# Predictions for dispersion measures of fast radio bursts through the epoch of reionization using CoDa II

Joshua J Ziegler, Paul R Shapiro, Taha Dawoodbhoy, Paz Beniamini, Pawan Kumar, Katherine Freese, Pierre Ocvirk, Dominique Aubert, Joseph S.W Lewis, Romain Teyssier, et al.

<figure><img src="image_1.png" /></figure>

## ► To cite this version:

Joshua J Ziegler, Paul R Shapiro, Taha Dawoodbhoy, Paz Beniamini, Pawan Kumar, et al. Predictions for dispersion measures of fast radio bursts through the epoch of reionization using CoDa II. Monthly Notices of the Royal Astronomical Society, 2025, 1518, pp.1531. 10.1093/mnras/staf1308. hal-04796119

HAL Id: hal-04796119

https://hal.science/hal-04796119v1

Submitted on 5 Sep 2025

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Predictions for dispersion measures of fast radio bursts through the epoch of reionization using CoDa II

Joshua J. Ziegler<sup>1,★</sup> Paul R. Shapiro,<sup>2</sup> Taha Dawoodbhoy,<sup>3</sup> Paz Beniamini<sup>1,4,5,6</sup> Pawan Kumar,<sup>2</sup>

Katherine Freese,<sup>1,7,8</sup> Pierre Ocvirk<sup>1,9</sup> Dominique Aubert,<sup>9</sup> Joseph S. W. Lewis,<sup>9</sup> Romain Teyssier,<sup>10</sup>

Hyunbae Park,<sup>11</sup> Kyungjin Ahn,<sup>12</sup> Jenny G. Sorce<sup>1,13,14,15</sup> Ilian T. Iliev<sup>1,16</sup> Gustavo Yepes<sup>1,17,18</sup>
and Stefan Gottlöber<sup>15</sup>

Accepted 2025 August 5. Received 2025 August 4; in original form 2024 November 6

## ABSTRACT

Dispersion measures (DM) of fast radio bursts (FRBs) probe the density of electrons in the intergalactic medium along their lines of sight, including the average density versus distance to the source and its variations in direction. While previous study focused on low redshift (e.g. $z \lesssim 5$), FRBs are potentially detectable out to high redshift, where their DMs can, in principle, probe the epoch of reionization (EoR) and its patchiness. We combine the cosmic density and ionization fields from large-scale radiation-hydro simulation Cosmic Dawn (CoDa) II of fully coupled galaxy formation and reionization to $z = 5.8$ (where reionization ended at $z_{re} \approx 6.1$), with those from N-body simulation CoDa II–Dark Matter for the fully ionized post-EoR, from $z = 5.8$ to the present. By calculating the mean and standard deviation of FRB DMs as functions of their source redshift, we are able to quantify the combined effect of self-consistent inhomogeneous density and ionization fields on FRB DMs for the first time. The mean and standard deviation of DM increase with redshift, and, assuming the CoDa reionization history, both reach a plateau by $z(x_{HII} \lesssim 0.25) \gtrsim 8$, i.e. well above $z_{re}$. The mean-DM asymptote $DM_{max} \approx 5900 \text{ pc cm}^{-3}$ reflects the end of the EoR and its duration. The standard deviation there is $\sigma_{DM,max} \approx 497 \text{ pc cm}^{-3}$, reflecting inhomogeneities of both patchy reionization and density. Inhomogeneities in ionization during the EoR contribute $O(1 \text{ per cent})$ of this value of $\sigma_{DM,max}$ from FRBs at redshifts $z \gtrsim 8$. Current estimates of FRB rates suggest this may be detectable within a few years of observation.

**Key words:** HII regions – dark ages, reionization, first stars – fast radio bursts.

## 1 INTRODUCTION

Photons emitted during a fast radio burst (FRB) with different wavelengths undergo different amounts of plasma dispersion, causing

them to be observed at different times. By measuring the spread of these arrival times, it is possible to determine the integrated electron density along the line of sight (LOS) from the FRB source to observation on Earth (Cordes & Chatterjee 2019; Petroff, Hessels & Lorimer 2019). Due to their high luminosity, FRBs are detectable out to great distance and correspondingly large look-back time, with some 50 FRBs localized to source galaxies at redshifts up to

© The Author(s) 2025.

Published by Oxford University Press on behalf of Royal Astronomical Society. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

<footnote n="1"><sup>1</sup>Department of Physics, The University of Texas at Austin, Austin, TX 78712, USA</footnote>

<footnote n="2"><sup>2</sup>Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA</footnote>

<footnote n="3"><sup>3</sup>Department of Physics, California Polytechnic State University, San Luis Obispo, CA, 93407, USA</footnote>

<footnote n="4"><sup>4</sup>Department of Natural Sciences, Astrophysics Research Center of the Open University (ARCO), The Open University of Israel, P.O Box 808, Ra'anana, 4353701, Israel</footnote>

<footnote n="5"><sup>5</sup>Department of Natural Sciences, The Open University of Israel, P.O Box 808, Ra'anana, 4353701, Israel</footnote>

<footnote n="6"><sup>6</sup>Department of Physics, The George Washington University, 725 21st Street NW, Washington, DC 20052, USA</footnote>

<footnote n="7"><sup>7</sup>Department of Physics, The Oskar Klein Centre, Stockholm University, AlbaNova, SE-10691, Stockholm, Sweden</footnote>

<footnote n="8"><sup>8</sup>Nordic Institute for Theoretical Physics (NORDITA), SE-10691, Stockholm, Sweden</footnote>

<footnote n="9"><sup>9</sup>Université de Strasbourg, CNRS, Observatoire astronomique de Strassbourg, Université de Strasbourg, UMR 7550, F-67000, Strasbourg, France</footnote>

<footnote n="10"><sup>10</sup>Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08540, USA</footnote>

<footnote n="11"><sup>11</sup>Center for Computational Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba 305-8577, Ibaraki, Japan</footnote>

<footnote n="12"><sup>12</sup>Department of Earth Sciences, Chosun University, 375 Seosuk-dong, Dong-gu, Gwangju 501-759, Korea</footnote>

<footnote n="13"><sup>13</sup>UMR 9189 CRISTAL, Université de Lille, CNRS, Centrale Lille, F-59000 Lille, France</footnote>

<footnote n="14"><sup>14</sup>Institut d'Astrophysique Spatiale, Université Paris-Saclay, CNRS, F-91405 Orsay, France</footnote>

<footnote n="15"><sup>15</sup>Leibniz-Institut für Astrophysik Potsdam (AIM), An der Sternwarte 16, D-14482 Potsdam, Germany</footnote>

<footnote n="16"><sup>16</sup>Astronomy Center, Department of Physics and Astronomy, University of Sussex, Pevensey II Building, Falmer, Brighton BN1 9QH, UK</footnote>

<footnote n="17"><sup>17</sup>Departmento de Física Teórica M-8, Universidad Autónoma de Madrid, Cantoblanco, E-28049 Madrid, Spain</footnote>

<footnote n="18"><sup>18</sup>Centro de Investigación Avanzada en Física Fundamental (CIAFF), Facultad de Ciencias, Universidad Autónoma de Madrid, Cantoblanco, E-28049 Madrid, Spain</footnote>

<footnote>*E-mail: jjziegler@utexas.edu</footnote>
$z = 1.017$ (Gordon et al. 2023; Lorimer, McLaughlin & Bailes 2024) and some hundreds to thousands of unlocalized FRBs including several believed to be produced at higher redshifts (Lorimer et al. 2024). Furthermore, while the sources of FRBs are not fully understood, most models identify black holes or neutron stars (especially magnetars) as likely candidates (Rees 1977; Popov & Postnov 2013; Katz 2016; Kumar, Lu & Bhattacharya 2017; Vieyro et al. 2017; Gupta & Saini 2018; Margalit & Metzger 2018; Wadiasingh & Timokhin 2019; Kumar & Bošnjak 2020; Beniamini & Kumar 2023). Because both black holes and neutron stars form primarily through the supernova of massive stars, FRBs may be present wherever stars form. If this is the case, measurements of the dispersion measures of FRBs can probe the density of ionized gas throughout most of the Universe's history, from the formation of the first stars and the epoch of reionization (EoR) to the present. In addition, since the stars believed responsible for most of the emission of the UV radiation that caused reionization are also the stars that undergo the supernovae that produce neutron stars and black holes (Wise 2019), it is a natural expectation that FRBs were emitted during the EoR.

For distant FRBs, the intergalactic medium (IGM)<fnref n="1" /> is expected to be the dominant contributor to the dispersion measure (Zhu & Feng 2021), and as such, studying the dispersion measures of these distant FRBs could allow us to probe the ionization history of the IGM.<fnref n="2" /> In particular, statistical variations in the dispersion measure could tell us something about the rise of structure in the universe and the resulting inhomogeneous nature of reionization (Beniamini et al. 2021). The simplest case would be if sufficient numbers of FRBs could be localized to a host galaxy so that their redshifts were known. However, even without FRB redshifts, identifiable properties in the distribution of dispersion measures could also be used. In either case, observations of FRB dispersion measures could be combined with modelling of the theoretical dependence of FRB dispersion measures on redshift to better understand and constrain the patchiness of reionization and the growth of structure more generally.

In what follows, we compute the dispersion measure through the universe along different LOSs, based upon cosmological simulations, to determine the mean and variance of this quantity as functions of redshift. As we are most interested in providing a proper account of the contribution from the EoR, including its reionization patchiness, for the first time, we take advantage of a recent fully coupled large-scale high-resolution radiation-hydrodynamical simulation of galaxy formation and reionization during the EoR by the Cosmic Dawn (CoDa) Project, the CoDa II simulation (Ocvirk et al. 2020), down to redshift 5.8, past the end of the EoR,<fnref n="3" /> for our analysis. By this redshift, the universe was almost fully ionized and, to a good approximation, may be assumed to be fully ionized, thereafter, with

the matter density field and the ionized gas density field tracing each other on large scales. To extend the LOS integrals of dispersion measure from $z = 5.8$ to $z = 0$ therefore we use a dark-matter-only companion simulation to CoDa II, an N-body simulation called CoDa II–Dark Matter, which was run with the same initial conditions as CoDa II (Ocvirk et al. 2020).

Previous work applied earlier cosmological simulations of varying degrees of physical realism to compute the FRB dispersion measures and their properties. Most of these previous analyses calculate dispersion measures from simulation results to redshifts below $z \simeq 6$, preventing them from probing the EoR. McQuinn (2013) used the results of a simulation by Faucher-Giguère, Kereš & Ma (2011) to model the mean and standard deviation of the dispersion measure out to redshift 1.4, and compared this to predictions from dark-matter-only halo density functions and simple top-hat models of structure formation. Dolag et al. (2015) used the Magneticum Pathfinder simulations to calculate the dispersion measures out to redshift 2. Jaroszynski (2019) used the results of the Illustris-3 simulation (Nelson et al. 2015) to calculate dispersion measures out to redshift 5. Pol et al. (2019) used the results of the MICE simulation (Fosalba et al. 2008) to calculate dispersion measures out to redshift 1.4. Batten et al. (2021) used the EAGLE simulation (Schaye et al. 2015) to calculate dispersion measures out to redshift 3. Walker et al. (2024) used the IllustrisTNG simulation (Nelson et al. 2019) to calculate dispersion measures out to redshift 5. Each of these simulation efforts involve different gas and dark matter mass resolutions, as well as different simulation volume sizes, which may contribute to slightly different results. In contrast to the studies detailed above, Zhang et al. (2021) do calculate the dispersion measure to a redshift of $z = 9$ using IllustrisTNG-300 data, and consequently are able to explore the EoR. Our current analysis differs from this previous work in the treatment of radiation in the two underlying simulations. Because the CoDa II simulation models UV radiation self-consistently, we can study the effects of the spatial inhomogeneities (patchiness) on the FRB dispersion measure. We also consider a larger range of redshift (out to $z \gtrsim 12$). Finally, a recent paper complements the work done in this paper by exploring more in depth how FRB dispersion measures in the EoR could offer a probe of the nature of ionization patches (Shaw et al. 2024). While the authors of this paper do consider numerical simulations to validate semi-analytic models, these numerical simulations are mostly limited to redshifts $z \ge 6$.

In addition to numerical simulations that tend to cover the universe predominantly after the EoR, there has also been recent work seeking to forecast the extent to which FRBs can provide information about the EoR using semi-empirical models to construct synthetic distributions of FRBs. For example, Pagano & Fronenberg (2021) are able to estimate the number of high-redshift (during the EoR) FRBs that would be needed to rule out certain reionization scenarios. Likewise, Heimersheim et al. (2022) estimate how many FRBs would be needed to constrain the optical depth and redshift at which the universe is half-ionized. Maity (2024) shows that combining high-redshift FRB observations with large-scale 21 cm data can provide an estimate for the shape of the ionization history and the redshift at which it occurs with a per cent-level precision with 100–1000 FRBs. Because these works rely upon semi-empirical methods, our current analysis complements these results.

In this paper, we compute the mean and standard deviation of the dispersion measure to FRBs as a function of the source redshift of those FRBs. In Section 2, we discuss the dispersion measure of cosmologically distant FRBs, and in Section 3, we describe how we can use the results of the CoDa II simulations to compute these

<footnote n="1">¹When discussing dispersion measures, it is often practical to consider contributions from the Milky Way, source galaxies, and everything in between. This latter category is what we identify as IGM, even though it also includes any haloes intervening between the source and Milky Way galaxies.</footnote>

<footnote n="2">²An example of an alternate way to constrain the ionization history of the IGM, using CMB data rather than FRBs, can be found in Paoletti et al. (2025) and references therein.</footnote>

<footnote n="3">³The specific definition used for the end of the EoR is not universal, but is usually based upon a certain value of the globally averaged neutral fraction of intergalactic hydrogen. In CoDa II, the EoR ends at $z = 6.1$, where the neutral fraction ends its sharp, exponentially steep drop from 0.99 at $z = 6.2$ to $10^{-4.6}$ at $z = 6.1$. (Ocvirk et al. 2020)</footnote>
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Globally averaged H ionized fraction, $\langle x_{HII} \rangle$, versus redshift $z$, from the CoDa II simulation. The black points show the hydrogen ionization fraction of snapshots at different redshifts from the CoDa II simulation, averaged over the simulation volume. The curve connecting them represents a cubic spline fit to these points. In this model, reionization is largely completed around redshift $z \approx 6$, and the universe is half-ionized around redshift $z \approx 7$.</figcaption></figure>

dispersion measures. We present our results in Section 4, particularly focusing on the mean, standard deviation, and ratio of mean to standard deviation of the dispersion measures we calculate. One of our primary results is that the patchiness/inhomogeneity of ionization during the EoR contributes an effect that is distinguishable at the $\mathcal{O}(1 \text{ per cent})$ level among FRBs from sources at redshifts $z \gtrsim 8$. Finally, in Section 5, we discuss how our results might change if the ionization history assumed in the CoDa II simulations were to be altered. We conclude in Section 6.

## 2 ARRIVAL TIME DELAY OF FRB SIGNALS AT COSMOLOGICAL DISTANCES

The presence of free electrons of number density $n_e$ along the LOS to a distant FRB introduces a frequency-dependent time delay, $t_d(v)$, in the arrival of the FRB signal observed at frequency $v$, relative to its arrival time in vacuum, due to plasma dispersion. If the source is located a distance $D$ from the observer, in a static medium and in the absence of cosmological redshift, the rate of change of this time delay with respect to observed frequency is given by

$$ \frac{dt_d}{dv} = - \frac{e^2}{\pi m_e c} \frac{\mathcal{D}M}{v^3}, \qquad (1) $$

where $e$ is the electric charge of the electron, $m_e$ is the mass of the electron, and $c$ is the speed of light in vacuum, and the dispersion measure $\mathcal{D}M$ is defined by (Petroff et al. 2019)

$$ \mathcal{D}M = \int_{0}^{D} n_e dl, \qquad (2) $$

where $dl$ is the differential distance along the LOS.

When we take into account cosmological effects, the dispersion measure can be rewritten as (Deng & Zhang 2014)

$$ \mathcal{D}M = c \int_{0}^{z_s} n_e \frac{1+z}{H(z)} dz, \qquad (3) $$

where we compute the electron density in comoving coordinates. The integral along the LOS is replaced with a corresponding integral over

redshift, from $z=0$ to the $z=z_s$, the redshift at which an FRB is emitted. The function $H(z)$ is the Hubble constant at a given redshift, which we approximate according to

$$ H^2 = H_0^2 [\Omega_{m,0}(1+z)^3 + \Omega_{\Lambda}]. \qquad (4) $$

Here, $H_0$ is the local Hubble constant, and $\Omega_{m,0}$ and $\Omega_{\Lambda}$ are the matter and dark energy density parameters evaluated at redshift $z=0$: $\Omega_i = \rho_i/\rho_c$ for the mass density $\rho_i$ in each component $i$, and critical density necessary to close the universe $\rho_c$.

For our purposes, it is useful to further rewrite the electron density. The free electron number density $n_e(z, \hat{n})$ at redshift $z$ along the LOS in direction $\hat{n}$ is related to the mean number density of H atoms at present, $n_{H,0}$, the number of free electrons per H atom, $x_e$, and the fractional overdensity, $\delta_H$, of H atoms at that location along the LOS, according to

$$ n_e(z, \hat{n}) = x_e(z, \hat{n}) n_{H,0} (1+z)^3 (1+\delta_H(z, \hat{n})). \qquad (5) $$

The mean number density of hydrogen atoms at redshift $z=0$ is defined as

$$ n_{H,0} = \left( \frac{3H_0^2 \Omega_{b,0}}{8\pi G m_p} \right) X, \qquad (6) $$

where $\Omega_{b,0}$ is the present density parameter of baryons and $X$ is the mass fraction of the baryons that are hydrogen. The number of free electrons will vary over redshift as a function of the fraction of hydrogen atoms that have ionized and of the fraction of helium atoms that have ionized singly or doubly:

$$ x_e = \frac{1}{X} \left[ X x_{HII}(z, \hat{n}) + \frac{1}{4} Y y_{HII}(z, \hat{n}) + \frac{1}{2} Y y_{HIII}(z, \hat{n}) \right]. \qquad (7) $$

Here, $X=0.76$ and $Y=0.24$ are the mass fractions of hydrogen and helium, respectively, and the three fractions $x_{HII}$, $y_{HII}$, and $y_{HIII}$ are the ionization fraction of hydrogen, singly ionized helium, and doubly ionized helium, respectively. We assume that the second ionization of helium occurs instantaneously at redshift $z=3$. Therefore, prior to $z=3$, $y_{HIII}=0$ and it is expected that $x_{HII} \simeq y_{HIII}$. Conversely for $z \le 3$, $x_{HII} = y_{HIII} = 1$. Throughout this paper, we present ionization fractions as the hydrogen ionization fraction $x_{HII}$ (which CoDa papers refer to as the mass-weighted ionized fraction $x_m$), and use these relations between the hydrogen and helium ionization fractions to compute the ionization fraction of helium. Using this convention, we can relate the hydrogen ionization fraction to the electron fraction per hydrogen atom $x_e$ as

$$ x_e = \begin{cases} 1.16x_{HII} & z \le 3 \\ 1.08x_{HII} & z > 3 \end{cases}. \qquad (8) $$

The history of the globally averaged H ionization fraction in the CoDa II simulation is plotted in Fig. 1.

Finally, by combining this with equation (3), we can write

$$ \begin{aligned} \mathcal{D}M(z_s, \hat{n}) ={}& cn_{H,0} \\ & \times \int_0^{z_s} dz \, x_e(z, \hat{n}) [1 + \delta_H(z, \hat{n})] \frac{(1+z)}{H(z)}. \end{aligned} \qquad (9) $$

Henceforth in what follows, we shall refer to the cosmological quantity defined in equation (9) when we use the term ‘dispersion measure’, so there is no confusion with the non-cosmological quantity for a static, non-expanding medium.

As a comparison for our later results, we can estimate the mean dispersion measure we would expect as a function of redshift, if the universe were fully ionized at all redshifts. In doing so, we
<table><caption>**Table 1.** Comparison of CoDa II and CoDa II–Dark Matter Simulation and Analysis Parameters. Throughout the paper, we use two simulations from the CoDa collaboration (Ocvirk et al. 2020). The simulation labelled as ‘CoDa II–Dark Matter’ is a dark-matter-only *N*-body simulation performed using the GADGET-2 code. Meanwhile, the simulation labelled as ‘CoDa II’ includes baryons as well as dark matter, and simulates hydrodynamics, radiative transfer, and non-equilibrium ionization using the RAMSES-CUDATON code. Both of these simulations use the same initial conditions, which were selected to produce an arrangement of galaxies like the local LSS including the Local Group at redshift $z = 0$ (Sorce & Tempel 2018). Some of the numerical parameters used in these two simulations are presented here. Of particular note, while the CoDa II simulation is performed at a higher resolution (4096³), it was smoothed for analysis to a coarsened resolution consistent with that of the CoDa II–Dark Matter simulation (2048³). In addition, the CoDa II simulation was unable to simulate to redshifts below $z = 5.8$, while the CoDa II–Dark Matter simulation was able to continue to redshift $z = 0$.</caption><thead><tr><th>Simulation</th><th>CoDa II</th><th>CoDa II–Dark Matter (CoDa II–DM2048)</th></tr></thead><tbody><tr><td>Redshift range</td><td>5.8–150</td><td>0–150</td></tr><tr><td>Evolution type</td><td>baryonic rad-hydro and dark matter gravity</td><td>dark-matter-only gravity</td></tr><tr><td>Box length</td><td>94.4 cMpc</td><td>94.4 cMpc</td></tr><tr><td>Number of cells (simulation)</td><td>4096³</td><td>−NA−</td></tr><tr><td>Number of cells (analysis)</td><td>2048³</td><td>2048³</td></tr><tr><td>Number of (dark matter) particles</td><td>4096³</td><td>2048³</td></tr></tbody></table>

use the values of cosmological parameters that we use throughout the paper, based on the choices made in the CoDa II simulations (Planck Collaboration XVI 2014; Ocvirk et al. 2020). In particular, the matter density parameter at the present day $\Omega_{m,0} = 0.307$, the dark energy density parameter $\Omega_{\Lambda} = 0.693$, the baryonic matter density parameter at the present day $\Omega_{b,0} = 0.048$, and the Hubble constant at the present day $H_0 = 67.7 \text{ km s}^{-1} \text{ Mpc}^{-1}$. With these values, for all redshifts where baryons can be approximated as fully ionized (i.e. at lower redshifts than the end of the EoR), equation (9) could be rewritten as

$$DM = 1741 \text{ pc cm}^{-3} \int_{0}^{z_s} \frac{1+z}{\sqrt{(1+z)^3 + 2.26}} dz. \quad (10)$$

### 3 METHODOLOGY

To evaluate the dispersion measure in equation (9) for a given LOS to an FRB, we must know the time- and spatially varying values of the 3D ionized fraction and overdensity fields, $x_e(r, t)$ and $\delta_H(r, t)$. For this, we use the detailed results of the large-scale radiation-hydrodynamical simulation of galaxy formation and reionization CoDa II (Ocvirk et al. 2020) through the end of reionization, down to $z = 5.8$. For the redshift range from $z = 5.8$ to $z = 0$, we use the collisionless, *N*-body simulation CoDa II–Dark Matter, which was run from the same initial conditions as CoDa II (i.e. the same linear-perturbation, plane-wave fluctuation density and velocity modes)(Ocvirk et al. 2020). Some details of the CoDa II and CoDa II–Dark Matter simulations can be found in Table 1. In CoDa II, the atomic gas density and ionized fraction fields were computed by modelling fully coupled galaxy formation and reionization self-consistently, solving the gas dynamical conservation equations, along with explicit treatment of radiative transfer of H-ionizing UV starlight and non-equilibrium ionization rate equations. In CoDa II–Dark Matter, the dynamics of collisionless dark matter was treated by a standard cosmological *N*-body method. Nevertheless, we can use the results from the CoDa II–Dark Matter simulation to model the electron density fields over redshifts $z < 5.8$. As $z = 5.8$ is later than the end of reionization ($z = 6.1$ as computed by the CoDa II simulation), we can assume that the hydrogen gas is fully ionized at all later redshifts, and the baryonic and electron density fields closely follow the total matter density field of the *N*-body simulation.

The CoDa II simulation was performed in a comoving cubic box of size 94.4 cMpc, utilizing 4096³ *N*-body particles of dark matter and a fixed Eulerian grid of 4096³ cells for both the baryonic gas and radiation fields. However, to reduce storage requirements, the raw simulation output was smoothed to a coarser grid of 2048³ cells. The CoDa II–Dark Matter simulation, meanwhile, is performed in the same 94.4 cMpc box, but utilized only 2048³ particles, and the particle data were used to create a grid-based density field. As a result, the data we used from either simulation’s output were in the form of a 2048³ grid-based field.

While not directly relevant to the analysis in this work, we can evaluate the CoDa II simulation’s performance against observations to justify its use in this analysis (Ocvirk et al. 2020). For example, the CoDa II optical depth is consistent with Planck 2018 data (Aghanim et al. 2020). Furthermore, because CoDa II simultaneously models star formation, we can compare star formation rates against observations by, for example, Bouwens et al. (2015), and show agreement. In addition, the UV luminosity function shows consistency with a number of observations (Bouwens et al. 2015, 2017; Finkelstein et al. 2015; Atek et al. 2018). Further discussions on the consensus that CoDa II can reach with observations can be found in Ocvirk et al. (2020). Based on the consistency between CoDa II simulation results and observations, we will treat the CoDa II ionization history as encapsulating a generally realistic ionization history throughout the remainder of this paper.

Since the analysis underlying this project was performed, the CoDa team has completed archival of a newer simulation, ‘CoDa III’. CoDa III improves upon CoDa II in multiple ways, including increased spatial (by a factor of 2) and mass resolution (by a factor of 8). However, this improvement resulted in higher storage requirements for each snapshot, and consequently fewer snapshots being archived. By using the CoDa II recorded snapshots, which were more finely spaced in time, we were able to update the evolution of the electron density more frequently, reducing a source of error throughout our calculations. However, one result of CoDa III is a later end to reionization ($z \sim 5.6$ for CoDa III compared to $z \sim 6.1$ for CoDa III). As we discuss later, the result of a later end to reionization generally is to lower the plateau values of the mean and standard deviation of the dispersion measure.

In order to calculate statistical properties of the dispersion measure from FRBs, we construct multiple LOSs from the CoDa II and CoDa II–Dark Matter simulation results. Specifically, we construct
2 million LOSs, but the results we describe in the next sections are relatively insensitive to the precise number of lines constructed.<fnref n="4" /> We construct each of these lines by following the path a photon would take through density and hydrogen ionization fraction fields generated using the CoDa simulations. Along this path, a photon can travel at most the box length, 94.4 Mpc, before it would sample a patch of the simulation volume that it has sampled previously. Without correction, this would introduce an unphysical correlation between regions of space that are separated by almost 100 Mpc. To minimize these correlations, each time light has travelled 94.4 Mpc, we randomize its direction so that it samples a new set of density and hydrogen ionization fractions. This establishes a natural segmentation of the light travel path into box-length pieces, and we can associate the ends of these segments with redshifts $z_{i,box}$ through the equation $L_{box} = c \int_{z_{i-1,box}}^{z_{i,box}} \frac{1}{(1+z)H(z)} dz$.

The CoDa simulations output density and ionization fields across the entire simulation volume at specific times (and corresponding redshifts). This introduces two challenges to directly using the CoDa outputs as the basis for the box-length segments of a photon path. First, the universe's evolution is continuous, while relying on the CoDa snapshots allows only discrete steps in the evolution. This effect increases with redshift, but even at the highest redshifts we consider, it contributes an error of less than 1 per cent. Furthermore, any error like this would be systematic to all of our results, so while it might slightly impact the overall magnitude of the dispersion measures we calculate, the impact on any comparisons will be even smaller. As a result, in our calculations, we ignore any effects of the discrete sampling of evolution. With that in mind, we can partially account for the cosmological expansion of the gas by describing the atomic H density at a redshift $z$ within one of the box-length segments as

$$n_H = n_{H,0}(1+z)^3(1+\delta_H), \quad (11)$$

where $\delta_H$ is the overdensity from the non-evolving snapshot data.

The second challenge introduced by using the outputs from the CoDa simulations is that those outputs are not aligned with the list of redshifts $z_{i,box}$ that define the box-length segments. In particular, the CoDa II snapshots are recorded more frequently than the segments ($\Delta z_{i,box} > \Delta z_{i,CoDaII}$), while the CoDa II–Dark Matter snapshots are recorded less frequently ($\Delta z_{i,box} < \Delta z_{i,CoDaIIIDM}$). Visually, this comparison can be represented in Fig. 2, which represents each snapshot as a box whose endpoints correspond to the redshift the snapshot is recorded and the redshift for which the horizon was larger smaller by one box-length. That is, in this figure, the redshift difference $\Delta z_i$ can be seen as the difference between the left sides of successive boxes. We treat each of these cases slightly differently, but with the common goal of generating for each box-length segment (between successive redshifts $z_{i,box}$), a distribution of contributions to the dispersion measure. More details on this process can be found in Appendix A.

## 4 DISPERSION MEASURES FROM THE CODA II AND CODA II–DARK MATTER SIMULATIONS

In this section we discuss the results of our simulations for FRBs in the redshift range $0 < z < 12$. In Fig. 3, we show the mean

<figure><img src="image_4.png" /><figcaption>**Figure 2.** Diagram comparing the time between CoDa II simulation snapshots and those in the CoDa II–Dark Matter simulation. From the CoDa II simulation, we averaged over multiple simulated snapshots in the time it takes light to cross any one simulation volume, while from the CoDa II–Dark Matter simulation, we interpolated between two consecutive simulation snapshots to construct a synthetic snapshot. Therefore, we approach the process of creating synthetic snapshots based off of these two simulations in slightly different manners, described in detail in the main text and shown diagrammatically in the sequences labelled ‘CoDa II’ and ‘CoDa II–Dark Matter’. Ultimately, our goal in both cases is to generate a series of synthetic snapshots based off the simulation results but spaced in time in such a way that the time between snapshots is exactly the time it takes light to travel across the simulation volume, as depicted in the sequence labelled ‘Target’.</figcaption></figure>

of the dispersion measures calculated from the column densities constructed above, and in Fig. 4, we show the standard deviation. In both figures, the solid blue curve shows the results from the FRB dispersion measures calculated using the CoDa II and CoDa II–Dark Matter simulations (using CoDa II–Dark Matter for $0 < z < 5.8$ and CoDa II for all higher redshifts). As a comparison, the red dot–dashed curves show the results of dispersion measures constructed again using the CoDa II and CoDa II–Dark Matter simulations, but where we have averaged the ionization fraction $x_{HII}$ over space in each of the CoDa II snapshots. As a result, the solid blue curve shows results that include inhomogenity (patchiness) of the ionization and the red dot–dashed curve shows results that assume homogeneous reionization and therefore include inhomogeneities only in the baryon density, and by comparing the two, we can infer the impact of the ionization inhomogeneities on the observable mean and standard deviation of the dispersion measures. As an addition comparison, the orange dashed curve shows results from the CoDa II–Dark Matter simulations alone. The latter simulations assume that the universe is fully ionized with hydrogen ionization fraction $x_{HII} = 1$ at all redshifts considered.<fnref n="5" />

The dominant feature that can be seen when comparing the means of the dispersion measures is the plateau that occurs when full ionization is not assumed. This is expected naturally because the dispersion measure is sensitive only to the density of charged particles, so when the universe is largely neutral the dispersion measure does not increase noticeably. The dispersion measure only begins to vary once appreciable reionization occurs, allowing the location of the plateau to be a useful measure of when the EoR

<footnote n="4">⁴To establish this, we calculate the mean and standard deviation from a numbers of LOSs, ranging from 10 000 to 4 million. The largest deviation, generated by using 10 000 LOSs, showed a deviation from the results of Fig. 4 of approximately 2 per cent at most.</footnote>

<footnote n="5">⁵We have also studied the case of CoDa II with $x_{HII} = 1$ everywhere to be able to compare the impact of CoDa II–Dark Matter versus CoDa II baryon densities, and found negligible differences compared to the difference due to the effect of the different ionizations shown in the figures.</footnote>
<figure><img src="image_5.png" /></figure>

**Figure 3.** Mean of dispersion measures modelled from CoDa II and CoDa II–Dark Matter simulations. Using the CoDa II and CoDa II–Dark Matter simulations, we can construct distributions of the dispersion measures from FRBs at different locations on the sky. Here, we plot the mean of this distribution as a function of the FRB source redshift $0 \le z < 12$. The solid blue curve shows the mean of the dispersion measures constructed from the CoDa II and CoDa II–Dark Matter simulation results without any modifications. The orange dashed curve is the mean constructed using the CoDa II–Dark Matter dark matter density but assuming full ionization at all redshifts we consider. The red dash-dotted curve is the mean constructed using the CoDa II and CoDa II–Dark Matter simulation results, but spatially homogenizing the ionization fraction, so that the average ionization history $\langle x_{HII} \rangle(z)$ is that of the CoDa II simulation, but without patchiness. As a result, by comparing the blue solid and red dash-dotted curves we can isolate which effects arise from incomplete ionization at high redshift, and which arise from the inhomogeneity of this partial ionization. The shaded regions represent the $1\sigma$ standard deviations calculated for each distribution. The means in these three cases are indistinguishable from $z=0$ out to the EoR, at which point a plateau appears in the CoDa II patchy and non-patchy reionization cases, where the globally averaged hydrogen ionized fraction drops below unity at $z \gtrsim 6$ (as shown in Fig. 1). We also plot the results from Jaroszynski (2019) as a comparison (black points) found using the data from the Illustris simulation and the results from Walker et al. (2024) (grey squares) and Zhang et al. (2021) (grey diamonds) found using data from the IllustrisTNG data. A slight difference between the results in Jaroszynski (2019) and the three results we construct for the mean arises because we include a period of helium reionization at redshift $z=3$. A more significant difference between the means from Zhang et al. (2021) and the means we calculate at redshifts higher than approximately $z \gtrsim 6$ seems indicative of a different ionization history during CoDa between the CoDa II and Illustris-TNG simulations.

occurs. Using the CoDa simulation results, the dispersion measure of this plateau is approximately $DM_{max} = 5900$ pc cm$^{-3}$.

Further, we plot results (black points) from the work of Jaroszynski (2019) found using data from the Illustris-3 simulation to a redshift of $z=5$. Comparing the mean dispersion measure curves from our work (solid blue and dashed orange curves) to the results of Jaroszynski (2019) also shows a small discrepancy that is to be expected. One of the assumptions we made was that helium reionizes from He II to He III at $z=3$. At earlier times ($z>3$), the number of free electrons contributed by each helium atom was assumed lower (since ionized H was accompanied by singly ionized He then), so our dispersion measure was likewise reduced compared with its value if He were assumed to be doubly ionized wherever H was ionized. The results in Jaroszynski (2019) assume He was doubly ionized over the entire range of redshifts, including $z>3$, so they show a

<figure><img src="image_6.png" /></figure>

**Figure 4.** Standard deviation of dispersion measures modelled from CoDa II/CoDa II–Dark Matter simulations. The blue solid, orange dashed, and red dash-dotted curves represent the same set of dispersion measures as in Fig. 3, but here the standard deviation rather than mean is plotted. The large difference between the standard deviation in the three results we construct and those from Jaroszynski (2019) seems to arise primarily from simulation differences between the CoDa II–Dark Matter and Illustris-3 simulations as well as differences in the analyses used in this work and that used in Jaroszynski (2019) (black dots). The standard deviation constructed from Illustris-TNG data (Walker et al. 2024) (grey squares) is more consistent with the results we find in this work, while the standard deviation from Zhang et al. (2021) (grey diamonds, also from Illustris-TNG data) shows consistency with this work for redshifts less than approximately $z \lesssim 4$ at which point the standard deviation constructed from the data of Zhang et al. (2021) remains relatively constant, while the standard deviation we find continues to increase. The inset figure zooms in on the time period around reionization. The most notable difference here is the increase in the standard deviation of the dispersion measures constructed using CoDa II (blue solid curve, including the effects of a spatially inhomogeneous ionization fraction that on average begins to drop below $x_{HII} = 1$ at $z \gtrsim 6$), relative to the dispersion measures constructed from the CoDa II simulation with the spatially averaged (but redshift dependent) ionization fraction (red dash-dotted curve), as well as to the dispersion measures constructed using the CoDa II simulation where the hydrogen ionization fraction has artificially been set to 1 throughout (orange dashed curve, and consequently no patchiness). The relative difference between the two cases of redshift dependent ionization fraction, with (blue solid curve) or without (red dash-dotted curve) patchiness, reaches $\mathcal{O}(1\%)$ of the total standard deviation at $z \sim 8$.

slight increase relative to our results. In addition, we can compare our results to those of Zhang et al. (2021), which use Illustris-TNG results to estimate the dispersion measure. Our results are consistent for redshifts $z \lesssim 6$, but the ionization model used in Zhang et al. (2021) is much more sudden than the one modelled from the CoDa II simulation, and consequently the maximum value we observe is higher.

In addition to the mean of the dispersion measure, we can look to the standard deviation of the dispersion measure to offer further information about the spatial inhomogeneity of the density and ionization fraction fields during the EoR. The paths taken by different FRB signals intersect regions with differing densities of electrons. As a result, the standard deviation of dispersion measures includes a factor due to the inhomogeneity in density. However, there is an additional contribution to the standard deviation of FRBs emitted at redshifts prior to the end of the EoR due to FRB signals passing
through inhomogeneously ionized regions of space. With the CoDa
II data, we can artificially homogenize this ionization field, allowing
us to estimate the role the patchiness of reionization takes. We expect
that the standard deviation of the distribution of dispersion measures
out to a given redshift will be higher when we consider the effects of
patchy or inhomogeneous reionization. In Fig. 4, we can see this by
comparing the blue solid curve to the red dot-dashed curve. The blue
solid curve shows the standard deviation we calculate when including
the effects of patchy reionization from the CoDa II simulation, as a
function of the redshift of a source FRB, while the red dot-dashed
curve shows standard deviation when we apply the spatially averaged
but time-evolving hydrogen ionization fraction to the CoDa II density
fields. That is, in both cases, the dark matter density fields are the
same, and the average ionization fraction across the box is the same at
every redshift, but the blue solid (inhomogenous) case allows patches
within the simulation volume to have different hydrogen ionization
fractions, while the red dot-dashed (homogeneous) case does not.

An important feature to observe in both the inhomogeneous and homogenous CoDa II standard deviations is the plateau that is reached at redshifts higher than the end of the EoR. This plateau can be explained similarly to the plateau in the mean of the dispersion measure discussed above: namely that when the universe is essentially entirely neutral, it does not contribute to the dispersion measure. In comparison, if we consider a scenario where the universe remains ionized to arbitrarily high redshifts (i.e. labeled here as the ‘fully ionized’ case), this plateau does not occur, and the standard deviation of the dispersion measure continues to rise. As a result, this case sets an upper limit to the standard deviation of the dispersion measure regardless of ionization history, but does not include any effects from inhomogeneous reionization. However, as illustrated in the inset of Fig. 4, while the standard deviation of the homogeneous case reaches a plateau at the same redshift as the mean, the standard deviation in the inhomogeneous case only does so at a higher redshift. Furthermore, the plateau in the standard deviation of dispersion measures in the inhomogeneous case is higher than in the homogeneous case, by a factor of approximately 1 per cent. These two effects, the higher redshift at which the plateau is reached and the increase in the standard deviation at that plateau, offer clues as to the nature of reionization. For example, if the patches are typically small but numerous, the effect of the patchiness will be similar across most LOSs, and there will be a smaller difference between the standard deviations of the homogeneous and inhomogeneous cases. Conversely, if reionizing patches are sparse and consequently grow to large size, different LOSs can experience dramatically different ionization fractions, and the difference in the standard deviations will be larger. For a more complete discussion of how differences in the nature of reionization patches correlate with FRB dispersion measures, see Shaw et al. (2024).

In addition, in Fig. 4, we provide two additional comparisons to
the two cases that use the CoDa II ionization history. The orange
dashed curve shows the standard deviation of dispersion measures if
the gas in the universe is fully ionized to redshifts of at least z = 12.
To construct this curve, we use the CoDa II densities, but set the
hydrogen ionization fraction artificially to be x<sub>HII</sub> = 1 everywhere.
In addition, we plot the standard deviations from (Jaroszynski 2019)
as black points. Jaroszynski's analysis extends only to a redshift
z = 5, so does not offer any comparison during the EoR, but provides
a point of comparison at lower redshifts. Of particular note, the
standard deviation we find is larger than the standard deviations
found by Jaroszynski by a factor of approximately 5/4. We discuss

this difference in more detail in Appendix B. We propose three
partial explanations for this difference, but leave it to future work to
determine which, if any, of these effects dominate. The CoDa II–Dark
Matter simulation has a higher mass resolution than the Illustris-3
simulation that Jaroszynski (2019) analysed. This higher resolution
allows many low-mass structures to form. Because the standard
deviation of the dispersion measure is sensitive to the high-dispersion
measure tail of the distribution, filling out this tail with more disper-
sion measures from LOSs that pass through small overdensities could
shift the standard deviation to higher values. Alternatively, in works
like McQuinn (2013), the standard deviation of dispersion measures
from dark-matter-only models were systematically higher than the
standard deviations from simulations including baryonic effects. The
Illustris simulation includes those baryonic effects, while the CoDa
II–Dark Matter simulation we employ at redshifts z < 5.8 is a dark-
matter-only simulation. This difference could also contribute to the
observed difference between the standard deviations we find in this
work and those from Jaroszynski (2019). We note, however, that
direct comparisons of the CoDa II–Dark Matter simulation at higher
redshifts with the CoDa II simulation that includes baryonic effects
show nearly identical standard deviations. Finally, authors including
Park, Alvarez & Bond (2018) have argued that the first generation of
the Illustris simulation, including Illustris-3, includes active galactic
nucleus (AGN) feedback that oversuppresses structure growth at
low redshifts, leading to an overall reduction in the variance of
contributions to the dispersion measure from low redshifts. Analyses
of other more recent simulations that do not have this exaggerated
AGN feedback, including Illustris-TNG, bear out standard deviations
that are comparable to or higher than the ones we find using the CoDa
simulation (Walker et al. 2024).

Our primary interest in this paper is to explore the EoR using FRB
dispersion measures, and this is achieved predominantly through
differences in the standard deviation of dispersion measures from
FRBs at prior to or during the EoR. From the results in Fig. 4, based
on the CoDa II simulation, we estimate the size of the effect of
patchy reionization on the standard deviation of dispersion measures
to be approximately 1 per cent, for FRBs at z ≳ 8. From this,
we can make a rough estimate of how challenging it would be to
observe an effect of this size, while we leave a more comprehensive
analysis to other work (see for example, Hashimoto et al. 2021 and
Shaw et al. 2024 for some analyses of the constraining power of
upcoming FRB observations to reconstruct the ionization history).
First, the results we present here show the statistics of the dispersion
measure as a function of redshift, which would only be directly
observable if the redshifts of FRB sources could be identified. As
we look to higher redshifts, it is likely to become more challenging
to precisely identify a redshift for any individual FRB source. As
such, observationally, the most practical statistic to look at is the
distribution of FRB dispersion measures. An example of a semi-
analytic calculation of this distribution can be seen in fig. 3 of
Beniamini et al. (2021). To explore the EoR, the most relevant feature
in the distribution of FRB dispersion measures is a build-up of FRBs
with the maximal dispersion measure, which occurs because all FRB
signals emitted before the EoR will have comparable dispersion
measures. The dispersion measure at which this peak occurs will
be related to what appears in our analysis as a plateau, so from
the CoDa II simulation we would expect the peak to occur around
5900 pc cm⁻³. The shape of this peak, particularly the width about
the mean, will be related to the standard deviation of the dispersion
measures for FRB signals emitted prior to the end of the EoR. With
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Standard deviation of $\log DM$. We plot the standard deviation of the log of the dispersion measure, with the blue solid, red dash-dotted, and orange dashed curves representing the same cases described in Fig. 3. As a comparison, we plot the coefficient of variance $\sigma(DM)/\langle DM \rangle$ from the ‘CoDa ion. inhomog.’ case as a thin grey line. The difference between this thin grey curve and the solid blue curve is indicative of the deviation between the dispersion measure distribution and an idealized lognormal distribution.</figcaption></figure>

this in mind, ignoring other sources of error,<fnref n="6" /> we can estimate that in order to detect a 1 per cent change in the standard deviation of the dispersion measure, we would need at least 1 per cent resolution on the width about this peak. Since standard error scales as $1/\sqrt{N}$, where N here is the number of FRBs observed, we anticipate needing to observe $\mathcal{O}(10^4)$ FRBs emitted prior to the end of the EoR, in order to determine the standard deviation to high enough precision to compare with the results we show here. Beniamini et al. (2021) report an estimate of approximately $10^4$ FRBs occur per day and estimate that approximately 0.1 per cent of detectable FRBs are produced prior to the end of reionization. Based on these estimates, if all FRBs could be observed, and other uncertainties could be ignored, it would take approximately 3 yr of observations to compare with our results. While these assumptions are idealized, this estimate offers hope that observational comparison can be achieved in the not-too-distant future.

Throughout this discussion, we have considered the arithmetic mean and standard deviation in our analysis. These are natural choices if the underlying distribution of dispersion measures is approximately Gaussian. Alternatively, if the underlying distribution can better be approximated by a lognormal distribution, a more natural choice would be to compare the arithmetic mean and standard deviation of the log of the dispersion measures. The arithmetic mean of the log of the dispersion measures is equivalent to the log of the geometric mean of the dispersion measures, and for the distributions of dispersion measures we consider the arithmetic and geometric means are approximately equal. We therefore take $(\log(DM)(z)) \approx \log((DM(z)))$. Furthermore, if the distribution of dispersion measures is approximately lognormal, and the standard deviation of the log of the dispersion measures $\sigma_{\log DM} \ll 1$, then $\sigma_{\log DM} \approx \sigma(DM)/\langle DM \rangle$. In Fig. 5, we plot $\sigma_{\log DM}$. We consider the three cases discussed above, but we compare these to the

<figure><img src="image_8.png" /><figcaption>**Figure 6.** Coefficient of variance of dispersion measures. We plot the ratio of standard deviation to mean for the scenarios described in Fig. 3 on a log-linear plot. The case where the gas is fully ionized throughout the redshift range $0 < z < 12$ (orange dashed curve) fits a power law, while the dispersion measures constructed from the combination of the CoDa II and CoDa II–Dark Matter simulations (blue solid curve) approximately fit a broken power law (dot-dashed black curve, with fit $f(z)$ as described inside the plot). Residuals between solid blue curve and the broken power law that best fits that curve (black dash-dotted curve) are also shown. While the difference is generally small, two peaks occur: one at the kink in the broken power law and the other at low redshifts for which the power law becomes a less reliable fit.</figcaption></figure>

quantity $\sigma(DM)/\langle DM \rangle$, evaluated from the simulation results that include inhomogeneous reionization. The standard deviation of the logarithmic dispersion measure is consistently smaller than the ratio of standard deviation to mean, a consequence of the dispersion measures not being exactly lognormally distributed. While the difference between these two statistics is small at high redshifts, it gradually increases with decreasing redshift. This pattern is consistent with the thicker high-density tail at lower redshifts discussed above.

Finally, we look in more detail at the quantity $\sigma(DM)/\langle DM \rangle$, which we identify throughout the remainder of this paper as the coefficient of variance, to distinguish it from the standard deviation along. We plot this quantity in Fig. 6 as a function of redshift. This ratio is well modelled by a power law in the CoDa II–Dark Matter case, and by a broken power law for the combined CoDa II and CoDa II–Dark Matter case. At lower redshifts, approximately $z \lesssim 0.5$, the ratio $\sigma(DM)/\langle DM \rangle$ rapidly decreases. The broken power law that best fits the coefficient of variance is

$$ \frac{\sigma(DM)}{\langle DM \rangle} \approx \begin{cases} 0.316z^{-0.677} & z < 3 \\ 0.310z^{-0.660} & 3 < z < 7.11 \\ 0.0851 & z \ge 7.11 \end{cases} \qquad (12) $$

Here, the first break is fixed at redshift $z = 3$, corresponding to the redshift at which we set the reionization of helium. The redshift of the second break is fitted, but corresponds to the redshift at which the IGM is half-ionized. As we will discuss below, power laws of this form may be particularly useful in distinguishing between different models of reionization, and the one described here offers a baseline to compare against.

To confirm the quality of the fit, we also plot the residuals between the coefficient of variance from the constructed electron densities from the CoDa II and CoDa II–Dark Matter simulations and the best-fitting broken power law above in Fig. 6. The (relative) difference is fairly small, mostly of the order of 0.5 per cent over all redshifts

<footnote n="6">⁶Uncertainties in the dispersion measure contribution from source galaxies in particular can be quite large, potentially substantially larger than the effect of the inhomogeneous ionization field.</footnote>
that we consider. However, there are two dominant features where the residual increases by a factor of a few. One in the redshift range $0 < z < 2$, and one around redshift $z = 7$. Both of these peaks hint at situations in which modelling the coefficient of variance as a broken power law is not accurate to the smooth nature of the data. The feature around redshift $z = 7$ arises because we are modelling a smooth plateau as a sharp kink. At lower redshifts, the coefficient of variance shows a more pronounced deviation from the power law. This may reflect a physical meaningful reduction in the coefficient of variance relative to a pure power law at redshifts $z < 0.5$. Finally, we can also compare to the corresponding coefficient of variance calculated from the Jaroszynski (2019) results. At every redshift for which a comparison can be made, our results show a higher coefficient of variance than in Jaroszynski (2019), consistent with our higher standard deviation. However, the results of Zhang et al. (2021), using Illustris-TNG rather than Illustris 3, show much closer agreement with our results.

## 5 EFFECTS OF ALTERNATE GLOBAL REIONIZATION HISTORY

Throughout this work, we use the results of the CoDa II simulation as the basis for our ionization fraction and density fields. However, all of these results are subject to modelling uncertainties (e.g. star formation efficiency), so in this section, we explore the extent to which one such variability can affect the results we discuss above. Namely, we alter the globally averaged reionization history (specifically, the redshift dependence of the spatially averaged ionized fraction $\langle x_{HII} \rangle$), but continue to use the CoDa II and CoDa II–Dark Matter density fields.<fnref n="7" /> As a result, in these alternate reionization histories, the ionization fraction may not be fully consistent with the density fields, but we can identify trends that arise from the different ionization histories.

In order to do this analysis, we first calculate the spatially averaged ionization fraction as a function of redshift from the CoDa II data. We then compute a fluctuating electron density field by multiplying the inhomogeneous matter density field from the CoDa II–Dark Matter simulation by this spatially averaged ionization fraction $((1 + \delta_H)\langle x_{HII} \rangle)$. We can then model different ionization histories by modifying the redshift function of ionization fraction from the CoDa II ionization history. However, because we are spatially averaging the ionization fraction, we cannot consider in this section the effects of inhomogeneities in the ionization fraction on the standard deviation, as we did in the previous section. As such, we expect that a similar ~1 per cent increase to the standard deviation in dispersion measures that we previously found when including inhomogeneity would be observed in addition to any differences in the standard deviation that arise from the alternate histories considered below.

We consider four different homogeneous CoDa II ionization histories: three new ones in addition to the previously studied case of ‘CoDa ion. homog.’ (homogeneous but time-dependent reionization). The ionization fractions in these four cases are plotted as a function of redshift in Fig. 7. We note that all models are the same for redshifts from today up to the end of reionization ($z = 6.1$ in the canonical case), with the differences playing a role at redshifts higher than that. These four modifications can be summarized as follows.

<figure><img src="image_9.png" /><figcaption>**Figure 7.** Average hydrogen ionization histories ($\langle x_{HII} \rangle(z)$) in each of the reionization histories we consider. As in Fig. 1, each curve represents the spatially averaged ionization fraction as a function of redshift. The red dash-dotted curve shows the hydrogen ionization history of the CoDa II simulation without modification, which shows the effects of a homogeneous non-patchy reionization. The magenta densely dotted (teal sparsely dotted) curve shows the ionization history if the Universe ionized earlier (later) than in the CoDa II simulation, so that the end of reionization is approximately $z = 7.1$ ($z = 5.6$). The indigo dash-dot-dotted curve shows the ionization history if reionization occurs in two phases, where 50 per cent of the Universe is ionized in each phase, but ionization fraction plateaus between the two phases.</figcaption></figure>

(i) CoDa II ionization history: This model ‘CoDa ion. homog’ (studied earlier in the paper) uses the mean hydrogen ionization fraction within each CoDa II simulation box to determine $x_{HII}(z)$. As such, it largely reproduces the CoDa II results as if there were no patchy reionization and the entire Universe reionized at the same rate everywhere. This is therefore perhaps the most direct comparison to the patchy reionization results.

(ii) Early reionization: In this model, we use a hydrogen ionization fraction that is shifted in redshift: $x_{HII}(z) = x_{HII,CoDaII}(z - 1)$. Effectively, this amounts to shifting the reionization to an earlier time, so that instead of ending at $z = 6.1$ it ends at $z = 7.1$. The hydrogen ionization history is still the same as the hydrogen ionization history from the CoDa II simulation, but shifted in redshift.

(iii) Late reionization: Similar to the early reionization model, we use a hydrogen ionization fraction that is shifted in redshift, but now to lower redshifts: $x_{HII}(z) = x_{HII,CoDaII}(z + 0.5)$. By shifting the ionization history to slightly later redshifts, we approximate more recent observational findings of a later end of the EoR. This model has the EoR ending at approximately $z = 5.6$, and the Universe being half-ionized at approximately $z \approx 6.5$.

(iv) Two-phase ionization history: In this model, the hydrogen ionization history is split into two phases. Again based on the hydrogen ionization history from the CoDa II simulation, we created two phases of reionization. When $x_{HII} < 0.5$, we use the CoDa II hydrogen ionization history directly, but when $x_{HII} > 0.5$, we use the shifted hydrogen ionization history: $x_{HII}(z) = x_{HII,CoDaII}(z - 1)$. Effectively, in this scenario the Universe ionizes until half ionized and then experiences a pause, before ionizing completely some time later. An example of a physical effect that results in the two-phase global reionization history is the inclusion of minihaloes – haloes with mass $M \lesssim 10^8 M_\odot$ – as sources of star formation (Ahn et al. 2012; Ahn & Shapiro 2021). Because minihaloes are the earliest haloes to form, they are also the first sources of UV starlight, and

<footnote n="7">⁷An example visualization of the spatial structure of these density fields can be found in fig. 1 of Ocvirk et al. (2020).</footnote>
<figure><img src="image_10.png" /><figcaption>**Figure 8.** Comparison of dispersion measure means (left) and standard deviations (right) during the EoR from various ionization histories. As a reference, the mean and standard deviation of dispersion measures assuming a fixed hydrogen ionization fraction of $x_{HII} = 1$ is plotted as the orange dashed curve. The blue solid curve represents the dispersion measures derived from the CoDa II ionization history, including patchy reionization. The remaining curves each involve the CoDa II ionization history, averaged over the simulation volume, and modified in various ways, described in the main text. In each case, both the mean and standard deviation have different plateau values that occur at redshifts prior to the end of reionization. For the mean dispersion measures shown in the left panel, one can see that the case of full ionization ($x_{HII} = 1$, orange dashed curve) is an upper bound on all of the models that include reionization, and that the plateau in the mean dispersion measure increases the earlier the EoR ends and for longer EoR durations. Likewise, the orange dashed curve is an upper bound on all of the standard deviations of dispersion measures calculated with spatially homogeneous ionization fraction. However, once inhomogeneities in the ionization are taken into account (blue solid curve), the standard deviation can increase above the orange curve. Furthermore, while different ionization histories predict different standard deviations of dispersion measures, the models we consider all predict standard deviations within $\mathcal{O}(1 \text{ per cent})$ of one another. This can be compared to the difference between the standard deviations calculated from the CoDa II simulation with homogeneous (red dash-dotted curve) and inhomogeneous (blue solid curve) ionization fractions, which is also $\mathcal{O}(1 \text{ per cent})$.</figcaption></figure>

they initially dominate reionization. By the time a significant fraction of the volume is reionized, however (e.g. as high as ~0.2–0.3), the ability of minihaloes to form stars, involving $H_2$ formation and cooling, is suppressed by the rising UV background of $H_2$-dissociating starlight in the Lyman–Werner bands, and the contribution of reionizing photons from minihaloes begins to diminish. At this point, reionization stalls until more massive atomic cooling haloes can form in abundance to take over the process and finish reionization. The combination of these two periods of reionization, the first dominated by minihaloes and the second dominated by atomic cooling haloes, produces an early tail to the ionization history, similar to the model we show here.

In Fig. 8, we compare the mean and standard deviation of the dispersion measures calculated using these alternate ionization histories to the dispersion measures using the CoDa II–Dark Matter (fully ionized) and combined CoDa II–Dark Matter and CoDa II (patchy reionization) models. As expected the mean dispersion measure is very similar between the patchy reionization history and the model based on the unmodified CoDa II ionization history ‘CoDa ion. homog’ with time-dependent ionization but without patchiness. The other models (all with spatially homogeneous ionization fractions), however, diverge from this model in different ways. In particular, the early ionization model plateaus at a higher redshift than the CoDa II model, and consequently reaches a higher mean dispersion measure. The two-phase reionization model follows the plateauing of the CoDa II reionization, but then diverges and continues increasing until plateauing at a lower dispersion measure compared to the early ionization history. In general, each of these mean dispersion measures follow the ionization fraction history of the model associated with them, as we would expect.

The standard deviations of the dispersion measures from each of these models likewise have features tied to the ionization history. A

particularly notable feature is the distinctly large difference between the standard deviations from the patchy reionization history and the toy model based on the unmodified CoDa II ionization history. Comparison of these two curves offers perhaps the most direct way to explore the effect of the patchy reionization versus the same reionization history without the patchiness. The difference in the maximum standard deviations achieved by these two models is, in fact, larger than the differences between any of our other reionization history models. This implies that the patchiness of the reionization specifically will produce a distinct pattern in the standard deviation history of FRBs. If sufficient numbers of FRBs can be detected at redshifts prior to the EoR, and their dispersion measures accurately extracted, the standard deviation of the dispersion measure could allow a measure of the typical spatial uniformity of the patches reionization occurs in.

In addition to the two comparisons separately, in Fig. 9, we compare the coefficients of variance as a function of redshift for the alternate reionization histories. At low redshifts, the coefficient of variance follows the power-law fit in the previous section, but at higher redshifts it can offer a probe of the EoR that is complementary to the standard deviation and mean separately. Like the mean, the coefficient of variance is more sensitive to differences in the average ionization history than to whether the ionization is homogeneous or inhomogeneous. Combining these statistical quantities therefore offers a potential way to constrain the timing and history of reionization. For example, the coefficient of variance follows the power law identified above at low redshifts, and deviates from this power law at a point determined by the end of the EoR. Meanwhile, at high redshifts, the coefficient of variance is constant and deviates from this plateau value at a redshift related to the beginning of the EoR. If enough FRBs can be observed to span this entire redshift range, from beginning to end of EoR, the standard deviation, mean, and coefficient of variance may offer a means to probe the evolution of
<figure><img src="image_11.png" /><figcaption>**Figure 9.** Comparison of the coefficient of variance of dispersion measures modelled using various ionization histories. The curves have the same meaning as in Fig. 8. While the coefficient of variance from each model can be described by a common power law at lower redshifts than reionization, the constant value at which the coefficient of variances plateaus is different for each model.</figcaption></figure>

the EoR. Even if sufficient numbers of FRBs cannot be observed at all redshifts in this range, FRBs in multiple parts of this range could probe different aspects of the EoR.

## 6 CONCLUSION

Using the results of the CoDa II simulations, we have predicted the mean and standard deviation of the dispersion measures of FRBs, as a function of the source redshift of those FRBs. In doing so, we observe and quantify the theorized plateaus in the mean dispersion measure and in the standard deviation around that mean at redshifts higher than the end of the EoR. We present these results in a way that can be most directly applied if FRBs are localized to a well-defined redshifts. However, even if well-constrained redshift information cannot be obtained, the plateaus we describe would lead to a peak in the distribution of dispersion measures. We predict that the plateau dispersion measure, and consequently the dispersion measure at which this peak occurs, is $DM_{peak} \approx 5900$ pc cm$^{-3}$. The value of this peak dispersion measure can be related to the redshift at which reionization ends, where a later end of the EoR leads to a lower peak dispersion measure, and to the duration of the EoR, where a longer period of reionization (e.g. in a two-phase ionization history) leads to a higher peak dispersion measure.

Additionally, we predict the standard deviation of dispersion measures at redshifts $z \gtrsim 8$, which will be related to the spread about the peak dispersion measure, to have a value of $\sigma(DM) \approx 497$ pc cm$^{-3}$, approximately 10 per cent of the mean value. This standard deviation is related to the evolution of inhomogeneities in both the ionized gas and overall gas density. We quantify how much of this standard deviation can be attributed to the inhomogeneity in ionization fraction due to the patchy nature of reionization by comparing the standard deviation with the inhomogenous ionization to the standard deviation with the same ionization history but a spatially uniform ionization fraction.

One of our main results can be found in the inset of Fig. 4 as described in Section 4, where we showed that evidence for patchy reionization may be found at the $O(1$ per cent) level once FRBs are found at $z \gtrsim 8$. Specifically we find that the patchiness of

reionization contributes approximately a 1 per cent increase to the standard deviation that would be observed from FRBs with redshift $z \gtrsim 8$. A back-of-the-envelope calculation suggests that this could optimistically be resolved if $O(10^4 - 10^5)$ FRBs emitted prior to the end of the EoR can be observed. Using estimates for the rate at which FRBs are produced, it may take as little as 3 yr for this number of FRBs to be produced, suggesting that even this 1 per cent effect is within observational reach.

We also compute standard deviation of $\log DM$, as a function of redshift, which is a more natural choice of statistic if the distribution of dispersion measures can be accurately approximated by a lognormal distribution. We compare this to the coefficient of variance, $\sigma(DM)/\langle DM \rangle$, finding good agreement especially at high redshifts. We then fit the coefficient of variance to a broken power law, as in equation (12). For redshifts $z \le 3$, when helium is fully ionized, the coefficient of variance can be represented by the power law $\sigma(DM)/\langle DM \rangle = 0.316z^{-0.677}$. For redshifts $3 < z \le 7.11$, $\sigma(DM)/\langle DM \rangle = 0.310z^{-0.660}$. And prior to the end of reionization, the coefficient of variance also plateaus at a value of $\sigma(DM)/\langle DM \rangle = 0.0851$. We hope that power laws like these can be used in future semi-analytical works to estimate the distribution of dispersion measures.

Finally, we explore how our results might shift if the ionization history is not identical to the one predicted from the CoDa II simulation. We consider three alternative models of the global ionization fraction, mimicking universes that see reionization start and end earlier than in the CoDa model, start and end later than in the CoDa model, and start earlier but end at the same time as the CoDa model. Each of these alternate ionization histories produces slightly different peak values in the mean of the dispersion measures, standard deviations about those peak values, and value of the coefficient of variance at redshifts higher than the end of the EoR. Therefore, combining measurements of these three properties offers a direct probe of the global ionization history.

## ACKNOWLEDGEMENTS

TD and PRS are grateful for the support of the NSF Graduate Research Fellowship Program under Grant No. DGE-1610403. This research used resources of the Oak Ridge Leadership Computing Facility (OLCF) at Oak Ridge National Lab, which is supported by the Office of Science of the US Dept. of Energy (DOE) under Contract No. DE-AC05-00OR22725. CoDa II was performed on Titan at OLCF under DOE INCITE 2016 award to Project AST031. Data analysis was performed using computing resources from NSF XSEDE grant TG-AST090005 and the Texas Advanced Computing Center at the University of Texas at Austin. PRS also acknowledges support from NASA under Grant No. 80NSSC22K175. KF is Jeff & Gail Kodosky Endowed Chair in Physics at the University of Texas at Austin, and KF and JJZ are grateful for support via this Chair. KF and JJZ acknowledge support by the U.S. Department of Energy, Office of Science, Office of High Energy Physics program under Award Number DE-SC-0022021 as well as support from the Swedish Research Council (Contract No. 638-2013-8993). PB is supported by a grant (no. 2020747) from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel, by a grant (no. 1649/23) from the Israel Science Foundation, and by a grant (no. 80NSSC 24K0770) from the NASA astrophysics theory program. PK is supported by an NSF grant AST-2009619, and a NASA grant 80NSSC24K0770 (PK and PB). KA is supported by NRF-2021R1A2C1095136 and RS-2022-00197685.
DATA AVAILABILITY

The data underlying this article are a product of The Cosmic Dawn (CoDa) Project. The derived data generated in this research will be shared upon reasonable request to the corresponding author.

REFERENCES

Aghanim N. et al., 2020, A&A, 641, A6

Ahn K., Shapiro P. R., 2021, ApJ, 914, 44

Ahn K., Iliev I. T., Shapiro P. R., Mellema G., Koda J., Mao Y., 2012, ApJ, 756, L16

Atek H., Richard J., Kneib J.-P., Schaerer D., 2018, MNRAS, 479, 5184

Batten A. J., Duffy A. R., Wijers N. A., Gupta V., Flynn C., Schaye J., Ryan-Weber E., 2021, MNRAS, 505, 5356

Beniamini P., Kumar P., 2023, MNRAS, 519, 5345

Beniamini P., Kumar P., Ma X., Quataert E., 2021, MNRAS, 502, 5134

Bouwens R. J. et al., 2015, ApJ, 803, 34

Bouwens R. J., Oesch P. A., Illingworth G. D., Ellis R. S., Stefanov M., 2017, ApJ, 843, 129

Cordes J. M., Chatterjee S., 2019, ARA&A, 57, 417

Deng W., Zhang B., 2014, ApJ, 783, L35

Dolag K., Gaensler B. M., Beck A. M., Beck M. C., 2015, MNRAS, 451, 4277

Faucher-Giguère C.-A., Kereš D., Ma C.-P., 2011, MNRAS, 417, 2982

Finkelstein S. L. et al., 2015, ApJ, 810, 71

Fosalba P., Gaztañaga E., Castander F. J., Manera M., 2008, MNRAS, 391, 435

Gordon A. C. et al., 2023, ApJ, 963, L34

Gupta P. D., Saini N., 2018, JA&A, 39, 14

Hashimoto T. et al., 2021, MNRAS, 502, 2346

Heimersheim S., Sartorio N. S., Fialkov A., Lorimer D. R., 2022, ApJ, 933, 57

Jaroszynski M., 2019, MNRAS, 484, 1637

Katz J. I., 2016, ApJ, 826, 226

Kumar P., Bošnjak Ž., 2020, MNRAS, 494, 2385

Kumar P., Lu W., Bhattacharya M., 2017, MNRAS, 468, 2726

Lorimer D. R., McLaughlin M. A., Bailes M., 2024, Ap&SS, 369, 59

Maity B., 2024, A&A, 689, A340

Margalit B., Metzger B. D., 2018, ApJ, 868, L4

McQuinn M., 2013, ApJ, 780, L33

Nelson D. et al., 2015, Astron. Comput., 13, 12

Nelson D. et al., 2019, Comput. Astrophys. Cosmol., 6, 2

Ocvirk P. et al., 2020, MNRAS, 496, 4087

Pagano M., Fronenberg H., 2021, MNRAS, 505, 2195

Paoletti D., Hazra D. K., Finelli F., Smoot G. F., 2025, Phys. Rev. D, 111, 043532

Park H., Alvarez M. A., Bond J. R., 2018, ApJ, 853, 121

Petroff E., Hessels J. W. T., Lorimer D. R., 2019, A&AR, 27, 4

Planck Collaboration XVI, 2014, A&A, 571, A16

Pol N., Lam M. T., McLaughlin M. A., Lazio T. J. W., Cordes J. M., 2019, ApJ, 886, 135

Popov S. B., Postnov K. A., 2013, preprint (arXiv:1307.4924)

Rees M. J., 1977, Nature, 266, 333

Schaye J. et al., 2015, MNRAS, 446, 521

Shaw A. K., Ghara R., Beniamini P., Zaroubi S., Kumar P., 2024, Asking Fast Radio Bursts (FRBs) for More than Reionization History, preprint (arXiv:2409.03255)

Sorce J. G., Tempel E., 2018, MNRAS, 476, 4362

Vieyro F. L., Romero G. E., Bosch-Ramon V., Marcote B., del Valle M. V., 2017, A&A, 602, A64

Wadiasingh Z., Timokhin A., 2019, ApJ, 879, 4

Walker C. R. H., Spitler L. G., Ma Y.-Z., Cheng C., Artale M. C., Hummels C. B., 2024, A&A, 683, A71

Wise J. H., 2019, Contemp. Phys., 60, 145

Zhang Z. J., Yan K., Li C. M., Zhang G. Q., Wang F. Y., 2021, ApJ, 906, 49

Zhu W., Feng L.-L., 2021, ApJ, 906, 95

APPENDIX A: ADDITIONAL METHODS

In the main text, we outline the method we use to calculate the dispersion measure using the CoDa simulations. In this section we expand upon some of the details in that discussion. Our goal is to approximate equation (9) by using overdensity $\delta_H$ and ionization fraction $x_{HII}$ fields that are defined only at discrete redshifts.

The first step of this analysis is to determine the factors $x_e(z_i)$ and $\delta_H(z_i)$ for appropriate values of $z_i$ along different LOSs. If we imagine tiling simulation volumes along the LOSs, then the redshifts $z_i$ should be such that $t(z_i) - t(z_{i-1}) = L_b/c$, or that the time it takes light to cross the box should ideally be the same as the time interval at which we evaluate $x_e$ and $\delta_H$. The CoDa II and CoDa II–Dark Matter simulations were not designed to have this requirement, so either have simulated snapshots either more closely spaced than the ideal (in the CoDa II simulation) or more sparsely spaced than the ideal (in the CoDa II–Dark Matter simulation), as depicted in Fig. 2. We therefore process these two simulations differently so as to achieve the desired spacing in time/redshift.

In the first case, the time separation between subsequent snapshots is smaller than the target (the CoDa II simulation). In this case, a photon travelling across a distance equal to the simulation box will experience multiple snapshots before fully crossing the box. Equivalently, a photon will move through only a fraction of the box described by one snapshot before the time of the succeeding snapshot. Therefore, when we calculate the contribution to the dispersion measure from one box-length segment of a LOS, we incorporate the updated density and ionization information that a photon travelling through the box would see.

The second case is the opposite extreme, where the time between simulation snapshots is longer than the target (the CoDa II–Dark Matter simulation). As a result, we seek to construct distributions of box-length contributions to the dispersion measure for redshifts between two simulation snapshots. For a given redshift, the distribution of the contributions to the dispersion measure is generated from the simulation snapshots immediately before and after that redshift, such that statistically the contributions to the dispersion measure are a linear interpolation between the contributions that would be calculated from those simulation snapshots. Effectively, this approach allows us to approximate changes to the density and ionization fraction, so that the universe ‘evolves’ at least once per box-length that a photon travels.

Our goal in constructing these distributions is only to achieve something that is statistically representative, rather than physically accurate. Consider a redshift $z$ between two simulation redshifts $z_l$ and $z_r$ such that $z_l < z < z_r$. We can construct a distribution from each of the simulation redshifts, by considering the contribution to the dispersion measure along each LOS as independent random elements of this distribution. Then, to calculate the distribution for the redshift $z$, we can average between the simulation distributions at $z_l$ and $z_r$. That is, we create a new distribution by selecting

$$N_l = \frac{z_r - z}{z_r - z_l} N \quad (A1)$$

elements from the simulation distribution at $z_l$. Here, $N$ is the total number of elements in the distribution we seek to create. Likewise, we also include $N_r = N - N_l$ elements randomly selected from the simulation distribution at $z_r$. While other approaches can be used to generate these distributions, this approach retains any skewness that the simulation distributions have, so long as a sufficient number of elements from each of the simulation distributions are chosen. See
Appendix B for an alternative approach, and how it compares to the one used throughout.

Once we have a distribution of contributions to the dispersion measure from each box-length segment, we combine them into a set of LOSs. As mentioned in the main text, we do not use the full $3 \times 2048^2$ possible contributions from each segment, but select 2 million from each. We combine these randomly, without repeating. Effectively, this amounts to each LOS following a photon path through a continuous stack of simulation volumes that are rotated and translated randomly, and independently of any other LOSs. This approach could limit any correlations that might exist between nearby LOSs over distance scales greater than the size of the simulation box. However, it is expected that there are very few correlations in density on scales greater than 100 Mpc, so we assume any limitation due to the randomization is negligible. Finally, segment length contributions are added together so that the total LOS reaches a given redshift.

## APPENDIX B: ILLUSTRIS DATA SET

Throughout this paper, we have used a methodology designed to complement the CoDa II simulation, as well as the associated CoDa II–Dark Matter simulation. To confirm that this approach is consistent with other methods in the literature, we attempted to match the results found in Jaroszynski (2019) based on the Illustris-3 simulation (Nelson et al. 2015). While successfully able to reproduce these results, and to show marked similarity between the dispersion measures calculated using the Illustris and CoDa II–Dark Matter simulations at low redshifts, we also identify a systematic difference between the approach we develop above and the approach that matches the results in Jaroszynski (2019), which may have significant repercussions.

Following Jaroszynski (2019), the Illustris data sets we used were the Illustris-3 snapshots at redshifts $z = 0.00, 1.00, 2.00, 3.01, 4.01$, and $5.00$. However, these data sets store data differently than the CoDa II simulations, with the Illustris data sets storing data about cells with variable volumes but a similar mass, while the CoDa II and CoDa II–Dark Matter data sets store data about fixed volume cells, which consequently contain differing amounts of mass. For our analysis, we converted the Illustris data set to the spatial grid utilized by CoDa II and CoDa II–Dark Matter by taking a simple sum of the dark matter and gas particle masses whose centres were located in each spatial grid subdivision. This simple summing procedure may have introduced inaccuracies to our reproduction of the Illustris data set, but it is believed that these inaccuracies are small.

Next, we use each snapshot to generate a distribution of the overdensity $\delta_H$ from the snapshot volume for both the Illustris and CoDa II–Dark Matter simulations. Fig. B1 shows a representation of these distributions, specifically $(1 + \delta_2)$, which is the mean of the quantity $(1 + \delta_H)$ along a line through the snapshot volumes. As elsewhere in the paper, this distribution can be approximated well by taking only lines perpendicular to one of the faces of the simulation volume and considering 2 million randomly selected elements of the 12 million independent values at each redshift. The approach taken in Jaroszynski (2019) uses the same simplification, but allows for a larger number of statistically independent lines. This difference in the number of lines has no impact on the statistical results we find.

As described in Appendix A, we can use these simulation snapshots to construct a series of synthetic snapshots that entirely cover the range, so as to probe the dispersion measure out to any given redshift $z$. The methodology of Jaroszynski (2019) uses a similar

approach, but with slight differences. In either case, given a target redshift $z$ bounded by redshifts $z_{left}$ and $z_{right}$, for which there are simulation snapshots, we calculate a factor

$$x = \frac{z - z_{\text{left}}}{z_{\text{right}} - z_{\text{left}}}, \qquad (B1)$$

which provides a measure for how close the redshift $z$ is to $z_{left}$ relative to $z_{right}$. Then, in the approach we use throughout this paper, we randomly select $xN$ elements from the snapshot at redshift $z_{left}$ and $(1-x)N$ elements from the snapshot at redshift $z_{right}$, where $N$ is the total number of elements considered in the distribution we consider at each redshift. Throughout this paper, we have used $N = 2000000$, but as long as this number is sufficiently large to be representative of the simulation snapshots, the value itself does not have a large impact on the results.

In order to compare against the approach used in Jaroszynski (2019), we followed the methodology outlined there, although minor adjustments needed to be made to accommodate the additional conversion of the Illustris data to a fixed grid. Beginning with the simulation snapshots, we generate a distribution $\delta(z_{snapshot})$, based on a histogram of the logarithm of the contribution to the dispersion measure from each snapshot. The histograms we use have 560 bins in the range $\log(DM/\langle DM \rangle) \in [-2.5, 3.1]$. It turns out that the results are highly sensitive to this range, with the bounds listed here being comparable to those used in Jaroszynski (2019). In that work, it was reported that this range contained all densities $\delta$, while our analysis presented a small number of densities outside this range; it is probable that these outliers arise from the pre-processing that we do to conform the Illustris data to the spatial gridding of the CoDa II simulation. In any case, with these distributions generated from each snapshot, we can construct the distributions associated with synthetic snapshots as

$$\delta(z) = x\delta(z_{\text{left}}) + (1-x)\delta(z_{\text{right}}). \qquad (B2)$$

We then randomly select $N$ elements from this distribution.

We calculate the total dispersion measure as detailed in the main body of the paper, and compare to the results from Jaroszynski (2019) in Fig. B2. While both approaches predict similar mean dispersion measures, there is a notable difference between the standard deviations calculated using each approach. This difference can be traced entirely to the choice of range used in generating the histograms. Widening the range of the histograms can increase the standard deviation at any redshift, while reducing the range can substantially reduce the standard deviation.

We choose to use the approach that involves randomly selecting elements from the two snapshot distributions directly, without approximating them with histograms, as this seems to better support highly skewed distributions. While the contributions to the dispersion measure from the Illustris simulations can be approximated well by lognormal distributions, with only a slight skew, the same cannot be said of the CoDa II–Dark Matter results, particularly at low redshifts. As seen in Fig. B1, the contributions to the dispersion measure from the CoDa II–Dark Matter simulation have a relatively large high-dispersion-measure tail, which may be related to the difference between the physics involved in the two simulations.

From this comparison of the analysis of Illustris data we can draw a few conclusions. First, and foremost, the majority of the difference between the standard deviations of the dispersion measure we predict from CoDa II–Dark Matter and that arises in Jaroszynski (2019) does not come from the small difference in our analysis methods. Instead, it is a result of real differences in the Illustris and CoDa II–Dark Matter data. Secondly, while the different approaches to
<figure><img src="image_12.png" /><figcaption>**Figure B1.** Comparison of the distribution of overdensities at low redshifts in the CoDa II–Dark Matter and Illustris simulations. We compare the distribution of the logarithm of the overdensity log(1 + δ₂) from the CoDa II–Dark Matter (left) and Illustris (right) simulation results at similar redshifts. While the Illustris overdensities are mostly lognormal even to low redshifts, the CoDa II–Dark Matter overdensities show distinct skewness, reflecting a higher number of high-density haloes in the dark-matter-only evolution. In both simulations, higher redshifts show less skew than z = 0.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Figure B2.** Mean and standard deviation of dispersion measures calculated from Illustris data. We compare the mean (left) and standard deviation (right) of the dispersion measures calculated in two ways using the Illustris data. The blue solid curve represents the results from our reproduction of the analysis technique used in Jaroszynski (2019), while the orange dashed curve represents the analysis technique used elsewhere in this paper, but applied to the Illustris data. For comparison, we plot the results from Jaroszynski (2019) as black points. The mean dispersion measure of all three are consistent, while the standard deviation from Jaroszynski (2019) and that calculated from our recreation of the same analysis are also consistent. However, there remains a ~8 per cent difference between the analysis technique used throughout this paper and the approach used in Jaroszynski (2019), compared to the ~18 per cent difference in Fig. 4. This remaining difference can be attributed to the result of using different analyses, while the difference in Fig. 4 can be attributed to a combination of using different analyses and different data sets.</figcaption></figure>

interpolating between simulation snapshots are not the dominant
source of difference in standard deviation, they do contribute.
However, because of the skewness in the distribution of (1 + δ₂) in
the CoDa II–Dark Matter results, the approach used in Jaroszynski

(2019) cannot be directly applied to the CoDa II–Dark Matter
data.

This paper has been typeset from a LaTeX file prepared by the author.