Do Mini-halos Affect Cosmic Reionization?

The role of unresolved structures ("mini-halos") in determining the consumption of ionizing photons during cosmic reionization remains an unsolved problem in modeling cosmic reionization, despite recent extensive studies with small-box high-resolution simulations by Park et al. and Chan et al., because the small-box studies are not able to fully sample all environments. In this paper these simulations are combined with large-box simulations from the"Cosmic Reionization On Computers"(CROC) project, allowing one to account for the full range of environments and to produce an estimate for the number of recombinations per hydrogen atom that are missed in large-scale simulations like CROC or Thesan. I find that recombinations in unresolved mini-halos are completely negligible compared to recombinations produced in large-scale cosmic structures and inside more massive, fully resolved halos. Since both Park et al. and Chan et al. studies have severe limitations, the conclusions of this paper may need to be verified with more representative sets of small-box high-resolution simulations.


INTRODUCTION
The consumption of ionizing photons in unresolved structures (commonly dubbed "mini-halos") during cosmic reionization has been a small but active area of research since the first attempts to model the reionization process numerically.In the pioneering study, Haiman et al. (2001) estimated that numerical simulations that do not resolve mini-halos may underestimate the required number of ionizing photos by factors as large as 2 to 10. Later studies produced wildly varied estimates for this number, with some confirming it (Shapiro et al. 2004;Iliev et al. 2005a,b;Emberson et al. 2013) and others finding significantly lower values (Park et al. 2016;D'Aloisio et al. 2020;Chan et al. 2023).This apparent inconsistency was, at least in part, due to variations in what counted as "unresolved".
The underlying physics of how mini-halos affect reionization is well understood: (a) ionizing photons are con-sumed to ionize the dense gas inside them; (b) minihalos are evaporated after being ionized since their virial temperatures are below the typical temperatures of photo-ionized gas (this is the definition of a "minihalo"); (c) their dense gas recombines during evaporation, requiring additional ionizing photons to ionize it again.If ionizing radiation is intense, a significant number of ionizing photons can be absorbed during the evaporation stage, while the gas density is still high enough.If, however, ionization proceeds slowly, just one ionizing photon per baryon may be sufficient to raise the gas temperature above the virial temperature of a mini-halo, and hydrodynamics does the rest (M.Haehnelt, private communication).
The challenge in evaluating the effect of mini-halos on cosmic reionization accurately is the very large mass scales required.While simulations that model mini-halo evaporation typically have box sizes of hundreds of kpc, modeling the overall process of cosmic reionization requires simulation volumes hundreds of times larger (Iliev et al. 2014).Hence, no single simulation can resolve this question yet, and combining large-scale and small-scale models is needed for accounting for mini-halos.To the best of my knowledge, the first time this approach was implemented by Ciardi et al. (2006), who used an analytical small-scale model and found a moderate, but not negligible effect.Similar studies were undertaken later only a handful of times (Raičević & Theuns 2011;Sobacchi & Mesinger 2014;Mao et al. 2020;Park et al. 2023).
In the last half a decade two major advances took place that warrant a revision to the question of photon consumption in mini-halos.First, large-scale, fully coupled simulation of reionization finally reached super-100-Mpc scales and 2000 3 mass dynamic range -the flagship examples of such simulations are CROC (Gnedin 2014), Coda (Ocvirk et al. 2016(Ocvirk et al. , 2020)), and Thesan (Kannan et al. 2022).All these simulation projects reach similar resolutions, so the concept of "unresolved" is welldefined for them.Second, two larger parameter studies for photo-evaporation of mini-halos became available (Park et al. 2016;Chan et al. 2023).The goal of this paper is to combine large box simulations from CROC with Chan et al. (2023) and Park et al. (2016) models to evaluate the effect of unresolved small-scale structure on CROC reionization histories.

METHODOLOGY
Largest CROC simulations modeled reionization in boxes of 80h −1 ≈ 117 cMpc on a side, with adaptive spatial resolution reaching 100 physical parsecs inside modeled galaxies but never exceeding 117/2048 cMpc ≈ 57 ckpc.One of the saved data products of CROC simulations is 1024 3 uniform grids covering the whole computational domain.These grids are frequently sampled in time, allowing accurate time integration, and each cell in these grids has a size of at least two resolution elements, hence offering an independently resolved piece of spatial information with a size of 114 ckpc.
In comparison, Chan et al. (2023) simulations were performed in 800 ckpc boxes, hence covering almost exactly a block of 7 3 cells from CROC uniform grids.Those simulations modeled the effects of instantaneous reionization down to, at least, 1.6 ckpc (their mean interparticle separation) with the maximum resolution set by the Plummer softening length of 0.1 ckpc.The two main parameters that specify instantaneous reionization in Chan et al. (2023) simulations are the redshift of reionization z i and the global photoionization rate Γ.For a set of these two parameters, Chan et al. (2023) computed the number of recombinations per hydrogen atom N REC /N H in the simulation volume as a function of time, and the ratio of that number to the number of recombinations in a uniform medium with the same median temperature, N REC /N UNI .Their results for the latter can be accurately fitted with the following simple relation, where t 100 is the time since the instant of reionization in units of 100 Myr.This fit becomes inaccurate for t 100 < 1.
The dependence on Γ in Chan et al. (2023) results is somewhat non-trivial.One can fit them with a powerlaw, where Γ −12 ≡ Γ/(10 −12 s −1 ), but the fit is not good enough.A more complex, double-power-law fit is significantly better, and I adopt it as a fiducial form, where α = 0.2 for Γ −12 > 0.15 and α = 0.32 for smaller photoionization rates.Finally, one would expect the dependence on Γ eventually saturate -when most of the gas is ionized, decreasing the neutral fraction from, say, 10 −2 to 10 −3 does not change the recombination rate, while the photoionization rate increases by a factor of 10.
The following form includes saturation and provides an extremely accurate fit to the Chan et al. (2023) results: A major limitation of Chan et al. (2023) simulations is that they all adopt the mean cosmic density as the mean density in the simulation box, and hence do not account for the density variation in 800 ckpc boxes.That variation is shown in Figure 1 at z = 5.If not accounted for, it will result in a significantly lower value for the number of recombinations.Fortunately, Park et al. (2016) performed an analogous simulation set in smaller, 300 ckpc boxes.Park et al. (2016) simulations used unrealistically large values for Γ and continue for only 150 ∼ M yr, which makes them less suitable for my purpose (and, as I show below, they substantially undercount recombinations).Fortunately, Park et al. (2016) did explore several boxes with mean densities deviating from the cosmic mean.Fitting the dependence on the mean density in these boxes, I find a highly expected result, Notice, that Park et al. ( 2016) fitted the difference between N REC /N H and N UNI /N H and found a slightly steeper dependence, ∝ (1+ δ) 2.5 .This fit is 3 times worse (in its p − value) than the two dependencies above, and obviously cannot extrapolate beyond their highest value for δ = 0.6.Nevertheless, I show their exact results later as well.
The final model for the number of recombinations unresolved in 800 ckpc boxes is thus obtained by multiplying equation ( 1) by (1 + δ800 ), where δ800 is the average overdensity in 800 ckpc volume.I call this model "Chan-Park", as it combines both small-scale simulation sets.
In CROC simulations not every 800 ckpc block is reionized instantly.For example, at z = 7 such a block has a length of 100 pkpc.A typical cosmological ionization front, moving at 3000 km/s, crosses this region in some 30 Myr.Hence, one needs to come up with a working definition of reionization for such a region.For the sake of simplicity, I call such a block reionized when its mass-weighted neutral hydrogen fraction falls below a given threshold, x i .
To use the fitting formula (1), one also needs to define the instantaneous photo-ionization rate Γ.Unfortunately, the uniform grid data from CROC simulations does not include that quantity, and saved full simulation snapshots are not sampled frequently enough for accurate time integration.I, thus, estimate the photo-ionization rate from the instantaneous reionization model, x HI (t) = exp(−Γ(t − t i )), where t i is the moment of reionization.Using two distinct moments t a and t b , one can obtain an estimate of Γ as Such an estimate includes a numerical error, so to account for it I use 3 different moments t 1 , t 2 , t 3 that corresponds to ⟨x HI ⟩ V = 0.1, 0.03, 0.01, and compute 2 estimates for Γ with (a, b) = (1, 2) and (a, b) = (2, 3).Using the two estimates, I can compute the mean estimate and its error.Hence, the whole computational procedure is as follows: for each 7 3 block in a 1024 3 uniform grid (the last 2 cells along each axis are ignored, so only 7×146 = 1022 cells used) I compute the estimate for the total number of recombinations N REC as the product of the fitting formula (1) and the number of recombinations in an equivalent uniform medium with the density, neutral fraction, and temperature averaged over the 7 3 from a CROC simulation.In order to check the dependence on the numerical parameter x i , I use x i = 0.3, 0.1, 0.03.
Figure 2 shows the total (i.e. the cosmic mean) number of recombinations per hydrogen atom in the fiducial CROC simulation (described in more detail below).Two panels show the effects of various methodological choices.I elect the x i = 0.1 with the double-powerlaw fit for ψ(Γ) (eq. 3) as "fiducial", and I also record the maximum and the minimum models to demonstrate the limitations of the chosen approach.The left panel of the figure also shows the calculation using Equation ( 22) from Park et al. (2016), with their steeper density dependence, as applied to 3 3 blocks from the CROC simulation.Park et al. (2016) fit under-counts the number of recombinations in comparison to the full Chan-Park model, primarily because their simulations do not extend beyond 150 Myr.The slope of −0.55 in the time dependence of Chan et al. (2023) simulation demonstrates that there is still a significant number of recombinations taking place well after 150 Myr.

RESULTS AND CONCLUSIONS
Fig. 2 would pretty much represent the results of this short paper if the CROC simulations had a resolution of 800 ckpc.However, CROC simulations cover the whole computational domain with cells of at most 57 ckpc on a side, and much smaller in highly resolved regions.With the easily available uniform grid data I can compute both the number of recombinations per hydrogen atom ion the full 1024 3 CROC uniform grid, and for the same grid with all physical quantities smoothed in 800 ckpc blocks (7 3 cells).
I do this for two CROC simulations from Gnedin (2022).Both of these simulations have sizes of 80h −1 ≈ 117 cMpc.The first of the two, which I call "fiducial" or "earlier reionization", is referred to as "DC=0" in Gnedin (2022).This simulation provides a reasonable match to the observed distribution of mean opacities  22) applied to CROC blocks of 3 3 cells (the best match to their 300 ckpc boxes).Right: the same quantity for 3 different fits to the Γ dependence, Equations (2-4). in 50h −1 Mpc skewers from Becker et al. (2015) and Bosman et al. (2018), but fails to match the observed distribution of "dark gaps" (regions in quasar absorption spectra without significant transmission) from Zhu et al. (2021).The second simulation, which I call "later reionization", is referred to as "DC=-1" in Gnedin (2022).That simulation matches the distribution of dark gaps very well but fails to match the distribution of mean opacities.Thus, the two simulations each match one of the two key observables and fail to match the other.Unfortunately, this is currently the state-of-the-art in reionization modeling (Gnedin & Madau 2022).
Figure 3 now shows the main result of this paper -the total number of recombinations per hydrogen atom esti- mated with the Chan-Park model (both the fiducial approach and the full range due to methodological choices described above) as compared with the actual number of recombinations computed in two CROC simulations.In both cases, the fiducial approach finds barely any recombinations from the structures unresolved in CROC.In the most extreme scenario, there may be up to 50% more recombinations produced that are captured in CROC 1024 3 uni-grids.However, the actual coarsest resolution of 80h −1 Mpc CROC boxes is twice higher, the equivalent of a 2048 3 uni-grid (with the adaptive resolution even higher in adaptively refined regions).Because of the of the available storage, only 1024 3 uni-grids were produced for 80h −1 Mpc CROC boxes, and the same size grids were also produced for 6 smaller, 40h −1 Mpc CROC boxes.Hence, these smaller boxes have twice higher resolution of their 1024 3 uni-grids.They can be used to check the effect of the uni-grid resolution on the results of this paper.Two 40h −1 Mpc CROC boxes happen to have together almost the same number of recombinations per hydrogen atom as the fiducial 80h −1 Mpc box when averaged with the same uni-grid resolution (512 3 , ∆L = 114 ckpc).The total number of recombinations per hydrogen atom from these two 40h −1 Mpc boxes without averaging (i.e. in original 1024 3 , ∆L = 57 ckpc blocks) is shown in Fig. 4, together with all other lines from the left panel of Fig. 3.The additional factor of 2 increase in resolution significantly increases the total number of recombinations per hydrogen atom, totally drowning any possible con-tribution from mini-halos.This increase should not be over-interpreted -as the resolution increases, the contributions from recombinations from CGM and ISM of galaxies increase, but with tens of ckpc resolution, these contributions are not captured accurately.In addition, at small enough scales, some of the recombinations are balanced by collisional ionizations, which are not accounted for in this work.The increase in the number of recombinations between 114 and 57 ckpc resolution simply implies that at such scales the counting of recombinations becomes complex and ambiguous, with recombinations from CGM, ISM, and those balanced by collisional ionizations requiring special treatment.
In conclusion, it appears that the mass and spatial resolution of the current generation of large reionization simulations, such as CROC or Thesan, is sufficient to account for all recombinations, and that unresolved recombinations in mini-halos are negligible for all practical purposes.This conclusion is based on Park et al. (2016) and (Chan et al. 2023) simulations.These simulation sets can be improved further by, for example, adding the density dependence to (Chan et al. 2023) simulations and improving the sampling of the photo-ionization rate dependence.If such extensions of the existing small box simulations become available, the conclusions of this paper will need to be revised.I am grateful to Tsang Keung Chan and Hyunbae Park for constructive comments that significantly improved the original manuscript.This work was supported in part by the NASA Theoretical and Computational Astrophysics Network (TCAN) grant

Figure 1 .
Figure1.The PDF of mean densities in 800 ckpc boxes at z = 5.The non-negligible spread in densities is not accounted for inChan et al. (2023) simulations but can be included usingPark et al. (2016) simulations.

Figure 2 .
Figure 2. Left: the total number of recombinations per hydrogen atom for the three choices of the reionization threshold in the full 117 Mpc CROC box.The translucent band shows the error in the mean as measured from 2 different numerical estimates of the photo-ionization rate Γ.The purple line shows the xi = 0.1 case for Park et al. (2016) (their Equation22) applied to CROC blocks of 3 3 cells (the best match to their 300 ckpc boxes).Right: the same quantity for 3 different fits to the Γ dependence, Equations (2-4).

Figure 3 .
Figure3.The total number of recombinations per hydrogen atom for two CROC simulations: fiducial "earlier reionization" (left) and "later reionization" (right).Blue lines with bands show the fiducial Chan-Park model with the full range (from minimum to maximum) of uncertainties due to methodological choices.Orange solid and dotted lines show NREC/NH and NUNI/NH from CROC simulations, which can be interpreted as the total number of recombination per hydrogen atom from uniform blocs of ∆L = 114 ckpc and ∆L = 800 ckpc respectively.

Figure 4 .
Figure4.The same as the left panel of Fig.3, and now also showing recombinations from 2 smaller, 40h −1 Mpc boxes with the same size, 1024 3 uni-grids, that offer twice higher uniform spatial resolution.With additional resolution, the contribution of unresolved strictures becomes negligibly small.