d_Trans = 187.5 h⁻¹ Mpc. In preliminary tests, we found an unsatisfactory performance for the reconstructions at nearby comoving distances of d_com ≲ 100 h⁻¹ Mpc, which we suspect is caused by the 700 deg² footprints of the 2m-z01 samples being insufficiently wide. In a real reconstruction, we expect to be able to use all-sky reconstructions of the local universe to cover d_com ≲ 100 h⁻¹ Mpc (e.g., Erdoǧdu et al 2006), so we begin our reconstructions at d_com = 100 h⁻¹ Mpc. Volume 1 thus spans a line-of-sight range of L_LOS = 100–1225 h⁻¹ Mpc, while Volume 2 covers L_LOS = 1125–2250 h⁻¹ Mpc. There is an overlap of 100 h⁻¹ Mpc between the two volumes for consistency. The coverage of the two reconstruction volumes on one of our mock light cones is illustrated in Figure 3. We reconstruct the individual volumes at their mean redshifts (see Equation (11)), which are ${\bar{z}}_{1}=0.25$ for Volume 1 and ${\bar{z}}_{2}=0.67$ for Volume 2.

For Volume 1 we combine the data of the three surveys 2m-z01, 4m-z03, and 8m-z10, whereas for Volume 2 we only use 8m-z10 data, as shown in Figure 3.

We model the angular and radial selection function of each mock survey, which we then combine with the survey completeness (see Equation (11)). In this paper we consider the angular footprint to not have any selection effect, i.e., we assume the angular footprints to be entirely uniform across their entire area as summarized in Table 1. The radial selection function, on the other hand, is more challenging to model. A natural way to model the expected galaxy numbers in radial direction can be done using the Schechter luminosity function (e.g., Erdoǧdu et al 2004). This is accurate if averaged over the whole sky. In our case, dealing with footprints of ${ \mathcal O }(1\,{\deg }^{2})$, we are prone to fluctuations in the large-scale structure. Therefore, we average the radial galaxy distribution from the six mock light cones extracted from H15 and calculate the radial selection function from it; see Figure 6.

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We seek a single bias fitting function to obtain unbiased density fields for all light cones extracted from H15. For an initial estimate we use linear bias measurements from completed galaxy surveys, in particular the bias values measured from SDSS (Tegmark et al. 2004) for the low-redshift contribution and VIPERS measurements (Di Porto et al. 2016) for the high-redshift contribution.

We find that a moderate increase of the galaxy bias with redshift z fits the data sufficiently well, yielding nearly unbiased density fields with respect to the H15 simulations:

$\begin{eqnarray}&&b(z)=b(z=0.33)+0.22z,\end{eqnarray} \tag{ 21 }$

where b(z = 0.33) = 1.24.

However, when we compute the integrated line-of-sight density, we still find a small bias in the resulting values in comparison with the "truth." Due to the time-consuming computational requirements to carry out the reconstructions, it is prohibitive to iterate over different bias models to find a more accurate bias model. Instead, we will correct for the residual bias as a post-processing step, which we describe in the following subsection.

Examples of the resulting reconstructed real-space matter density fields are shown in Figure 7. The Volume 1 and Volume 2 maps are shown in the top and bottom panels, respectively. On top of the density slices we plot the inferred real-space positions of the galaxies. For each reconstruction, we run the HMC sampler for up to 10,000 iterations. However, the posterior samples are found to have correlation lengths of ∼100–150; thus, we extract 50 iterations of each density field for analysis with the selected samples separated by at least 150 iterations on the chain. This allows us to estimate not only the integrated line-of-sight DM but also its uncertainty.

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ARGO reconstructs the real-space matter density field based on input catalogs of galaxy positions and redshifts, but for a given localized FRB with known redshift, we do not have a priori information regarding its real-space position along the line of sight. We therefore have to convert the FRB redshift into a line-of-sight comoving distance using the Hubble law in order to determine its coordinate within the reconstruction box. In other words, we are forced to ignore the line-of-sight peculiar velocity of the FRB host galaxy when computing its integrated line-of-sight DM. This, however, is typically of the order of several hundred kilometers per second, corresponding to an error of up to several comoving Mpc along the line of sight. For FRBs at cosmological distances, the error from ignoring the peculiar velocity contribution is negligible in comparison with the overall line-of-sight paths of hundreds or thousands of megaparsecs.

We first smooth the reconstructed density field by R_sm =0.7 h⁻¹ Mpc to match the smoothing scale we adopt for the Millennium density field. Next, we calculate DM_argo, a version of the discretized line-of-sight integral in Equation (4) in which the smoothed matter density ${\delta }_{m}^{\mathrm{sm}}$ is evaluated on the reconstructed matter density grid from ARGO rather than the "true" Millennium density field as in Equation (4). For the purposes of the analysis in this section only, we will adopt f_igm = 1 for simplicity, although it will be considered a free model parameter in our forecasts of Section 5. As noted earlier, our reconstructions skip the first 100 h⁻¹ Mpc of the line of sight from the observer toward the FRB, so we simply add in a fixed contribution of DM_igm(d_com < 100 h⁻¹ Mpc∣ f_igm = 1) = 33 pc cm⁻³ to all the derived DM_igm values.

Since we know the "true" DM_igm values for each mock FRB sight line from the H15 Millennium simulations, we can directly assess the accuracy and precision of the DM_argo relative to the true DM_igm. These are shown in Figure 8, where the DMs are evaluated for an ensemble of random "FRB" sight lines at 0.1 < z < 0.8 with random angular positions within 12 ' of the light-cone field centers. We see that the reconstructed DM_igm deviate slightly from an exact one-to-one relationship with the true DM_igm, while also having some scatter. We therefore fitted a quadratic offset as a function of the median reconstructed DM_igm of each sight line to correct this small bias—this was done separately for sight lines contained within Volume 1, and for the combined DM_igm of Volumes 1 and 2 in the case of sight lines that originate in Volume 2. The orange points in Figure 8 show the reconstructed DM_igm after implementing this correction. In real observations, similar corrections can also be carried out if realistic simulations and mock catalogs that reflect the observational data are available.

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Since it is the goal of this paper to carry out Fisher parameter forecasts, it is necessary to check whether the distribution of the reconstructed DM_igm from the ARGO realizations reflects the uncertainty of the line-of-sight integrated density. To do this, we compute the "pull" distribution, (DM_igm,argo − DM_igm,true)/σ_argo. If the scatter of the ARGO realizations accurately reflects the true error in our estimate of DM_igm, then the pull should be distributed as a Gaussian with a standard deviation of 1. We find the pull distributions of DM_igm,argo to be approximately Gaussian, but with standard deviations of 1.63 and 1.55 for Volume 1 and Volume 1 + 2, respectively. There is a small skew to the pull distributions, with slightly more outliers at negative values where DM_igm,argo is underestimated relative to DM_igm,true. This suggests that the ARGO reconstructions do not currently recover the full amplitude in some of the density peaks. In an actual likelihood analysis on real data, this would need to be corrected to avoid biasing the results. For the purpose of our subsequent Fisher forecast, however, we will treat the pull distribution as Gaussian but broaden the DM_igm,argo distributions of each sight line around its median by the aforementioned factors of 1.63 and 1.35. This rescales the scatter in each sight line so as to be more accurate representations of the uncertainty in the DM_igm measurement from the foreground galaxy redshift measurement.

We illustrate the efficacy of the ARGO reconstructions in Figure 9, which shows the global 〈DM_igm〉 and its central 95% distribution (i.e., the range between the 2.5th and 97.5th percentiles of the distribution), as a function of FRB redshift. We also show as crosses the DM_igm corresponding to some randomly selected sight lines at 0.1 < z < 0.8 selected from the light cones. The error bars show the corresponding constraints on those sight lines obtained by applying ARGO on the foreground spectroscopic surveys described in Section 3.2. The ARGO reconstructions are effective in bracketing the true DM_igm to within an uncertainty that is ∼2–3× smaller than that from cosmic variance. The efficacy does appear to degrade somewhat toward the higher-redshift end (z ≳ 0.7), with broader errors and catastrophic failures where the reconstructed range falls outside the true value. There is ongoing work to improve the performance of ARGO, but in the meantime we will proceed with assuming this performance for our Fisher forecasts.

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In all sight lines, the ARGO reconstructions allow us to constrain DM_igm, assuming a fixed f_igm = 1, with better precision than if the global DM_igm distribution were to be assumed. Conversely, if f_igm was an unknown parameter, the density fluctuations estimated by ARGO would then be used to fit for the most likely f_igm that leads to the observed DM_igm. This is the scenario that will be adopted in the Fisher forecasts of the following section.

We will make specific forecasts for two hypothetical FRB samples: (i) N_frb = 30 spanning 0.1 < z < 0.5, and (ii) N_frb = 96 over 0.1 < z < 0.8. The N_frb = 30 sample approximately reflects the goals for the ongoing FLIMFLAM survey as described in Section 6, while foreground data sets covering N_frb = 96 should be achievable with the advent of 8 m class massively multiplexed spectrographs.

Early samples of localized FRBs show a roughly uniform redshift distribution (e.g., Heintz et al. 2020). However, the number of known localized FRBs is increasing rapidly and will soon outstrip the number that can be targeted for foreground mapping. At that point, the redshift distribution of FRBs to be targeted for foreground mapping will become a deliberate choice. We therefore choose to adopt a uniform redshift distribution for our mock samples, which is a trade-off between more expensive observational requirements of higher-redshift FRBs and better constraints on ${\overline{\mathrm{DM}}}_{\mathrm{host}}$ offered by a wider redshift range. In the longer term (≳2–3 yr), as samples of localized FRBs increase into the hundreds or thousands, the true underlying FRB redshift distribution might turn out not to be uniform with redshift.

From the resulting Fisher matrices, we compute the 68% and 95% error ellipses using standard techniques (usefully summarized in Coe 2009). Figure 10(a) shows the forecasted confidence intervals for the N_frb = 30 sample. With foreground spectroscopy from both wide-field surveys (described in Section 3.2) and deeper but narrow-field observations of intervening galaxies, we find that the 68.3% confidence constraints are f_igm = 0.80 ± 0.11, ${r}_{\max }/{r}_{200}=1.40\pm 0.35$, f_hot = 0.75 ± 0.15, and ${\overline{\mathrm{DM}}}_{\mathrm{host}}=100.0\pm 29.21\ \mathrm{pc}\,{\mathrm{cm}}^{-3}$.

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We also experimented with drawing several different random mock samples with the same number of sight lines spanning the same redshift range, and we find that there is some heterogeneity in the forecast between the different mock samples with N_frb = 30. This is caused by the variance in the number of intervening halos in the mock sight lines: "clean" sight lines with few intervening halos tend to be more useful for constraining f_igm and DM_host, whereas sight lines with more intervening galaxies are better for constraining ${r}_{\max }$ and f_hot. This is reflected in the Fisher terms contributed by the individual sight lines (compare Equation (22) and Appendix C), where we find that sight lines with more intervening halos have larger values of $\partial {\mathrm{DM}}_{\mathrm{halos}}/\partial {r}_{\max }$ and ∂DM_halos/∂f_hot than those without. We also note, from experimenting with the redshift range of the sight-line distribution, that a broader redshift distribution helps break the degeneracy between f_igm and DM_host thanks to the (1 + z_host) denominator associated with DM_host (Equation (3)). As expected, lower-redshift FRBs have greater sensitivity toward DM_host. These trends should help guide the choice of FRB sight lines to follow up with foreground spectroscopy once upcoming surveys start delivering large numbers of localized FRBs.

With N_frb = 96 sight lines over 0.1 < z < 0.8 (Figure 10(b)), we forecast 68% constraints of f_igm = 0.80 ± 0.06, ${r}_{\max }/{r}_{200}=1.40\pm 0.15$, f_hot = 0.75 ± 0.09, and ${\overline{\mathrm{DM}}}_{\mathrm{host}}=100.0\pm 18.36\ \mathrm{pc}\,{\mathrm{cm}}^{-3}$, i.e., all the IGM and CGM parameters are constrained to within ∼10%. This shows the high precision that should, in principle, be achievable once the next generation of massive-multiplexed spectroscopic facilities become available.

We can also make Fisher forecasts of constraints on the same set of parameters assuming only the redshift and DM_eg from the FRBs, i.e., without any knowledge of the foreground structures or galaxies. In practical terms, this is equivalent to trying to fit a theoretical Macquart relation to the extragalactic DMs and host galaxy redshifts from an observed sample of localized FRBs. The parametric model we need to differentiate for the Fisher matrix then becomes

$\begin{eqnarray}\begin{array}{rcl}{f}_{m}\left(p\right)\equiv {\mathrm{DM}}_{\mathrm{eg},m}^{\mathrm{model}} & = & \langle {\mathrm{DM}}_{\mathrm{igm}}({f}_{\mathrm{igm}})\rangle ({z}_{m})\\ & & +\,\langle {\mathrm{DM}}_{\mathrm{halos}}({r}_{\max },{f}_{\mathrm{hot}})\rangle ({z}_{m})\\ & & +\,{\overline{\mathrm{DM}}}_{\mathrm{host}}{\left(1+{z}_{m}\right)}^{-1}.\end{array}\end{eqnarray} \tag{ 25 }$

Instead of the per-sight-line model calculations that are feasible with foreground spectroscopic data, we are now forced to use the global 〈DM_igm〉(z) and 〈DM_halos〉(z) (i.e., Figure 1(a)), evaluated at the redshift of each FRB (see Appendix C for the partial derivatives). Similarly, for the σ_m calculation (Equation (24)) we insert the global scatter for σ_igm and σ_halos (see Figure 1(b)) instead of bespoke calculations based on the foreground data from each sight line. The forecasted parameters for the same mock sample of N_frb = 96—but without any foreground information—is shown in Figure 11. The constraining power clearly is much weakened in the absence of foreground data, with confidence intervals ∼2–5× broader in all the parameters. In addition, the degeneracy between ${r}_{\max }$ and DM_host appears to have increased when foreground data are unavailable.

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How much information do the foreground data bring to cosmic baryon census, in addition to that offered by the localized FRBs and their measured DM_eg values alone? We can attempt to quantify this by rescaling the elements of the no-foreground Fisher matrix (which yields the forecast in Figure 11) until the resulting constraints are roughly the same as if foreground data are available (Figure 10(b)) for the same sight lines. This is equivalent to increasing the number of sight lines available for analysis by that factor. We find that to approximately match the constraints of N_frb = 96 in Figure 10(b), we had to rescale the no-foreground Fisher matrix elements by ∼25×. In other words, the foreground data improve the Fisher information content such that every localized sight line that has a foreground map is equivalent to ∼25 localized FRBs where only their redshift and total DM is known. Without foreground data, samples of N_frb ≳ 2000 localized FRBs with known redshifts would be required to achieve similar constraints to N_frb ∼ 100 localized FRBs with foreground data. This is qualitatively consistent with the recent analysis of Batten et al. (2021a), who found that ∼10³ localized FRBs are needed to distinguish between their no-feedback simulations and those with feedback (i.e., in effect, detect the existence of the CGM), whereas ∼10⁴ localized FRBs would be needed to distinguish between galaxy feedback models that lead to different CGM properties.

Our forecasts are, on the surface, more pessimistic than previous efforts in the literature. Walters et al. (2019), for example, carried out an MCMC likelihood forecast and argued that N_frb = 100 localized FRB sight lines without any foreground data should be able to constrain total diffuse baryon fraction (i.e., combined IGM and CGM components) f_d to within a few percent at 95% confidence level. Meanwhile, Ravi (2019) assumed that the spectroscopic observations would be available for intervening galaxies within several arcminutes of each FRB sight line and argued that f_igm could be constrained to within ∼2%–5% at 95% confidence with N_frb = 100. Both of these previous forecasts were built on analytic models. In particular, they both assumed that the uncertainty in the DM contribution of the cosmic web or diffuse IGM is given by a standard deviation of σ_igm = 10 pc cm⁻³. This value, which was taken from analytic estimates by Shull & Danforth (2018), is a significant underestimate of the σ_igm compared with simulation results by a factor of >5 (see Figure 1; see also Batten et al. 2021b). Such a small σ_igm is indeed even tighter than our ARGO constraints on individual line-of-sight DMs as seen in Figure 9. Apart from this discrepancy, it is hard to make direct comparison with the earlier works since we have also adopted simultaneous constraints on galaxy halo parameters and the host DM contribution. Both Walters et al. (2019) and Ravi (2019) assumed a fixed DM_host with some uncertainty, while Ravi (2019) assumed fixed halo parameters for the intervening foreground galaxies. We therefore argue that our forecasts are a more realistic determination of the precision that can be achieved with upcoming FRB samples.

First, we calculate the partial derivatives in our default case, where we have information on the foreground matter distribution and intervening galaxies. In this case, the observable ${f}_{m}\left(p\right)$ for each individual FRB in the sample is given by the model of the total extragalactic DM component

$\begin{eqnarray}\begin{array}{rcl}{f}_{m}\left(p\right) & = & {\mathrm{DM}}_{\mathrm{igm}}\left({f}_{\mathrm{igm}}\right)+{\mathrm{DM}}_{\mathrm{halo}}\left({r}_{\max },{f}_{\mathrm{hot}}\right)\\ & & +\,{\mathrm{DM}}_{\mathrm{host}}\times {\left(1+z\right)}^{-1}.\end{array}\end{eqnarray} \tag{ C1 }$

For the first parameter f_igm, we find

$\begin{eqnarray}&&\frac{\partial {f}_{m}}{\partial {p}_{1}}=\frac{\partial \,{\mathrm{DM}}_{\mathrm{igm}}\left({f}_{\mathrm{igm}}\right)}{\partial {f}_{\mathrm{igm}}}=\frac{ \langle {\mathrm{DM}}_{\mathrm{argo}} \rangle }{{f}_{\mathrm{igm}}},\end{eqnarray} \tag{ C2 }$

where 〈DM_argo〉 is the mean of DM_igm values calculated from the N = 50 ARGO reconstructed density realizations. Since 〈DM_argo〉 is linear in f_igm, taking its derivative is equivalent to setting f_igm = 1.

Next, the partial derivative for DM_host is given by

$\begin{eqnarray}&&\displaystyle \frac{\partial {f}_{m}}{\partial {p}_{2}}=\displaystyle \frac{\partial \left({\mathrm{DM}}_{\mathrm{host}}\times {\left(1+{z}_{m}\right)}^{-1}\right)}{\partial \,{\mathrm{DM}}_{\mathrm{host}}}={\left(1+{z}_{m}\right)}^{-1},\end{eqnarray} \tag{ C3 }$

where z_m is the redshift of the FRB.

In order to find the partial derivative for r_max,

$\begin{eqnarray}&&\displaystyle \frac{\partial {f}_{m}}{\partial {p}_{3}}=\displaystyle \frac{\partial {\mathrm{DM}}_{\mathrm{halos}}\left({r}_{\max },{f}_{\mathrm{hot}}\right)}{\partial {r}_{\max }},\end{eqnarray} \tag{ C4 }$

where we numerically differentiate the modified NFW profiles (see Section 3.1) of each mock foreground galaxy in the line of sight of the FRB in question with respect to r_max.

Lastly, for f_hot the final partial derivative is given by

$\begin{eqnarray}&&\displaystyle \frac{\partial {f}_{m}}{\partial {p}_{4}}=\displaystyle \frac{\partial {\mathrm{DM}}_{\mathrm{halos}}\left({r}_{\max },{f}_{\mathrm{hot}}\right)}{\partial {f}_{\mathrm{hot}}}=\displaystyle \frac{{\mathrm{DM}}_{\mathrm{host}}({r}_{\max })}{{f}_{\mathrm{hot}}},\end{eqnarray} \tag{ C5 }$

computed at the fiducial value of r_max = 1.40 for all the halos within the sight line. Similar to the relation between f_igm and DM_igm, DM_host is linear with respect to f_hot, and therefore this is equivalent to setting f_hot = 1.

When no information on the foreground IGM structure and distribution of galaxies along the sight line is available, we have to rely on global estimates 〈DM_igm〉(z) and 〈DM_halos〉(z), evaluated at the redshift of each FRB (see Section 5 for details). In this case, the model ${f}_{m}\left(p\right)$ is given by

$\begin{eqnarray*}\begin{array}{rcl}{f}_{m}\left(p\right) & = & \langle {\mathrm{DM}}_{\mathrm{igm}}({f}_{\mathrm{igm}})\rangle ({z}_{m})+\langle {\mathrm{DM}}_{\mathrm{halos}}({r}_{\max },{f}_{\mathrm{hot}})\rangle ({z}_{m})\\ & & +\,{\mathrm{DM}}_{\mathrm{host}}{\left(1+{z}_{m}\right)}^{-1}.\end{array}\end{eqnarray*}$

Consequently, we find that the partial derivatives of f_m with respect to each model parameter are as follows.

For f_igm, the partial derivative becomes

$\begin{eqnarray}&&\displaystyle \frac{\partial {f}_{m}}{\partial {p}_{1}}=\displaystyle \frac{\partial \langle {\mathrm{DM}}_{\mathrm{igm}}\left({f}_{\mathrm{igm}}\right)\rangle ({z}_{m})}{\partial {f}_{\mathrm{igm}}}=\displaystyle \frac{\langle {\mathrm{DM}}_{\mathrm{igm}}\rangle ({z}_{m})}{{f}_{\mathrm{igm}}},\end{eqnarray} \tag{ C6 }$

which again is equivalent to setting the global IGM contribution to a value of f_igm = 1 (see Figure 1) since 〈DM_igm〉 is linear with respect to f_igm.

The partial derivative over DM_host does not change with respect to our default case (Equation (C3)), i.e., $\partial {f}_{m}/\partial {p}_{2}={\left(1+{z}_{m}\right)}^{-1}$.

Finally, in order to calculate the partial derivatives over parameters r_max and f_hot, we estimate the global contribution of the foreground galaxies along the FRB sight line as a function of FRB redshift, $\langle {\mathrm{DM}}_{\mathrm{host}}\left({r}_{\max },{f}_{\mathrm{hot}}\right)\rangle ({z}_{m})$ (see discussion of global DM contributions in Section 3.1).

First, we estimate $\langle {\mathrm{DM}}_{\mathrm{host}}\left({r}_{\max },{f}_{\mathrm{hot}}\right)\rangle ({z}_{m})$ at three values of ${r}_{\max }=\{1.38,1.40,1.42\}$ and compute the partial derivative with respect to r_max, $\partial {f}_{m}/\partial {r}_{\max }$, using the finite differences method. We evaluated this numerically at a grid of redshifts separated by Δ(z) = 0.1 and then fit the result with a linear function to find

$\begin{eqnarray}\begin{array}{rcl}\frac{\partial {f}_{m}}{\partial {p}_{3}} & = & \frac{\partial \langle {\mathrm{DM}}_{\mathrm{host}}\left({r}_{\max },{f}_{\mathrm{hot}}\right) \rangle ({z}_{m})}{\partial {r}_{\max }}\\ & = & 371.25\cdot {z}_{m}-30.88.\end{array}\end{eqnarray} \tag{ C7 }$

Second, since parameter f_hot enters Equation (7) as a multiplicative constant, the partial derivative over the parameter f_hot is given by

$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\partial {f}_{m}}{\partial {p}_{4}} & = & \displaystyle \frac{\partial \langle {\mathrm{DM}}_{\mathrm{host}}\left({r}_{\max },{f}_{\mathrm{hot}}\right)\rangle ({z}_{m})}{\partial {f}_{\mathrm{hot}}}\\ & = & \displaystyle \frac{\langle {\mathrm{DM}}_{\mathrm{host}}\left({r}_{\max },{f}_{\mathrm{hot}}\right)\rangle ({z}_{m})}{{f}_{\mathrm{hot}}},\end{array}\end{eqnarray} \tag{ C8 }$

where $\langle {\mathrm{DM}}_{\mathrm{host}}\left({r}_{\max }\right)\rangle ({z}_{m})$ is the global estimate of the DM_halos calculated as a function of FRB redshift. Given our fiducial value ${r}_{\max }=1.40$, we average over large numbers of mock sight lines at a redshift grid of Δ(z) = 0.1 to a find a reasonable fitting function of

$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\partial {f}_{m}}{\partial {p}_{4}} & \equiv & \displaystyle \frac{\langle {\mathrm{DM}}_{\mathrm{host}}\left({r}_{\max }\right)\rangle ({z}_{m})}{{f}_{\mathrm{hot}}}\\ & = & 150.21\cdot {z}_{m}^{2}+520.07\cdot {z}_{m}-1.21.\end{array}\end{eqnarray} \tag{ C9 }$

We also include priors on each of the parameters in the Fisher matrix calculations described in Section 5. In order to do that, we simply add the corresponding variances ${\sigma }_{\mathrm{prior},p}^{2}$ of the parameter p estimated from its prior distribution to the corresponding diagonal elements of the 4 × 4 Fisher matrix given by Equation (22). Because the free parameters that we use in our model have not been previously well constrained, we adopt flat uniform priors on each of them within the following limits:

$\begin{eqnarray*}&&\begin{array}{c}0\leqslant {f}_{\mathrm{igm}}\leqslant 1\\ 0\leqslant {\mathrm{DM}}_{\mathrm{host}}\leqslant 500\ \left(\mathrm{pc}\ {\mathrm{cm}}^{-3}\right)\\ 0\leqslant {r}_{\max }/{r}_{200}\leqslant 5\\ 0\leqslant {f}_{\mathrm{hot}}\leqslant 1.\end{array}\end{eqnarray*}$

The corresponding variance ${\sigma }_{\mathrm{prior},p}^{2}$ for each parameter p is then given by

$\begin{eqnarray}&&{\sigma }_{\mathrm{prior},p}^{2}=\displaystyle \frac{1}{12}{\left({p}_{\max }-{p}_{\min }\right)}^{2},\end{eqnarray} \tag{ C10 }$

where ${p}_{\min }$ and ${p}_{\max }$ are the lower and upper bounds of the uniformly distributed priors for a given parameter p, respectively.