The Dispersion Measure and Scattering of Fast Radio Bursts: Contributions from the Intergalactic Medium, Foreground Halos, and Hosts

We use the cosmological hydrodynamical simulation code RAMSES (Teyssier 2002) to track the evolution of cosmic matter in a cubic box with side length of $100{h}^{-1}$ Mpc. We adopt a ΛCDM cosmology with parameters Ω_m = 0.317, Ω_Λ = 0.683, h = 0.671, σ₈ = 0.834, Ω_b = 0.049, and n_s = 0.962 (Planck Collaboration et al. 2014). The simulation contains 1024³ dark matter particles, corresponding to a mass resolution of $1.03\times {10}^{8}{M}_{\odot }$. We use a 1024³ root grid, and the maximum level of adaptive mesh refinement is ${l}_{\max }=17$. The spatial resolutions of the coarse and finest grids are $97.6{h}^{-1}$ kpc and $0.763{h}^{-1}$ kpc, respectively. When the number of dark matter particles in a grid cell is greater than eight, this grid cell will be refined. This simulation starts at z = 99 and ends at z = 0. At redshift z = 8.5, a uniform UV background according to the model in Haardt & Madau (1996) is switched on. Gas cooling, star formation, and stellar feedback are implemented, but supermassive black hole and active galactic nucleus (AGN) feedback are not modeled. Star formation is triggered in regions where the number density of hydrogen exceeds $0.1\,{\mathrm{cm}}^{-3}$. We find that 10% of the newly formed stars would explode as supernovae, by assuming a Salpeter initial mass function. A total of 10⁵¹ erg energy is injected into the gas by each supernova, of which the average progenitor mass is assumed to be 10 M_⊙. The stellar feedback is implemented using the feedback module in RAMSES.

A total of 44 snapshots are stored during the simulation. We search for dark matter halos in snapshots after simulation using the HOP algorithm (Eisenstein & Hut 1998). For the sake of reliability, we only consider halos that have more than 1200 dark matter particles, that is, ${M}_{h}\gt 1.2\times \ {10}^{11}{M}_{\odot }$, in the following study. We identify about 35,000–40,000 dark matter halos with ${M}_{h}\gt 1.2\times \ {10}^{11}{M}_{\odot }$ in our snapshots at low redshifts.

Figure 1 shows the projected gas density around a massive halo at z = 0.03 in our simulation. It clearly indicates that there are many gaseous clumps with various sizes residing in the diffuse IGM. These clumps are associated with halos with different masses. It is expected that some of the light paths toward FRB events would pass through some of these gaseous clumps. Cordes et al. (2016) has included such clumps in their models of the dispersion and scattering of FRB events. Here, we decompose the dispersion and scattering of FRBs contributed by the intervening medium along the lines of sight (LOSs) into three parts: the medium in the host halos, the gaseous halos of foreground galaxies that had been passed through, and the IGM but excluding the medium in the second term. We will refer to them as the host halos, the foreground halos, and the IGM hereafter. We evaluate the DM and scattering measure caused by these three components according to procedures described in the following subsections. Note that, since we cannot resolve the gas density below ∼1 kpc, we will not take the effect of the local medium surrounding the sources into account.

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2.2.1. The IGM and Foreground Halos

For an FRB source residing at redshift z_f, the sum of the DM contributed by the IGM and foreground halos, denoted as ${\mathrm{DM}}_{\mathrm{IGM}}$ and ${\mathrm{DM}}_{\mathrm{halos}}$, respectively, is given by (McQuinn 2014; Deng & Zhang 2014)

$$\begin{eqnarray}&&{\mathrm{DM}}_{\mathrm{IGM}}({z}_{f})+{\mathrm{DM}}_{\mathrm{halos}}({z}_{f})={\int }_{0}^{{z}_{f}}\displaystyle \frac{{n}_{e}(z)}{1+z}{dl},\end{eqnarray} \tag{ 1 }$$

where n_e(z) is the number density of electrons at z along the LOS. In this work, we assume that all of the intervening gaseous medium is fully ionized, and the density power spectrum of these inhomogeneous media along the LOS toward FRBs follows the form

$$\begin{eqnarray}&&P(k)={C}_{N}^{2}{k}^{-\beta }{e}^{-{{kl}}_{0}},k\gt {L}_{0}^{-1}\end{eqnarray} \tag{ 2 }$$

in the turbulent range, where L₀ and l₀ are the outer and inner scales of turbulence. For a medium with index $\beta \gt 3$, ${C}_{N}^{2}(z)$ can be approximately related to the density variance of electrons $\langle \delta {n}_{e}^{2}(z)\rangle$ as

$$\begin{eqnarray}&&{C}_{N}^{2}(z)\approx \displaystyle \frac{\beta -3}{2{\left(2\pi \right)}^{4-\beta }}\langle \delta {n}_{e}^{2}(z)\rangle {L}_{0}^{3-\beta }.\end{eqnarray} \tag{ 3 }$$

Consequently, the sum of the effective scattering measure due to the IGM and foreground halos would be (e.g., Macquart & Koay 2013; Xu & Zhang 2016)

$$\begin{eqnarray}&&{\mathrm{SM}}_{\mathrm{eff},\mathrm{IGM}}({z}_{f})+{\mathrm{SM}}_{\mathrm{eff},\mathrm{halos}}({z}_{f})={\int }_{0}^{{z}_{f}}\displaystyle \frac{{C}_{N}^{2}(z){d}_{{\rm{H}}}(z)}{{\left(1+z\right)}^{3}}{dz},\end{eqnarray} \tag{ 4 }$$

where d_H(z) indicates the Hubble radius. We further assume that the turbulence in the IGM, foreground, and host halos fulfills the Kolmogorov turbulence model, that is, $\beta =11/3$, and take $\langle \delta {n}_{e}^{2}(z)\rangle \sim {n}_{e}^{2}(z)$. Therefore, Equation (4) can be expanded as

$$\begin{eqnarray}&&\begin{array}{l}{\mathrm{SM}}_{\mathrm{eff},\mathrm{IGM}}({z}_{f})+{\mathrm{SM}}_{\mathrm{eff},\mathrm{halos}}({z}_{f})\\ \quad \approx 1.42\times {10}^{-13}{\left(\displaystyle \frac{{{\rm{\Omega }}}_{b}}{0.049}\right)}^{2}{\left(\displaystyle \frac{{L}_{0}}{1\mathrm{pc}}\right)}^{-2/3}{{\rm{m}}}^{-20/3}\\ \qquad \times {\displaystyle \int }_{0}^{{z}_{f}}{\left({\rho }_{b}(z)/{\bar{\rho }}_{b}(z)\right)}^{2}{\left(1+z\right)}^{3}{d}_{{\rm{H}}}(z){dz}\\ \quad \approx 1.31\times {10}^{13}\,{{\rm{m}}}^{-17/3}\cdot \displaystyle \frac{1}{h}{\left(\displaystyle \frac{{{\rm{\Omega }}}_{b}}{0.049}\right)}^{2}{\left(\displaystyle \frac{{L}_{0}}{1\mathrm{pc}}\right)}^{-2/3}\\ \qquad \times {\displaystyle \int }_{0}^{{z}_{f}}{\left({\rho }_{b}(z)/{\bar{\rho }}_{b}(z)\right)}^{2}\displaystyle \frac{{\left(1+z\right)}^{3}}{{\left[{{\rm{\Omega }}}_{{\rm{\Lambda }}}+{{\rm{\Omega }}}_{m}{\left(1+z\right)}^{3}\right]}^{1/2}}{dz}.\end{array}\end{eqnarray} \tag{ 5 }$$

We adopt the following procedures to evaluate ${\mathrm{DM}}_{\mathrm{IGM}}$, ${\mathrm{DM}}_{\mathrm{halos}}$, ${\mathrm{SM}}_{\mathrm{eff},\mathrm{IGM}}$, and ${\mathrm{SM}}_{\mathrm{eff},\mathrm{halos}}$. From the outputs of simulations at different redshifts, the gas densities on a uniform 4096³ grid, corresponding to a grid level of 12 in the simulation, are constructed. Grids at this level have a spatial resolution of $24.4{h}^{-1}$ kpc. For each grid cell at different snapshots, we check whether it is within any dark matter halo more massive than $1.2\times {10}^{11}{M}_{\odot }$ or not. If a cell is located in a halo, we flag it as a cell of foreground halos. Otherwise, this cell will be flagged as a cell of the IGM.

Following conventional methods (e.g., Gnedin & Jaffe 2001; Dolag et al. 2015), we stack the gas density on the 4096³ grids of the simulation box at different redshifts with rotating and flipping, and we construct lines of sight to perform the integration in Equation (5). For ${\mathrm{DM}}_{\mathrm{halos}}$ and ${\mathrm{SM}}_{\mathrm{eff},\mathrm{halos}}$, the integration is only performed for those cells belonging to foreground halos. Meanwhile, only those cells belonging to the IGM are taken into account when measuring ${\mathrm{DM}}_{\mathrm{IGM}}$ and ${\mathrm{SM}}_{\mathrm{eff},\mathrm{IGM}}$. Due to the limited number of snapshots generated by our cosmological simulation, the stacking and integration are ended at redshift z = 0.82. At higher redshifts, we lack enough snapshots to continuously sample the distribution of the IGM and foreground halos.

2.2.2. Host Halos

If there is an FRB source sitting at the center of a halo at z_h, the DM caused by the host halo is evaluated as

$$\begin{eqnarray}&&{\mathrm{DM}}_{\mathrm{host}}={\int }_{0}^{r200}\displaystyle \frac{{n}_{e}(r)}{1+{z}_{h}}{dr}=\displaystyle \frac{1}{1+{z}_{h}}{\int }_{0}^{r200}{n}_{e}(r){dr},\end{eqnarray} \tag{ 6 }$$

where r200 is the virial radius of the dark matter halo. The scattering caused by the halo gas is defined as

$$\begin{eqnarray}&&\begin{array}{l}{\mathrm{SM}}_{\mathrm{eff},\mathrm{host}}({z}_{h})\approx 1.31\times {10}^{13}{m}^{-17/3}\cdot \\ \,\displaystyle \frac{1}{h}{\left(\displaystyle \frac{{{\rm{\Omega }}}_{b}}{0.049}\right)}^{2}{\left(\displaystyle \frac{{L}_{0}}{1\mathrm{pc}}\right)}^{-2/3}\\ \,\times {\displaystyle \int }_{0}^{r200}{\left({\rho }_{b}(r)/{\bar{\rho }}_{b}({z}_{h})\right)}^{2}\displaystyle \frac{{\left(1+{z}_{h}\right)}^{3}}{{\left[{{\rm{\Omega }}}_{{\rm{\Lambda }}}+{{\rm{\Omega }}}_{m}{\left(1+{z}_{h}\right)}^{3}\right]}^{1/2}}{dr}.\end{array}\end{eqnarray} \tag{ 7 }$$

For each dark matter halo with mass ${M}_{h}\gt 1.2\times {10}^{11}{M}_{\odot }$ in different snapshots, we randomly select 40 radial paths from the halo center to the halo boundary to produce 40 sets of ${\mathrm{DM}}_{\mathrm{host}}$ and ${\mathrm{SM}}_{\mathrm{eff},\mathrm{host}}$. Each radial path is evenly divided into 50 segments. The baryon density at each segment is evaluated from the distribution of baryonic gas in the halos that are resolved in our AMR simulation. By virtue of the AMR technique, the density fields in halos can be resolved on fine grids with spatial resolution of a few kiloparsecs. Then if the total number and the redshift distribution function of mock FRBs are given, we will assign these mock FRBs to halos in different snapshots. Each mock FRB is randomly associated with one of these 40 radial paths of the assigned halo and corresponding ${\mathrm{DM}}_{\mathrm{host}}$ and ${\mathrm{SM}}_{\mathrm{host}}$.

We randomly sampled one million lines of sight to estimate the DM and SM caused by the IGM and foreground halos. Due to the limited number of snapshots produced by our simulation, the LOSs are ended at z = 0.82. In the top panel of Figure 2, we show the mean and median values of $\mathrm{DM}$ caused by the IGM and foreground halos as a function of the redshift, ${\mathrm{DM}}_{\mathrm{IGM}}(z)+{\mathrm{DM}}_{\mathrm{halos}}(z)$, and the corresponding standard deviation at $0\lt z\lt 0.82$ by solid, dotted–dashed, and short dashed lines, respectively. In comparison, the results in McQuinn (2014, hereafter M14) and Zhu et al. (2018, hereafter Z18) are shown by gray and red lines, respectively. Note that the contribution from foreground halos is not separated from the contribution of the IGM in M14 and Z18.

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The magnitude of ${\mathrm{DM}}_{\mathrm{IGM}}(z)+{\mathrm{DM}}_{\mathrm{halos}}(z)$ in this work is lower than the corresponding result in Z18 by about 20%. The major reason is that star formation is modeled in this work but not included in Z18. However, the stellar component is somehow overproduced in our AMR simulation, probably due to the lack of AGN feedback. If we lower the mass fraction of stars by hand, that is, calibrate it by the observed cosmic star density (Pérez-González et al. 2008), ${\mathrm{DM}}_{\mathrm{IGM}}(z)+{\mathrm{DM}}_{\mathrm{halos}}(z)$ could be increased to $\sim 90 \%$ of Z18. If we further assume that there was no star formation, the expected mean value of ${\mathrm{DM}}_{\mathrm{IGM}}(z)\,+{\mathrm{DM}}_{\mathrm{halos}}(z)$ would be slightly lower than that of Z18, and is about ∼98%–100% of Z18 at $z\lt 0.2$, and ∼90%–95% of Z18 at $0.2\lt z\lt 0.82$. This minor discrepancy might result from variance along LOSs, and the different number of LOSs, and different simulation resolutions.

Inferred from our simulation, the dependence of the median value of ${\mathrm{DM}}_{\mathrm{IGM}}(z)+{\mathrm{DM}}_{\mathrm{halos}}(z)$ is found to be well fitted by

$$\begin{eqnarray}&&{\mathrm{log}}_{10}\mathrm{DM}(z)=2.92+1.13{\mathrm{log}}_{10}(z)-0.03{\left({\mathrm{log}}_{10}(z)\right)}^{2}.\end{eqnarray} \tag{ 8 }$$

For the mean values of ${\mathrm{DM}}_{\mathrm{IGM}}(z)+{\mathrm{DM}}_{\mathrm{halos}}(z)$, we found that it can adopt a similar form of fitting formula but with an alternative set of coefficients (2.96, 1.07, 0.01) on the right-hand side of the equation. In comparison, the corresponding coefficients are (3.02, 1.00, −0.01) while fitting the results in Z18 with the similar formula.

In Figure 3, we show the contributions to the sum of ${\mathrm{DM}}_{\mathrm{IGM}}(z)+{\mathrm{DM}}_{\mathrm{halos}}(z)$ from matter in different structures. On average, foreground halos approximately account for ∼20%–30% of ${\mathrm{DM}}_{\mathrm{IGM}}(z)+{\mathrm{DM}}_{\mathrm{halos}}(z)$, and the rest is caused by the IGM. As in Z18, we further assign the grid cells into four categories of cosmic large-scale structures, using the method given in Zhu & Feng (2017). At z = 0, halos residing in nodes/clusters, filaments, and walls contribute $\sim 20 \%$, $\sim 10 \%$, and $\sim 0.4 \%$ of ${\mathrm{DM}}_{\mathrm{IGM}}(z)+{\mathrm{DM}}_{\mathrm{halos}}(z)$, respectively. Those fractions decrease gradually as redshift increases. In contrast, the IGM residing in nodes, filaments, walls, and voids contributes $\sim 3.5 \%$, $\sim 28 \%$, $\sim 23 \%$, and $\sim 15 \%$ of ${\mathrm{DM}}_{\mathrm{IGM}}(z)+{\mathrm{DM}}_{\mathrm{halos}}(z)$, respectively, at z = 0.

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The standard deviation of DM in our AMR simulation is around 1.5 times that in Z18 and M14. This feature should be due to the higher resolution in this work. The spatial resolution at the refinement level of 12 in our AMR simulation is $24.4{h}^{-1}$ kpc, while the simulation B050 in Z18 has a resolution of $48.8{h}^{-1}$ kpc. In Z18, we have demonstrated that a relatively poor resolution will underestimate the baryon density and mass fraction in a highly overdense region, leading to a lower standard deviation of the DM. On the other hand, this work shows that the standard deviation of the DM contributed by the IGM and foreground halos could be larger than 200 $\mathrm{pc}\,{\mathrm{cm}}^{-3}$ at $z\gt 0.2$.

Based on the matter density in the simulation Illustris-3, Jaroszynski (2019) obtained values of $\sigma ({\mathrm{DM}}_{\mathrm{IGM}})=115\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$ at z = 1 and $\sigma ({\mathrm{DM}}_{\mathrm{IGM}}+{\mathrm{DM}}_{\mathrm{hlaos}})\sim \,200\ \mathrm{pc}\,{\mathrm{cm}}^{-3}$. The latter is consistent with McQuinn (2014) and Zhu et al. (2018), but is smaller than the result at z = 0.80 of this work. This should be mainly caused by the difference in resolutions among these simulations. Illustris-3 provides the density of 455³ gas cells in a cubic box with side length $75/h$ Mpc, which may underestimate the variance among different lines of sight due to limited resolution. Based on the halo model, Prochaska & Zheng (2019) demonstrated that there are large scatters in the DM caused by foreground halo gas, depending highly on the impact parameter and halo mass. Particularly, the larger scatters are attributed to massive halos, because galaxy groups and clusters can give rise to DMs as large as several hundreds to thousands of $\mathrm{pc}\,{\mathrm{cm}}^{-3}$.

To illustrate the distribution of ${\mathrm{DM}}_{\mathrm{IGM}}$ and ${\mathrm{DM}}_{\mathrm{halos}}$ as a function of redshift, and their contribution to the significant variance of ${\mathrm{DM}}_{\mathrm{IGM}}(z)+{\mathrm{DM}}_{\mathrm{halos}}(z)$ among different LOSs, we present the probability density distributions in Figure 4. We find that both the IGM and gaseous halos of foreground galaxies can give rise to a significant variance. However, the variance caused by the gaseous halos of foreground galaxies is much larger than that caused by the IGM. In the redshift range we studied, the standard deviation of ${\mathrm{DM}}_{\mathrm{halos}}$ can be larger than that of ${\mathrm{DM}}_{\mathrm{IGM}}$ by a factor of ∼3–4, despite the fact that the median value of ${\mathrm{DM}}_{\mathrm{halos}}$ is only about one-third of the median value of ${\mathrm{DM}}_{\mathrm{IGM}}$. At z = 0.82, $\sigma ({\mathrm{DM}}_{\mathrm{IGM}})\simeq 80\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$, while $\sigma ({\mathrm{DM}}_{\mathrm{halos}})\simeq 300\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$. This striking feature results from different lines of sight having different impact paths to halos of foreground galaxies, and those halos differ from each other significantly in properties such as mass, radius, and density distribution. Our result is basically in agreement with Prochaska & Zheng (2019). This result suggests that it is important to verify whether the line of sight to an FRB event has passed through any foreground halos, in order to precisely estimate the redshift of this event from its DM.

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Figure 5 presents the scattering measure caused by the IGM and foreground halos, ${\mathrm{DM}}_{\mathrm{IGM}}+{\mathrm{DM}}_{\mathrm{halos}}$, as a function of redshift. Not surprisingly, the increased resolution in our AMR simulation in comparison with Z18 has enhanced the effective scattering measure dramatically. This trend is consistent with the results of simulations with different resolutions in Z18. As demonstrated in Z18, the effective scattering measure is very sensitive to grid cells with very large overdensity. Simulations with finer grids can better resolve the density field in a highly nonlinear regime.

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The bottom panel of Figure 5 shows that the halos residing in nodes and filaments contribute about $\sim 80 \%$ and $\sim 15 \%$ of ${\mathrm{SM}}_{\mathrm{IGM}}+{\mathrm{SM}}_{\mathrm{halos}}$. In contrast, the contribution from the IGM in nodes and filaments is merely around 1%. The IGM and foreground halos in the wall and void regions contribute very little to the scattering. In Z18, we found that the medium in nodes and filaments contributes about ∼65%–80% and ∼20%–30% of the effective SM. Here, our investigation further indicates that the gaseous halos in nodes and filaments play a dominant role in the scattering of FRB signals throughout the path between the host halos and the Milky Way. This result is consistent with the theoretical investigation in Macquart & Koay (2013). The standard deviation of ${\mathrm{SM}}_{\mathrm{eff}}$ is larger than the mean value of ${\mathrm{SM}}_{\mathrm{eff}}$ by almost two dex. Again, foreground halos are the primary factor that results in tremendous differences between different lines of sight, as shown by Figure 6. Lines of sight that have passed through the most inner region of halos can significantly enhance the standard deviation.

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The contribution from the medium in the hosting halos is also an important component of the total DM and SM of FRB events. However, it remains very difficult to estimate those values for a particular event at the present moment. It is usually assumed to follow a log-normal distribution for the disk galaxy, based on results of previous theoretical studies (e.g., Xu & Han 2015). We tackle this issue statistically as follows. We calculate the DM and effective SM along each of the 40 randomly radial trajectories for each halo with ${M}_{h}\gt 1.2\times {10}^{11}{M}_{\odot }$ at certain redshift snapshots using Equations (6) and (7). The number of halos over $1.2\times {10}^{11}{M}_{\odot }$ slightly decreases from ∼39,500 at z = 0.82 to ∼34,700 at z = 0. Thus, we have about 140,000–160,000 radial trajectories within host halos for each simulation snapshot. Here, we assume the halos in a particular snapshot have the same redshift. The red solid lines in Figure 7 show the distributions of DM and effective SM associated with these trajectories at z = 0.

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Significant variations can be found among different radial trajectories of different halos, which should be due to the inhomogeneity of the gas distribution in host halos. At z = 0.0, ${\mathrm{DM}}_{\mathrm{host}}$ spans a wide range of values, $\sim 1\mbox{--}3000\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$, for most of the radial trajectories, and its distribution obviously deviates from a log-normal one, but exhibits an almost even distribution in the range $\sim 3\mbox{--}30\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$ and then continues with a bump peaked at $\sim 300\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$. The median value is $\sim 100\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$, in agreement with previous theoretical models (e.g., Xu & Han 2015; Luo et al. 2018; Walker et al. 2020). However, the distribution of ${\mathrm{DM}}_{\mathrm{host}}$ extracted from our simulation displays a significant difference from a simple log-normal form that is usually assumed in theoretical models.

This discrepancy may result from the following factors. In our simulation study, radial trajectories can be associated with host halos of various masses, morphology types, and randomly selected view angles. In the previous theoretical study based on scaled models of smooth electron distributions in galaxies (e.g., Xu & Han 2015), the distribution of ${\mathrm{DM}}_{\mathrm{host}}$ for a galaxy with a particular stellar mass, morphology type, and view angle can be fitted by a log-normal function. But the fitting parameters of the log-normal function will vary notably if either the galaxy mass or galaxy type, as well as view angle, is changed. Moreover, as shown in Figure 1, the gas distribution in halos is not smooth and contains high-density clumps, which may contribute to this discrepancy.

The scattering measure of hosts exhibits a distribution similar to ${\mathrm{DM}}_{\mathrm{host}}$ in the range ${\mathrm{SM}}_{\mathrm{eff},\mathrm{host}}\sim 1\mbox{--}{10}^{8}\times {10}^{12}{\left({L}_{0}/1\mathrm{pc}\right)}^{-2/3}{m}^{-17/3}$, with a median value $\sim {10}^{5}\times {10}^{12}{\left({L}_{0}/1\mathrm{pc}\right)}^{-2/3}\,{m}^{-17/3}$. The results for DM and SM caused by hosts at z = 0.82, shown with blue lines in Figure 7, are basically similar to that at z = 0.0, but with larger median values of DM and SM. This is not surprising, actually, because we selected halos from the same simulation, and the evolution of those halos is relatively slow at low redshifts. On the other hand, a higher physical density at high redshifts should have shifted the overall distribution toward larger dispersion and scattering measures. Note that all of the radial propagation paths within the hosts in our calculation are assumed to start from the halo center, neglecting the spatial distribution of FRB events. This may mildly overestimate the contribution to both DM and SM by host halos. Since our simulation has a limited spatial resolution in halos, further investigation with higher-resolution simulations is required to resolve the finer structure of host halos and give more reliable results.

With the mock FRB sources, we give a scatter plot of the total extragalactic DM, ${\mathrm{DM}}_{\mathrm{exg}}={\mathrm{DM}}_{\mathrm{host}+\mathrm{halos}+\mathrm{IGM}}\,={\mathrm{DM}}_{\mathrm{host}}\,+{\mathrm{DM}}_{\mathrm{halos}}+{\mathrm{DM}}_{\mathrm{IGM}}$, against their redshifts in the left panel of Figure 8. For a given redshift, the total DM displays significant variations. We further measure the variances caused by different components and present the result in the middle panel of Figure 8. Clearly, the host halo overwhelmingly dominates the variance in the total DM, ${\sigma }^{2}({\mathrm{DM}}_{\mathrm{exg}})$, accounting for nearly 99% at $z\lt 0.2$, and the fraction remains at around $\sim 90 \%$ until z = 0.8. The contribution to ${\sigma }^{2}({\mathrm{DM}}_{\mathrm{exg}})$ from foreground halos amounts to only $\sim 1 \%$ at $z\lt 0.2$ and increases gradually to $\sim 10 \%$ at $z\gt 0.4$. In addition, the IGM makes a contribution about 1/10 of that from the foreground halos to ${\sigma }^{2}({\mathrm{DM}}_{\mathrm{exg}})$.

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To date, the redshifts of eight localized FRB events have been available (Tendulkar et al. 2017; Bannister et al. 2019; Ravi et al. 2019; Macquart et al. 2020; Marcote et al. 2020). We plot them with filled symbols in the left panel of Figure 8, assuming that the contribution from the Milky Way's halo is $30\ {\mathrm{cm}}^{-3}\,\mathrm{pc}$ following Cordes et al. (2016). Except for FRB 190611, most of them show minor or moderate deviations, $\delta z\lesssim 0.1$, from the median DM–z relation of our mock sources. Therefore, for those observed FRB events without any information on the host, the median DM–z relation of our mock sources allows us to give a crude estimation of their redshifts. The fitting result for the median DM as a function of redshift is as follows:

$$\begin{eqnarray}&&{\mathrm{log}}_{10}{\mathrm{DM}}_{\mathrm{exg}}(z)=3.04+1.00{\mathrm{log}}_{10}(z)+0.25{\left({\mathrm{log}}_{10}(z)\right)}^{2}.\end{eqnarray} \tag{ 9 }$$

Meanwhile, the median DM caused by the foreground gaseous halos and the IGM of mock sources fulfills the fitting formula of Equation (8). If we assign the dispersion induced by host halos the same value as the median value obtained in Section 3.3, ${\mathrm{DM}}_{\mathrm{host}}=100\,{\mathrm{cm}}^{-3}\,\mathrm{pc}$, the redshift of observed events can be estimated using Equation (8). The right panel of Figure 8 demonstrates that the redshifts of five localized FRB events can be recovered with error $\delta z\lesssim 0.05$ following this scheme. The errors for the three other localized events are about $\delta z\sim 0.15$. The uncertainty in the dispersion measure caused by the host should be the primary source of such errors. For instance, the redshift of the event FRB 121102 is predicted to be ∼0.25 and ∼0.35 according to Equations (9) and (8), respectively, which are both higher than the real value of z = 0.1927 (Tendulkar et al. 2017). In comparison, if the contribution from the host is not taken into account, the redshift of this event estimated from the DM can be as high as 0.68 (e.g., Pol et al. 2019). The uncertainty in the dispersion measure arising from foreground halos would be the secondary source of errors in estimating redshift from ${\mathrm{DM}}_{\mathrm{exg}}$.

The left panel in Figure 9 compares the probability density function (PDF) and cumulative density functions (CDFs) of the total extragalactic DM of mock sources with the observed events, and it indicates a good agreement between the simulation and observations, though the mock sample has a slight higher fraction of sources with larger values of $\mathrm{DM}\gt 600\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$. This excess can partly be attributed to mock samples with larger ${\mathrm{DM}}_{\mathrm{host}}$. It suggests that a number of FRB events at redshift $z\lt 1.0$ but with larger DM may have been missed by current observations. Alternatively, the discrepancy can be alleviated by placing a lower value of maximum redshift z_max for mock sources. Currently, the highest redshift of FRB events is still unclear. Due to the limited number of stored snapshots, the maximum redshift of our mock sources has been set to 0.82. In comparison, we plot the results with ${z}_{\max }=0.6$ in Figure 9, which can provide better agreement with the observation.

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The relative importance to the extragalactic DM of different components of the intervening medium is crucial for the understanding of FRBs and its application to probing the cosmic baryons. We carry out an overall analysis on the relative importance in our mock samples. Figure 9 shows the mean and median fractions of DM contributed by the three types of medium as a function of the total extragalactic DM. For sources with ${\mathrm{DM}}_{\mathrm{exg}}$ either less than 20 or larger than $800\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$, the host halos make a dominant contribution to ${\mathrm{DM}}_{\mathrm{exg}}$. For these sources, the secondary contribution to ${\mathrm{DM}}_{\mathrm{exg}}$ arises from the IGM. For sources with ${\mathrm{DM}}_{\mathrm{exg}}$ in the range $20\mbox{--}800\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$, the primary contributor is the IGM, followed by the host halos and foreground halos in turn. The mean fraction of ${\mathrm{DM}}_{\mathrm{exg}}$ caused by the IGM declines monotonically from $\sim 60 \%$ at ${\mathrm{DM}}_{\mathrm{exg}}=20\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$ to $\sim 50 \%$ at ${\mathrm{DM}}_{\mathrm{exg}}=800\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$. Simultaneously, the mean contribution from the host halos drops from 35% to 30%.

It should be noted that the transition of relative importance between the IGM and host halos at ${\mathrm{DM}}_{\mathrm{exg}}\sim 800\mbox{--}900\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$ should result from the maximum redshift of mock sources ${z}_{\max }=0.82$ that was placed by hand. As shown in Figure 2, the mean and median DMs induced by the IGM and foreground halos are increasing with redshift and will be around $600\mbox{--}900\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$ at z = 0.82. Hence, our mock sources with ${\mathrm{DM}}_{\mathrm{exg}}\geqslant 600\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$ are inevitably biased by those with relatively large ${\mathrm{DM}}_{\mathrm{host}}$. If the upper limit on the redshift of sources was increased, it would be expected that the behavior of fractions of DMs found in the range $20\mbox{--}800\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$ can be extended to higher DM ends. If we naively extrapolate the fitting curve within the range $20\mbox{--}800\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$ to a higher DM range linearly, the IGM would still dominate the total DM even at ${\mathrm{DM}}_{\mathrm{exg}}=5000\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$. Meanwhile, the average contribution from host halos may drop below $\sim 30 \%$ of the ${\mathrm{DM}}_{\mathrm{exg}}$ and keep declining slowly as ${\mathrm{DM}}_{\mathrm{exg}}$ increases, and it would stand as the secondary contributor until ${\mathrm{DM}}_{\mathrm{exg}}\sim 2000\mbox{--}3000\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$. However, it should be emphasized here that these fractions are obtained by taking averages over a large number of mock sources. For a single event, a striking variance is expected. In addition, the contribution from the local surrounding medium is not taken into account in our models.

We shall first introduce the models and calculation of the scattering time τ before describing the DM–τ relation of mock sources. Assuming that the turbulent medium can be described by the Kolmogorov turbulence model, the temporal broadening time caused by a medium extending between z and $z+{\rm{\Delta }}z$ for a source at z_f is given by (e.g., Macquart & Koay 2013)

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}\tau & = & 3.32\times {10}^{-4}{\left(1+z\right)}^{-1}{\left(\displaystyle \frac{{\lambda }_{0}}{30\mathrm{cm}}\right)}^{4}\left(\displaystyle \frac{{D}_{\mathrm{eff}}}{1\,\mathrm{Gpc}}\right)\\ & & \times \left(\displaystyle \frac{{\rm{\Delta }}{\mathrm{SM}}_{\mathrm{eff}}}{{10}^{12}{{\rm{m}}}^{-17/3}}\right){\left(\displaystyle \frac{{l}_{0}}{1\mathrm{au}}\right)}^{-1/3}\,\mathrm{ms},{r}_{\mathrm{diff}}\lt {l}_{0},\end{array}\end{eqnarray} \tag{ 10a }$$

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}\tau & = & 9.50\times {10}^{-4}{\left(1+z\right)}^{-1}{\left(\displaystyle \frac{{\lambda }_{0}}{30\mathrm{cm}}\right)}^{22/5}\left(\displaystyle \frac{{D}_{\mathrm{eff}}}{1\,\mathrm{Gpc}}\right)\\ & & \times {\left(\displaystyle \frac{{\rm{\Delta }}{\mathrm{SM}}_{\mathrm{eff}}}{{10}^{12}{{\rm{m}}}^{-17/3}}\right)}^{6/5}\,\mathrm{ms},{r}_{\mathrm{diff}}\gt {l}_{0},\end{array}\end{eqnarray} \tag{ 10b }$$

where ${\rm{\Delta }}{\mathrm{SM}}_{\mathrm{eff}}$ is the effective scattering measure of this medium between z and $z+{\rm{\Delta }}z$; ${\lambda }_{0}$ is the observed wavelength; ${D}_{\mathrm{eff}}$ is defined as $={D}_{L}{D}_{{LS}}/{D}_{S}$, with D_L, D_S, and D_LS being the angular diameter distance of the screen of the scattering medium (centered at $z+{\rm{\Delta }}z/2$), of the FRB source (at z_f) to us, and of the source to the screen of the scattering medium, respectively. For each mock source, we measure the value of ${\rm{\Delta }}\tau$ at successive redshift intervals, using the gas density and scattering measure along the chosen LOS and trajectory toward it, and then sum the individual ${\rm{\Delta }}\tau$ to estimate the total scattering time τ. For each gas cell along a light path, we could identify which components it belongs to, that is, either the IGM, the foreground halos, or the host halos. Accordingly, we can directly know the separate contribution to τ from those three components. Note that the geometric effect due to ${D}_{\mathrm{eff}}$ is precisely calculated in this work, which is a substantial improvement with respect to Z18.

The diffractive length scale ${r}_{\mathrm{diff}}$ for Kolmogorov turbulence with index $\beta =11/3$ is given by (Macquart & Koay 2013)

$$\begin{eqnarray}\begin{array}{rcl}{r}_{\mathrm{diff}} & = & 2.7\times {10}^{10}{\left(\displaystyle \frac{{\lambda }_{0}}{30\mathrm{cm}}\right)}^{-1}{\left(\displaystyle \frac{{\mathrm{SM}}_{\mathrm{eff}}}{{10}^{12}{{\rm{m}}}^{-17/3}}\right)}^{-1/2}\\ & & \times {\left(\displaystyle \frac{{{l}}_{0}}{1\mathrm{au}}\right)}^{1/6}\,{\rm{m}},{r}_{\mathrm{diff}}\lt {{l}}_{0},\end{array}\end{eqnarray} \tag{ 11a }$$

$$\begin{eqnarray}\begin{array}{rcl}{r}_{\mathrm{diff}} & = & 1.6\times {10}^{10}{\left(\displaystyle \frac{{\lambda }_{0}}{30\mathrm{cm}}\right)}^{-6/5}{\left(\displaystyle \frac{{\mathrm{SM}}_{\mathrm{eff}}}{{10}^{12}{{\rm{m}}}^{-17/3}}\right)}^{-3/5}\,{\rm{m}},\\ & & {r}_{\mathrm{diff}}\gt {{l}}_{0}.\end{array}\end{eqnarray} \tag{ 11b }$$

Considering the results for effective scattering measure probed in the last section, the typical diffractive length scale in the host and foreground halos and the IGM is likely to be about $\sim {10}^{9}\mbox{--}{10}^{10}\,{\rm{m}}$ for radio signals of wavelength $\lambda \sim 30$ cm.

As shown by the above equations, the scattering timescale τ depends highly on the outer and inner scales, L₀ and l₀, of the turbulence and the relative sizes of ${r}_{\mathrm{diff}}$ and l₀. The plausible values of the inner and outer scales of the interstellar medium in the Milky Way are around $200\mbox{--}1000\,\mathrm{km}$ and 1–100 pc, respectively (e.g., Cordes et al. 2016). The outer and inner scales in the host halos and foreground halos might be comparable to or moderately larger than that in the Milky Way. The central regions of these halos would resemble the Milky Way to some extent, especially for those halos containing a large fraction of gas in the central region. On the other hand, more and more observations in the past decades reveal that the CGM is in a complex and multiphase state, containing subkiloparsec gas clumps (e.g., Rudie et al. 2012; Crighton et al. 2015; Werk et al. 2016; Tumlinson et al. 2017). Meanwhile, recent simulations using a delicate designed refinement strategy show that the circumgalactic medium exhibits significant turbulent behavior and contains more cool, dense filaments and clumps and eddies when the resolution is boosted to a few kiloparsecs, and even subkiloparsecs (e.g., Hummels et al. 2019; Suresh et al. 2019; Bennett & Sijacki 2020). Star formation can occur in those dense filaments and clumps in the CGM (e.g., Bennett & Sijacki 2020). Consequently, stellar evolution and feedback may drive turbulence in the CGM with an outer scale shorter than 1 kpc.

Currently, there is barely any solid constraint on the inner and outer scales of the turbulence in the IGM. The driving scale could be tens of kiloparsecs to a few megaparsecs, depending on the specific energy-injection mechanism (see discussions in Luan & Goldreich 2014 and Zhu et al. 2018). But information on the state of the IGM below tens of kiloparsecs is absent from both simulation and observation works. To simplify the problem, we assume the same value of outer scale L₀ in the IGM and host halos, as well as foreground halos, in most of our models. The reason behind this assumption is that, as demonstrated in Figure 5 and Zhu et al. (2018), the scattering caused by the IGM mainly comes from those media residing in filaments and clusters (nodes). It is probably dominated by the clumps located in filaments and clusters, that is, belonging to halos less massive than $1.2\times {10}^{11}{M}_{\odot }$.

Here, we study the scattering time of FRB events in four models. For the first three models, the outer scales in the three types of intervening media are assumed to be the same. We set the inner scales in different media to some fixed values, and then adjust the outer scales to produce a scattering timescale comparable to the observations. In the first model, denoted "Model A," the inner scale of the host halos and foreground halos is 1000 km, and the inner scale of the IGM is 1 au. In the second and third models, denoted "Model B" and "Model C," all of the inner scales in the three types of intervening media are 1000 km and 1 au, respectively. As the real values of the outer and inner scales in the foreground halos and the IGM are poorly constrained so far and the scattering time depends highly on these scales, we also include the fourth model, denoted "Model D," in which these turbulence scales in the foreground halos and IGM are much larger than those in the host halos. We list the values of l₀ and L₀ in different models in Table 1. A inner scale ${l}_{0}=1000$ km would be much shorter than the typical ${r}_{\mathrm{diff}}$ in those three types of media discussed here, while for an inner scale ${l}_{0}=1$ au, it is probably longer than ${r}_{\mathrm{diff}}$. We will adopt the corresponding equation to estimate τ while using these two values of l₀.

Table 1.  Inner and Outer Scales of Turbulence in Different Models for Calculating τ

Model	Medium	Inner Scale	Outer Scale
(5)	(6)	(7)	(8)
	Host	1000 km	5.0 pc
Model A	Halos	1000 km	5.0 pc
	IGM	1 au	5.0 pc
	Host	1000 km	10.0 pc
Model B	Halos	1000 km	10.0 pc
	IGM	1000 km	10.0 pc
	Host	1 au	0.2 pc
Model C	Halos	1 au	0.2 pc
	IGM	1 au	0.2 pc
	Host	1000 km	5.0 pc
Model D	Halos	${10}^{6}\,\mathrm{km}$	50.0 pc
	IGM	1 au	500.0 pc

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The pink circles in Figure 10(a) show the extragalactic DM–τ relations of 500 randomly selected mock sources adopting Model A, against the observed events that have either a reported τ (solid symbols) or the upper limit of τ (downward arrows). The outer scale in the three types of medium is L₀ = 5.0 pc. In Figure 10(a), each panel indicates the contribution to the extragalactic DM and τ of a mock source by a particular component, or by the sum of two or three components. In the bottom right panel, where both the DM and τ of mock sources are the sum of all three types of intervening medium, we can see that the distribution of mock sources in the DM–τ space basically overlaps the observed events. Most of the sources are below the DM–τ relation of pulsars in the Milky Way to some extent (Cordes et al. 2016). Comparing the bottom right panel to the other five panels in Figure 10(a), we conclude that every component of the intervening medium is an important ingredient in explaining the observed DM–τ relation.

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The results of Model B, Model C, and Model D are presented in Figures 10(b) and 11. In both Model B and Model C, the relative importance of the IGM to τ is enhanced, as the inner scale of the turbulence in the IGM now equals that of the host and foreground halos. In comparison to Model A, the improvement on ${\tau }_{\mathrm{IGM}}$ in Model B can relax the requirement on the outer scale L₀ up to ∼10 pc in order to match the observation. However, in Model C a much shorter L₀, ∼0.2 pc, is needed to reproduce the observed DM–τ relation if the inner scale l₀ is 1 au in all of the intervening medium. For Model D, the inner and outer scales of turbulence in the host are set to the same values as in Model A. The foreground halos can still cause a significant scattering time of mock events as large as tens of milliseconds, if ${L}_{0,\mathrm{halos}}=50\,\mathrm{pc}$ and ${l}_{0,\mathrm{halos}}={10}^{6}\,\mathrm{km}$. The IGM, however, can only cause a scattering time smaller than 0.1 ms at 1 GHz for most mock sources, if the inner and outer scales of turbulence in the intergalactic region are 1 au and 500 pc, respectively.

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Figure 12 shows the PDF and CDFs of τ in four models, in comparison with the observed events. The τ distribution of mock sources is somewhat similar to that of the 74 observed events with reported τ (or upper limit) at $\tau \gt 0.03\,\mathrm{ms}$. However, there are two major differences. First, the τ distribution of observed events shows notable peaks and dips, which should partly be caused by fluctuations due to limited sample size. Second, there are many mock sources with very small τ. Note that there are 50 observed FRB events having neither a measured τ nor an upper limit on τ. Some of these events may have scattering on a timescale of a few to tens of microseconds, which need observations with high time resolutions to resolve (e.g., Farah et al. 2018; Day et al. 2020). We find that if we hypothetically assign a small τ, randomly distributed in the range ${10}^{-2.5}\mbox{--}{10}^{-1.5}\,\mathrm{ms}$, to each of these 50 events and include them in the τ distribution, the discrepancy between mock sources and observation for the τ distribution would be largely relieved.

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On the other hand, Figure 12 indicates that in Model A and Model B the contribution from the foreground halos to the scattering time is comparable to that of host halos and is much stronger than that of the IGM. In Model C, foreground halos dominate the time broadening of mock sources, while the contribution from host halos is comparable to the IGM. In Model D, the foreground halos remain an important contributor to the scattering time. The IGM, however, plays a very limited role in the scattering. In Z18, where the role of foreground halos is not separated from the diffuse IGM, we found that the medium residing in cosmic knots and filaments may induce significant scattering. With improved simulation resolution and tools, our models in this work suggest that foreground halos may play an important role in the scattering of FRBs.

Finally, we measure the relative importance of the IGM, foreground halos, and host halos to the scattering time τ in our models. Figure 13 shows the mean and median fractions of τ contributed by the different media as a function of the total extragalactic DM. Similar to the case of ${\mathrm{DM}}_{\mathrm{exg}}$, the host halos dominate the τ for sources with ${\mathrm{DM}}_{\mathrm{exg}}\lt 20\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$ or ${\mathrm{DM}}_{\mathrm{exg}}\gt 800\mbox{--}1000\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$. The secondary contribution to τ is made by the IGM and foreground halos for sources with ${\mathrm{DM}}_{\mathrm{exg}}\lt 20\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$ and ${\mathrm{DM}}_{\mathrm{exg}}\gt 800\mbox{--}1000\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$, respectively.

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For sources with total extragalactic DM in the range $20\mbox{--}800\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$, the mean fraction of τ contributed by the IGM declines monotonically from ∼35%–60% at ${\mathrm{DM}}_{\mathrm{exg}}\,=20\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$ to ∼5%–20% at ${\mathrm{DM}}_{\mathrm{exg}}=800\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$ in the four different models considered here. In contrast, the contribution to τ from foreground halos increases monotonically from 5% to 50%–60%, while those from host halos decrease slowly from 40%–55% to 15%–20%. For sources with $20\lesssim {\mathrm{DM}}_{\mathrm{exg}}\lesssim 200\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$, the IGM is the primary contributor to τ in the first three models, but the host dominates in Model D. For sources with $200\lesssim {\mathrm{DM}}_{\mathrm{exg}}\lesssim 800\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$, the foreground halos are the primary contributors to the time scattering in all four models. However, the secondary contributor varies from model to model, depending on the values of l₀ and L₀.

For the same reason stated in Section 4.1, our mock sources with ${\mathrm{DM}}_{\mathrm{exg}}\geqslant 800\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$ are biased by those with relatively large ${\mathrm{DM}}_{\mathrm{host}}$ arising from a limited value of z_max. The, if z_max was increased, the foreground halos would be expected to contribute more than 60% of the scattering time at $\mathrm{DM}\geqslant 800\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$, based on our models. Meanwhile, the average contribution from host halos to τ may drop below ∼15%–20% and keep declining slowly with the total DM increasing. Note that these fractions only give the average values. There could be significant variations between different sources. Once again, we should remind the reader that the turbulent scales in the foreground halos and the IGM are barely known so far. Given recent progress made in the study of CGM, Model D might be more favorable currently. If the inner and outer scales of turbulence in the foreground halos and IGM are much larger than the assumed values in our models, the relative importance of foreground halos and IGM to scattering will be further decreased.