First Structure Formation under the Influence of Gas-Dark Matter Streaming Velocity and Density: Impact of the Baryons-trace-dark matter Approximation

The impact of the streaming motion between gas and dark matter on the first structure formation has been actively explored by recent studies. Here we investigate how much the key results are affected by two approximations made in many of those studies. One is to implement the streaming motion by accounting for only the relative velocity between baryons and dark matter while assuming"baryons trace dark matter"spatially at the initial conditions of the simulation. This assumption neglects the impact on the gas density taking place before the initialization. In our simulation initialized at $z=200$, this approximation overestimates the gas power spectrum up to 30\% at $k\approx10^2~h~\mbox{Mpc}^{-1}$ at $z=20$. However, the halo mass function and the gas fraction in halos are minimally affected. The other approximation is to artificially amplify the density/velocity fluctuations in a cosmic mean density volume at the initialization to simulate the first minihalos. This approximation gives a head start to the halo growth and the subsequent growth rate is as fast as in the mean density. The growth in actual overdensity, however, is accelerated gradually in time. For example, increasing $\sigma_8$ by 50\% effectively transforms $z\rightarrow\sqrt{1.5}z$ in the halo mass growth history while in 2-$\sigma$ overdensity the acceleration is described by $z\rightarrow{z}+4.8$. Halos in the former case are more grown than in the latter before $z\approx27$ and vice versa after. The gas fraction in halo remains unchanged by this second approximation as well, suggesting that those approximations do {\it{not}} bias the PopIII star-formation rate significantly.


INTRODUCTION
Formation of the first stars (population III or "Pop III" star) is an important milestone in the cosmic history, where the primordial density fluctuations from the cosmic inflation (Guth 1981;Linde 1982) started collapsing ambient baryons into bound objects from z ∼ 30 and producing ultra-violet radiation into space for the first time in the cosmic history (e.g., Bromm 2013; Barkana & Loeb 2001). According to the standard ΛCDM model of the structure formation, the structures begin collapsing from small scales followed by their assembly into larger structures. Low-mass dark matter halos with ∼ 10 4 -10 8 M (i.e., minihalos) are considered as the formation site of the first collapsed objects. The details of the collapse involve highly nonlinear physics and are an active field of numerical astrophysics (e.g., Yoshida et al. 2003).
Recently, Tseliakhovich & Hirata (2010) pointed out that the residual velocity fluctuations from the baryonic acoustic oscillation (BAO) result in a strong relative motion of typically ∼ 30 km/s between baryons and dark matter at the cosmic recombination. This motion decays in time, but this is strong enough to induce the streaming of gas through dark matter potential wells and thus make it more difficult for minihalos to grow their masses and accrete gas at the time of the first star formation. Sub-sequent numerical studies confirmed that the baryonic fraction in minihalos is highly suppressed by the streaming motion (Greif et al. 2011;Naoz et al. 2013;Richardson et al. 2013;Asaba et al. 2016). Moreover, the supersonic motion shock heats the gas to make cold gas even rarer inside halos (Schauer et al. 2019a).
The global impact of the streaming motion on the cosmic reionization is being actively explored. The beginning of the reionization is expected to be significantly delayed (Maio et al. 2011;Schauer et al. 2019b) although the impact considered to be limited at the late stage of reionization (z ∼ 6) driven by more massive ( 10 8 M ) atomic-cooling halos (Stacy et al. 2011;Fialkov et al. 2014b). In semi-numerical models of starformation and reionization, the streaming is considered to raise the minimum halo mass that can form Pop III stars (e.g., Greif et al. 2011;Muñoz 2019;Visbal et al. 2020). Also, the effect is expected to vary spatially because the streaming velocity is known to fluctuate at the BAO scale (∼ 140 Mpc). It is an interesting possibility that large-scale fluctuations in the Pop III star-formation rate can leave an imprint on the spin temperature of atomic hydrogen Visbal et al. 2012;Muñoz 2019), which may be proved by upcoming 21cm survey such as the Hydrogen Epoch of Reionization Array (HERA) and Square Kilometre Array (SKA: Mellema et al. 2013;Fialkov et al. 2014a;DeBoer et al. 2017).
There also are attempts to explain existing tensions between the standard cosmology and observation using the streaming motion. Regarding the mystery of highredshift (z ∼ 6) supermassive blackholes with ∼ 10 9 M (Mortlock et al. 2011;Wu et al. 2015;Bañados et al. (d0v2) and the right panel shows the case without streaming velocity (d0v0). The density field is color-coded so that overdensity is shown in red and underdensity in blue.

2018)
, several numerical studies showed that the streaming can induce the formation of direct collapse blackholes (DCBH) of ∼ 10 5−6 M at z ∼ 30 (Tanaka & Li 2014;Hirano et al. 2017) to give a head start to the blackhole growth although there are counter-arguments to this scenario (Latif et al. 2014;Visbal et al. 2014). Some studies attempt to explain the formation mechanisms of missing satellites and globular clusters based on the fact that the streaming separates dark matter and baryons (Bovy & Dvorkin 2013;Naoz & Narayan 2014;Popa et al. 2016;Chiou et al. 2019).
Given the increasing number of numerical simulation studies about the streaming motion, it is worth investigating the validity of approximations frequently made at the initialization of simulation. The first approximation to test is the assumption that baryons trace dark matter (BTD) at the initial conditions. Commonly used initial condition generators mostly assume the initial density/velocity field of baryonic matter is the same as that of the dark matter at the initialization. The actual amplitude of baryon density fluctuation is smaller than that of dark matter at z 100, but these two amplitudes are known to converge toward each other due to gravity before the first objects start forming. Thus, many numerical studies applied the streaming effect by simply adding a constant velocity to the baryon velocity field in the initial conditions, while using the same density field for both baryons and dark matter. This baryons-tracedark matter assumption, however, likely to break down in the presence of the streaming velocity that would in-terrupt the coupling of the two components. Also, the streaming effect should be stronger at higher redshift, but the approximation misses the effect taking place between the cosmic recombination and the initialization of simulation.
Another approximation to test is to artificially increase σ 8 , which will amplify density fluctuation in all scales, to mimic an overdense patch of the universe. This method is used to assimilate the biased formation of the first structures in dense regions of the universe. For example, Hirano et al. (2017) used this method to simulate the extremely rare objects, only a few of which form in a (Gpc) 3 volume. We shall examine how the structure growth compares between such an artificial case and the actual overdense case.
To provide self-consistent initial conditions with the streaming motion in overdensity, Ahn (2016) developed a quasi-linear perturbation theory of small-scale fluctuations under the influence of the large-scale overdensity and streaming-velocity environment. Ahn & Smith (2018) then developed an initial condition generator BC-COMICS 6 (Baryon-Cold dark matter COsMological Initial Condition generator for Small-scales), which calculates the perturbation equations of Ahn (2016) and generates corresponding 3-dimensional initial conditions of dark matter and baryons. BCCOMICS treats a given overdense (underdense) patch as a separate universe with positive (negative) curvature and provides a set of "lo-cal cosmology parameters" to account for the local expansion rate different from the mean cosmic expansion rate. Ahn & Smith (2018) used BCCOMICS to generate a suite of initial conditions for varying streamingvelocity and density environments and then performed N-body and hydrodynamic simulations to explore the cosmic variance of high-redshift structure formation.
This study is a continuation of the efforts by Ahn (2016) and Ahn & Smith (2018) to explore the dual impact of the streaming motion and overdensity with correctly generated initial conditions, and an extension of these work to compare the self-consistent approach quantitatively to the two common approximations used in generating initial conditions. Therefore, this work partially revisits the work of O' Leary & McQuinn (2012) that tested the baryons-trace-dark matter assumption by providing more results on key statistics of the Pop III star formation.
The paper is organized as follows. In Section 2, we introduce our numerical methods used in this study. In Section 3, we show our results. In Section 4, we summarize our results and make conclusions. For the rest of this paper, we assume ΛCDM cosmology consistent with the WMAP 9-year results (Hinshaw et al. 2013): Ω m,0 = 0.276, Ω b,0 = 0.045, h = 0.703 and n s = 0.961.

METHODOLOGY 2.1. Basics of Streaming Motion
In the absence of the streaming velocity, the perturbation equation for the overdensity δ and the peculiar velocity v of matter is given by where θ ≡ a −1 ∇ · v with the scale factor a and the gradient in the comoving frame ∇, H is the Hubble parameter, and Ω m is the cosmic matter fraction of the universe at given cosmic time t. Common IC generators use the solution from the above equation to set the density and velocity fluctuation amplitudes of both baryons and dark matter. The perturbation equation in the presence of baryondark matter streaming velocity in the CDM-rest frame (V c = 0). Here, /Ω m are the global baryon and dark matter fraction in matter, respectively, x e is the global ionized fraction, t γ = 1.17 × 10 12 yrs, andT γ = 2.725(1 + z) K is the mean temperature of the cosmic microwave background at redshift z. The bulk quantities of a patch ∆ b , ∆ c and ∆ T denote the overdensity of baryons, the overdensity of dark matter and the baryon temperature fluctuation, respectively, and their values in the Fourier space are identical to the real-space values. ∆ b and ∆ c defined at the length scale 4 h −1 Mpc are tightly correlated at z 200 and almost uncorrelated at z ∼ 1000 (Ahn 2016;Ahn & Smith 2018). In this work, we shall run several simulations with ∆ c = 0.0323 and ∆ b = 0.027 at z = 200, which corresponds to 2-σ overdensity for a 4 h −1 Mpc box at that redshift.
The streaming velocity fluctuates spatially at the BAO scale (∼ 140 Mpc). Thus, the streaming velocity V cb can be treated as a constant drift within 10 Mpc. |V cb | follows the Boltzmann distribution with the standard deviation of σ = 28 km/s at z = 1000, which decays as (1 + z) with cosmic expansion.

Simulation Parameter Choice
The list of the simulation parameter choices for the simulations in this work is given in Table 1. The fiducial case, d0v0, has the cosmic mean density and zero streaming velocity in a 1 h −1 Mpc box. Several cases are run with a streaming velocity of 56(z/1000) km/s, which is twice the r.m.s. of the streaming velocity distribution. We run a streaming case in a cosmic mean density (d0v2) and in a 2-σ overdensity (d2v2; ∆ = 3.14 × 10 −2 ). We also run one simulation with 2-σ streaming velocity and 2-σ overdensity in a bigger box of 4 h −1 Mpc to obtain statistics of higher mass halos (d2v2L) that cannot be captured in a 1 h −1 Mpc box.
We make two cases that reproduce the approximations made in other studies. In d0v2 BTD, we apply the "baryons trace dark matter" assumption in the initial conditions by assigning the same density/velocity field to both baryons and dark matter. The amplitude of the density/velocity fluctuations is given by Equation (1) in this case. We also run a 4 h −1 Mpc box simulation with the same set-up (d0v2L BTD). In d0v2 IS, we artificially boost the normalization of the initial density power spectrum in d0v2 by raising σ 8 from 0.8 to 1.2. This method was adopted by Hirano et al. (2017) to simulate ∼ 10 6 M halos at z = 30, which can only exist in highly overdense regions.

Simulation Setup
We adopt the smoothed particle hydrodynamics (SPH) code GADGET-2 (Springel et al. 2001;Springel 2005) to follow the structure formation from z = 200 to ∼ 15. BCCOMICS solves Equation (2) to create the initial conditions in the ENZO ) simulation format. Thus, we convert the output of BCCOMICS into the GADGET format. We initialize the gas and dark matter particles on two separate grids that are offset by half the average particle distance.
The initial conditions are generated in five 1 h −1 Mpc boxes and two 4 h −1 Mpc boxes with 512 3 dark matter particles and the same number of gas particles at z = 200. Simulations with the same box-size are initialized with the same set of random phases to exclude the cosmic variance effect in the comparison.
We note that we use separate transfer functions for gas and dark matter by definition, except for the BTD cases. At the initialization redshift z = 200, the gas density power spectrum is much smaller than the linear matter density power while the dark matter power is slightly larger. In the case of zero streaming velocity (d0v0), their weighted average power spectrum agrees with the linear power spectrum as described by the black dotted line in Figure 1. Both power spectra converge toward the linear power spectrum until nonlinear growth takes off. However, generating gas and dark matter density fields from the same transfer function is known to cause mild systematic effects (Yoshida et al. 2003), which may become more severe in the presence of the streaming velocity separating gas density peaks with respect to dark matter density peaks. The density power spectrum of gas (solid lines), dark matter (dashed lines) and the total matter (dotted lines) field for the case with (d0v2; cyan) and without the streaming motion (d0v0; black) at z = 200. The linear matter-density power spectrum is shown as a grey line for a reference.

Simulating Overdense Region
The first collapsed objects are likely to appear in overdense patches of the universe. Some numerical studies rely on a multi-resolution adaptive-refinement scheme (e.g., MUSIC; Hahn & Abel 2011) to start from a large box with the cosmic mean density and zoom into a denser subregion where structures develop earlier than in other parts of the box. We simulate one case in an overdense region (d2v2) to study the accelerated growth of structure in overdensity.
In this work, we present a complementary method referred to as the "separate universe" approach that starts from initial conditions of overdense volume using solutions of Equation (2) for a non-zero local overdensity ∆. An advantage of this method is that one can easily create extremely rare density peaks, only a few of which appear in a (Gpc) 3 volume. Creating such an extremely overdense patch would require an excessive number of refinements with the adaptive-resolution scheme.
In overdense regions, the cosmic expansion rate is locally slower than the global rate. We capture this effect by modifying the cosmology parameters in the simulation setup. BCCOMICS provides the local cosmology parameters for volumes with non-zero overdensity. In an overdense volume, cosmology parameters of a closed universe are used to describe the expansion rate. The derivation of the local cosmology parameters is given in Section 3 of Ahn & Smith (2018). A detailed description of this method can also be found from Sirko (2005) and for cases without the streaming velocity.
Our method does not capture higher-order gravitation effects like the shear and tidal force from missing largescale structures. However, such effects should be negligible at the time of first structure formation. A similar approach was used by Goldberg & Vogeley (2004) in simulations of underdense regions to study cosmic voids.
In volumes with non-zero overdensity, the redshift evolves differently from the global value due to the modified expansion rate. Thus, one must keep track of the relation between the true and the local redshift in the simulation. We define the local redshift (z) so that it becomes zero at the end of the simulation: in this work simulations end at the global redshift z = 20. Then, the initial value of the local redshift in d2v2 becomes z i = 7.72 while the true redshift is z = 200. In Figure 2, we show howz and the box size in the global comoving scale evolve in time. The local cosmological parameters of the d2v2 case areΩ Λ,0 = 3.58 × 10 −4 ,Ω m,0 = 1.43, Ω b,0 = 0.27 andh = 31.6, where tildes are used to denote that the parameters are the local values. Note that the "present" values denoted by the subscript "0" of these cosmological parameters are evaluated atz = 0.
It is worth noting that the simulation volume expands by a factor ofz i + 1 = 8.72 between z = 200 and 20 while the universe globally expands by a factor of (200 + 1)/(20 + 1) = 9.57. As a result, the comoving box-size of the overdense simulation at z = 20 is smaller by ∼ 9% (see also Fig.2) and the mean density of the simulation is (9.57/8.72) 3 = 1.32 times higher than the cosmic mean at z = 20.

Halo Identification
We use the publicly available version of Amiga Halo Finder 7 (AHF; Gill et al. 2004;Knollmann & Knebe 2009) to identify halos from the simulation output at z = 20. AHF outputs a list of gas, dark matter and total mass of identified halos. These quantities are used for obtaining the halo mass function and baryonic fraction in halos for a given halo mass, which is considered highly relevant to the Pop III star formation rate.
As usual, the virial radius of a halo is chosen to make the mean density within halo ∆ th = 200 times the cosmic mean. In the overdense case, the mean density of the simulation box grows increasingly when compared to the cosmic mean. We thus compensate for the overdensity by re-scaling the density threshold parameter ∆ th , which is in the unit of the mean density of the simulation box as described in Figure 2. For example, the virial radius of a halo in the overdense simulation at z = 20 is defined by ∆ th = 200/1.32 = 152 times the mean density. The details of this mapping process is described in Ahn & Smith (2018). 7 http://popia.ft.uam.es/AHF/Download.html 3. RESULTS Figure 1 describes the effect of the streaming motion in the initial conditions of the no-streaming case (d0v0) and streaming case (d0v2) at z = 200. They share similar large-scale patterns due to the same random phases used in the initial conditions, but the small density fluctuations are strongly suppressed in the steaming case. This trend is also shown in Figure 3, where gas density power spectrum of d0v0 and d0v2 are similar to each other at k < 10 2 h Mpc −1 while they diverge at k > 10 2 h Mpc −1 due to an extra suppression in the case of d0v2.
We present our results around z = 20 in the subsequent subsections. We explore the impact of the baryons-tracedark matter assumption on structure formation in Section 3.1 and mimicking an overdense region by boosting σ 8 in Section 3.2.

Streaming Effect with Baryons Tracing Dark
Matter In this section, we examine the baryons-trace-dark matter assumption by comparing the z = 20 snapshots of the no-streaming case (d0v0), the cases where the streaming is applied with the approximate assumption (d0v2 BTD and d0v2L BTD), and the case that correctly accounts for the streaming effect (d0v2 & d0v2L), where all the cases except d0v0 have the streaming velocity of 56(z/1000) km/s. The gas particle maps are shown in Figure 4, the gas and dark matter density power spectra are shown in the left panel of Figure 5, the accumulated halo mass functions are shown in Figure 6, and the baryonic mass fraction for given halo mass in Figure 7.
The baryons-trace-dark matter assumption ignores the streaming effect taking place between the Cosmic recombination (z ∼ 1000) and the beginning of the simulation (z = 200). It is thus expected that the smoothing effect on the gas density field is underestimated in d0v2 BTD. The gas density maps (Fig. 4) describe how the gas density field is smoothed in both d0v2 and d0v2 BTD compared to in d0v0 while the difference between in d0v2 and in d0v2 BTD is not evident.
100 h Mpc −1 in d0v2 BTD. The difference between the two cases peaks at around k = 100 h Mpc −1 , where the dimensionless power spectrum (∆ δδ ≡ k 3 P δδ (k)/(2k 2 )) is suppressed by 30% in d0v2 while only by 3.5% in d0v2 BTD. The difference decays towards both the large and small-k limits and disappears at k 20 or k 500 h Mpc −1 .
The suppression of the halo mass function (See Fig. 6) is slightly stronger in d0v2 than in d0v2 BTD. For example, the number halos with M h > 3 × 10 5 M in d0v2 is reduced to 45.1% of that in d0v0 while to 52.1% in d0v2 BTD. A similar trend is seen from the comparison of dark matter density power spectrum at k 300 h Mpc −1 , which should be closely related to the halo abundance.
Despite the bias in other statistics discussed above, our results show that the assumption does not affect the halo baryon-to-mass fraction f b,h to significant level. f b,h in d0v2 described by the cyan lines in Figure 7 agrees with that in d0v2 BTD well within the 1-σ uncertainty at the mass range of 3 × 10 4 M h /M 10 6 . Similarly, f b,h in d0v2L agree with that in d0v2L BTD at 10 6 M h /M 3 × 10 7 . Therefore, we confirm that f b,h is insensitive to the assumption for a wide halo mass range of 3 × 10 4 M h /M 3 × 10 7 . The suppression in f b,h is often considered as the most direct impact of the streaming motion on the Pop III star formation. f b,h has been repeatedly modeled with the streaming motion by previous works, most of which relied on the baryons-trace-dark matter assumption. It is naively expected that the assumption would underestimate the suppression effect. However, we do not find any evidence for the underestimation in this work.
3.2. Growth of Structure with Increased σ 8 In this section, we compare the overdense case (d2v2) to the case that we artificially boosted the initial density/velocity fluctuations in a mean density volume by increasing σ 8 (d0v2 IS). We take d0v2 as the fiducial case so that all three cases mentioned have the 2-σ level  streaming velocity (V bc = 56(z/1000) km/s). The density power spectrum (right panel of Fig. 5) and the halo mass function (left panel of Fig. 6) show how much the structure growth is enhanced in d2v2 and d0v2 IS compared to in d0v2 at z = 20. The enhancement appears stronger in d2v2: the number density of halos with M h > 10 6 M /h is increased by a factor of 8 and 4.2 in d2v2 and in d0v2 IS at z = 20, respectively. However, halo function comparison at higher redshifts in the right panel of Figure 6 shows d0v2 IS has more halos than d2v2 does at z 30.
The substantial difference between d2v2 and d0v2 IS indicates that self-consistent initial conditions are crucial in studying the effect of local overdensity on structure formation. Note that both the density power spectrum and the halo mass function are evaluated in the global comoving frame, and thus the result from d2v2 simulation reflects the fact that the local patch has been detached and shrunken from the global comoving frame. Even though the boosted σ 8 of e.g. d0v2 IS case can mimic the expedited formation of structures in overdense regions, this scheme cannot reproduce the density bias of halo clustering correctly because the simulation volume still has the expansion rate same as the global value. In terms of the peak-background split scheme (Mo & White 1996) however, a boost of σ 8 (e.g., d0v2 IS) only affects the linear density threshold for halo formation, but a locally collapsing patch (e.g., d2v2) affects both the haloformation density threshold and the clustering scale of halos.
To compare the time evolution of the halo mass function in a convenient manner, we define a mass M 10 in a way that the number of halos above that mass is fixed to a certain number density: (3) In Figure 6, this would be the x-coordinate of the intersection of mass function and the second grey horizontal grid line from the bottom. Since the halo mass function grows monotonically in time, M 10 can be used to indicate how much halos are grown in the simulation. We plot M 10 as the function of redshift for d0v2, d2v2, and d0v2 IS in Figure 8. M 10 in d2v2 is smaller than in d0v2 IS at z 27 and larger at z 27, which is in agreement with the mass function comparison mentioned above. This can be understood from the difference in how the structure formation is enhanced between the two cases. The structure growth in d0v2 IS is given a headstart in the beginning and then proceeds just as fast as in the mean density case later on. On the contrary, the structure growth in d2v2 starts with nearly the same as in the mean density case (d0v2) and is gradually accelerated by locally slower cosmic expansion.
Transforming z to √ 1.5z in M 10 (z) in d0v2 reproduces M 10 (z) in d0v2 IS quite precisely (see Fig. 8). This is explained by the growth rate of structure during the matter dominated era: P δδ (k) ∝ a 2 ≈ z −2 . A factor of 1.5 increment in the initial density power spectrum results in the structure growths happening earlier in the way √ 1.5 is multiplied to the redshift.
Interestingly, M 10 (z) in d0v0 is similar to transforming z to √ 1.15z in M 10 (z) in d0v2. According to the above finding, the impact of the 2-σ level streaming velocity is similar to lowering σ 8 by a factor of 1.15 at the initialization.
In actual overdense case (d2v2), the structure formation is accelerated by a nearly a constant in redshift. M 10 (z) in d2v2 is close to M 10 (z − 4.8) in d0v2 throughout the range we explored ( 15 z 35; see Fig. 8). This constant shift is smaller than the multiplicative shift in d0v2 IS until z ≈ 27, but is larger at lower redshifts.
The impact of local overdensity on the baryonic fraction of halo is generally negligible throughout this study. Moreover, we find that all the cases with 2-σ level streaming velocity follow similar trends within uncertainty ( Fig. 7) suggesting that f b,h depends only on the streaming velocity.

SUMMARY AND DISCUSSION
Recently, a number of simulation studies have been performed in the context of assessing the impact of the baryon-dark matter streaming motion on first collapsed objects. In this study, we have examined the approximations frequently made in their simulations. One is the baryons-trace-dark matter assumption that ignores the smoothing effect of the streaming motion in gas density before the initialization of simulation. The other is boosting the initial amplitude of density/velocity fluctuations to represent an overdense volume that forms the first collapsed objects.
The baryons-trace-dark matter assumption underestimates the suppression in both gas and dark matter density power spectrum. Because of this, the number of minihalos is slightly overestimated. Despite those biases, the gas fraction in halos, which is considered to be directly related to the Pop III star-formation is not affected to a significant level. This finding supports the previous studies of Pop III formation with streaming velocity based on the approximation.
Boosting σ 8 at the initialization of simulation gives a head start in the structure growth while the growth in an actual overdense region is gradually accelerated over time. In that case, the structure growth is overestimated in the early time and underestimated in the late time, and the halo clustering is underestimated. This can bias the growth history of halo mass. For example, 10 6 M minihalo at z ≈ 30 from simulations with increased σ 8 in Hirano et al. (2017) likely have a different mass growth history and halo-clustering scale from reality.
We find that the impact of streaming motion on the halo mass function is similar to the impact of lowering σ 8 . The amount of suppression in mass function by streaming velocity of V cb = 54[(1 + z)/1000] km/s is similar to what we expect from lowering σ 8 by 13%. We presume this is because of the characteristic decay of the streaming velocity. The suppression of halo mass growth should be stronger at an earlier time due to faster stream velocity. That suppression effect should cease by the Pop III formation epoch (z ∼ 30) where the streaming velocity has dropped to ∼ 1 km/s and the structure growth should proceed in the normal rate with some amount of delay from the earlier epoch.
We note that our simulations do not include chemical cooling needed to distinguish the cold component in the minihalo gas, which should be more relevant to starformation (Schauer et al. 2019a). In this sense, our conclusion about the impact on Pop III formation is subject to change in our future studies.