Dispersion measures of fast radio burst host galaxies derived from IllustrisTNG simulation

Fast radio bursts (FRBs) are radio transients with short duration and high dispersion measures (DMs). After the first FRB was discovered (Lorimer et al., 2007), a few hundred FRBs have been observed by different telescopes to date (Thornton et al., 2013; Petroff et al., 2016; CHIME/FRB Collaboration et al., 2019a; Shannon et al., 2018). Observationally, there are two types of FRBs, repeating FRBs and non-repeating FRBs. Till now, twenty repeating FRBs have been published (Spitler et al., 2016; CHIME/FRB Collaboration et al., 2019b, c; Fonseca et al., 2020). For comparison, most FRBs are apparently non-repeating events. The data of published FRBs can be found in the FRB catalog (Petroff et al., 2016).

The physical origin of FRBs is still under debate (for recent reviews, see Platts et al., 2019; Cordes & Chatterjee, 2019; Petroff et al., 2019). More recently, a radio burst from Galactic magnetar SGR 1935+2154 was discovered (The CHIME/FRB Collaboration et al., 2020; Bochenek et al., 2020), which supports the conjecture that some FRBs are produced by magnetars. The merger of compact binary is also a promising model for non-repeating FRBs, which has been discussed by many works (Totani, 2013; Wang et al., 2016).

The large DMs of FRBs are well beyond the contribution from the Milky Way galaxy, which indicates an extragalactic origin. The localization of FRB 121102 (Tendulkar et al., 2017), FRB 180924 (Bannister et al., 2019), FRB 190523 (Ravi et al., 2019), FRB 181112 (Prochaska et al., 2019), and FRB 180916 (Marcote et al., 2020) support their cosmological origin. Therefore, FRBs can be used as cosmological probes, such as constraining the cosmological parameters (Gao et al., 2014; Zhou et al., 2014; Walters et al., 2018; Jaroszynski, 2019; Li et al., 2018), measuring the cosmological proper distance (Yu & Wang, 2017), constraining the baryon number density (Deng & Zhang, 2014; Wei et al., 2019; Li et al., 2019b; Macquart et al., 2020), testing dark matter models (Muñoz et al., 2016; Wang & Wang, 2018), measuring Hubble parameter (Wu et al., 2020), testing Einstein equivalent principle (Wei et al., 2015; Yu et al., 2018) and testing hydrogen and helium reionization histories (Fialkov & Loeb, 2016; Caleb et al., 2019) .

The main difficulty in using FRBs for cosmological purpose is how to determine their redshifts. Theoretically, the redshift of FRB can be derived from the observed DM, which is the integration of free electron number density along a given line of sight. According to their origin, the observed DM can be divided into several parts:

which includes the contributions from the interstellar medium of the Milky Way (DM_MW), the halo of the Milky Way (DM_halo), the intergalactic medium (IGM) (DM_IGM), the host galaxy (DM_host) and the source (DM_source). DM_MW can be derived from the NE2001 model (Cordes & Lazio, 2002) or YMW16 model (Yao et al., 2017). We have a poor understanding of DM_halo. Dolag et al. (2015) found the typical value of DM${}_{\mathrm{halo}}\sim 30$ pc cm^-3 from numerical simulations. Prochaska & Zheng (2019) estimated that DM_halo is about $50\sim 80$ pc cm^-3. The relation between DM_IGM and redshift $z$ has been studied by some works (Ioka, 2003; Deng & Zhang, 2014), which can be used to compute the pseudo redshifts of FRBs. The source contribution DM_source depends on the unclear central engine of FRBs. For example, if FRBs are produced by the collapses of massive stars, the values of DM_source are large (Piro, 2016). On the other hand, if FRBs are produced by mergers of binary neutron stars (Wang et al., 2016; Zhang, 2020), they have a small DM_source (Margalit et al., 2019; Wang et al., 2020). If the central engine is known, the value of DM_source can be derived analytically under some assumptions (Piro, 2016; Yang & Zhang, 2017; Wang et al., 2020). In our discussion, this term is ignored. The value of DM_host is hard to determine. For FRB 121102, Tendulkar et al. (2017) obtained the value of host galaxy DM as $55\leq\mathrm{DM}_{\mathrm{host}}\leq 225$ pc cm^-3, which may consist of large contribution from the sources. Marcote et al. (2020) estimated the DM_host of FRB 180916 is less than 70 pc cm^-3. For non-repeating FRBs, Bannister et al. (2019) found DM${}_{\mathrm{host}}\simeq 30-81$ pc cm^-3 for FRB 180924.

Until now, only ten FRBs have been localized, including the Galactic FRB 200428. Using Very Large Array (VLA), FRB 121102 has been localized to a dwarf galaxy with low metallicity and low star formation rate (SFR) (Chatterjee et al., 2017; Tendulkar et al., 2017). FRB 180916 is localized to a spiral galaxy with high stellar mass and SFR (Marcote et al., 2020). For non-repeating FRBs, Ravi et al. (2019) localized the host galaxy of FRB 190523 through Deep Synoptic Array ten-antenna prototype (DSA-10). Using Australian Square Kilometer Array Pathfinder (ASKAP), the host galaxies of FRB 180924 (Bannister et al., 2019) and FRB 181112 (Prochaska et al., 2019) have also been determined. Recently, four ASKAP localized FRBs have been released (Macquart et al., 2020). The properties of these host galaxies are listed in Table 1.

In this paper, we use the IllustrisTNG simulation to investigate the DM_host of FRBs. The IllustrisTNG project is a successor of the Illustris project. Using the moving mesh code AREPO, they perform a cosmological magnetohydrodynamical simulation (Pillepich et al., 2018a). Multiple physical processes that derive galaxy formation are implemented in AREPO. Several simulations are performed in this project. Each simulation contains a large number of simulated galaxies, and detailed information on each one of these galaxies is provided.

This paper is organized as follows. In section 2, we introduce the IllustrisTNG simulation and give the method to derive DM_host. We will present the DM_host distributions for repeating FRBs like FRB 121102 in section 3, repeating FRBs like FRB 180916 in section 4 and non-repeating FRBs in section 5. In section 6, the effect of selection criteria of galaxies is discussed and the redshifts of simulated FRBs are derived based on our results. Finally, conclusions are given in section 7.

The DM in the FRB rest frame can be derived through

where $n_{e}$ is the electron number density and $dl$ is the light path. If the electron distribution of a galaxy is known, the DM_host can be derived through Equation 2. However, the electron distribution of a galaxy can hardly be obtained from observations. Therefore, the IllustrisTNG simulation is used to derive the electron distribution.

The IllustrisTNG project is a successor of the Illustris project. It is a large-volume, cosmological and magnetohydrodynamical simulation. It consists of TNG50, TNG100, and TNG300, which mean that the lengths of the simulation boxes are about 50 Mpc, 100 Mpc, and 300 Mpc, respectively. The large simulation box can be used to study the large-scale structure of the Universe, while the small simulation box can provide a better resolution of galaxies, which is important when we are interested in the structure of galaxies. Therefore, the TNG50 simulation is the best choice to derive the distribution of DM_host. However, as the data of TNG50 has not been released, we choose the data of TNG100 to perform the analysis, which has been released (Nelson et al., 2018; Pillepich et al., 2018b; Naiman et al., 2018; Nelson et al., 2018; Marinacci et al., 2018; Springel et al., 2018). The TNG100 includes three runs with different numbers of particles, which are called TNG100-1, TNG100-2, and TNG100-3, respectively. Among these three runs, we choose the TNG100-1, which contains the most particles. The side length of this run is 110.7 Mpc and it contains 6,028,568,000 dark matter particles. A good particle resolution can be achieved in this run. The initial conditions of this run are consistent with the Planck 2015 results (Planck Collaboration et al., 2016). The same cosmological parameters are also used in this paper.

Employing the friends-of-friends (FoF) algorithm, the dark matter particles in the TNG simulation are divided into different FoF halos (groups). In each halo, the particles are separated into subhalos using the subfind algorithm and the subhalos correspond to galaxies. For each subhalo, the TNG project provides detailed information, such as the stellar mass, SFR, gas metallicity, half mass radius, etc. This information can help us to select similar galaxies to the observed FRB host galaxies. Repeating FRBs and non-repeating FRBs may occur in different types of galaxies. According to mass and SFR, we select galaxies that are similar to the host galaxies of localized FRBs. The details will be shown in the following sections. The data of each subhalo is divided into several parts, including gas, dark matter, tracers, black holes, stars, and wind particles. We download the gas part of the selected subhalos. The TNG simulation uses Voronoi tessellation to construct the geometry of finite volume. The information of each Voronoi cell is given by the TNG simulation. According to the simulation, the electron number density of each cell can be derived.

The electron number density of the non-star-forming cells can be obtained as follows. The TNG simulation provides the information needed to calculate the electron number density. According to the simulation, the electron number density $n_{e}$ can be derived by

where $\eta_{e}$ is the electron abundance, $\rho$ is the mass density, $m_{H}$ is the mass of hydrogen atom, and $X_{H}$ is hydrogen abundance of each Voronoi cell. But for star-forming cells, the electron abundance $\eta_{e}$ given by the TNG simulation is not reliable. In this case, we consider the subgrid model proposed by Springel & Hernquist (2003), which is also adopted by the TNG simulation. According to this model, the multi-phase interstellar medium is comprised of a cold cloud and hot gas. In our calculation, we assume that the cold cloud is entirely neutral and the hot gas is totally ionized. This assumption has been used in many previous works (Pakmor et al., 2018; Stevens et al., 2019). The fraction of hot gas is calculated according to Springel & Hernquist (2003). The method has been discussed in Springel & Hernquist (2003) and the appendix of Stevens et al. (2019). Assuming that the hot gas is totally ionized, we derive the electron number density of star-forming cells. The electron distribution in one Voronoi cell is assumed to be uniform, so each region in one Voronoi cell shares the same electron number density. Therefore, the DM_host in the local frame can be written as

where $i$ refers to the $i$th Voronoi cell, $l_{i}$ is the length of the light travel in the $i$th cell, and $N$ is the total number of the cells which the light travels through.

Because the TNG simulation does not provide a clear boundary of subhalo, we use the following method to set the boundary of integration. When we try to get the DM_host of a subhalo, all the cells in parent halo are selected to analyze. The light of an FRB is emitted from a cell of this subhalo and propagates in a random direction. If the light travels into a cell which does not belong to this subhalo, we stop the integration.

The location of FRB in the host galaxy is important. Repeating FRBs and non-repeating FRBs may have different locations in their host galaxies. The positions of FRBs may strongly depend on the progenitor models. We will discuss the positions in detail later.

The TNG simulation contains 100 snapshots at different redshifts for each run. Among these snapshots, 20 snapshots are full and the remaining 80 are mini. Only the full snapshots are considered in our calculation. The current observations show that more than 90% of FRBs have the pseudo redshifts $z<1.5$. Thus, only the snapshots with $z<1.5$ are selected to analyze. According to these criteria, we select the snapshots with redshifts $z=0.1,0.2,0.3,0.4,0.5,0.7,1.0$ and $1.5$. In each snapshot, 1000 subhalos are selected to derive DM_host for repeating FRBs like FRB 121102. But for non-repeating FRBs and repeating FRBs like FRB 180916, in order to save calculation time, 200 subhalos are selected. For each subhalo, we simulate the positions of 500 FRBs and each 10 random propagating directions are simulated for each FRB. The details are discussed in the following sections. The evolution of DM_host as a function of redshift is also studied. In our calculation, we use PyTorch and GPU to accelerate the calculation.

So far, twenty repeating FRBs have been published (Spitler et al., 2014, 2016; CHIME/FRB Collaboration et al., 2019b, c; Fonseca et al., 2020). FRB 121102 is the first observed repeating FRBs. It has been localized to a dwarf galaxy with stellar mass about $4-7\times 10^{7}M_{\odot}$ and SFR about $0.4\,M_{\odot}$ yr^-1 (Tendulkar et al., 2017). Recently, repeating FRB 180916 was localized to a spiral galaxy (Marcote et al., 2020), which is dramatically different from FRB 121102. The stellar mass of this galaxy is about $10^{10}M_{\odot}$ and the SFR of the region around FRB 180916 is greater than 0.016 $M_{\odot}$ yr^-1. The properties of these two galaxies are dramatically different, so we will discuss them separately. In this section, we choose the host galaxy of FRB 121102 as a typical host galaxy. The case of FRB 180916 will be discussed in the next section. We select 1000 subhalos with the stellar mass $1-50\times 10^{7}M_{\odot}$ and the SFR $0.1-0.7\,M_{\odot}$ yr^-1. The total masses of the selected galaxies are also collected, which include the mass contributed by gas, stars, dark matter, and black holes. Among these components, dark matter is the main contributor. The median and the 1$\sigma$ error of the total mass are given in Table 2.

It is difficult to determine the location of FRB in the galaxy. Some models suggest that repeating FRBs originate from the newborn neutron stars (Lyubarsky, 2014; Metzger et al., 2017). Recently, a faint FRB 200428 has been discovered associated with a Galactic magnetar SGR 1935+2154 (The CHIME/FRB Collaboration et al., 2020; Bochenek et al., 2020). It supports the magnetar origin of FRBs. Newborn neutron stars are mainly produced by the core-collapse supernovae and the formation rate may trace the SFR. Therefore, we assume that the probability of FRBs occurring in a cell is proportional to the SFR. For simplicity, we also assume that FRBs only occur at the centers of cells. According to this scenario, 500 positions of FRBs are simulated in each selected galaxy. For each simulated FRB, ten propagation directions are randomly selected in its rest frame. As a result, about 5$\times 10^{6}$ DM_host are derived at each redshift. Considering the cosmological evolution, the observed DM_host can be computed. The DM_host distribution at $z=0.2$ is shown in Figure 1 as an example. As for the DM_host distributions at other redshifts, they have a similar shape but different parameters. We only list the medians and the 1$\sigma$ errors of these distributions in Table 2. Besides, the evolution of DM_host with redshifts is important. We show the median and the 1$\sigma$ error at each redshift in the left panel of Figure 2 as the blue line.

From Figure 1, it is obvious that the distribution of DM_host has a long tail. Due to this reason, the median of DM_host is more representative than the mean value. Tendulkar et al. (2017) obtained the DM_host of FRB 121102 between 55 and 225 $\mathrm{pc}\,\mathrm{cm}^{-3}$ at redshift $z=0.193$. We show this DM_host with the blue shaded region in Figure 1. In our simulation, the DM_host at $z=0.2$ is $42.81^{+50.73}_{-26.09}$ pc cm^-3, which is consistent with the observation at $1\sigma$ confidence level. It must be noted that the value of DM_source of FRB 121102 may be large. From observations, the rotation measure (RM) of FRB 121102 is very large, which is about 10⁵ rad m^-2 (Michilli et al., 2018). This large RM originates from the local environment, which suggests a large DM_source. The DM_host given by Tendulkar et al. (2017) includes the contribution from the environment around the source. Thus, the DM contributed by the host galaxy must be smaller. Meanwhile, it has been shown that the host galaxy of FRB 121102 is atypical among nearby FRBs (Li et al., 2019a). The possible DM_host of repeating FRB 180814 is given in their work, which is DM${}_{\rm{host}}<80$ pc cm^-3. This is also consistent with our results. The distribution of DM_host has a long tail, which indicates some light paths pass through the whole galaxy. The values of DMs along these directions are very large. We use the log-normal function to fit the distribution of DM_host,

where $\mu$ and $\sigma$ are free parameters. The mean and variance of this distribution are $e^{\mu}$ and $e^{(2\mu+\sigma^{2})}[e^{\sigma^{2}}-1]$, respectively. In Figure 1, the red line is the best-fitting result and the fitting parameters are listed in Table 3.

We also find that DM_host increase as redshifts increase. The evolution of the median of DM_host can be fitted by

The best fitting parameters are $A=34.72^{+17.77}_{-14.47}$ pc cm^-3, and $\alpha=1.08^{+0.87}_{-0.70}$. The fitting result is shown as the red dashed line in the left panel of Figure 2. In order to investigate this evolution, we show the comoving electron number density at different redshifts in Figure 3. At each redshift, we collect the electron number density of each cell for the selected subhalos. The red solid line and the red shaded region are the median and 1$\sigma$ uncertainty of the electron number density, respectively. The comoving electron number density decreases slowly with the increases of redshift. This evolution is so small that we can safely neglect it. Considering the cosmological evolution, the local electron number density at high redshift is much higher. Besides, we also should consider the factor $(1+z)$ when the local DM_host is converted to the observed one. These effects cause the redshift evolution of DM_host. We also count the number of gas cells in each subhalo and show the result in Figure 3. The blue dashed line is the median and the blue shaded region is the 1$\sigma$ error of the number. The number of cells remains constant for all redshifts. Thus, it has no contribution to the DM_host evolution.

The mass resolution of IllustrisTNG100 is $7.5\times 10^{6}~{}M_{\odot}$ in dark matter particles and $1.4\times 10^{6}~{}M_{\odot}$ in gas particles ¹¹1https://www.tng-project.org/about/. In our work, it is important to test the completeness of the halo catalogue we used at the low-mass end. To do this, we compare the measured halo mass function from the simulation with the analytical fitting formula proposed by Tinker (Tinker et al., 2008). As the Tinker mass function only works for a dark matter only simulation, in this test we use the IllustrisTNG100 dark matter only run instead of the full baryonic ones. Figure 4 compares the halo mass function of the IllustrisTNG100 simulation (black solid line) with the Tinker’s fitting formula (blue solid line). The results agree very well at the low mass end, which indicates that the resolved halos in the IllustrisTNG100 are complete down to $10^{9}M_{\odot}$.

The host galaxy of FRB 180916 is dramatically different from FRB 121102. It indicates that repeating FRBs can occur in various galaxies. In this section, we investigate the distribution of DM_host for the host galaxies like FRB 180916. This host galaxy is a spiral galaxy with stellar mass about $10^{10}M_{\odot}$ (Marcote et al., 2020). This repeating FRB was born in a star-forming region. The total SFR of this galaxy is not given in their work, but they give the SFR of a small region, which is greater than 0.016 $M_{\odot}$ yr^-1. In our analysis, the subhalos with stellar mass about $(0.1-10)\times 10^{10}M_{\odot}$ are selected. The total SFR of the host galaxy is not clear, therefore, we adopt a wide range of SFR 0.01-10 $M_{\odot}$ yr^-1. In order to save the calculation time, only 200 subhalos are selected for each redshift. Again, 500 FRB positions are simulated and 10 propagating directions are randomly selected for each FRB. FRB 180916 is born in a star-forming region. Therefore, we assume that the probability of FRBs borning in a cell traces the SFR of this cell. This assumption agrees with the case of repeating FRBs like FRB 121102. The distribution of DM_host at $z=0.1$ is shown in Figure 5 and the medians at different redshifts are shown in the middle panel of Figure 2 as the blue solid line. We also list the median and 1$\sigma$ range at different redshifts in Table 2. The shapes of distributions at other redshifts are similar to the distribution at $z=0.1$. The log-normal function (Equation 5) is used to fit the distribution of DM_host and the best-fitting parameters are listed in Table 3.

In this case, the DM_host at $z=0.1$ is about 110 pc cm^-3 and the 1$\sigma$ region is about $30-250$ pc cm^-3. Marcote et al. (2020) estimate the DM_host less than 70 pc cm^-3 for FRB 180916, which is consistent with our results at 1$\sigma$ level. In Figure 5, the blue shaded region denotes the DM_host of FRB 180916. Due to face-on view (Marcote et al., 2020), the DM_host of FRB 180916 is very small. Besides, we find that the DM_host is larger than the results of host galaxies like FRB 121102. It is caused by the larger subhalos. The subhalos with larger stellar mass are selected in this case. They have more cells and the light travel paths in host galaxies are much longer. The similar evolution of DM_host with redshifts is also found. We use Equation (6) to fit the evolution of DM_host. The red dashed line in the middle panel of Figure 2 is the best fitting with $A=96.22^{+50.10}_{-42.26}$ pc cm^-3, and $\alpha=0.83^{+0.87}_{-0.58}$. This evolution is also caused by the cosmological effect. Similar to the previous case, the galaxies with high redshifts are much denser and the electron number densities at high redshifts are much higher. Besides, the evolution factor $(1+z)$ needs to be considered when convert the local DM_host to the observed one. These cosmological effects cause the evolution of DM_host.

Recently, seven apparently non-repeating FRBs have been localized (Bannister et al., 2019; Ravi et al., 2019; Prochaska et al., 2019; Macquart et al., 2020). The host galaxies of these FRBs are similar. The stellar mass is in the range $10^{9}-10^{11}M_{\odot}$ and the SFR is in the range $0-2M_{\odot}$ yr^-1. Based on these observations, we select 200 subhalos with stellar mass between $10^{9}M_{\odot}$ and $2\times 10^{11}M_{\odot}$, and SFR in the range $0-2M_{\odot}$ yr^-1. Similar to repeating FRBs, the locations of non-repeating FRBs are important and difficult to determine. Their progenitor model is different from the repeating FRBs. There are many works suggesting that the non-repeating FRBs originate from the merger of compact binaries (i.e., Totani, 2013; Wang et al., 2016). According to these models, the positions of non-repeating FRBs should follow the mergers of compact binaries. From observations, the positions of these FRBs are far from the centers of host galaxies (Bannister et al., 2019; Ravi et al., 2019; Prochaska et al., 2019; Macquart et al., 2020), which also supports the conjecture that they are produced by the binary neutron star mergers. We adopt the results of Wang et al. (2020). Assuming that FRBs are produced by the mergers of binary neutron stars, they calculated the locations of FRBs in different types of galaxies through the population synthesis method (Wang et al., 2020). Because of the long merger time of binary neutron stars and the large kick velocity, the merger locations are far from the galaxy centers. We adopt their results for massive spiral galaxies with $10^{11}M_{\odot}$ case. According to their results, we simulate 500 positions of FRBs in each galaxy. In the rest frame of each FRB, 10 random propagating directions are simulated.

For non-repeating FRBs, we show the distributions of DM_host in Figure 6. The distributions at the redshifts which close to the observed FRBs are shown. The median and 1$\sigma$ error at all redshifts are listed in Table 2. The long tails are shown in these distributions, which is similar to repeating FRBs. The log-normal function (Equation 5) is used to fit these distributions and the fitting results are shown as the red lines in Figure 6. We also list the best-fitting parameters in Table 3. In this case, the stellar mass is similar to the case of repeating FRBs like FRB 180916, but the value of DM_host is different, which is caused by the different locations. For non-repeating FRBs, we simulate the positions according to Wang et al. (2020). Thus, the simulated FRBs are located at the edges of the galaxies and the DM_host is small. Based on the observations of host galaxies, some DM_host have been given in previous works (Ravi et al., 2019; Bannister et al., 2019; Chittidi et al., 2020). We show these values in Figure 6 with the blue shaded regions. Assuming a narrow luminosity distribution, Yang et al. (2017) gave the relation between DM_host, observed flux and luminosity. They found DM_host to be $267.00^{+172.53}_{-110.68}$ pc cm^-3, which is different from our results. This may be caused by their strong assumption on the luminosity distribution. They assumed that the luminosity function is very narrow. However, Luo et al. (2018) found that the luminosity function spans a large range. This wide range of luminosity function has also been found by Agarwal et al. (2019). Li et al. (2019a) also give similar results for some non-repeating FRBs. In their calculation, some possible DM_host for the selected FRBs are given, which are about $10-120$ pc cm^-3. We also show the relation between DM_host and redshift in the right panel of Figure 2 with the blue solid line. In this case, DM_host increases as redshift increases. We use equation 6 to fit this evolution. The fitting result is shown in the right panel of Figure 2 with the red dashed line and the best-fitting parameters are $A=32.97^{+23.23}_{-17.65}$ pc cm^-3 and $\alpha=0.84^{+0.93}_{-0.60}$. Like the previous cases, this evolution is also caused by the cosmological effects.

In this paper, we use the IllustrisTNG simulation to derive the distributions of DMs contributed by the host galaxies of FRBs. Based on the observations of localized FRBs, we select galaxies to calculate the host galaxy DMs of FRBs. For repeating FRBs like FRB 121102, we select the galaxies with stellar mass in the range $1-50\times 10^{7}M_{\odot}$ and SFR in the range $0.1-0.7\,M_{\odot}$ yr^-1. Assuming the formation rate of repeating FRBs in a given cell is proportional to the SFR of this cell, 500 positions of FRBs are simulated in each galaxy. For each FRB, ten random propagating directions are selected. We derive the distributions of $\mathrm{DM}_{\rm{host}}$ at different redshifts and show the result at z = 0.2 in Figure 1. The observed $\mathrm{DM}_{\rm{host}}$ of FRB 121102 is also shown in Figure 1 with the blue shaded region. In addition, the evolution of the median with redshifts is shown in Figure 2. We find that the value of $\mathrm{DM}_{\rm{host}}$ increases as redshift increases. We also select subhalos according to the host galaxy of repeating FRB 180916. The host galaxy of this FRB is dramatically different from FRB 121102. Therefore, we analyze it separately. The result is shown in Figure 5 and we compare our results with the observation. In this case, the $\mathrm{DM}_{\rm{host}}$ also increases as redshift increases, which is caused by the cosmological evolution.

For non-repeating FRBs, we select the galaxies with masses in the range $10^{9}-2\times 10^{11}M_{\odot}$ and SFRs in the range $0-2M_{\odot}$ yr^-1, which are similar to the observed non-repeating FRB host galaxies. Assuming that the burst sites are the locations of binary neutron star mergers, we generate 500 FRBs positions in each galaxy. The distributions of $\mathrm{DM}_{\rm{host}}$ are shown in Figure 6 and listed in Table 2. The value of $\mathrm{DM}_{\rm{host}}$ increases as redshift increases, which is caused by the cosmological evolution.

The effect of the selection criteria for galaxies is also discussed. We use a narrow range of the stellar mass and the SFR to investigate the distribution of DM_host. The distribution in this case is consistent with the results of the wide-parameters case. Thus, if the ranges of the stellar mass and the SFR are compatible with the observation, the width of parameter range would have little effect on the results. We also use our results to infer the pseudo redshifts of simulated FRBs and show that we can recover the distribution of simulated redshifts.