# The Role of Radiation and Halo Mergers in Pop III Star Formation

Lilia Correa Magnus,1 Britton D. Smith,1 Sadegh Khochfar,1 Brian W. O’Shea,
2,3,4,5 John H. Wise, 6 Michael L. Norman, 7,8 and Matthew J. Turk 9,10
1Institute for Astronomy, University of Edinburgh, Royal Observatory,
Edinburgh EH9 3HJ, UK
2Department of Physics and Astronomy, Michigan State University, East Lansing,
MI 48824, USA
3Department of Computational Mathematics, Science, and Engineering, Michigan
State University, East Lansing, MI 48824, USA
4National Superconducting Cyclotron Laboratory, Michigan State University,
East Lansing, MI 48824, USA
5JINA-CEE: Joint Institute for Nuclear Astrophysics-Center for the Evolution
of the Elements, USA
6Center for Relativistic Astrophysics, School of Physics, Georgia Institute of
Technology, Atlanta, GA, 30332, USA
7San Diego Supercomputer Center, University of California, San Diego, 10100
Hopkins Drive, La Jolla, CA 92093, USA
8Center for Astrophysics and Space Sciences, University of California, San
Diego, 9500 Gilman Dr, La Jolla, CA 92093, USA
9School of Information Sciences, University of Illinois, Urbana-Champaign, IL,
61820, USA
10Department of Astronomy, University of Illinois, Urbana-Champaign, IL,
61820, USA

###### Abstract

We present a study of the co-evolution of a population of primordial star-
forming minihalos at Cosmic Dawn. In this study, we highlight the influence of
individual Population III stars on the ability of nearby minihalos to form
sufficient molecular hydrogen to undergo star formation. In the absence of
radiation, we find the minimum halo mass required to bring about collapse to
be $\sim$10${}^{5}\ \mathrm{M_{\odot}}$, this increases to $\sim$10${}^{6}\
\mathrm{M_{\odot}}$ after two stars have formed. We find an inverse
relationship between halo mass and the time required for it to recover its
molecular gas after being disrupted by radiation from a nearby star. We also
take advantage of the extremely high resolution to investigate the effects of
major and minor mergers on the gas content of star-forming minihalos. Contrary
to previous claims of fallback of supernova ejecta, we find minihalos
evacuated after hosting Pop III stars primarily recover gas through mergers
with undisturbed halos. We identify an intriguing type of major merger between
recently evacuated halos and gas-rich ones, finding that these “mixed” mergers
accelerate star formation instead of suppressing it like their low redshift
counterparts. We attribute this to the gas-poor nature of one of the merging
halos resulting in no significant rise in temperature or turbulence and
instead inducing a rapid increase in central density and hydrostatic pressure.
This constitutes a novel formation pathway for Pop III stars and establishes
major mergers as potentially the primary source of gas, thus redefining the
role of major mergers at this epoch.

###### keywords:

stars: Population III — galaxies: star formation — galaxies: high redshift —
early Universe

††pubyear: 2023††pagerange: The Role of Radiation and Halo Mergers in Pop III
Star Formation–The Role of Radiation and Halo Mergers in Pop III Star
Formation

##  1 Introduction

The first stars to form in the universe, classified as Population III (Pop
III), triggered an important transition in the early evolution of structure
formation. The environment at this epoch, being mostly composed of hydrogen
and helium – greatly contrasts the diverse and increasingly complex universe
observed at present (see reviews by Ciardi & Ferrara 2005; Glover 2012 with
Bromm 2013). The birth of these first luminous objects initiated reionisation
and their supernovae enriched the intergalactic medium (IGM) and enabled the
formation of subsequent generations of stars. Understanding the conditions
that lead to their creation is therefore crucial to fully comprehend the
Cosmic Dawn era and the seeding of the first galaxies.

The discussion of this epoch also requires a brief delve into the current
concordance model to establish some cosmological context. Under the
$\Lambda$CDM framework, dark matter is assumed to be cold and weakly
interacting. This introduces a hierarchical growth system whereby dark matter
halos grow from small to large via a series of mergers (Couchman & Rees 1986;
Khochfar & Burkert 2008). The halos that hosted Pop III stars are therefore
assumed to have been small – usually referred to as minihalos – and had virial
temperatures of around $10^{2}-10^{3}$ K. Studies such as Couchman & Rees
(1986) and Tegmark et al. (1997) established these initial virial temperatures
and highlighted the unique scenario minihalos presented in the formation of
Pop III stars. The environment provided by these young halos directly
determined the formation channels available to the first stars. The metal-free
gas present alongside virial temperatures below the atomic hydrogen cooling
limit meant a cooling mechanism was required for gas to effectively collapse.
At high redshifts, the dominant cooling channel in primordial gas is via
molecular hydrogen transitions (Peebles & Dicke 1968; Omukai 2001). Tegmark et
al. (1997) also presented analytic methods to model this early collapse period
alongside analysis of the chemical processes involved in the production of
$\mathrm{H_{2}}$. Because minihalos do not readily contain enough
$\mathrm{H_{2}}$ to efficiently cool, they first require a period of
$\mathrm{H_{2}}$ build-up. At high redshifts, $\mathrm{H_{2}}$ is created
primarily via the $\mathrm{H^{-}}$ channel
($\mathrm{H}+\mathrm{H^{-}}\rightarrow\mathrm{H_{2}}+\mathrm{e^{-}}$) and the
$\mathrm{H_{2}^{+}}$ channel
($\mathrm{H_{2}^{+}}+\mathrm{H}\rightarrow\mathrm{H_{2}}+\mathrm{H^{+}}$).
Studies of this early period (Abel et al. 2002; Yoshida et al. 2003; Reed et
al. 2005; Wise & Abel 2007; O’Shea & Norman 2007; Yajima & Khochfar 2017)
generally agree that the first stars formed at $z\sim 20-30$, once the first
halos reached a high enough $\mathrm{H_{2}}$ fraction. The fragile nature of
$\mathrm{H_{2}}$, however, means this process is not as straightforward at
slightly later epochs. Molecular hydrogen is easily photo-dissociated by UV
radiation in the Lyman-Werner (LW) bands (11.2-13.6 eV), which is emitted by
Pop III stars and can reach the cores of nearby halos if these are not yet
large enough for self-shielding to take place (O’Shea & Norman 2008; Agarwal &
Khochfar 2015). The presence of nearby star formation can therefore set-back
collapse by lowering $\mathrm{H_{2}}$ levels below what is needed to cool
efficiently (Johnson et al. 2013; Agarwal et al. 2019).

Over the past 20 years cosmological simulations have been used to study this
high redshift period and works have focused heavily on the effect of LW
radiation has on the halo mass scale of star forming minihalos. Early research
work such as Tegmark et al. (1997), Machacek et al. (2001) and Yoshida et al.
(2003) attempted to establish a function for the critical halo mass
$\mathrm{M_{crit}}$ above which minihalos are able to form stars. They also
established LW radiation as an effective suppressant of star formation,
agreeing on the heavy impact this type of feedback had, pushing
$\mathrm{M_{crit}}$ up from $\sim 10^{5}$ $\mathrm{M_{\odot}}$ under no
radiation to $10^{6}$ $\mathrm{M_{\odot}}$ or even $10^{7}$
$\mathrm{M_{\odot}}$ for irradiated halos. More recent studies such as Park et
al. (2021), Schauer et al. (2021), Kulkarni et al. (2021) and Incatasciato et
al. (2023) have continued to investigate this topic, with additional
considerations such as an X-ray background or baryon-dark matter streaming
velocities alongside LW radiation. Although more nuanced – due to the
additional factors considered – these works also found delayed collapse
scenarios in environments where halos were exposed to UV photons. The
evolution of the $\mathrm{M_{crit}}$ function is still a debated topic, with
many finding a redshift dependence: Tegmark et al. (1997); Trenti & Stiavelli
(2009); Kulkarni et al. (2021); and others contesting this claim: Machacek et
al. (2001); Schauer et al. (2019); Schauer et al. (2021) and Incatasciato et
al. (2023). Despite this tension, there is a strong consensus that LW
radiation effectively suppresses and delays star formation at early epochs by
lowering the $\mathrm{H_{2}}$ content inside minihalos. However, these works
have focused on the large-scale environmental impact of UV radiation via a
steadily building LW background. Little attention has been paid to the smaller
scale impact of discrete populations of nearby Pop III stars.

Building on the chemical model presented in Tegmark et al. (1997), some works
have also attempted to account for other coolants such as $\mathrm{HD}$
(Yoshida et al. 2006; Ripamonti et al. 2007; Greif et al. 2008). The advantage
of cooling via HD is its asymmetry, which provides a permanent dipole moment
and allows gas to cool down to CMB temperatures. This cooling channel
contrasts the $\mathrm{H_{2}}$ rotational transitions, which become
inefficient below $\sim$200 K, and above number densities of $\sim$103-4 cm-3,
when populations are dictated by local thermodynamic equilibrium (LTE). As
discussed in studies that looked at molecule formation in the early universe
such as Galli & Palla (1998) and Stancil et al. (1998), production of HD under
Cosmic Dawn conditions is only possible via the enhancement of the
$\mathrm{H_{2}}$ formation reaction itself. Given the reaction:
$\mathrm{H_{2}}+\mathrm{D^{+}}\rightarrow\mathrm{HD}+\mathrm{H^{+}}$, gas
clouds that contained a higher HD fraction at this epoch are also expected to
be highly ionized. Based on this idea, Shchekinov & Vasiliev (2006) produced a
simplified one-dimensional model to find whether mergers could boost the
$\mathrm{H^{-}}$ and $\mathrm{e^{-}}$ content, helping catalyse the formation
of $\mathrm{H_{2}}$ and $\mathrm{HD}$. Later studies such as Prieto et al.
(2012) and Bovino et al. (2014), would expand on this by producing more
realistic merger scenarios in hydro-cosmological simulations. They both found
turbulence and shock-waves from such dynamical events could increase HD levels
and provide an alternative pathway to star formation.

These theories could initially be seen as contradictory to studies that assert
dynamical events such as major mergers delay star formation (Yoshida et al.
2003; Fernandez et al. 2014; Wise et al. 2019). Starting from the Rees-
Ostriker relation (Rees & Ostriker, 1977): $t_{\mathrm{dyn}}\gg
t_{\mathrm{cool}}$; which requires cooling timescales to be much smaller than
dynamical times for collapse to occur, any mechanism that can invert this
relation results in the opposite effect. Fast accretion, in the form of major
mergers provides such a scenario. Increased turbulence and shock-heating from
these events is thought to be able to greatly reduce dynamical timescales and
therefore suppress star formation. Wise et al. (2019) find important delays in
their star forming halos, as turbulence maintains gas pressure-supported
against collapse. Moreover, Latif et al. (2022) find that the cosmic web
collapse provides enough turbulence to prohibit star formation, even at
moderate growth rates of halos. In their research of HD cooling, Prieto et al.
(2012) and Bovino et al. (2014) do see a large increase in gas turbulence and
heating, which is required to increase the abundance of $\mathrm{H_{2}}$ and
boost the HD content. Although these works don’t specifically mention delays
in star formation, none of their collapse scenarios occur whilst a major
merger is ongoing. Further studies on the subject such as Hirano et al. (2014)
additionally suggest a slow collapse mode where gas clouds are rotationally
supported can provide the right conditions for HD cooling to become effective.
Both concepts are not mutually exclusive and could occur in the same merger
events, though the possibility of HD enhancement adds complexity to the
effects of major mergers on Pop III star formation.

While there is general consensus that Pop III star formation is heavily
influenced by environmental factors (i.e., radiation and structure formation),
few works have investigated the scenario of minihalos co-evolving amidst the
formation of discrete stellar populations. In this paper, we study the
conditions that lead to the formation of metal-free, star-forming gas with a
cosmological simulation that includes the radiative feedback of individual Pop
III stars. The simulation resolves halos that are many orders of magnitude
smaller than the canonical minimum halo mass for metal-free star formation,
thus allowing us to robustly probe the influence of minor mergers. The paper
is structured as follows: the details of the simulation code and setup are
introduced in Section 2, our results are presented in Section 3, followed by a
discussion alongside the implication of our findings in Section 4. Our final
summary and conclusions are then discussed in Section 5.

##  2 Methods

The simulation analyzed in this work was first presented in Smith et al.
(2018). It is the second in the Pop2Prime series of simulations, first
presented in Smith et al. (2015). We refer readers to that work for a detailed
discussion of the methods employed. The simulation was run using the open-
source, adaptive mesh-refinement + N-body code Enzo (Bryan et al., 2014;
Brummel-Smith et al., 2019). Enzo has been used extensively to study Pop III
star formation at high redshift (e.g., Abel et al., 2002; O’Shea & Norman,
2007; Turk et al., 2009; Bovino et al., 2014; Skinner & Wise, 2020). A more
thorough explanation of this framework is presented in Bryan et al. (2014),
although our overview should be enough to cover the functionality most
relevant to our work. The analysis of the simulation snapshots and merger
trees was performed using the yt (Turk et al., 2011) and ytree (Smith & Lang,
2019) packages. Halo finding and merger tree generation were performed with
the Rockstar halo finder (Behroozi et al., 2013a) and consistent-trees
(Behroozi et al., 2013b).

###  2.1 Enzo

Enzo is a hydrodynamics and N-body cosmological simulation code that uses
adaptive mesh-refinement (AMR) to model structure formation under the
$\Lambda$CDM framework. It uses AMR to introduce varying levels of refinement
within the simulation grid depending on the amount matter content and detail
required in the volume (Berger & Colella, 1989). As a result, the coarseness
of the grids is either refined or de-refined such that the grid is
increasingly subdivided to obtain higher detail in the denser regions. Within
this basis, hydrodynamics and adaptive particle-mesh gravity solvers are
implemented, in this case the Piecewise Parabolic Method from Woodward &
Colella (1984) is used alongside an N-body adaptive particle-mesh gravity
solver (Efstathiou et al., 1985). Finally, when a radiation source is present
in the volume, the Moray adaptive radiation transport method from Wise & Abel
(2011) is applied to simulate the radiating field.

We make use of the precursor to the Grackle library (Smith et al., 2017),
built-in to the Enzo code, to follow non-equilibrium primordial chemistry and
cooling for the $\mathrm{e^{-}}$, $\mathrm{H}$, $\mathrm{D}$, and
$\mathrm{He}$ species as well as their ionic and molecular variants:
$\mathrm{H^{+}}$, $\mathrm{H^{-}}$, $\mathrm{H_{2}}$, $\mathrm{H_{2}^{+}}$,
$\mathrm{D}$, $\mathrm{D^{+}}$, $\mathrm{He^{+}}$, $\mathrm{He^{++}}$,
$\mathrm{HD}$. We also include the cooling from heavy elements with tables of
rates pre-computed with the Cloudy (Ferland et al., 2013), which assume a
solar abundance pattern and collisional ionization equilibrium (Smith et al.,
2008).

We simulate the formation of individual Pop III stars using the method from
Wise et al. (2012), where dense gas is followed until it reaches the criteria
detailed below. The criteria for Pop III star formation is exactly that
detailed in Smith et al. (2015) except with a higher metallicity threshold of
10-4 Z⊙, instead of 10-6 Z⊙. The specific star particle creation criteria are
as follows:

  1. 1.

The proper baryon number density, $n_{\mathrm{b}}>10^{7}\mathrm{cm}^{-3}$.

  2. 2.

The gas flow is convergent ($\nabla\cdot v_{\mathrm{gas}}<0$).

  3. 3.

The molecular hydrogen mass fraction; $f_{\mathrm{H_{2}}}$ is greater than
$5\times 10^{-4}$, where
$f_{\mathrm{H_{2}}}\equiv(\rho_{\mathrm{H_{2}}}+\rho_{\mathrm{H_{2}^{+}}})/\rho_{\mathrm{b}}$

  4. 4.

The metallicity is below $10^{-4}$ $\mathrm{Z_{\odot}}$

All Pop III star particles produced in the simulation are identical, with
masses of $\mathrm{M}=40$ $\mathrm{M_{\odot}}$ and lifetimes of $3.86$
$\mathrm{Myr}$, and all ending as core-collapse supernovae with energies of
$10^{51}$ ergs. These properties were taken from the stellar evolution models
presented in Schaerer (2002). Each star is treated as a point source that
radiates from the moment it is placed in its halo. The ionizing radiation from
the star particles is modelled using the Moray adaptive ray-tracing method and
following 3 energy groups that ionize: H with $\mathrm{E}=28$ eV and
$\mathrm{L_{\gamma}}=2.47\times 10^{49}\ s^{-1}$; He with $E=30$ eV and
$\mathrm{L_{\gamma}}=1.32\times 10^{49}\ s^{-1}$; $\mathrm{He^{+}}$ with
$E=58$ eV and $\mathrm{L_{\gamma}}=8.80\times 10^{46}\ s^{-1}$.
$\mathrm{H_{2}}$ photo-dissociating radiation is also emitted with
$\mathrm{L_{\gamma}}=2.90\times 10^{49}\ s^{-1}$ and modelled to decline as
$\mathrm{r^{-2}}$ alongside the $\mathrm{H_{2}}$ “Sobolev-like” self-shielding
model from Wolcott-Green et al. (2011). The unique approach in this simulation
combined the absence of a uniform background radiation with a high resolution
capacity, allowed us to treat the formation of each star as an individual
radiative event.

###  2.2 Simulation Environment

A 500 kpc/h simulation box was initialized using the music initial conditions
(Hahn & Abel, 2011) with the WMAP-7 (Komatsu et al., 2011) cosmological
parameters $\Omega_{\mathrm{m}}=0.266$, $\Omega_{\mathrm{\Lambda}}=0.732$,
$\Omega_{\mathrm{b}}=0.0449$, $\mathrm{H_{0}}=71.0$ km/s/Mpc,
$\sigma_{8}=0.801$, and $n_{\mathrm{s}}=0.963$. The simulation zooms in on a
halo reaching 107 M⊙ at $z\sim 10$ with 5123 cells/particles and two initial
levels of refinement, for an effective resolution of $2048^{3}$. This
corresponds to an initial spatial and dark matter resolution of 0.244
$\mathrm{kpc/h}$ and $\sim 1$ $\mathrm{M_{\odot}}$, respectively. The
simulation was designed to model the conditions leading to the formation of
the first Pop II stars assuming the original critical metallicity based on
gas-phase metal cooling (Bromm & Loeb, 2003). When dense gas above this
metallicity forms, the simulation comes to an end. This occurs at $z\sim
11.8$. The simulation is the exact same realization presented in Smith et al.
(2015), only with the higher threshold metallicity, allowing it to run for an
additional $\sim$250 Myr. We find a total of 12 Pop III stars within 9
distinct halos, all of which are studied for this paper.

##  3 Results

Halo label | Formation Time (Myr) | Halo Mass ($\mathrm{M_{\odot}}$) | Double star (Y/N) | Metallicity ($\mathrm{Z_{\odot}}$) | Major mergers (#)
---|---|---|---|---|---
|  |  |  |  | (mixed merger before star? Y/N)
Halo 1 | 144 | $1.2\times 10^{5}$ | No | 0 | 0 (N)
Halo 2 | 211 | $2.0\times 10^{5}$ | No | 0 | 0 (N)
Halo 3 | 240 | $9.8\times 10^{5}$ | No | $2.6\times 10^{-5}$ | 3 (Y)
Halo 4 | 249 | $5.2\times 10^{5}$ | No | 0 | 2 (N)
Halo 5/6 | 279 | $1.1\times 10^{6}$ | Yes | 0 | 1 (N)
Halo 7/8 | 289 | $1.8\times 10^{6}$ | Yes | 0 | 1 (N)
Halo 9 | 312 | $8.6\times 10^{5}$ | No | 0 | 0 (N)
Halo 10/11 | 344 | $6.8\times 10^{5}$ | Yes | 0 | 1 (N)
Halo $12^{*}$ | 379 | $7.3\times 10^{6}$ | No | $5.2\times 10^{-5}$ | 4 (Y)
Table 1: Table detailing the star formation times, halo mass, halos that
hosted two stars, metallicity within the densest gas cell before star
formation, number of major mergers that occurred before a star was created and
whether a mixed merger occurred before star formation for each of the halos.

Before delving into the results, we define the following the halo labelling
system. Each minihalo will henceforth be referred to by the star it formed
such that Halo 1 is the halo that formed the first star, Halo 2 the second and
so on. Three of the halos produced two stars a few thousand years apart; in
this case the $5^{\mathrm{th}}$ and $6^{\mathrm{th}}$ stars formed inside the
same halo, as did the $7^{\mathrm{th}}$ and $8^{\mathrm{th}}$ stars and
$10^{\mathrm{th}}$ and $11^{\mathrm{th}}$ stars. We treat them as the same
radiative event due to their proximity in space and time and no distinction
between the stellar objects is made in our analysis, we therefore label these
minihalos as Halos 5/6, 7/8 and 10/11. We also give a special label to the
halo that forms the final star: Halo $12^{*}$, since it is the same halo that
previously formed the $3^{rd}$ star. Table 1 provides an overview of the star
formation events and their associated halos. We note the occurrence of three
events in which two star particles formed in the same halo within a few
thousand years of each other. We did not identify any obvious distinguishing
characteristics of these halos, but will examine this further in a future
work.

Figure 1: Mass growth of halos from the beginning of the simulation up to the
moment they formed a star, labelled with the halo they formed in. The grey
dashed line shows the evolution of the halo that forms the final star. Because
it merges into Halo 2 (orange line) the halo finder assumes it is formed there
instead, so the final star is labeled as $12^{*}$. The initial time of the two
mixed major mergers that occur in the simulation are marked with a red
triangle. Overplotted in blue are fits of $\mathrm{M_{crit}}$ for different
levels of background radiation ($\mathrm{J_{21}}$) assuming no streaming
velocities, taken from Kulkarni et al. (2021).

###  3.1 The Impact of Radiation

Past works have discussed the clear connection between radiative feedback and
star formation, with photo-dissociation reducing or even halting the cooling
of gas inside halos. We therefore investigated $\mathrm{H_{2}}$ content
alongside halo growth to better understand when radiation effectively delays
star birth during halo evolution.

In Figure 1 we present the mass growth of the star forming minihalos starting
from very low masses at z = $30-40$ up to the point of star formation. The
first halo to undergo star formation is also the first to reach a mass of
$10^{5}$ $\mathrm{M_{\odot}}$ at $z\sim 24$. Halo 1 experiences no radiation
feedback during this period and undergoes no major or minor mergers; it is
therefore an example of star formation under no external influence. Halo 2 is
the only other halo that forms a star while still having a relatively small
mass: $2\times 10^{5}\mathrm{M_{\odot}}$, although it undergoes a small merger
event that introduces a significant amount of gas before its star forms
($\sim$1/4 of its total gas content at $\sim$150 Myr). After two sets of
radiation bursts, none of the other halos form a star near this mass
threshold; instead, they undergo star birth once they approach a mass of
$\sim$106 $\mathrm{M_{\odot}}$ with the final star forming inside one of the
largest halos in the system: $7\times 10^{6}$ $\mathrm{M_{\odot}}$.

Figure 1 leads to a great amount of insight into the effects of radiation, as
we find that only two events are needed to increase the mass of a star forming
halo by an order of magnitude. Subsequent stars therefore take longer to form
and Halos $3-11$ experience close star formation events as they all approach
this higher mass threshold at similar times. Another unique scenario is also
reflected in this figure: the last star that forms inside Halo 3, which had
previously merged with Halo 2, technically forms inside a subhalo structure
created after a major merger. This merging halo has a shallower growth curve,
as seen from the dashed line, since it is the last halo to reach a mass of
$10^{5}$ $\mathrm{M_{\odot}}$ we naturally expect it to form a star at a much
later epoch than the rest of the halos.

Figure 2: Evolution of the $\mathrm{H_{2}}$ fraction alongside the intensity
of the LW radiation in units of $10^{-21}$ erg $\mathrm{s^{-1}\ cm^{-2}\
sr^{-1}\ Hz{{}^{-1}}}$ at the densest point in Halo 7/8’s gas core. The red
stars in the x-axis mark the time of each star’s birth.

It should be noted, that despite Halo 3 also reaching $\mathrm{M_{crit}}$ at a
similar time to Halo 2 (see Figure 1) it would take an extra $20$ Myr for it
to form a star. We attribute this to its accelerated growth slowing the
efficiency of its cooling enough for Halo 2 to form a star first. Radiation
feedback and the blastwave from the second star further exacerbated this
delay, but we will delve into Halo 3’s unique evolution with more detail in
Section 3.2.

The standard effect of LW radiation on our halos is shown in Figure 2. It
presents the $\mathrm{H_{2}}$ fraction measured at the densest gas cell in
Halo $7/8$ up until the formation of its stars. Figure 2 also highlights how
the simulation has no background radiation and instead models individual
“bursts” produced by the stars. Halos are generally exposed to
$\mathrm{J_{21}}$ values of $\sim 0.1$ to 1 while other stars are shining. By
analyzing this Figure, we find that $\mathrm{H_{2}}$ fractions are more
heavily reduced when the mass of the halo is still below or near $10^{5}$
$\mathrm{M_{\odot}}$. We can see however, that the effectiveness of photo-
dissociation diminishes with time, with faster recoveries following each
stellar event. Halo $7/8$ (light purple line in Figure 1) takes about $100$
Myr to grow from $10^{5}$ $\mathrm{M_{\odot}}$ to $10^{6}$
$\mathrm{M_{\odot}}$, throughout this time, the amount of $\mathrm{H_{2}}$
dissociated by each star that forms in the system diminishes as does the time
taken to recover back to star forming levels. This indicates a clear relation
between halo growth and the total delay caused by nearby star formation.
Although only Halo $7/8$ is included for clarity, similar $\mathrm{H_{2}}$
fraction curves are observed for all halos that form stars.

To better quantify the relation between photo-dissociation and halo growth we
looked more closely at recovery times. In Figure 3 we present the calculated
and estimated recovery times for halos in the simulation. The halos included
in this plot had to fullfill the following criteria:

  1. 1.

Within the halo’s progenitor line, the halo at some point in its evolution
must exceed a dark matter mass of $10^{4}$$\mathrm{M_{\odot}}$.

  2. 2.

At the time of a radiative event, the halo must have a molecular hydrogen
fraction $\mathrm{f_{H_{2}}}\geq 10^{-4}$ within its densest grid cell and a
dark matter to baryon mass ratio above 1% of the cosmic mean
($\Omega_{\mathrm{baryon}}\ /\ \Omega_{\mathrm{DM}}$).

  3. 3.

The halo must not be a subhalo.

With this criteria we attempted to remove all the subhalos, gas poor halos and
halos that didn’t have a high enough $f_{\mathrm{H_{2}}}$ to be recovered in
the first place. To measure the recovery time itself we found the highest
$\mathrm{H_{2}}$ fraction reached by each halo before a star formed in the
system. We measured the $\mathrm{H_{2}}$ fraction in the densest cell of each
halo since this is where we would expect star formation to occur. We defined
this as $\mathrm{f_{H_{2}}^{\ 0}}$ and found the amount of time (after the
star’s death) it took the halos to recover half of their original peak
molecular hydrogen content: $\frac{1}{2}\mathrm{f_{H_{2}}^{\ 0}}$. The halo
mass was taken at the time of $\mathrm{f_{H_{2}}^{\ 0}}$, as this established
the initial $\mathrm{H_{2}}$ production rate. Many of the halos never managed
to reach $\frac{1}{2}\mathrm{f_{H_{2}}^{\ 0}}$ before the next star formed, in
these cases we approximated the recovery curve to a logarithmic function since
the form of $f_{\mathrm{H_{2}}}(t)$ is well fit by this as can be seen from
the recovery curves in Figure 2. We therefore made use of the function in
equation 1 to find the extrapolated times.

$f_{\mathrm{H_{2}}}(t)=A+Blog_{10}(Ct)+Dt$ (1)

Where A, B, C and D were free parameters to be fit. The parameters found for
each halo varied due to the redshift, mass and accumulated radiation events.
Using equation 1, we estimate the time it would have taken these halos to
reach $\frac{1}{2}\mathrm{f_{H_{2}}^{\ 0}}$ had the subsequent radiative event
not occurred. The detriment of using such a simple model is evident in Figure
3, since these extrapolated points created a large scatter. Despite this, the
general mass-recovery trend still holds, with halo growth clearly reducing the
$\mathrm{H_{2}}$ recovery times.

Figure 3: Halo $\mathrm{H_{2}}$ recovery times against their dark matter mass
at times just after star formation for halos that met the criteria.
Extrapolated points are plot as open circles. Mean recovery times for unevenly
spaced mass bins: $10^{3}-5\times 10^{3}$; $5\times 10^{3}-10^{4}$;
$10^{4}-5\times 10^{4}$; $5\times 10^{4}-10^{5}$; $10^{5}-5\times 10^{5}$;
$5\times 10^{5}-10^{6}$; $10^{6}-10^{7}$; and their standard deviations are
plot in red. Figure 4: Plot of baryon mass against dark matter for each of the
halos that fulfilled the criteria. The cosmic mean is shown with a black
dashed line and the fit to the data in red (equation 2). Figure 5: Total
molecular hydrogen mass inside the halos that met the criteria as a function
of their dark matter content, measured just after a radiative event. Lines of
different cosmic $\mathrm{H_{2}}$ fractions are shown with different coloured
dashed lines.

Figure 3 therefore presents a clear mass-recovery relation, with recovery
times being reduced exponentially with increased halo mass. Halos with a dark
matter mass of around $10^{5}$ $\mathrm{M_{\odot}}$ seem to experience an
important turning point, as the scatter in recovery times is heavily reduced
and limited to times below $\sim$30 Myr. As halos reach masses $\geq 10^{6}$
$\mathrm{M_{\odot}}$, the overall trend falls to $\sim 10$ Myr. Since the
recovery times reflect $\mathrm{H_{2}}$ production rates, we can understand
that the first two stars to form did so in halos that had masses of $\sim$105
$\mathrm{M_{\odot}}$ because production rates at this threshold became
efficient enough to reduce cooling times below Hubble time. From Figure 4 we
also find that halos that reach this mass shift abruptly towards the cosmic
mean baryon fraction, $\Omega_{\mathrm{baryon}}/\Omega_{\mathrm{DM}}$, being
significantly baryon-poor at lower masses. By the time of the second star’s
radiation event, a larger number of halos in the system had reached or were
approaching this threshold mass, leading subsequent star formation to occur
within very small time delays. This is seen as the closely packed stars
($3^{\mathrm{rd}}$– $11^{\mathrm{th}}$) in Figure 1.

To further investigate the nature of the halos presented in the recovery
graph, we looked at their baryon and molecular hydrogen masses. Figures 4 and
5 show the baryon mass and $\mathrm{H_{2}}$ mass of the halos right before a
radiation event. Figure 4 reflects an interesting behaviour in halo evolution,
gas-poor halos dominate the lower masses and fall far below the cosmic mean,
as would be expected in a $\Lambda$CDM universe. Once they approached the
cosmological Jeans mass ($\mathrm{M_{DM}}$ $\sim 10^{5}$ M⊙, Barkana & Loeb,
2001), they align with the cosmic mean and become potential Pop III star
hosts. The important behavioural pattern highlighted in this figure is
therefore the line break at $\mathrm{M_{DM}}$$\sim 10^{5}$
$\mathrm{M_{\odot}}$, which coincides with the critical mass previously
mentioned. We parameterised this behaviour by fitting a piecewise function to
the data (seen in equation 2), where $x$ in this case is the dark matter mass
of the halo, $f(x)$ the total baryon mass and $x_{\mathrm{th}}$ the threshold
dark matter mass.

$f(x)=\begin{cases}Ax^{\alpha 1},&\text{if }x\leq x_{\mathrm{th}}\\\
Bx^{\alpha 2}+Ax^{\alpha 1-\alpha 2},&\text{if }x>x_{th}\\\ \end{cases}$ (2)

A total of 5 parameters were fit: $A=2.43$, $B=1.97$, $\alpha 1=0.065$,
$\alpha 2=0.0824$ and $x_{\mathrm{th}}=9.8\times 10^{4}$ $\mathrm{M_{\odot}}$.
From the fitted function we therefore find the break in power laws occurs at
around $\mathrm{M_{DM}}$$\sim 1\times 10^{5}$ $\mathrm{M_{\odot}}$ which is
just below the cosmological Jeans mass: $\sim 1.5\times 10^{5}$
$\mathrm{M_{\odot}}$. Using this, in conjunction with the previous recovery
graph (Figure 3) we see there are two distinct star formation epochs. At high
redshifts, before a star has emitted any radiation in the system, gas poor
minihalos dominate; and a race to meet the cosmic mean ensues. The first halos
to approach the cosmic mean ($\Omega_{\mathrm{baryon}}\ /\
\Omega_{\mathrm{DM}}$) will be those that achieve masses of
$\mathrm{M_{DM}}\sim 10^{5}$ $\mathrm{M_{\odot}}$ and are able to subsequently
bring about collapse. Once the system has experienced a few radiation bursts,
in our case two star formation events, star formation is suppressed. The halos
that are aligned with the cosmic mean but have not yet formed a star
experience a consistent photo-dissociation and $\mathrm{H_{2}}$ production
cycle until their recovery times become smaller than the times between star
formation. This growth and feedback relation is what forces the next set of
stars to form within halos that grow to $\mathrm{M_{DM}}\sim 10^{6}$
$\mathrm{M_{\odot}}$, a whole order of magnitude above the initial
$\mathrm{M_{crit}}$. Figures 4 and 5 also explain the lower left clump of
halos found in Figure 3, these seem to be very gas and $\mathrm{M_{H_{2}}}$
poor halos that were included due to our criteria only considering
$\mathrm{{H_{2}}}$ fractions inside halo cores, and not the overall amount.

Continuing the analysis of these halos with Figure 5, we see that baryon mass
is a good indicator of $\mathrm{H_{2}}$ content, as larger halos shift towards
higher constant $f_{\mathrm{H_{2}}}$ cosmic mean lines. These are simply
calculated as: $f_{\mathrm{H_{2}}}\times(\Omega_{\mathrm{baryon}}\ /\
\Omega_{\mathrm{DM}})$. Halos that reside around or above the
$f_{\mathrm{H_{2}}}=10^{-4}$ line are at the star formation stage, whereas
halos below this would still need some time to produce more $\mathrm{H_{2}}$.
The break around $\mathrm{M_{DM}}\sim 1\times 10^{5}$ $\mathrm{M_{\odot}}$ is
less evident in this figure, but still present since no halos above this mass
fall below the $f_{\mathrm{H_{2}}}=10^{-5}$ line. We also find that despite
larger halos having a higher baryon mass content, they don’t necessarily have
a large enough amount of molecular hydrogen to form a star at that point in
time. This is because these halos exist mainly at lower redshifts (z $<$ 17),
and are experiencing successive star formation events that consistently lower
their overall $\mathrm{M_{H_{2}}}$. These larger halos are therefore expected
to quickly move up towards the $f_{\mathrm{H_{2}}}=10^{-4}-10^{-3}$ area
during their recovery phase in order to meet the conditions for star
formation. Given that these halos were observed right after star formation
events, we are also able find the potential star-hosting candidates at each
redshift using Figure 5.

By examining the recovery times, baryon mass and molecular hydrogen content
together we are able to understand the critical masses around which star
formation becomes available to halos. We are also able to see at what mass
scales and redshifts feedback delays are reduced to a minimum. Given that
halos above $2\times 10^{6}$ $\mathrm{M_{\odot}}$ move permanently above the
$f_{H_{2}}=10^{-4}$ cosmic mean and their recovery times (Figure 3) remain
well below 20 Myr, we can see that such halos in our system are able to
recover $\mathrm{H_{2}}$ fast enough and contain a high enough content of
$\mathrm{M_{H_{2}}}$ already to efficiently form a star after a radiation
event. We can also infer that at $\mathrm{M_{vir}}\simeq 10^{7}$
$\mathrm{M_{\odot}}$, halos have short recoveries ($\sim 10$ Myr). It should
be stressed however that these recoveries are calculated from the moment that
a star dies in the system and no halo in our simulation forms a star while
another is already present. In other words, we find the recovery process can’t
counteract LW feedback directly and even the larger halos in the simulation
take time after a star has stopped emitting to produce enough $\mathrm{H_{2}}$
to cool efficiently.

###  3.2 The Role of Major Mergers

Having looked at radiative events, we explored other environmental factors
that could affect star formation. We focused on mergers within a range of mass
ratios in the main progenitor lines of our star forming halos.

Figure 6: Baryon mass introduced from a range of merger types throughout Halo
3’s evolution. The red bars are placed at the time the halos meet and are
extended in a light red up to where the halo finder starts treating them as
the same object. Note that the red bars are placed behind the yellow and blue
bars and can become partially covered, such as is the case with the third
major merger at $t=220$ Myr. The $2^{\mathrm{nd}}$, $3^{\mathrm{rd}}$ and
$12^{\mathrm{th}}$ star formation times are marked with black stars on the
x-axis. The white dots on the dashed curve show the times at which the panels
in Figures 9 and 11 were taken. Figure 7: This is the same type of plot as
Figure 6 but for Halo 5/6. The time of the $5/6^{th}$ star’s formation is
marked with a black star on the x-axis. Figure 8: This is the same type of
plot as Figures 6 and 7 but for Halo 9. The time of the $9^{th}$ star’s
formation is marked with a black star on the x-axis.

Merger types are generally divided into two categories: major mergers, which
are often restricted to ratios of 5-1:1 and minor mergers, that lie at a lower
range of 10-5:1. However, due to the high resolution of the simulation, we
extended these limits down to 20:1 to account for the prominence of small
halos in our system. Even smaller mergers were also included in the analysis,
with the smallest merging halos having masses of $900$ $\mathrm{M_{\odot}}$
since it is around and below this mass that the halo objects become poorly
defined, making measurements increasingly unreliable. This categorisation of
mergers resulted in a very small number of minor mergers being found.
Potentially expanding the mass regime of these mergers down to 25-30:1 would
lead to minor mergers being a more common event; however, the purpose of this
particular analysis was to show the gas contribution of different merger
types, not their statistical relevance. We found that the contribution of many
small halos can lead to a considerable reintroduction of gas, although a
single major merger is enough to outweigh their importance in the recovery
process of halos.

Figure 9: Above: Dark matter particle projections centered on Halo 3 along the
x-axis. The panels show its merger with the recently evacuated Halo 2. The
plot scale is of 1kpc and the depth of the projection is 1kpc. Halo 2 is
observed to move in from the top of the plots. Below: Gas density slice plot
centered on Halo 3 along the x-axis, the panels correspond to the particle
projections shown above. The times chosen for the panels correspond to the
time before the halo cores merged ($z=17.8$), the time during the merger
($z=17.12$) and the moment before star formation ($z=16.66$). The full and
dashed circles represent Halos 2 and 3’s virial radii respectively. The effect
of the second star’s supernova blastwave is observed as Halo 2 moves into Halo
3. Note the halo finder algorithm places Halo 2 inside Halo 3 before the cores
have settled as can be observed when comparing particle and density slices.
Figure 10: Mass-weighted, spherically averaged gas quantities: density,
temperature, hydrostatic pressure, $x_{\mathrm{HD}}$/$x_{\mathrm{H_{2}}}$,
cooling rate and metallicity; as a function of distance from the center of
Halo 3. The cooling rate is further split into $\mathrm{H_{2}}$, HD and
metals. The profiles are taken at different times around the merger event:
pre-merger (blue), merger (green) and pre-star formation (purple). Note that
because no HD photo-dissociation is implemented, the
$x_{\mathrm{HD}}$/$x_{\mathrm{H_{2}}}$ ratio at times during and just after
star formation is inaccurate, this effect is seen in the lower right graph,
where a large spike at outer radii is observed in the highest redshift line
(blue).

In Figure 6 we present the gas evolution as well as the merger history of Halo
3, past the formation of its first star up until the end of the simulation. We
chose to analyse this halo more in depth specifically due to its unique
evolution and formation history. Throughout the simulation it undergoes four
major mergers, two of which occur around star formation times. The first star
it forms (the third in the simulation) is seen as the dip in the dashed line
at $\sim 255$ Myr in Figure 6. This is a direct effect of the stars’ supernova
depleting and photoevaporating most of the gas content inside the halo. After
a period of around 100 Myr it then hosts the final star, soon after this, the
simulation ends at $\sim$380 Myr. The coloured bars below the dashed curve
show the total baryon mass that has directly entered Halo 3’s progenitor line
from merging objects at each timestep. For the smallest mergers recorded, in
this case below the 20:1 regime but above the $900$ $\mathrm{M_{\odot}}$ mass
limit, multiple objects tend to enter the halo within each snapshot, leading
to the sometimes high cumulative gas contribution observed from the blue bars.
It should also be noted that for the smallest objects, as well as early
mergers, the bars mark the moment the merging object fell in and settled into
the halo. Larger objects however, take longer to settle down, the ensuing
process of dynamical mixing after the halos coalesce, leads to the formation
of subhalo structures that first “slosh” around before eventually falling into
the core. This is represented as the extended bars in the plots, with larger
halos taking longer to settle, as can be seen from the increasingly long
shaded regions in Figures 6 and 7. The halo finder considered phase-space
occupation, not just physical-space locality, which allowed halos to persist
beyond the point at which they are spatially merged. Hence, we considered the
mergers to be complete once the halo finder took both objects to be the same.

Examining further halo merger histories provides insight into role of major
mergers and gas recovery. As seen from Figure 6, recently evacuated halos can
quickly recover gas via major mergers. Another example of this is seen in Halo
5’s evolution and merger history. From Figure 7, we find a similar gas
recovery history as Halo 3, with Halo 5/6 also experiencing a set of major
mergers that introduce large amounts of gas soon after its own star and
subsequent supernova depleted it of its gas contents. This large accretion
event meant it was ready for star formation by the end of the simulation. This
greatly contrasts Halo 9, that undergoes no major mergers and has a final
total baryon mass that is an order of magnitude lower than Halo 3 and 5/6
(Figure 8). Major mergers in these cases can therefore accelerate metal-
enriched star formation by quickly reintroducing gas back into evacuated
minihalos. Despite the rarity of the event (when compared to smaller mergers
that occur consistently), a single major merger during a halo’s evolution can
re-introduce up to 100% of the halo’s gas content, as is seen in the merger
histories for Halos 3 and 5/6. These events are also important in the recovery
process after star formation, since we find they can introduce over 10% of the
halos previous peak gas content, accelerating recovery and later star
formation.

###  3.3 Mixed Mergers and Star Formation

Figure 11: Above: Dark matter particle projection centered around Halo 2
showing the merger of halo halo full of gas with the recently evacuated Halo
3. The core of the incoming halo can be seen entering Halo 3 from the lower
right corner. It passes through Halo 3’s center and continues towards the
upper left corner. The times chosen for the panels follow the same idea as
those in Figure 9 with a slightly larger plot scale of 2 kpc and depth of
projection of 1 kpc. Below: Gas density slices corresponding to the particle
projections shown above. The overplotted dashed and full line circles
correspond to Halo 2 and the merging halos’ virial radii respectively. Figure
12: Mass-weighted, spherically averaged quantities: density, temperature,
hydrostatic pressure and $x_{\mathrm{HD}}$/$x_{\mathrm{H_{2}}}$; as a function
of distance from the densest point in the halo merging into Halo $12^{*}$.
Times were taken pre-merger (blue), during merger (green) and pre-star
formation (purple).

During our analysis of Halo 3’s merger history, we found the major merger seen
starting at $\sim 225$ Myr and ending at $\sim$275 Myr (in Figure 6) was an
event that involved Halo 2. We can observe their proximity by the direct
effect of Halo 2’s supernova on Halo 3 as the flattening of the dashed curve.
We better illustrate this merger with particle projection plots and gas
density slice plots in Figure 9. Halo 2 first approaches Halo 3 at around
$z=17$, just before these objects make contact, the second star forms and the
shockwave from its supernova causes visible disruption inside Halo 3 (Bottom
Figure 9). Halo 2 then “sloshes” inside Halo 3 for around $50$ Myr, during
which the third star is formed. This merger event is well within the classical
regime that defines major mergers, with Halo 2 and 3 having a dark matter mass
ratio of $\sim$1:4, yet we surprisingly find no delay in star formation. We
attribute this to the evacuated nature of Halo 2 and define it as a “mixed”
merger in analogy with low redshift galaxy mergers.

The radial profiles in Figure 10 illustrate the physical changes observed in
Halo 3’s gas density, HD fraction, hydrostatic pressure, temperature,
metallicity and cooling rates for different times throughout its merger with
Halo 2. These profiles show an increasingly favourable set of conditions for
collapse all while Halo 2 continues to move inside Halo 3. An increase in HD
fraction is paralleled by a consistent drop in temperatures and increase in HD
cooling, illustrating how this type of cooling become more present at the
merger occurs. It should be noted that although HD photo-dissociation is not
included in the simulation, the absence of nearby star formation occurring
during the mixed merger and subsequent collapse means this does not affect our
results. Another feature of the mixed merger that is not unexpected is the
presence of metals in the low density gas surrounding the two merging halos.
These have been expelled during the supernova associated with the now gas-poor
halo. As Figure 10 (bottom, right) shows, some of these are able to mix into
the gas-rich halo. However, the bottom left panel of Figure 10 also shows the
cooling from metals to be negligible until just prior to star formation, and
hence not a contributing factor when the gas cools the most. Thus, we maintain
that the rise in HD in a notable feature of these mixed mergers.

Noting that electron fraction is high in relic HII regions, we understand that
a higher HD fraction will be observed in halos that have already hosted a
star. However, in this case, because Halo 3 is so close to Halo 2 during its
SNe event, the increase in HD is also attributed to the large amount of
turbulence caused by the blastwave). From Figure 10 we find that as the merger
occurs, the cooling from HD quickly rises with cooling from metals only
contributing near the end of the merger. Within its densest cell (where the
star particle will eventually be placed) Halo 3 bottoms out at a temperature
of around $78$ K, far below the limit of $\mathrm{H_{2}}$’s LTE. The continued
plummeting of gas temperatures may seem surprising since the merger with Halo
2 is still ongoing, however, the evacuated state of Halo 2 minimizes the shock
heating that would occur from gaseous collisions. This unique scenario means
the presence of Halo 2 inside Halo 3 is ineffective at shock heating the gas
and instead helps create a favourable environment for collapse. Hydrostatic
pressure also increases as the dark matter mass content rises with the
presence of Halo 2. More specifically, it ends up being about an order of
magnitude higher than before Halo 2 entered Halo 3. Such mass growth also
helps deepen the central potential well and create a denser core which maxes
out at $10^{-21}$ $\mathrm{g\ cm^{-3}}$, a peak around which runaway collapse
is triggered.

After the formation of the third star, the newly evacuated Halo 3 (henceforth
refered to as Halo $12^{*}$ since it forms the final star) remains gas-poor
for close to $100$ Myr, until $z\sim$13.8, when a halo of $\mathrm{M_{DM}}\sim
7\times 10^{5}$ $\mathrm{M_{\odot}}$ and $\mathrm{M_{baryon}}\sim 10^{4}$
$\mathrm{M_{\odot}}$ enters its virial radius. This event leads to the
formation of the final Pop III star, although soon after its birth the
simulation ends. The star forms within the subhalo within Halo 3, as the
merger does not finish before runaway collapse occurs. As seen from the dark
matter and gas density projection plots in Figure 11, the smaller halo moves
into the virial radius of Halo 3 continuing past its core onto the other side.
The dynamics of this interaction are quite interesting since the merging halo
brings with it a significant amount of gas as the merging halo is at the star-
formation stage. This particular merger is slow, with the subhalo taking
roughly $40$ $\mathrm{Myr}$ to make its first crossing through Halo $12^{*}$.
The overall timescale of the merger is uncertain since the simulation ends
before the subhalo can settle down at the core. However, we can establish it
takes over 80 Myr for the merging halo to settle. Such prolonged mixing is
expected due to the massive nature of both these halos, roughly
$\mathrm{M_{DM}}\sim 9\times 10^{5}$ $\mathrm{M_{\odot}}$ and $3\times 10^{6}$
$\mathrm{M_{\odot}}$ at the time of initial contact. When considering the
free-fall and sound-crossing times ($t_{ff}=\sqrt{3\pi/32G\rho}$ and
$t_{cs}=R\sqrt{m_{h}/\gamma k_{b}T}$ where $\gamma$ is the adiabatic index and
$k_{b}$ the Boltzmann constant) of the larger merging halo at $z=13.9$, we
find these are 25 Myr and 37 Myr respectively, further highlighting the length
of this merger.

To further establish the parallels of this mixed merger with the previous one
and assert the nature of these particular events, we again analysed the
specific changes that occurred to the gas quantities before collapse.
Following a similar analysis to that made with Halo 2 and 3’s radial profiles,
profiles for this event, shown in Figure 12 were produced. It is important to
note these are centered around the merging halo, not Halo $12^{*}$. We again
find HD enhancement occurring alongside a temperature drop and increased
central pressure and densities. Although in this case there was a smaller
enhancement of the HD fraction, something that is expected since Halo $12^{*}$
had no gas-rich major or minor mergers nor SNe blastwaves that could provide
any prior turbulence. We therefore attribute the increase in HD to be caused
by the long merger timescale. We find that metal cooling contributes at a
similar level to HD early on as the metallicity is higher here than at
analogous times for Halo 3. However, by the final time shown, HD cooling is
higher than metal cooling throughout most of the inner halo despite a higher
metallicity than in Halo 3. Hence, while present, the metals are not necessary
to drive the cooling of the gas. The smaller HD fraction increase in this halo
may also indicate that only a relatively small amount of HD production is
required for HD cooling to take over from its less efficient precursor. The
large mass scales involved in this merger are also well reflected in the
hydrostatic pressure curves, as the core pressure increases by just under two
orders of magnitude. Accordingly, the core densities rose to a central peak of
$10^{-22}$ $\mathrm{g\ cm^{-3}}$ and would proceed to increase to $10^{-21}$
$\mathrm{g\ cm^{-3}}$ before the final collapse.

We therefore find that mixed mergers present a unique scenario in early star
formation. The evacuated nature of one of the merging halos reduces the
gaseous collisions that would normally stir turbulence and cause significant
shock heating. The presence of a gas-poor halo, however, still becomes evident
as central densities and hydrostatic pressure increases, accelerating
$\mathrm{H_{2}}$ production and creating more optimal conditions for collapse.
Such events are not a rarity in our simulation, since they occur multiple
times in our small sample of halos. What is crucial with these types of
mergers, however, is the timing. An example of a major merger that ended up
becoming a dry merger due to its timing is seen with the Halo 4 and 5/6 pair.
From Figure 7, we find that Halo 5/6 undergoes a merger event with another
very gas-poor halo, this is Halo 4. (It can be seen in Figure 7 as the small
shaded red bar starting at $\sim$330 Myr with a height of $\sim 2\times
10^{3}$ $\mathrm{M_{\odot}}$). The dry merger in this case occurs after Halo 4
and 5/6 have each formed their respective stars, leading this event to have no
effect in accelerating collapse itself but instead results in fast gas
accretion. This new, larger halo may have formed a star quite quickly had the
simulation kept going. Despite this timing requirement, a system of early
star-forming halos would be expected to undergo “mixed” mergers frequently
enough to take these events as potential avenues for collapse.

##  4 Discussion

In this work we have analyzed the evolution of 9 minihalos that formed Pop III
stars in the Pop2Prime simulation. We focused on the environmental factors, in
the form of radiative feedback and major mergers, that directly affected these
halos’ ability to effectively cool and form stars. As mentioned in Section 2,
the simulation analyzed here was the same as the high-resolution simulation in
Smith et al. (2018). This simulation was an extension of an earlier simulation
presented in Smith et al. (2015) that set a lower threshold for the
metallicity of Pop III stars. They therefore found only 2 Pop III star-forming
halos before metal-enriched collapse occurred and the simulation ended. These
are Halos 1 and 2 in this work. Our higher threshold allowed the simulation to
carry on longer, thus providing a larger sample of minihalos and enabling us
to follow the evolution of halos long after they had hosted a star.

###  4.1 Comparison with Previous Works

We initially followed much of the established work on radiative feedback
(Machacek et al. 2001; Abel et al. 2002; Yoshida et al. 2003, Reed et al.
2005; O’Shea & Norman 2008; Greif et al. 2010), and focused on the effect of
LW radiation in isolation as a factor in delaying collapse. Because our
simulation did not implement a uniform background field, instead turning
point-source radiation on or off depending on the presence of a star, we were
able study feedback as a discrete event. Despite this unique scenario, our
findings are in close agreement with works that implement a constant
background radiation; in the absence of LW photons, as is the case for Halo 1,
we observe star formation occurring in halos with $\mathrm{M_{DM}}=10^{5}$
$\mathrm{M_{\odot}}$, while repeatedly radiated halos undergo delayed collapse
and do not form stars until reaching masses close to or above
$\mathrm{M_{DM}}=10^{6}$ $\mathrm{M_{\odot}}$.

We find quite strong agreement with the mentioned past works, though more
recent studies such as Skinner & Wise (2020), Schauer et al. (2021) and
Kulkarni et al. (2021) find slightly larger critical masses:
$\mathrm{M_{crit}}\sim 2-3\times 10^{5}$$\mathrm{M_{\odot}}$ for no radiation
and $\mathrm{M_{crit}}\sim 1-3\times 10^{6}$$\mathrm{M_{\odot}}$ with
radiation. While Park et al. (2021), that implement an X-ray background, found
that this effect could significantly lower a halos critical mass in weak LW
backgrounds. We find however their results are still within reasonable
agreement with our findings. Despite the different simulation setups between
works; with other studies such as Schauer et al. (2021) and Kulkarni et al.
(2021) also including streaming velocities and self-shielding effects, as well
as modelling radiation as a smooth varying background, the overall consensus
on halo critical masses remains consistently around $\mathrm{M_{crit}}\sim
10^{5}$ $\mathrm{M_{\odot}}$

Unlike our work, many of the studies considering radiative feedback have also
attempted to parametrize the evolution of the $\mathrm{M_{crit}}$. We have not
done this here since our sample is so small. We instead compared predictions
for $\mathrm{M_{crit}}$ from Kulkarni et al. (2021) with our own observations
and find quite reasonable agreement for multiple halo masses within a range of
$\mathrm{J_{21}}$ intensities. As previously mentioned, under no radiative
influence, Kulkarni et al. (2021) predict a halo mass that is in good
agreement with Halos 1 and 2 here. Halo 4’s star formation mass is in
agreement with their prediction for $\mathrm{M_{crit}}$ when
$\mathrm{J_{21}}=1$ and Halo 9 and 10/11 sit right above and below this fit.
Halo 9 experienced no major mergers before star formation and so was subject
only to radiative delays. Having an $\mathrm{M_{crit}}$ that falls near the
predicted masses for $\mathrm{J_{21}}=1$ may imply that under radiation
effects alone, halos will form stars at masses that follow this curve. Halo 4
and 10/11 both experienced a major merger event but with halos that were far
below cosmological Jeans mass, with the largest halos they merged with having
$\mathrm{M_{DM}}=8\times 10^{3}$ $\mathrm{M_{\odot}}$ and $3\times 10^{4}$
$\mathrm{M_{\odot}}$ respectively. In contrast Halos 5/6 and 7/8 experienced
major mergers with much larger halos that were close to the cosmological Jeans
mass, making these events more disruptive since the halos involved had enough
gas to potentially form a star. We therefore find that the cumulative effect
of radiation and major mergers can steepen a halos $\mathrm{M_{crit}}$ curve.
Halo 3 is an extreme example of this since it undergoes major mergers, hosts
SNe blastwaves, experiences radiative feedback, and forms the final star in
the simulation at an $\mathrm{M_{crit}}$ that is nearly two orders of
magnitude above the first halo, aligning with the $\mathrm{J_{21}}=10$ curve.
This further implies that dynamical feedback can produce $\mathrm{M_{crit}}$
curves that align with more extreme background intensities. In order to
accurately parameterize $\mathrm{M_{crit}}$, the contribution from mergers and
SNe blastwaves should be considered.

We also centered our analysis on $\mathrm{H_{2}}$ recovery and its relation to
halo growth, as this directly dictates whether efficient cooling and therefore
collapse will take place. We find, as is expected from the analytic models in
Tegmark et al. (1997), that radiative feedback becomes less effective as halos
grow. With recovery times decreasing exponentially with increased halo mass
and halos above $10^{6}$ $\mathrm{M_{\odot}}$ tending to have recovery times
that remain below 30 Myr. We also predict that larger halos reach a minimum
recovery time around 10 Myr, with larger halos potentially pushing this to
even smaller times. We also find a crucial growth period, as halos approach
$10^{6}$ $\mathrm{M_{\odot}}$, where star formation becomes increasingly
accessible to a larger proportion of halos and collapse can only be brought
about when recovery times become smaller than the competing halos’ collapse
times. As we do not observe any instances of co-existing Pop III stars (apart
from the double systems), we see how the LW radiation forces halos to
repeatedly compete with each other in order to bring about collapse.

$\mathrm{H_{2}}$ recovery times are also heavily dependent on the gas content
and total $\mathrm{H_{2}}$ mass within a halo. Understanding the global trends
of these quantities during halo growth and comparing them to the cosmic mean
provides a clearer picture on why certain halos were able to host the correct
environment for star formation to be possible. In our simulation we find two
clear epochs: at high redshifts, halos remain gas-poor until they reach masses
close to the cosmological Jeans mass $\sim 10^{5}$ $\mathrm{M_{\odot}}$. At
$z\geq 17$ a sufficient number of halos have reached this critical mass for
continual and successive star formation to take place, at this stage halos
require cooling times that fall below the average collapse timescales. This
leads to the $\mathrm{H_{2}}$ dissociation and formation cycle presented
extensively in Tegmark et al. (1997) and is what pushed $\mathrm{M_{crit}}$ to
$\sim$106 $\mathrm{M_{\odot}}$. These findings are in close agreement with all
the aforementioned studies, although other factors such as self-shielding and
baryon-dark matter streaming velocity have been shown to push this mass to
even higher thresholds, $\sim 10^{7}$ $\mathrm{M_{\odot}}$ (Trenti & Stiavelli
2009; Schauer et al. 2021; Kulkarni et al. 2021).

###  4.2 A New Major Merger Scenario

We find and establish a novel merger type for Pop III star forming halos in
the form of “mixed” mergers. These can trigger collapse in halos that would
otherwise have taken longer to do so. Mergers between gas-poor galaxies,
defined as “dry” mergers, have already been studied in early galaxy formation
scenarios, since they enable significant mass growth without altering stellar
populations (Bell et al. 2006; Khochfar & Silk 2006; Khochfar & Silk 2009). We
find a similar situation here, since halos that end up being depleted of their
gas content after a SNe type II event maintain their dark matter content so
have high central densities. They can therefore provide their gas-rich
neighbours with a deeper potential well, causing the required increase in
central densities to bring about collapse without the major disruptions that
would be expected from regular mergers. Shchekinov & Vasiliev (2006) suggested
that under the correct conditions, halo mergers could boost the HD content in
a halo because of the increased electron abundance in post-shocked gas. This
enables a new and more efficient cooling mechanism to take over and lower
temperatures to be reached. Norman et al. (2010) addressed this cooling
channel in Pop III star formation, establishing that relic HII regions, such
as our recently evacuated halos, can form a higher amount of HD due to the gas
in these halos forming from initial conditions that have been thermally and
chemically altered. Prieto et al. (2012) explored this idea, focusing on the
effect of turbulence created by mergers increasing the HD content. Bovino et
al. (2014) would also present similar results by following collapse to higher
densities and showed significant ionization and HD enhancement from a major
merger event. They found temperatures dropped down to $60-70$ K once HD
cooling was triggered much like what we observed in the halos that undergo a
mixed merger event. In contrast, our non-merging halo example, Halo 9, cools
down to only $\sim$240 K.

These works agreed that shocks and turbulence from major mergers significantly
boost $\mathrm{H_{2}}$ and HD in halos, enabling cooling to temperatures close
to the CMB limit. However the sustained high turbulence measured from such
mergers, nearly $10$ $\mathrm{km\ s^{-1}}$ in the case of Bovino et al.
(2014), alongside the significant shock-heating inside the gas cores, would
delay star formation in halos where $\mathrm{H_{2}}$ cooling was still
inefficient, as is the case for our minihalos. Although these past works
aren’t able to follow the evolution of the halos down to star formation and
past this, they don’t expect star birth to be triggered until after the
mergers have finished. We instead find collapse occurs whilst the mixed
mergers are still ongoing, making such events a direct catalyst for star
formation. Mixed mergers are also not a rare occurrence in our simulation,
though the timing of these is crucial for them to help bring about collapse.
An example of this is the Halo 4 and 5 system, they exist close to each other
($\sim$0.3 kpc between their centers), however Halo 4 starts merging into Halo
5 after they have both hosted their corresponding stars. Making this dry
merger redundant in the formation of Pop III stars but potentially very useful
in the creation of subsequent stellar generations.

Despite our small sample size, we can still assess the potential impact of
mixed mergers within the broader context of early universe star formation. In
order to do this however, we need to also consider the nature of the
simulation or model being analysed. Mixed mergers, like any other phenomenon
that occurs due to the physics framework applied, cannot be isolated from the
rest of the simulation. However, it is possible that low resolution
simulations that do not adequately resolve the central potential in minihalos
might not see star formation induced by such events. In terms of star
formation rates (SFR), our small sample size again limits the conclusions we
can make about the impact of mixed mergers. It does, however, leave open
avenues for future work to build on. Mixed mergers may alter SFRs in denser
regions compared to mean or void areas. Denser populated zones in the universe
are expected to have a larger abundance of evacuated halos these could
accelerate star formation early on. However, these regions also have smaller
clustering lengths, leading halos to experience high levels of LW radiation
from the first star forming halo. Whether mixed mergers have outweigh this
effect would be an interesting avenue to take. Semi-analytic models based on
merger-trees from N-body simulations (Magg et al., 2022) could incorporate the
phenomenon of mixed mergers to estimate its overall impact on SFRs.

Though mixed mergers are present in early star formation, they may have a
limited timeframe within which they help produce Pop III stars. Because they
are relic HII regions that have seen star formation, they undergo small
amounts of enrichment. In our case, the close SNe encounter leads to
additional metals remaining in Halo 3 and therefore Halo $12^{*}$. The nature
of mixed mergers means halos involved in these experience small levels of
metal enrichment and metal cooling. While metals are likely to be present in
the environment of most mixed mergers, we have demonstrated that they are not
explicitly necessary to enhance cooling and induce star formation, as this can
be accomplished by HD and $\mathrm{H_{2}}$. What is less clear is how the
metals will mix into the dense gas where the stars next will form. In both
cases we study, the metallicity only marginally exceeds that necessary for
fragmentation due to dust (Schneider et al., 2012) and hence the possibility
remains that star formation in the manner of Pop III (i.e., with no
appreciable metal influence) occurs in these events.

###  4.3 Major mergers and recovery

Another important result that should be highlighted, despite already being
mentioned previously, is the ability major mergers have to inject large
amounts of gas into halos. The notion of recovery via fallback of supernova
ejecta presented in works such as Jeon et al. (2014) is completely absent in
over half of our halos, since they recover most of their gas via major mergers
instead of smooth accretion after star formation. Defining the recovery time-
scale as the time delay between a Pop III SN explosion and the replenishing of
cold, dense gas within the halo, Jeon et al. (2014) found that halos around
$5\times 10^{5}$ $\mathrm{M_{\odot}}$ have recovery times of $90$ Myr. We find
this is true for halos that recover gas via a major merger after star
formation, however those that do not (such as Halo 9) barely reach half of
their original maximum after this time. Major mergers are therefore a crucial
component of gas recovery for halos in the $5\times 10^{5}-10^{6}$
$\mathrm{M_{\odot}}$ mass range. Minor mergers in this case play a less
significant role in mass growth compared to major mergers. Aside from bringing
in less baryon mass, their dynamical friction time scales are longer, making
minor mergers impact halo mass growth at later times. We therefore find major
mergers are a more dominant factor in the formation of structure in the early
Universe.

##  5 Summary & Conclusion

We analyzed a cosmological radiation hydrodynamics simulation run with the
Enzo AMR + N-body code to study the effects of LW radiation and major mergers
on Pop III star formation. The reduced halo sample size along with the
simulations’ high resolution capacity allowed a detailed analysis of each
star-forming halo and the interaction between these. Moreover, by not
implementing a uniform background field and instead turning point-source
radiation on and off during each star’s lifetimes, we were also able to focus
on the effects individual stars had in the system. We therefore focused
heavily on the interplay of halo evolution and star formation to produce a
comprehensive narrative of the creation of the first stars. Our final
conclusion are summarized as follows:

  * •

We found a clear relation between $\mathrm{H_{2}}$ recovery times and halo
mass. With larger halos recovering faster and therefore experiencing
increasingly smaller delays. Without radiation, the halo mass for star
formation was, $\mathrm{M_{crit}}\sim 10^{5}\ \mathrm{M_{\odot}}$, this
increased to roughly $10^{6}\ \mathrm{M_{\odot}}$ after only two radiative
events. We expect to find that at a high enough halo mass, $\sim 10^{7}\
\mathrm{M_{\odot}}$, the delay caused by feedback effects reaches a minimum
that is around 10 Myr.

  * •

We also established a scaling relation in the form of a piecewise function for
the baryon mass and dark matter content inside minihalos, finding a break in
the power law at $\mathrm{M_{DM}}\sim 10^{5}\ \mathrm{M_{\odot}}$. This
inflection point coincides with the cosmological Jeans mass, which is the mass
around which previously gas-poor halos move towards the cosmic mean.

  * •

We presented a novel major merger scenario that can accelerate Pop III star
formation and define these as “mixed” merger events. These catalysts for star
formation can provide the correct local environmental changes to bring about
collapse while avoiding the detrimental effects associated with major mergers
at low redshift. The statistical relevance of such mergers as well as a closer
look at the halo dynamics at play would be an interesting direction to take in
future studies.

  * •

We additionally found major mergers as important processes for gas recovery in
halos, being especially important in the recovery of halos that have
previously hosted a star and are in a depleted state. We find that major
mergers can introduce up to 100% of the halo’s current gas content before star
formation and can reintroduce up to 10% of an evacuated halos previous maximum
gas mass.

In this work we have attempted to clarify the series of events that lead to
Pop III star formation. Focusing on the radiative effect of individual stars,
we reached similar conclusions to past works that implemented a background
radiation; while also allowing us to focus on a different feature resulting
from feedback effects, the recovery of a halo’s molecular content. Our
findings fit well within the established theories and we have attempted to
further deepen the understanding of the radiation and recovery process by
providing insight into the relations between the baryon and dark matter
content and $\mathrm{H_{2}}$ mass of these early minihalos. The high
resolution of the simulation also allowed us to get a closer look at halo-on-
halo interaction, with mixed mergers being our novel addition to the major
merger narrative. While major mergers may sometimes be effective suppressants
of star formation, we have shown that there is more to their role in early
star formation than has yet been presented in works so far.

## Acknowledgements

For the purpose of open access, the author has applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version arising
from this submission. We thank the anonymous referee for their comments, which
greatly improved the manuscript. BDS and SK are supported by STFC Consolidated
Grant RA5496. JHW is supported by NSF grants OAC-1835213 and AST-2108020 and
NASA grants 80NSSC20K0520 and 80NSSC21K1053. BWO acknowledges support from NSF
grants 1908109 and 2106575 and NASA grants NNX15AP39G, 80NSSC18K1105, and
80NSSC21K1053. The Pop2Prime simulations were performed on Blue Waters,
operated by the National Center for Supercomputing Applications with PRAC
allocation support by the NSF (award number ACI-0832662).

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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