# The statistical properties of stars at redshift, $z=5$, compared with the
present epoch

Matthew R. Bate1
1 Department of Physics and Astronomy, University of Exeter, Stocker Road,
Exeter EX4 4QL, United Kingdom E-mail<EMAIL_ADDRESS>
(Accepted by MNRAS)

###### Abstract

We report the statistical properties of stars and brown dwarfs obtained from
three radiation hydrodynamical simulations of star cluster formation with
metallicities of 1, 1/10 and 1/100 of the solar value. The star-forming clouds
are subjected to cosmic microwave background radiation that is appropriate for
star formation at a redshift $z=5$. The results from the three calculations
are compared to each other, and to similar previously published calculations
that had levels of background radiation appropriate for present-day ($z=0$)
star formation. Each of the calculations treat dust and gas temperatures
separately and include a thermochemical model of the diffuse interstellar
medium. We find that whereas the stellar mass distribution is insensitive to
the metallicity for present-day star formation, at $z=5$ the characteristic
stellar mass increases with increasing metallicity and the mass distribution
has a deficit of brown dwarfs and low-mass stars at solar metallicity compared
to the Galactic initial mass function. We also find that the multiplicity of
M-dwarfs decreases with increasing metallicity at $z=5$. These effects are a
result of metal-rich gas being unable to cool to as low temperatures at $z=5$
compared to at $z=0$ due to the hotter cosmic microwave background radiation,
which inhibits fragmentation at high densities.

###### keywords:

binaries: general – hydrodynamics – radiative transfer – stars: abundances –
stars: formation – stars: luminosity function, mass function.

##  1 Introduction

Around a decade ago, numerical studies of present day star formation advanced
to the point that individual radiation hydrodynamical calculations could
produce stellar groups or clusters containing more than a hundred stars and
brown dwarfs. With such numbers of stars and brown dwarfs, the statistical
properties of the stellar populations can be meaningfully compared with those
of observed Galactic stellar populations. Bate (2012) published the first
radiation hydrodynamical calculation to reproduce a wide variety of observed
statistical properties of Galactic stellar systems. The calculation produced
in excess of 180 stars and brown dwarfs, including 40 multiple systems, while
also resolving young discs down to radii of $\approx 10$ au. The stellar mass
function was found to be in good agreement with the observed Galactic initial
mass function (IMF), the stellar multiplicity as a function of primary mass
was in agreement with the results from field star surveys, and many of the
observed characteristics of multiple stellar systems were reproduced.
Moreover, Bate (2018) analysed the mass and radius distributions of the young
discs that were produced by the Bate (2012) calculation and found that the
distribution of disc radii are in good agreement with the observed radii of
discs in Galactic star-forming regions, and the mass distributions are similar
to the estimated mass distributions of very young (Class 0) protostellar
systems (see Tychoniec et al., 2018; Tobin et al., 2020).

In addition to comparing the results of star cluster formation calculations to
observed Galactic stellar populations, the ability to perform numerical
simulations that produce clusters of stars gives us the ability to directly
predict how the statistical properties of stellar systems depend on
environment and initial conditions, and to investigate the roles of various
physical processes. Thus far, hydrodynamical calculations that either include
radiative transfer and/or radiative heating from protostars have been used to
explore the dependence of stellar properties on metallicity (Myers et al.
2011; Bate 2014, 2019; He, Ricotti & Geen 2019; Guszejnov et al. 2022),
molecular cloud density (Bate, 2009b; Jones & Bate, 2018b; Tanvir et al.,
2022; Guszejnov et al., 2022), protostellar outflows (Krumholz, Klein & McKee
2012; Mathew & Federrath 2021; Tanvir, Krumholz & Federrath 2022; Grudić et
al. 2021; Grudić et al. 2022; Guszejnov et al. 2021; Guszejnov et al. 2022),
variations in turbulence (Lomax, Whitworth & Hubber 2015; Nam, Federrath &
Krumholz 2021; Mathew, Federrath & Seta 2022; Guszejnov et al. 2022), and
magnetic fields (Myers et al. 2013, 2014; Krumholz et al. 2016; Cunningham et
al. 2018; Grudić et al. 2021; Guszejnov et al. 2022). Some of the most recent
calculations have even begun exploring how the non-ideal magnetohydrodynamic
(MHD) effects of ambipolar diffusion, Hall conduction, and Ohmic resistivity
affect the formation of stellar groups, albeit only for small numbers of stars
as yet (Wurster et al., 2019).

The study of the dependence of star formation on metallicity of Bate (2019)
employed the method of Bate & Keto (2015) to model the thermodynamics. This
method combines radiative transfer with a thermochemical model of the diffuse
interstellar medium (ISM), with gas and dust temperatures treated separately.
The thermochemical model includes simple chemical models for the atomic and
molecular forms of hydrogen and whether carbon is in the form of C+, neutral
carbon, or CO. The ISM model is similar to that used by Glover & Clark (2012)
to study molecular cloud evolution, although the chemical model employed by
Bate & Keto is greatly simplified. Bate (2019) found that the stellar
properties did not vary greatly for present-day star formation if the
metallicities ranged from 1/100 to 3 times the solar value. For example, the
stellar mass functions were statistically indistinguishable. However, the
greater cooling rates at high gas densities due to the lower opacities at low
metallicities were found to increase the fragmentation on small spatial scales
which resulted in an anti-correlation between the close binary fraction of
low-mass stars and metallicity similar to that which is observed (Badenes et
al. 2018; El-Badry & Rix 2019; Moe, Kratter & Badenes 2019). The fraction of
protostellar mergers at low metallicities also increased due to the higher
prevalence of small-scale fragmentation. An indication was found that at lower
metallicity close binaries may have lower mass ratios and the relative
abundance of brown dwarfs to stars may increase slightly, but these effects
were quite weak. Elsender & Bate (2021) also studied the discs produced by the
calculations and found that the characteristic radii of protostellar discs
decrease with decreasing metallicity and that the discs and orbits of pairs of
protostars tend to be less well aligned at lower metallicity.

In this paper, we report results from three new radiation hydrodynamical
calculations of stellar cluster formation that are identical to those reported
by Bate (2019) except that the background interstellar radiation field (ISRF)
is modified so as to include a cosmic microwave background radiation component
that is appropriate for a redshift $z=5$ rather than $z=0$. We follow the
collapse of each of the molecular clouds to form a cluster of stars and then
compare the properties of the stars and brown dwarfs, both between each of the
new calculations, and to the results from the present-day ($z=0$) calculations
of Bate (2019). In Section 2, we summarise the method and initial conditions.
The results from the calculations are presented in Section 3. In Section 4, we
discuss observational evidence for variation of the stellar mass function in
the context of the results presented in this paper. Section 5 contains the
conclusions.

##  2 Method

The calculations were performed using the smoothed particle hydrodynamics
(SPH) code, sphNG, based on the original version from Benz (1990; Benz et al.
1990), but substantially modified using the methods described in Bate et al.
(1995), Price & Monaghan (2007), Whitehouse, Bate & Monaghan (2005),
Whitehouse & Bate (2006), Bate & Keto (2015) and parallelised using both MPI
and OpenMP.

Gravitational forces and particle nearest neighbours were calculated using a
binary tree. The smoothing lengths of particles were set such that the
smoothing length of each particle $h=1.2(m/\rho)^{1/3}$ where $m$ and $\rho$
are the SPH particle’s mass and density, respectively (see Price & Monaghan,
2007, for further details). A second-order Runge-Kutta-Fehlberg integrator
(Fehlberg, 1969) was employed, with individual time steps for each particle
(Bate, Bonnell & Price, 1995). The Morris & Monaghan (1997) artificial
viscosity was used to reduce numerical shear viscosity, with
$\alpha_{\rm{}_{v}}$ varying between 0.1 and 1 and setting $\beta_{\rm
v}=2\alpha_{\rm v}$ (see also Price & Monaghan, 2005).

###  2.1 Radiative transfer and the diffuse ISM model

The calculations used the combined radiative transfer and diffuse ISM
thermochemical model developed by Bate & Keto (2015). A brief summary of the
main elements of the method is provided by Bate (2019); the reader is directed
to those two papers for the details of the method since the calculations
performed for the present paper are identical to the calculations presented by
Bate (2019) except for the form of the interstellar radiation field.

Briefly, the radiation hydrodynamical calculations are solved in a frame co-
moving with the fluid, assuming local thermal equilibrium (LTE), and can be
written

$\frac{{\rm D}\rho}{{\rm D}t}+\rho\nabla\cdot\mbox{\boldmath$v$}=0~{},$ (1)
$\rho\frac{{\rm D}\mbox{\boldmath$v$}}{{\rm D}t}=-\nabla
p+\frac{\mbox{${\chi}\rho$}}{c}\mbox{\boldmath$F$}~{},$ (2)
$\begin{split}\rho\frac{{\rm D}}{{\rm
D}t}\left(\frac{E}{\rho}\right)=-&\nabla\cdot\mbox{\boldmath$F$}-\nabla\mbox{\boldmath$v${\bf:P}}-ac\kappa_{\rm
g}\rho\left(T_{\rm r}^{4}-T_{\rm g}^{4}\right)-\\\ &ac\kappa_{\rm
d}\rho\left(T_{\rm r}^{4}-T_{\rm d}^{4}\right),\end{split}$ (3)
$\begin{split}\rho\frac{{\rm D}u}{{\rm
D}t}=-&p\nabla\cdot\mbox{\boldmath$v$}+ac\kappa_{\rm g}\rho\left(T_{\rm
r}^{4}-T_{\rm g}^{4}\right)-\\\ &\Lambda_{\rm gd}-\Lambda_{\rm
line}+\Gamma_{\rm cr}+\Gamma_{\rm pe}+\Gamma_{\rm H2,g}~{},\end{split}$ (4)

(Bate & Keto, 2015), where, ${\rm D}/{\rm D}t\equiv\partial/\partial
t+\mbox{\boldmath$v$}\cdot\nabla$ is the convective derivative. The symbols
$\rho,$ $v$ and $p$ represent the material mass density, velocity, and scalar
isotropic pressure, respectively, and $c$ is the speed of light. The total
frequency-integrated radiation density, momentum density (flux) and pressure
tensor are represented by $E$, $F$, and P, respectively. The flux-limited
diffusion approximation is employed, such that
$\mbox{\boldmath$F$}=-c\lambda\nabla E/(\chi\rho)$ and ${\bf P}={\bf f}E$, in
which the flux limiter, $\lambda$, of Levermore & Pomraning (1981) is used and
the Eddington tensor, ${\bf f}$, is computed using the gradient of $E$.
Equations 3–4 are used to evolve the continuum radiation field and the
specific internal energy of the gas, $u$, separately from the dust. The gas,
dust, and continuum radiation fields all have their own associated
temperatures: $T_{\rm g}$, $T_{\rm d}$, and $T_{\rm r}$, respectively. The
dust temperature is set by assuming that the dust is in thermal equilibrium
with the combined ISRF and continuum radiation field, $E$, taking into account
the transfer of energy between the gas and the dust:

$\rho\Lambda_{\rm ISR}+ac\kappa_{\rm d}\rho(T_{\rm r}^{4}-T_{\rm
d}^{4})+\Lambda_{\rm gd}=0,$ (5)

where $\Lambda_{\rm ISR}$ is the heating rate per unit mass of the dust due to
the ISRF. The terms at the end of equation 4 are $\Lambda_{\rm gd}$ which
accounts for the thermal interaction due to collisions between the gas and the
dust, $\Lambda_{\rm line}$ which is the cooling rate per unit volume due to
atomic and molecular line emission, $\Gamma_{\rm cr}$ which is the heating
rate per unit volume due to cosmic rays, $\Gamma_{\rm pe}$ which is the
heating rate per unit volume due to the photoelectric effect, and $\Gamma_{\rm
H2,g}$ which is the heating rate per unit volume due to the formation of
molecular hydrogen on dust grains. We also define the radiation constant
$a=4\sigma_{\rm B}/c$, where $\sigma_{\rm B}$ is the Stefan-Boltzmann
constant. The Rosseland mean gas opacity $\kappa_{\rm g}$ is only important
above the dust sublimation temperature, and the mean dust opacity is
$\kappa_{\rm d}$. The total opacity, $\chi$, in equation 2 and the equation
for $F$ is set to the sum of the gas and dust opacities and ignores
scattering. It may be noted that the radiation pressure term (the last term in
equation 2) and the second term on the right-hand side of equation 3 are both
negligible in all of the calculations discussed in this paper.

Figure 1: A comparison of the form of the interstellar radiation field (ISRF)
that is assumed at redshift $z=5$ compared to that at $z=0$. It is assumed
that the only difference is due to the cosmic microwave background radiation
(CMBR) which has a greater temperature at $z=5$ (the contributions of the CMBR
to each ISRF are shown separately).

The hydrogen and helium mass fractions were taken to be $X=0.70$ and $Y=0.28$,
respectively, with the solar metallicity abundance set to be ${\rm
Z}_{\odot}=0.02$. We employed simple chemical models to treat the evolution of
hydrogen and carbon. The abundances of C$+$, neutral carbon, CO, and the
depletion of CO on to dust grains were computed using the model of Keto &
Caselli (2008). For hydrogen, we evolved the atomic and molecular hydrogen
abundances using the same molecular hydrogen formation and dissociation rates
as those used by Glover et al. (2010).

The clouds were assumed to be bathed in an ISRF. This contributes to both the
heating rate of dust grains and photoelectric heating of the gas. The ISRF is
attenuated due to dust extinction inside a molecular cloud, with opacities set
in the same way as in Bate (2014, 2019). To describe the ISRF at the present
epoch (redshift $z=0$) the analytic form of Zucconi, Walmsley & Galli (2001)
is used, but modified to include the ‘standard’ UV component from Draine
(1978) in the energy range $h\nu=5-13.6$ eV. For the calculations presented in
this paper, we modified the functional form of the component of the ISRF that
is due to the cosmic microwave background radiation. Specifically, we take the
temperature of the cosmic microwave background radiation (CMBR) to scale as

$T_{\rm CMBR}(z)=(1+z)T_{\rm CMBR}(0),$ (6)

where $T_{\rm CMBR}(0)=2.73$ K, so that at $z=5$, $T_{\rm CMBR}(z)=16.4$ K. In
Figure 1, we plot the forms of the ISRF that Bate (2019) used for the present
day (redshift $z=0$) and the form that we use for redshift $z=5$. The
contributions of the CMBR to each of the ISRFs are also plotted in the figure.
Note that we assume that the other contributions to the ISRF do not change
with redshift. Essentially we are assuming that the star-forming clouds are in
similar radiative environments, except for the contribution from the CMBR.
This need not be the case – for example, if a star-forming cloud was in a
galactic starburst environment there may also be a stronger high-frequency
component to the ISRF. However, making such additional changes to the ISRF
would add more free parameters to the models. Therefore, for the present paper
we only consider the effect of the hotter/stronger CMBR.

###  2.2 Sink particles

As in Bate (2019), the calculations followed the hydrodynamic collapse of each
protostar through the first core phase and into the second collapse phase
(that begins at densities of $\sim 10^{-7}$ g cm-3) due to molecular hydrogen
dissociation (Larson, 1969). Sink particles (Bate et al., 1995) were inserted
when the density exceeded $10^{-5}$ g cm-3. This density is only a factor of
one hundred lower than the density at which stellar core begins to form
(density $\sim 10^{-3}$ g cm-3) and the free-fall time at this density is only
one week.

The sink particle accretion radius was set to $r_{\rm acc}=0.5$ au. Sink
particles interact with the gas only via gravity and accretion, and the
gravitational interaction between sink particles is not softened. Sink
particle trajectories are integrated using the same Runge-Kutta-Fehlberg
integrator that is used for the SPH particles, but with a much lower error
tolerance so that even the closest orbits (semi-major axes less than 1 au) are
maintained for the period of the simulations (i.e. $\sim 10^{5}$ yrs). The
sink particles themselves do not contribute radiative feedback (see Bate,
2012; Jones & Bate, 2018a, for discussion of this limitation). Sink particles
are merged together if they pass within 0.03 au (i.e., $\approx 6$ R⊙).

Redshift | Initial Gas | Metallicity | No. Stars | No. Brown | Mass of Stars & | Mean | Mean | Median | Stellar
---|---|---|---|---|---|---|---|---|---
| Mass |  | Formed | Dwarfs | Brown Dwarfs | Mass | Log-Mass | Mass | Mergers
| M⊙ | Z⊙ |  |  | M⊙ | M⊙ | M⊙ | M⊙ |
$z=5$ | 500 | 0.01 | $\geq 89$ | $\leq 43$ | 52.4 | $0.40\pm 0.05$ | $0.16\pm 0.02$ | 0.15 | 17
| 500 | 0.1 | $\geq 81$ | $\leq 33$ | 56.5 | $0.50\pm 0.07$ | $0.20\pm 0.03$ | 0.19 | 7
| 500 | 1.0 | $\geq 65$ | $\leq 2$ | 67.4 | $1.01\pm 0.17$ | $0.49\pm 0.08$ | 0.40 | 1
$z=0$ | 500 | 0.01 | $\geq 90$ | $\leq 52$ | 49.8 | $0.35\pm 0.04$ | $0.15\pm 0.02$ | 0.14 | 17
| 500 | 0.1 | $\geq 125$ | $\leq 49$ | 73.4 | $0.42\pm 0.05$ | $0.17\pm 0.02$ | 0.15 | 11
| 500 | 1.0 | $\geq 171$ | $\leq 84$ | 90.1 | $0.35\pm 0.04$ | $0.16\pm 0.01$ | 0.15 | 14
| 500 | 3.0 | $\geq 193$ | $\leq 65$ | 92.0 | $0.36\pm 0.03$ | $0.18\pm 0.01$ | 0.17 | 6
Table 1:  The parameters and overall statistical results for each of the three
new radiation hydrodynamical calculations performed for this paper, and the
four calculations from Bate (2019). All calculations were run to 1.20 $t_{\rm
ff}$. Brown dwarfs are defined as having final masses less than 0.075 M⊙ [Note
that the equivalent table in Bate (2019) mistakenly gave incorrect numbers of
stars and brown dwarfs, although the sums of these numbers (the total numbers
of objects) were correct.]. The numbers of stars (brown dwarfs) are lower
(upper) limits because some brown dwarfs were still accreting when the
calculations were stopped. At $z=0$, different metallicities result in no
significant difference in the mean and median masses of the stellar
populations, but with $z=5$ the solar-metallicity calculation produces a much
heavier population of stars. At both redshifts, less gas is converted into
stars in calculations with lower metallicities and the average number of
stellar mergers per star consistently increases with decreasing metallicity.
This table and caption are comparable to Table 1 in Bate (2019). Figure 2: The
star formation rates obtained for each of the three radiation hydrodynamical
calculations. We plot: the total stellar mass (i.e., the mass contained in
sink particles) versus time (left panel), the number of stars and brown dwarfs
(i.e., the number of sink particles) versus time (centre panel), and the
number of stars and brown dwarfs versus the total stellar mass (right panel).
The different line types are for metallicities of 1/100 (magenta, dot-dashed),
1/10 (red, dashed), and 1 (black, solid) times solar. Time is measured from
the beginning of the calculation in terms of the free-fall time of the initial
cloud (bottom) or years (top). The star formation is delayed in the two low-
metallicity calculations, due to the warmer gas temperatures, but after
$t\approx 1.08$ $t_{\rm ff}$ the star formation rates in terms of the amount
of gas converted to stars are very similar for all of the calculations. There
is also a clear progression such that more objects are produced from a given
amount of gas for lower metallicities (right panel). This figure and caption
are comparable to Fig. 1, Bate (2019).

###  2.3 Initial conditions and resolution

The initial density and velocity structure for each calculation were identical
to those used by Bate (2012, 2014, 2019) – this allows close comparison of the
results with those of Bate (2019) because only one or two parameters are
changed at a time (i.e. the metallicity and/or the redshift). We only provide
a brief description of the initial conditions here – see Bate (2012) for a
more complete description. In short, for each calculation the initial
conditions consisted of a uniform-density, spherical cloud containing 500 M⊙
of molecular gas, with a radius of 0.404 pc (83300 au). This gives an initial
density of $1.2\times 10^{-19}$ g cm-3 ($n_{\rm H}\approx 6\times 10^{4}$
cm-3) and an initial free-fall time of $t_{\rm ff}=6.0\times 10^{12}$ s or
$1.90\times 10^{5}$ years. Although each cloud was uniform in density, we
imposed an initial supersonic ‘turbulent’ velocity field in the same manner as
Ostriker, Stone & Gammie (2001) and Bate, Bonnell & Bromm (2003). The power
spectrum of the divergence-free random Gaussian velocity field was
$P(k)\propto k^{-4}$, where $k$ is the wavenumber. The velocities were set on
a $128^{3}$ uniform grid and the SPH particle velocities were interpolated
from the grid. The velocity field was normalised so that the kinetic energy of
the turbulence was equal to the magnitude of the gravitational potential
energy of the cloud (this results in the initial root-mean-square Mach number
of the turbulence being ${\cal M}=11.2$ at 15 K). The same velocity field was
used for each calculation.

As in Bate (2019), the initial gas and dust temperatures were set such that
the dust was in thermal equilibrium with the local interstellar radiation
field. The gas was in thermal equilibrium such that heating from the ISRF and
cosmic rays was balanced by cooling from atomic and molecular line emission
and collisional coupling with the dust. This produces clouds with dust
temperatures that are warmest on the outside and coldest at the centre. For
the solar metallicity cloud, the initial dust temperature ranges from $T_{\rm
dust}=16.3-17.9$ K. For $Z=0.1~{}{\rm Z}_{\odot}$, $T_{\rm dust}=16.8-19.1$ K
and for the lowest metallicity, $T_{\rm dust}=18.6-19.9$ K. The mean initial
gas temperatures for the three calculations were $T_{\rm gas}\approx
16.5,17.6,14.3$ K for metallicities $Z=1,0.1,0.01~{}{\rm Z}_{\odot}$,
respectively.

The calculations used $3.5\times 10^{7}$ SPH particles to model the cloud (the
same resolution as that used by Bate 2019). This resolution is sufficient to
resolve the local Jeans mass throughout the calculation, which is necessary to
model fragmentation of collapsing molecular clouds correctly (Bate & Burkert
1997; Truelove et al. 1997; Whitworth 1998; Boss et al. 2000; Hubber, Goodwin
& Whitworth 2006). Currently there is no consensus as to the resolution that
is necessary and sufficient to capture disc fragmentation (see Section 2.3 of
Bate 2019 for discussion of this point and references for further reading).
The fact that the criteria for disc fragmentation are not well understood
should be kept in mind as a caveat for the results presented below.

Figure 3: Column density and temperature snapshots at three different times
($t=1.00,1.10,1.20$ $t_{\rm ff}$ ) for the $z=5$ calculation with a
metallicity of 1/100 times solar. From top to bottom, the rows give column
density, and the mass-weighted gas, dust, and protostellar radiation
temperatures. The colour scales are logarithmic. The column density scale
covers $0.03-30$ g cm-3, and the temperature scales cover $5-100$K. The stars
and brown dwarfs are plotted using white circles. Gas temperatures in the
dense parts of the cloud are much hotter than the dust temperature due to the
low metallicity and poor gas-dust coupling. This figure and caption are
comparable to Fig. 2, Bate (2019). Animations of the evolution shown in this
figure are provided in the online Additional Supporting Information that
accompanies this paper.  Figure 4: Column density and temperature snapshots at
three different times ($t=1.00,1.10,1.20$ $t_{\rm ff}$ ) for the $z=5$
calculation with a metallicity of 1/10 times solar. From top to bottom, the
rows give column density, and the mass-weighted gas, dust, and protostellar
radiation temperatures. The colour scales are logarithmic. The column density
scale covers $0.03-30$ g cm-3, and the temperature scales cover $5-100$K. The
stars and brown dwarfs are plotted using white circles. The gas temperatures
in the dense parts of the cloud tend to be hotter than the dust temperature,
but less than in the 1/100 metallicity case. This figure and caption are
comparable to Fig. 3, Bate (2019). Animations of the evolution shown in this
figure are provided in the online Additional Supporting Information that
accompanies this paper.  Figure 5: Column density and temperature snapshots at
three different times ($t=1.00,1.10,1.20$ $t_{\rm ff}$ ) for the $z=5$
calculation with solar metallicity. From top to bottom, the rows give column
density, and the mass-weighted gas, dust, and protostellar radiation
temperatures. The colour scales are logarithmic. The column density scale
covers $0.03-30$ g cm-3, and the temperature scales cover $5-100$K. The stars
and brown dwarfs are plotted using white circles. The gas temperatures are
generally lower than in the low metallicity calculations. This figure and
caption are comparable to Fig. 4, Bate (2019). Animations of the evolution
shown in this figure are provided in the online Additional Supporting
Information that accompanies this paper.

##  3 Results

###  3.1 Cloud evolution

Each of the three radiation hydrodynamical calculations was evolved to
$t=1.20~{}t_{\rm ff}$ (228,300 yrs). During this period they produced
different numbers of stars and brown dwarfs (modelled using sink particles
with accretion radii of 0.5 au). The initial conditions and the statistical
properties of the stars and brown dwarfs produced by each $z=5$ calculation
are summarised in Table 1, along with the properties of the $z=0$ calculations
that were previously published by Bate (2019).

At both redshifts there is a clear trend that lower metallicity clouds have
converted less gas into stars at the same time. For the present-day
calculations, Bate (2019) showed that this was due to a delay in the star
formation that was caused by more metal-poor gas at intermediate densities
being warmer (i.e. greater thermal pressure support). Fig. 2 shows how the
star formation rates evolve with time for the $z=5$ calculations, both in
terms of the number of stars and brown dwarfs and their total mass. As was the
case in the $z=0$ calculations, it is clear from the left panel of these
graphs that the trend of lower total stellar mass with lower metallicity at
the end of the calculations ($t=1.20~{}t_{\rm ff}$) arises primarily because
the star formation is delayed with lower metallicity. After $t=1.08~{}t_{\rm
ff}$ (205,500 yrs), the star formation rates (the slopes of the lines in the
left panel) are almost identical. But there is a delay in the star formation
getting started that increases for lower metallicities. At the end of the
calculations, almost 1/7 of the initial mass has been transformed into stars
in the solar metallicity calculation, but in the lowest metallicity
calculation only 1/10 of the mass has been converted to stars. Note that for
the same metallicity, the $z=5$ calculations have generally converted less gas
into stars after the same amount of time than the $z=0$ calculations – the
amount of mass is similar for $Z=0.01~{}{\rm Z}_{\odot}$ (actually 5 per cent
higher), but substantially less for $Z\geq 0.1~{}{\rm Z}_{\odot}$. As we will
see below, this is because the gas temperatures at the same metallicity tend
to be higher for $z=5$ than for $z=0$ due to the enhanced CMBR.

However, while the trend of the total stellar mass with metallicity is the
same for the $z=0$ and $z=5$ calculations, the dependence of the number of
objects with metallicity actually reverses for $z=5$ compared to $z=0$.
Whereas the present-day star formation calculations of Bate (2019) produced
more objects at higher metallicities at $t=1.20~{}t_{\rm ff}$, in the $z=5$
calculations higher metallicity results in fewer objects being formed. The
greater numbers of objects at higher metallicity in the present-day star
formation calculations is due to lower gas temperatures caused by the
combination of greater extinction shielding the inner parts of the cloud from
the ISRF and enhanced dust cooling. However, the latter effect starts to
saturate at very high metallicity ($Z\approx 1-3~{}Z_{\odot}$) when the cloud
is highly optically thick and begins to trap its own infrared radiation, so
that at $z=0$ the calculations with $Z=Z_{\odot}$ and $Z=3~{}Z_{\odot}$
produced similar numbers of objects: 255 & 258, respectively (Table 1; Bate
2019). By contrast, at redshift $z=5$ the ability of the gas to cool is
diminished by the higher CMBR temperature. Moreover, at higher temperatures
the clouds are more optically thick to their own radiation (i.e. the Rossland
mean opacity increases with temperature). These effects combine to reverse the
present-day metallicity trend. The $Z=0.01~{}{\rm Z}_{\odot}$ and
$Z=0.1~{}{\rm Z}_{\odot}$ calculations produce reasonably similar numbers of
objects (132 and 114, respectively), but the $Z={\rm Z}_{\odot}$ produces only
67 objects.

Figure 6: Phase diagrams of temperature vs gas density at the end of each of
the $z=5$ calculations ($t=1.20$ $t_{\rm ff}$). The upper row of panels gives
the gas temperature, while the lower row gives the dust temperature. The
colour scale in the upper panels gives the mean abundance of gas phase CO
relative to hydrogen at that temperature and density. Note that the
calculations include a prescription for the freeze out of CO onto dust grains,
but they do not treat thermal desorption of CO from dust grains at $T_{\rm
dust}\mathrel{\hbox{\hbox to0.0pt{\lower 2.36806pt\hbox{$\sim$}\hss}
\kern-3.00003pt\raise 1.72218pt\hbox{$>$}}}20$ K. In the dust temperature
plots, the grey scale is proportional to the logarithm of the number of fluid
elements (i.e., darker regions contain more SPH particles). The low-density
gas ($n_{\rm H}\mathrel{\hbox{\hbox to0.0pt{\lower 2.36806pt\hbox{$\sim$}\hss}
\kern-3.00003pt\raise 1.72218pt\hbox{$<$}}}10^{8}~{}{\rm cm}^{-3}$) tends to
be warmer at low metallicity than with high metallicity due to the poor
cooling, but the high-density gas ($n_{\rm H}\mathrel{\hbox{\hbox
to0.0pt{\lower 2.36806pt\hbox{$\sim$}\hss} \kern-3.00003pt\raise
1.72218pt\hbox{$>$}}}10^{10}~{}{\rm cm}^{-3}$) tends to be cooler at low
metallicity due to the reduced optical depths. The dust is almost all warmer
than 10 K, as opposed to the present-day star formation calculations of Bate
(2019) in which dust temperatures as low as $5-6$ K were reached with
metallicity $Z\mathrel{\hbox{\hbox to0.0pt{\lower 2.36806pt\hbox{$\sim$}\hss}
\kern-3.00003pt\raise 1.72218pt\hbox{$>$}}}{\rm Z}_{\odot}$. Since dust
cooling is crucial for gas cooling at high densities and metallicities, this
leads to significantly greater gas temperatures for $Z\mathrel{\hbox{\hbox
to0.0pt{\lower 2.36806pt\hbox{$\sim$}\hss} \kern-3.00003pt\raise
1.72218pt\hbox{$>$}}}0.1~{}{\rm Z}_{\odot}$ at $z=5$ than at $z=0$. This
figure and caption are comparable to Fig. 7, Bate (2019).

The mean and median masses of the objects that are produced in each of the
clouds are provided in columns 7–9 of Table 1. Bate (2019) found that these
statistical properties of the stars/brown dwarfs produced by the calculations
at $z=0$ were statistically indistinguishable over the full range of
metallicities from $Z=0.01-3~{}{\rm Z}_{\odot}$. Although at a given time more
mass was converted into stars at higher metallicity, more objects were
produced from that gas leading to a protostellar mass function that was
independent of the metallicity. However, at $z=5$ although at a given time
more mass is again converted into stars at higher metallicity (left panel of
Fig. 2), the number of objects produce by that gas consistently decreases with
increasing metallicity (centre and right panels of Fig. 2). The result is that
the typical stellar mass increases with increasing metallicity (Table 1).
There is only a small difference between the stellar populations produced by
the $Z=0.01~{}{\rm Z}_{\odot}$ and $Z=0.1~{}{\rm Z}_{\odot}$ calculations, but
the characteristic mass doubles (either the mean or median mass) between the
$Z=0.1~{}{\rm Z}_{\odot}$ and $Z={\rm Z}_{\odot}$ calculations.

In all of the calculations there are some protostellar mergers (see the last
column of Table 1). These occur when two sink particles pass within $6~{}{\rm
R}_{\odot}$ of each other. This merger radius was chosen because low-mass
protostars that accrete at high rates ($\dot{M}_{*}\sim 10^{-6}-10^{-5}$ M⊙
yr-1) are believed to have radii of $2-3~{}{\rm R}_{\odot}$ (e.g. Larson,
1969; Hosokawa & Omukai, 2009). At both redshifts the average number of
stellar mergers per star consistently increases with decreasing metallicity.
This is due to the lower opacities at lower metallicity that allow faster
cooling of very high-density gas and, therefore, more small-scale
fragmentation (c.f. Bate, 2019).

In the right panel of Fig. 2, we plot the number of stars and brown dwarfs
versus their total mass. There appears to be a change in the nature of the
star formation in the solar-metallicity calculation when $\approx 45$ M⊙ of
gas has been converted into stars in that beyond this point fewer new objects
are created per unit mass of gas. This is reminiscent of the calculations of
Krumholz, Klein & McKee (2011) in which the star-forming clouds were so dense
and the protostars so close together that the radiative feedback from the
young protostars heating the high-density gas inhibited the formation of new
protostars, resulting in a top-heavy mass function. The same effect seems to
be happening in the solar-metallicity calculation here, but for different
reasons: the combination of the inherently hotter clouds due to the enhanced
CMBR and more metal-rich high-density gas trapping radiation and being less
able to cool is inhibiting the formation of new protostars. Thus, the
collapsing gas primarily gets accreted by existing protostars, boosting their
mass, rather than producing new objects.

The reason for the delay of the star formation at low metallicities is that
the intermediate-density, metal-poor gas is hotter. In Figs. 3 to 5 we provide
images of the column densities and mass-weighed temperatures from the three
$z=5$ calculations. The times ($t=1.00,1.10,1.20$ $t_{\rm ff}$) have been
chosen to cover the main periods of star formation. In each figure, the
second, third, and fourth rows give the separate gas, dust, and radiation
temperature distributions, respectively. The radiation temperature is that of
the radiation field from the collapsing gas and dust, modelled using flux-
limited diffusion; the ISRF that also heats the dust is treated separately.

For $z=0$, Bate (2019) found a clear progression that both gas and dust
temperatures were hotter at lower metallicities for $Z\leq{\rm Z}_{\odot}$.
For example, in the $z=0$ calculations at higher metallicities the inner parts
of the clouds were more shielding from the ISRF, resulting in typical dust
temperatures on large spatial scales that decreased with increasing
metallicity. This is not the case at $z=5$. The dust temperatures are much
more uniform due to the heating from the hotter CMBR (which is long wavelength
radiation that is much more able to penetrate the clouds than the high
frequency components of the ISRF). This is further illustrated in lower panels
of Fig. 6 which shows phase diagrams of the gas and dust temperatures
functions of density, that can be directly compared to those in Bate (2019).
The upper panels of Fig. 6 provide the gas-phase CO abundance relative to
hydrogen using a colour scale. Note that the chemical model treats the freeze
out of CO on to dust grains and desorption of CO by cosmic rays, but it does
not treat the thermal desorption that occurs at dust temperatures above 20 K
(e.g. in protostellar discs). This is neglected because the only role of the
chemistry in the calculations is to provide realistic gas temperatures and the
primary coolant at the densities above which CO freeze out becomes important
is usually the dust rather than the CO. In the lower panels of Fig. 6 the grey
scale is proportional to the logarithm of the amount of dust at each
temperature and density. In the $z=5$ calculations, the dust temperatures at
low and intermediate densities are $T_{\rm dust}\approx 15-20$ K and even in
the solar metallicity calculation the coldest dust barely reaches down to
$T_{\rm dust}\approx 10$ K. By contrast, at $z=0$ the dust was able to cool to
much lower temperatures of $T_{\rm dust}\approx 10$ K at $Z=0.01~{}{\rm
Z}_{\odot}$, $T_{\rm dust}\approx 8$ K at $Z=0.1~{}{\rm Z}_{\odot}$, $T_{\rm
dust}\approx 6$ K at $Z\geq{\rm Z}_{\odot}$ (Bate, 2019).

The inability of the dust to cool in the $z=5$ calculations means that the gas
is also unable to cool as it did in the $z=0$ calculations. At low gas
densities ($n_{\rm H}\mathrel{\hbox{\hbox to0.0pt{\lower
2.36806pt\hbox{$\sim$}\hss} \kern-3.00003pt\raise 1.72218pt\hbox{$<$}}}10^{3}$
cm-3) the spread of gas temperatures (Fig. 6) is reasonably similar in the
$z=0$ and $z=5$ calculations (c.f. Fig. 7 of Bate, 2019). At both redshifts,
at intermediate gas densities ($n_{\rm H}\approx 10^{4}-10^{7}$ cm-3) the
hotter gas tends to be a bit hotter at lower metallicity (see Fig. 6) because
metal-poor gas is less able to cool than metal-rich gas due to the reduction
of the atomic and molecular abundances (that dominate the cooling at low
densities). When shocks and other compressive motions in the clouds increase
the thermal energy, the low-metallicity gas cannot radiate this heat away as
easily and, thus, the gas is significantly hotter. This hotter gas gives
greater gas pressure support against gravity to the metal-poor clouds and
delays the collapse of the low metallicity clouds, as seen in Fig. 2.
Comparing the temperature ranges of gas at intermediate densities at the same
metallicity but different redshift, the hottest gas is very similar, but the
coldest gas is colder in the $z=0$ calculations than in the $z=5$
calculations.

At higher densities this difference in the lower range of gas temperatures
between calculations with the same metallicity but at different redshifts
becomes even more significant, particularly at high metallicity. The cooling
of high-density molecular gas with $Z\mathrel{\hbox{\hbox to0.0pt{\lower
2.36806pt\hbox{$\sim$}\hss} \kern-3.00003pt\raise
1.72218pt\hbox{$>$}}}0.01~{}Z_{\odot}$ is primarily dominated by dust
continuum cooling; gas and dust are thermally coupled by collisions. Dust and
gas temperatures become similar above number densities

$n_{\rm H}\approx 10^{6}\left(\frac{Z}{Z_{\odot}}\right)^{-1}~{}\mbox{\rm
cm}^{-3},$ (7)

as can be seen in Fig 6 (see also Omukai, 2000; Glover & Clark, 2012; Bate,
2019). In the calculations of present-day star formation ($z=0$), the cooler
dust temperatures at higher metallicities produced substantially lower gas
temperatures at higher metallicity. This trend with increasing metallicity is
almost completely absent in the $z=5$ calculations. In Fig. 6 at high
densities ($n_{\rm H}\geq 10^{7}$ cm-3) there is very little difference in the
spread in $T_{\rm gas}$-nH for different metallicities because the dust
temperatures are very similar and the much warmer dust is not as effective at
cooling the gas.

It is this comparatively ineffective dust cooling that is fundamentally
responsible for the differences in the stellar properties between the $z=0$
and $z=5$ calculations. For the same metallicity ($Z\geq 0.1~{}{\rm
Z}_{\odot}$), the dust embedded in the high-density gas is substantially
warmer in the $z=5$ calculations than in the $z=0$ calculations. Furthermore,
the magnitude of this difference increases with increasing metallicity because
at $z=0$ the high-density gas becomes colder and colder at higher metallicity
(see Fig. 7 of Bate, 2019). Thus, the gas is less prone to fragment into
protostars in the $z=5$ calculations, producing fewer protostars for the same
amount of gas that is converted into stars and, therefore, the characteristic
stellar masses are higher in the $z=5$ calculations than in the $z=0$
calculations with the same metallicity. Since the $z=0$ calculations produced
protostars with a characteristic mass that was independent of metallicity,
this change in the thermodynamics means that the $z=5$ calculations instead
produce a characteristic stellar mass that increases with increasing
metallicity.

As the star formation proceeds, the radiation fields generated by the gas
collapsing into the gravitational potential wells of the protostars become
stronger (bottom rows of Figs. 3 to 5). These primarily heat the gas and dust
with the highest densities. The effect of this protostellar feedback is much
less obvious on the gas and dust temperatures in Figs. 3 to 5 than it was in
the equivalent figures of Bate (2019) at $z=0$ because of the generally higher
dust and gas temperatures due to the hotter CMBR (i.e. the protostellar
radiation fields tend to be swamped on scales $\mathrel{\hbox{\hbox
to0.0pt{\lower 2.36806pt\hbox{$\sim$}\hss} \kern-3.00003pt\raise
1.72218pt\hbox{$>$}}}0.02$ pc by the greater overall level of radiation from
the CMBR).

Figure 7: Snapshots of the mass-weighted abundances of atomic hydrogen,
ionised carbon, neutral carbon, and CO at the end of each of the three
calculations with differing metallicity ($t=1.20$ $t_{\rm ff}$). There are
three groups of panels, one for each metallicity of 1/100, 1/10, and 1 times
solar. For each group, the top left panel gives the relative abundance HI/H,
the top-right panel the relative abundance C+/C, the lower-left panel the
relative abundance C0/C, and the lower-right panel the relative abundance of
gas phase CO to the total carbon abundance. The stars and brown dwarfs are
plotted using white dots. The colour scale of relative abundances is
logarithmic ranging from $10^{-3}$ to unity. Generally, the outer parts of the
clouds contain atomic hydrogen and C+. The inner parts of the cloud are
primarily composed of H2 and carbon is in the form of CO. At the highest
densities, CO freezes out onto grains (hence the drop in gas-phase abundance).
At lower metallicities, there is a greater fraction of atomic hydrogen and
more of the carbon is in C+ because there is less extinction of the ISRF by
dust and more self-shielding of H2. This figure and caption are comparable to
Fig. 6, Bate (2019).

Figure 8: Histograms giving the distributions of the masses of the stars and
brown dwarfs produced by the three radiation hydrodynamical calculations, each
at $t=1.20t_{\rm ff}$. The dark blue histograms are used to denote those
objects that have stopped accreting (defined as accreting at a rate of less
than $10^{-7}$ M⊙ yr-1), while those objects that were still accreting when
the calculations were stopped are plotted using light blue. The Kroupa (2001)
and Chabrier (2005) parameterisations of the IMF are also plotted. The
vertical dashed line marks the stellar/brown dwarf boundary. The two low-
metallicity calculations produce mass functions that are in reasonable
agreement with the Chabrier (2005) fit to the observed IMF for individual
objects, but the solar metallicity calculation is noticeably deficient in
brown dwarfs and low-mass stars. This figure and caption are comparable to
Fig. 8, Bate (2019).

###  3.2 Chemistry

The present-day star formation calculations of Bate (2019) were the first to
combine a model of the diffuse ISM with radiative transfer to treat the
radiation produced by the collapsing gas. The calculations keep track of
whether the hydrogen is in atomic or molecular form, and whether carbon is in
the form of ionised carbon, C+, neutral carbon, C0, or molecular in the form
of CO. There is also a model for the freezing out of CO onto dust grains (see
Bate & Keto, 2015). Modelling the chemistry is important because the formation
of molecular hydrogen is a potential source of heating, while carbon atoms and
molecules are important coolants.

Following Bate (2019), in Fig. 7 we provide snapshots of the chemical make up
of the clouds at the end of each of the three $z=5$ calculations. The
chemistry varies most in the low-density gas and in the outer parts of the
clouds, so we show the large scales. The carbon abundances are scaled relative
to the total carbon abundance in each calculation (i.e. the absolute carbon
abundance is 10 times lower in the $Z=0.1~{}{\rm Z}_{\odot}$ calculation than
in the calculation at solar metallicity).

If the panels in Fig. 7 are compared with the corresponding panels in Figure 6
of Bate (2019), very little difference is seen for the same metallicity. This
is because it is the higher energy part of the ISRF that dictates the
chemistry of the low-density gas, not the CMBR at $z=0-5$. In both sets of
calculations, the hydrogen deep within the clouds is almost entirely
molecular. The atomic hydrogen abundance is only high in the outer parts of
the clouds where there is low extinction of the ISRF by dust and less self-
shielding provided by the molecular hydrogen. Although the atomic hydrogen
abundance within the cloud is a little higher at low metallicity, even with
$Z=0.01~{}{\rm Z}_{\odot}$ the hydrogen is 97.8 percent molecular at the end
of the calculation.

Similarly, carbon is almost entirely in the form of C+ in the outer parts of
the clouds, because it is exposed to the ISRF, while deep within the clouds
the main gas phase form of carbon is CO. Neutral atomic carbon is found at
intermediate depths, but it is never very abundant. The C+ fraction has a
greater dependence on metallicity than the fraction of atomic hydrogen.

###  3.3 The statistical properties of the stellar populations

In this section, we compare the statistical properties of stars and brown
dwarfs produced by the three calculations. We also compare our results with
the those from the present-day star formation calculations of Bate (2019). As
in Bate (2019), we study the stellar mass distributions, stellar
multiplicities, the distributions of separations of multiple stellar systems,
and the mass ratio distributions of binary systems. In earlier papers
performing calculations of a similar scale (e.g. Bate, 2009a, 2012, 2014),
other statistics such as the orbital eccentricity distributions of multiple
systems, triple system orbits, stellar accretion histories and kinematics, and
stellar closest encounters were also examined. We do not present data on these
properties here, however, that we find no evidence that these other properties
vary significantly with metallicity or redshift between $z=0$ and $z=5$.

####  3.3.1 The initial mass function

In Fig. 8, we compare the differential mass functions at the end of each of
the three radiation hydrodynamical calculations with different metallicities.
We compare them with parameterisations of the observed Galactic IMF, given by
Chabrier (2005), and Kroupa (2001). There is a clear difference between the
mass distributions, with the solar metallicity case producing stars with a
higher typical mass than the two sub-solar metallicity calculations and very
few brown dwarfs.

Figure 9: The cumulative stellar/brown dwarf mass distributions produced by the three radiation hydrodynamical calculations with metallicities of $Z=0.01~{}{\rm Z}_{\odot}$ (magenta dot-dashed line), $Z=0.1~{}{\rm Z}_{\odot}$ (red dashed line), and $Z={\rm Z}_{\odot}$ (black solid line). We also plot the Chabrier (2005) IMF (black dotted line). The vertical dashed line marks the stellar/brown dwarf boundary. The form of the stellar mass distribution is similar for all three calculations, but the median mass increases with metallicity, particularly for the solar metallicity calculation which produces no low-mass ($M_{*}\mathrel{\hbox{\hbox to0.0pt{\lower 2.36806pt\hbox{$\sim$}\hss} \kern-3.00003pt\raise 1.72218pt\hbox{$<$}}}0.05$ M⊙) brown dwarfs at all. This figure and caption are comparable to Fig. 9, Bate (2019).  Mass Range [M⊙] | Single | Binary | Triple | Quadruple
---|---|---|---|---
Redshift $z=5$, Metallicity $Z=0.01~{}{\rm Z}_{\odot}$
$M<0.03$ | 11 | 0 | 0 | 0
$0.03\leq M<0.07$ | 20 | 1 | 0 | 0
$0.07\leq M<0.10$ | 6 | 0 | 0 | 0
$0.10\leq M<0.20$ | 11 | 2 | 1 | 0
$0.20\leq M<0.50$ | 9 | 2 | 1 | 4
$0.50\leq M<0.80$ | 4 | 0 | 3 | 1
$0.80\leq M<1.2$ | 2 | 1 | 1 | 0
$M>1.2$ | 2 | 3 | 1 | 2
Redshift $z=5$, Metallicity $Z=0.1~{}{\rm Z}_{\odot}$
$M<0.03$ | 9 | 0 | 0 | 0
$0.03\leq M<0.07$ | 16 | 0 | 0 | 0
$0.07\leq M<0.10$ | 6 | 0 | 0 | 0
$0.10\leq M<0.20$ | 11 | 2 | 1 | 0
$0.20\leq M<0.50$ | 9 | 0 | 2 | 1
$0.50\leq M<0.80$ | 1 | 1 | 0 | 2
$0.80\leq M<1.2$ | 1 | 1 | 0 | 1
$M>1.2$ | 0 | 1 | 6 | 2
Redshift $z=5$, Metallicity $Z={\rm Z}_{\odot}$
$M<0.03$ | 0 | 0 | 0 | 0
$0.03\leq M<0.07$ | 2 | 0 | 0 | 0
$0.07\leq M<0.10$ | 5 | 0 | 0 | 0
$0.10\leq M<0.20$ | 10 | 1 | 0 | 0
$0.20\leq M<0.50$ | 7 | 2 | 1 | 0
$0.50\leq M<0.80$ | 2 | 1 | 0 | 0
$0.80\leq M<1.2$ | 3 | 0 | 1 | 0
$M>1.2$ | 3 | 2 | 3 | 2
All masses, 3 calculations | 150 | 20 | 21 | 15
Table 2:  The numbers of single and multiple systems for different primary mass ranges at the end of the three $z=5$ radiation hydrodynamical calculations with different metallicities. This table and caption are comparable to Table 2, Bate (2019).  Object Number | Mass | $t_{\rm form}$ | Accretion Rate
---|---|---|---
| [M⊙] | [$t_{\rm ff}$] | [M⊙ yr-1]
1 | 2.4253 | 0.8606 | $4.00\times 10^{-5}$
2 | 7.1796 | 0.8616 | $9.83\times 10^{-5}$
3 | 7.1896 | 0.8699 | $9.88\times 10^{-5}$
4 | 2.1061 | 0.8801 | $8.23\times 10^{-6}$
5 | 3.6424 | 0.9140 | $1.28\times 10^{-4}$
Table 3:  For each of the three calculations, we provide online tables of the
stars and brown dwarfs that were formed, numbered by their order of formation,
listing the mass of the object at the end of the calculation, the time (in
units of the initial cloud free-fall time) at which it began to form (i.e.
when a sink particle was inserted), and the accretion rate of the object at
the end of the calculation (precision $\approx 10^{-7}$ M⊙ yr-1). The first
five lines of the table for the $z=5$ solar metallicity calculation are
provided above. This table and caption are comparable to Table 3, Bate (2019).

In Fig. 9, we compare the cumulative mass functions at the end of the three
calculations, along with the parameterisation of the observed Galactic IMF of
Chabrier (2005). There is a consistent trend of higher stellar masses being
produced at higher metallicity, although there is only a small difference
between the mass distributions with $Z=0.01~{}{\rm Z}_{\odot}$ and
$Z=0.1~{}{\rm Z}_{\odot}$. In the most metal-poor calculation, approximately
1/3 of the objects are brown dwarfs, while in the most metal-rich calculation
only 2 out of 67 objects (3%) are brown dwarfs, and one of these is still
accreting when the calculation is stopped (we take brown dwarfs to have masses
$M<0.075~{}{\rm M}_{\odot}$).

Performing Kolmogorov-Smirnov tests on each pair of distributions shows that
although both the mean and median masses are slightly higher in the
$Z=0.1~{}{\rm Z}_{\odot}$ calculation than in the $Z=0.01~{}{\rm Z}_{\odot}$
calculation (Table 1), formally, the two lowest metallicity distributions are
consistent with random sampling from a single underlying distribution
(probability 38%). However, the solar metallicity calculation is significantly
different, having probabilities of $4\times 10^{-4}$ and $6\times 10^{-5}$ of
being randomly sampled from the same distribution as the $Z=0.1~{}{\rm
Z}_{\odot}$ and $Z=0.01~{}{\rm Z}_{\odot}$ calculations, respectively.

The two low-metallicity distributions are both in reasonable agreement with
the Chabrier (2005) IMF. Kolmogorov-Smirnov tests that compare the numerical
distributions to Chabrier’s parameterisation give probabilities of 2.6%
($Z=0.01~{}{\rm Z}_{\odot}$) and 20% ($Z=0.1~{}{\rm Z}_{\odot}$) of the
numerical distributions being randomly drawn from the Chabrier (2005) IMF.
However, the solar metallicity distribution is inconsistent with being drawn
from the Chabrier (2005) IMF, with a Kolmogorov-Smirnov probability of only
$1\times 10^{-4}$. The differences essentially arise from the fact that the
median mass of the $Z=0.1~{}{\rm Z}_{\odot}$ distribution is in agreement with
that of the Chabrier (2005) IMF (0.2 M⊙), while the median mass for the
$Z=0.01~{}{\rm Z}_{\odot}$ distribution is a little lower and the median of
the $Z={\rm Z}_{\odot}$ distribution is much greater (see Table 1). Of course
in these probabilities no account is taken of the observational uncertainty in
the Galactic median mass.

Finally, as noted by Bate (2019), the calculations produce protostellar mass
functions (PMFs) rather than IMFs (Fletcher & Stahler, 1994a, b; McKee &
Offner, 2010) because many of the objects are still accreting when the
calculations are stopped and the star formation has not finished (Fig. 8).
This is particularly true of the solar metallicity calculation. Bate (2012)
showed that the form of the distribution of stellar masses in a calculation
similar to those performed here (but not treating the diffuse interstellar
medium or having separate gas and dust temperatures) did not change
significantly with time during his calculation. Although the maximum stellar
mass and the total number of stars both increased with time, the
characteristic (median) mass and the form of the mass function remained
similar due to the production of new protostars. This approximate invariance
with time is also true of the calculations of Bate (2019) and the calculations
presented here, even in the $z=5$ solar metallicity case.

Figure 10: Multiplicity fraction as a function of primary mass at the end of each of the calculations with different metallicities. The thick solid lines give the continuous multiplicity fractions from the calculations computed using a sliding log-normal average and the shaded areas give the approximate $1\sigma$ confidence intervals around the solid line. The blue squares with error bars and/or upper/lower limits give the observed multiplicity fractions from the surveys of Close et al. (2003), Basri & Reiners (2006), Fischer & Marcy (1992), Raghavan et al. (2010), Duquennoy & Mayor (1991), Kouwenhoven et al. (2007), Rizzuto et al. (2013), Preibisch et al. (1999) and Mason et al. (1998), from left to right. Note that the error bars of the Duquennoy & Mayor (1991) results have been plotted using light blue since this survey has been superseded by Raghavan et al. (2010). The observed trend of increasing multiplicity with primary mass is reproduced by all calculations. There is no strong difference between the multiplicities obtained with different metallicities, but there seems to be a consistent trend that low-mass stars ($0.1-0.5$ M⊙) have somewhat higher multiplicities with decreasing metallicity. This figure and caption are comparable to Fig. 10, Bate (2019).  Object Numbers | No. of | No. in | $M_{\rm max}$ | $M_{\rm min}$ | $M_{1}$ | $M_{2}$ | $q$ | $a$ | P | $e$ | Relative Spin | Spin1 | Spin2
---|---|---|---|---|---|---|---|---|---|---|---|---|---
| Objects | System |  |  |  |  |  |  |  |  | or Orbit | -Orbit | -Orbit
|  |  |  |  |  |  |  |  |  |  | Angle | Angle | Angle
|  |  | [M⊙] | [M⊙] | [M⊙] | [M⊙] |  | [au] | [yr] |  | [deg] | [deg] | [deg]
22, 30 | 2 | 4 | 1.240 | 0.870 | 1.240 | 0.870 | 0.702 | 0.70 | 0.40 | 0.029 | 19 | 63 | 48
36, 42 | 2 | 3 | 0.876 | 0.614 | 0.876 | 0.614 | 0.701 | 3.23 | 4.76 | 0.195 | 54 | 59 | 18
( 36, 42), 43 | 3 | 3 | 0.876 | 0.393 | 1.490 | 0.393 | 0.264 | 17.15 | 51.74 | 0.071 | 20 | – | –
( 24, 34), ( 22, 30) | 4 | 4 | 1.400 | 0.870 | 2.738 | 2.110 | 0.771 | 417.97 | 3880 | 0.576 | – | – | –
Table 4:  For each of the three calculations, we provide online tables of the
properties of the multiple systems at the end of each calculation. The
structure of each system is described using a binary hierarchy. For each
‘binary’ we give the masses of the most massive star $M_{\rm max}$ in the
system, the least massive star $M_{\rm min}$ in the system, the masses of the
primary $M_{1}$ and secondary $M_{2}$, the mass ratio $q=M_{2}/M_{1}$, the
semi-major axis $a$, the period $P$, the eccentricity $e$. For binaries, we
also give the relative spin angle, and the angles between orbit and each of
the primary’s and secondary’s spins. For triples, we give the relative angle
between the inner and outer orbital planes. For binaries, $M_{\rm max}=M_{1}$
and $M_{\rm min}=M_{2}$. However, for higher-order systems $M_{1}$ gives the
combined mass of the most massive sub-system (which may be a star, binary, or
a triple) and $M_{2}$ gives the combined mass of the least massive sub-system
(which also may be a star, a binary, or a triple). Multiple systems of the
same order are listed in order of increasing semi-major axis. As examples, we
provide selected lines from the table from the $z=5$ solar metallicity
calculation. This table and caption are comparable to Table 4, Bate (2019).

####  3.3.2 Multiplicity as a function of primary mass

The formation mechanisms of multiple systems and the evolution of their
properties (e.g. separations) has been discussed in some detail by Bate (2012)
and Bate (2018) and will not be repeated here. Our primary purpose is to
determine whether or not variation of the metallicity and/or redshift alters
stellar properties significantly.

As in Bate (2009a, 2012), and subsequent papers, to quantify the fraction of
stars and brown dwarfs that are in multiple systems, we use the multiplicity
fraction, $mf$, defined as a function of stellar mass as

$mf=\frac{B+T+Q}{S+B+T+Q},$ (8)

where $S$ is the number of single stars within a given mass range and, $B$,
$T$, and $Q$ are the numbers of binary, triple, and quadruple systems,
respectively, for which the primary has a mass in the same mass range. As
discussed by Hubber & Whitworth (2005) and Bate (2009a), this measure of
multiplicity is relatively insensitive to both observational incompleteness
(e.g., if a binary is found to be a triple it is unchanged) and further
dynamical evolution (e.g., if an unstable quadruple system decays the
numerator only changes if it decays into two binaries). We also use the same
method for identifying multiple systems as that used by Bate (2009a) and Bate
(2012). We identify binary, triple, and quadruple stellar systems, but we
ignore higher-order multiples (e.g. a quintuple system consisting of a triple
and a binary orbiting one another is counted as one triple and one binary). We
choose to stop at quadruple systems since it is likely that many higher order
systems would be unstable and would eventually decay.

Table 2 provides the numbers of single and multiple star and brown dwarf
systems produced by each calculation. Bate (2019) provided electronic ASCII
tables of the properties of each of the stars, brown dwarfs, and multiple
systems produced by the calculations discussed in that paper, and we do the
same here. We present tables of the masses, formation times, and final
accretion rates of the stars and brown dwarfs (see Table 3 for an example).
The tables are given file names such as Table3_Stars_z5_Metal01.txt for the
$Z=0.1~{}{\rm Z}_{\odot}$ calculation. We also produce tables listing the
properties of each multiple system (see Table 4 for an example). These tables
are given file names such as Table4_Multiples_z5_Metal1.txt for the $Z={\rm
Z}_{\odot}$ calculation.

The overall multiplicities for stars and brown dwarfs of all masses from each
of the calculations are 26, 27, and 29 per cent for the calculations with
metallicities of 1/100, 1/10, 1 times solar, respectively. The typical
$1\sigma$ uncertainties are $\pm 5$ per cent. Therefore, there is no evidence
for a significant dependence of the overall multiplicity on metallicity.
However, because the mass distributions vary between the calculations and
multiplicity varies with stellar mass this can be misleading.

In Fig. 10, for each of the three simulations we compare the multiplicity
fraction of the stars and brown dwarfs as functions of the stellar mass of the
primary with the values obtained from various Galactic observational surveys
(see the figure caption). The results from each of the calculations have been
plotted using a thick solid line that gives the continuous multiplicity
fraction computed using a sliding log-normal-weighted average. The width of
the log-normal average is half a decade in stellar mass. The shaded region
gives the approximate $1\sigma$ (68%) uncertainty on the sliding log-normal
average.

Figure 11: The distributions of separations (semi-major axes) of multiple
systems with stellar primaries ($M_{*}>0.1$ M⊙) produced by the calculations
with different metallicities. The dark, medium, and light histograms give the
orbital separations of binaries, triples, and quadruples, respectively (each
triple contributes two separations, each quadruple contributes three
separations). The solid curve gives the M-dwarf separation distribution
(scaled to match the same area as each of the histograms) from the M-dwarf
survey of Janson et al. (2012), and the dashed curve gives the separation
distribution for solar-type primaries of Raghavan et al. (2010). Note that
since most of the simulated systems are low-mass, the distributions are
expected to match the Janson et al. distribution better than that of Raghavan
et al. The vertical dotted line gives the resolution limit of the calculations
as set by the accretion radii of the sink particles (0.5 au). This figure and
caption are comparable to Fig. 11, Bate (2019).

All of the calculations produce multiplicity fractions that increase strongly
with increasing primary mass, in qualitative agreement with observed stellar
systems. The actual values of the multiplicities are in good agreement with
the observed multiplicities of Galactic field stars for the lowest metallicity
calculation, and they are also similar to the multiplicities obtained for all
of the present-day star calculations of Bate (2019) with metallicities ranging
from $Z=0.01-3$ Z⊙. However, unlike the present-day star formation
calculations of Bate (2019), the multiplicities of the three calculations at
redshift $z=5$ do seem to depend on metallicity. There is no consistent trend
of the multiplicities for solar-mass and more massive stars, but for primaries
with masses $M_{1}\mathrel{\hbox{\hbox to0.0pt{\lower
2.36806pt\hbox{$\sim$}\hss} \kern-3.00003pt\raise
1.72218pt\hbox{$<$}}}0.1-0.5$ M⊙ (i.e. M-dwarfs) the multiplicities
consistently increase with decreasing metallicity such that the multiplicities
at metallicity $Z=0.01$ Z⊙ are approximately twice as high as for $Z={\rm
Z}_{\odot}$.

Only a single very-low-mass (VLM; $M_{1}<0.1~{}{\rm M}_{\odot}$) multiple
system is produced across all three simulations (see Table 2), and this is in
the lowest metallicity calculation. Fig. 10 shows that the VLM multiplicity
for metal-poor ($Z=0.01~{}{\rm Z}_{\odot}$) star formation at redshift $z=5$
is similar to the Galactic value. However, at higher metallicities we find
that the VLM multiplicity produced by star formation at redshift $z=5$ is
lower than both Galactic field value and that obtained from the present-day
star formation calculations of Bate (2019).

Figure 12: The cumulative semi-major axis (separation) distributions of the
multiple systems produced by the three $z=5$ calculations with different
metallicities. All orbits are included in the plot (i.e. two separations for
triple systems, and three separations for quadruple systems), and the lines
for different metallicities include both stars and brown dwarfs. The vertical
dashed line marks the resolution limit of the calculations as determined by
the accretion radii of the sink particles. Performing Kolmogorov-Smirnov tests
on pairs of distributions shows that the distributions for the different
metallicities are statistically indistinguishable. This figure and caption are
comparable to Fig. 12, Bate (2019).

####  3.3.3 Separation distributions of multiple systems

In Fig. 11, we present the separation (semi-major axis) distributions of the
stellar ($M_{1}>0.1$ M⊙) multiples (there is only one VLM multiple, the binary
in the $Z=0.01~{}{\rm Z}_{\odot}$ calculation). The distributions are compared
with the log-normal distributions from the surveys of M-dwarfs by Janson et
al. (2012) and solar-type stars by Raghavan et al. (2010). Although binaries
as close as 0.03 au (6 R⊙) can be modelled without being merged, gas is not
modelled within the sink particle accretion radii of 0.5 au. The lack of gas
to provide dissipation on small scales inhibits the formation of very close
systems (Bate, Bonnell & Bromm, 2002).

There is no obvious dependence of the shape of the distributions on
metallicity (in all cases the peak occurs in the $10-100$ au range of semi-
major axis). The absolute numbers are lower with increasing metallicity due to
the smaller number of stars produced at higher metallicity (Table 1). The peak
occurs at a similar separation to the peak in the Galactic population of
stellar multiples. The spread appears slightly wider than for Galactic M-dwarf
systems and narrower than Galactic solar-type stars, consistent with a mixture
of primary stellar masses that is dominated by low-mass stars.

In Fig. 12, we provide the cumulative separation distributions for each of the
calculations (this time including the VLM binary from the $Z=0.01~{}{\rm
Z}_{\odot}$ calculation). Kolmogorov-Smirnov tests performed on each pair of
distributions shows that they are statistically indistinguishable. They are
also each in good agreement with the separation distribution of M-dwarfs as
determined by Janson et al. (2012), and they tend to have slightly closer
separations than the solar-type stars (Raghavan et al., 2010). In the
$Z=0.01~{}{\rm Z}_{\odot}$ calculation, most of the multiple systems have
primary masses $M_{1}=0.1-0.8~{}{\rm M}_{\odot}$, while in each of the other
two calculations roughly half of the multiples are low-mass and half have
primary masses $M_{1}>0.8~{}{\rm M}_{\odot}$.

For present-day star formation, Bate (2019) found a weak possible trend for
multiple systems to be preferentially tighter at lower metallicities, though
the small numbers of systems limited the confidence in the result. There is no
sign of such a trend for the systems formed at redshift $z=5$.

Figure 13: The cumulative mass ratio distributions of pairs of objects
produced by the three $z=5$ calculations with different metallicities. Pairs
include both binaries and bound pairs that are components of triple or
quadruple systems. We also plot the observed mass ratio distribution of solar-
type stars from Raghavan et al. (2010). There is a clear and consistent trend
for the more metal-rich systems to have pairs consisting of more equal-mass
components. However, performing Kolmogorov-Smirnov tests on each pair of the
simulated distributions, only the $Z=0.01~{}{\rm Z}_{\odot}$ and $Z={\rm
Z}_{\odot}$ distributions are inconsistent with being drawn from the same
underlying distribution (Kolmogorov- Smirnov probability of 1.3 percent) due
to the relatively small numbers of pairs that are produced by the
calculations. This figure and caption are comparable to Fig. 14, Bate (2019).

####  3.3.4 Mass ratio distributions of pairs

In Fig. 13 we plot the cumulative mass ratio distributions for each
calculation (including all primary masses, and including the single VLM binary
from the $Z=0.01~{}{\rm Z}_{\odot}$ calculation). These distributions include
binary systems, and pairs that are the inner components of triple and
quadruple systems. A triple system composed of a binary with a wider companion
contributes the mass ratio from the closest pair, as does a quadruple composed
of a triple with a wider companion. A quadruple composed of two pairs orbiting
each other contributes two mass ratios – one from each of the pairs. We also
plot the cumulative observed mass-ratio distributions for M-dwarfs from Janson
et al. (2012), and for solar-type primaries from Raghavan et al. (2010).

There is a clear progression in the mass ratio distributions from the $z=5$
calculations such that the more metal-rich calculations tend to have pairs
consisting of more equal-mass components. The mass ratio distribution of the
$Z=0.1~{}{\rm Z}_{\odot}$ is most similar to the mass ratio distribution of
M-dwarfs from Janson et al. (2012). The $Z=0.01~{}{\rm Z}_{\odot}$ calculation
has a greater fraction of $q<0.5$ pairs, while the $Z={\rm Z}_{\odot}$
calculation has almost no pairs with $q<0.5$. Despite the clear progression
with metallicity, formally only the $Z=0.01~{}{\rm Z}_{\odot}$ and $Z={\rm
Z}_{\odot}$ distributions are inconsistent with being drawn from the same
underlying distribution (Kolmogorov-Smirnov probability of 1.3 percent) due to
the relatively small numbers of pairs that are produced by the calculations.

For present-day ($z=0$) star formation, Bate (2019) found that all of the mass
ratio distributions from the four calculations ranging in metallicity from
$Z=0.01-3~{}{\rm Z}_{\odot}$ were statistically indistinguishable, although
again the lowest metallicity calculation had the greatest fraction of low
mass-ratio systems. All of the distributions were formally consistent with
being drawn from the Raghavan et al. (2010) distribution, except for the
highest metallicity calculation ($Z=3~{}{\rm Z}_{\odot}$) which had a deficit
of low mass ratio pairs. Thus, the general trends found (a greater fraction of
low mass ratio pairs in lower metallicity calculations) are the same for both
present-day star formation and for redshift $z=5$. We have also performed
Kolmogorov-Smirnov tests on the mass ratio distributions for the $z=0$ and
$z=5$ distributions that were obtained using the same metallicity (i.e.
$Z=0.01,0.1,1~{}{\rm Z}_{\odot}$). In each case the mass ratio distributions
are indistinguishable at the two redshifts, although again the sensitivity of
this test is limited by the relatively small numbers of pairs that are
produced by the $z=5$ simulations.

Figure 14:  The frequencies of close companions (semi-major axes $a<10$ au)
for low-mass stars with masses $M_{*}=0.1-1.5$ M⊙ that are produced by the
three $z=5$ calculations with different metallicities (magenta filled circles
and errorbars). We compare the results with the equivalent frequencies from
the numerical simulations of present-day ($z=0$) star formation by Bate (2019)
(blue open squares with errorbars) and Bate (2014) (red triangles and dashed
errorbars), and with the observed values for stellar masses $M_{*}=0.6-1.5$ M⊙
from Moe et al. (2019) (black filled squares and errorbars). The error bars
for the numerical simulations give 95 percent confidence intervals. The values
of the metallicity have been slightly offset horizontally for the Bate (2014,
2019) results for clarity. The results from the simulations are consistent
with the observed anti-correlation between the frequency of close companions
and metallicity, and the numerical values from the present-day calculations
are in reasonable agreement with the observed values. For the redshift $z=5$
calculations, the frequencies are on average approximately a factor of two
lower and the anti-correlation is less clear. This figure and caption are
comparable to Fig. 15, Bate (2019).

####  3.3.5 Close binaries

Figure 15:  The mass ratio distributions of stellar pairs with primary masses
$M_{1}=0.1-1.5$ M⊙ that are produced by the four calculations with different
metallicities. In the left panel, we give the distributions for pairs with
semi-major axes $a<10$ au, while in the right panel we give the distributions
for pairs with $a>10$ au. Due to the small numbers of objects, the probability
of any two pairs of distributions being drawn from the same underlying
distribution never falls below 9 percent in the left plot. However, the close
pairs (left panel) do display a consistent trend of more low mass ratio
systems as the metallicity is reduced, whereas there is no clear trend for the
wider pairs (right panel). This figure and caption are comparable to Fig. 16,
Bate (2019).

Bate (2019) found that although the stellar multiplicities for present-day
star formation with metallicities ranging from $Z=0.01-3~{}{\rm Z}_{\odot}$
were indistinguishable when considering all separations, if the frequency of
close pairs with semi-major axes less than 10 au were considered then the
binary frequency was anti-correlated with metallicity. Bate (2019)
investigated this particular separation range because three observational
papers that had been published during the preceding year had found that the
close binary fractions for solar-type stars were anti-correlated with
metallicity (Badenes et al., 2018; El-Badry & Rix, 2019; Moe et al., 2019).
This type of anti-correlation had been claimed before (Grether & Lineweaver,
2007; Raghavan et al., 2010), but the small numbers of stars involved and
potential observational biases had limited the confidence in the results.

In addition, Bate (2019) found that close pairs (semi-major axes, $a<10$ au)
displayed a consistent trend such that there were more low mass-ratio systems
for lower metallicities, but that there was no such trend for wider pairs.

In the redshift $z=5$ calculations analysed in this paper, we have already
seen that the multiplicities of low-mass stars (M-dwarfs) appear to be lower
at higher metallicity (Section 10) even considering all separation ranges.
Here we examine just the close pairs ($a<10$ au). We also restrict our study
to systems with primary with masses $M_{1}\approx 0.1-1.5$ M⊙, which is larger
than the range of solar-type primaries, $M_{1}\approx 0.6-1.5$ M⊙, that was
considered by Moe et al. (2019). We use a larger range because we have a
comparatively small number of systems (this larger range was also used by Bate
2019).

In Fig. 14, we present the frequencies of close pairs (semi-major axes $<10$
au) in systems with primary masses $M_{1}=0.1-1.5$ M⊙ for each of our
simulations. We compare these to the fractions that were presented in the
equivalent figure by Bate (2019) for the the present-day star formation
simulations analysed by that paper, the earlier calculations of Bate (2014),
and also to the observational results of Moe et al. (2019). The results from
the $z=5$ simulations are consistent with an anti-correlation between close
binary frequency and metallicity, but the results are less convincing than for
the Bate (2014, 2019) simulations. The close companion fractions are generally
lower for the $z=5$ calculations than for the present-day star formation
simulations. In other words, star formation at $z=5$ results in a lower
frequency close pairs than present-day star formation by about a factor of two
for the same metallicity. On top of this, there is evidence for an anti-
correlation with metallicity since the frequency in the solar-metallicity
calculation is about 1/3 of that obtained from the $Z=0.1~{}{\rm Z}_{\odot}$.
However, the frequencies are similar for the two lowest metallicity
calculations. Overall, the results are consistent with an anti-correlation
between close binary frequency and metallicity for star formation at both
redshift $z=0$ and $z=5$, but larger numbers of systems are required to be
certain.

Following Bate (2019), in Fig 15, we give the mass ratio distributions of
pairs with low-mass stellar primaries ($M_{1}=0.1-1.5$ M⊙) for close pairs
(semi-major axes $a<10$ au; left panel) and wider pairs (right panel). For the
close pairs, the lowest metallicity calculation produces more low-mass ratio
systems than the intermediate metallicity calculation, whereas there is no
clear difference for the wider systems. This is the same trend that was found
by Bate (2019) for present-day star formation. But performing Kolmogorov-
Smirnov tests on all pairs of cumulative distributions for the $z=5$
calculations shows that, in each case, the two distributions are consistent
with being drawn from the same underlying distribution due to the small
numbers.

As discussed by Bate (2019), the fact that anti-correlation between close pair
frequency and metallicity was found in both the Bate (2014) and Bate (2019)
sets of calculations implies that the physical mechanism originates from the
metallicity dependence of the opacity used in the radiative transfer (which
applied in both sets of calculations), rather than from the separate treatment
of gas and dust temperatures or the model of the diffuse interstellar medium
that was employed in the calculation of Bate (2019) and in this paper. Bate
(2019) found that two different effects of the dust opacities combined to
produce the anti-correlation. First, at higher metallicities the higher dust
opacities gave less small-scale fragmentation because dense gas is more
optically thick and less able to cool quickly. Second, at higher metallicities
the higher dust opacities result in slower cooling rates of first hydrostatic
cores (FHSCs) and, thus, longer lifetimes. Since FHSCs typically have radii of
$\approx 5$ au (Larson, 1969), FHSCs need to undergo collapse to form stellar
cores before close binaries ($a<10$ au) can be formed. Longer lifetimes mean
that even if small scale fragmentation produces multiple FHSCs, those with
high-metallicity have longer to merge into a single FHSC, thereby producing a
single protostar rather than a close binary. Conversely, low-metallicity FHSCs
can quickly collapse to produce stellar cores, potentially avoiding merging
with another nearby FHSC and producing a close pair. It is also worth noting
that Bate (2019) found that most close pairs were formed by cloud or filament
fragmentation, not by disc fragmentation.

In the redshift $z=5$ calculations, there is still evidence for an anti-
correlation between close binary frequency and metallicity, but it may be
weaker than for present-day star formation and the close binary frequency may
be lower (by approximately a factor of two) at a given metallicity at $z=5$
than at $z=0$. This may be because the first of the two mechanisms leading to
the anti-correlation is weakened at higher redshift. In other words, the
slower cooling rates of the FHSCs at higher metallicity leading to more FHSC
merges at high metallicity would still operate (giving fewer close pairs at
higher metallicity). But with the higher CMBR temperature there is less small-
scale fragmentation of high-density gas because the dense gas is less able to
cool, particularly at high metallicity.

##  4 Discussion

Bate (2019) performed radiation hydrodynamical calculations of present-day
star formation in molecular clouds whose metallicity was varied from
$Z=0.01-3~{}{\rm Z}_{\odot}$ and found that the statistical properties of the
stars were largely independent of the metallicity (see also Myers et al.,
2011; Bate, 2014). The resulting stellar mass distributions, overall stellar
multiplicity, and mass ratio distributions of bound pairs were statistically
indistinguishable. Furthermore, the stellar mass distributions were all very
similar to the parameterisation of the Galactic IMF of Chabrier (2005). The
only clear differences that were found was an anti-correlation between the
frequency of close ($a<10$ au) protostellar pairs and metallicity, in
qualitative agreement with observations of Galactic field stars, a preference
for a greater fraction of unequal-mass close pairs with decreasing
metallicity, and an increase in the number of protostellar mergers per star
with decreasing metallicity.

By contrast, in this paper we have presented results from simulations that are
identical to those of Bate (2019) except that the cosmic microwave background
radiation was hotter because they were performed at a redshift of $z=5$. With
the hotter CMBR we find that the statistical properties of the stars do vary
with metallicity. In particular, at $Z=0.01~{}{\rm Z}_{\odot}$ the stellar
mass distributions obtained at redshifts $z=0$ and $z=5$ are indistinguishable
(and both are similar to the observed Galactic IMF), but at $z=5$ increasing
the metallicity results in an increase in the characteristic stellar mass to
the point that brown dwarfs are very rare at solar metallicities. Furthermore,
the multiplicities of M-dwarfs are approximately a factor of two lower at
solar metallicity than at $Z=0.01~{}{\rm Z}_{\odot}$. In agreement with the
present-day star formation calculations, the mass ratios of stellar pairs are
more unequal and there are more stellar mergers per star at lower metallicity.
The anti-correlation between the frequencies of close pairs and metallicity
still seems to be present, but it is less clear than in the $z=0$
calculations.

Perhaps the most surprising result from the two papers is that the different
level of CMBR qualitatively changes the dependence of the stellar mass
function on metallicity: the characteristic stellar mass is predicted to be
independent of metallicity at $z=0$, but increase with metallicity at $z=5$.
Or, alternately, at low metallicity ($Z=0.01~{}{\rm Z}_{\odot}$) the stellar
mass function doesn’t vary with redshift, but at solar-metallicity the
characteristic stellar mass increases with increasing redshift due to the
hotter CMBR.

In the following sections, we examine the predictions of past theoretical
studies for the dependence of the IMF on redshift, we discuss the implications
of these results for how the IMF may vary continuously with redshift and
metallicity, and we examine the observational evidence for variations in the
IMF and how these results may aid in their interpretation.

###  4.1 Comparison with previous theoretical models

It is common in the literature to postulate that the characteristic stellar
mass is proportional to the typical Jeans mass in a molecular cloud. The Jeans
mass depends on temperature: for example, it can be expressed as varying with
temperature and density as $\propto T^{3/2}\rho^{-1/2}$, or with temperature
and pressure as $\propto T^{2}P^{-1/2}$, or with temperature and surface
density as $\propto T^{2}\mu^{-1}$ (Larson, 1998). Given the strong dependence
of the Jeans mass on temperature, if the characteristic stellar mass scales
with the Jeans mass it is natural to postulate that the characteristic mass of
the IMF should increase with redshift because the cosmic microwave background
radiation scales with redshift as $(1+z)$ (equation 6) and this exceeds the
lowest temperatures found in Galactic molecular clouds ($\sim 6-8$) K by
redshift $z=2-3$. Similarly, because metal-poor gas is less able to cool than
metal-rich gas when it is optically thin, one can make the argument that
metal-poor gas is generally expected to be hotter and, therefore, the
characteristic stellar mass would also increase with decreasing metallicity.
Indeed, both these arguments are made by Larson (1998) when he discusses the
possibility of a time-varying IMF, and assuming that metallicity generally
decreases with increasing redshift both of these affects act together to
suggest that the characteristic stellar mass should increase with increasing
redshift (see also Baugh et al. 2005; Narayanan & Davé 2012).

However, as we have seen from the results in this paper the dependence of the
characteristic stellar mass is apparently more complicated than this. For one
thing, although metal-poor gas is a poor at cooling compared metal-rich gas at
low densities, at high densities when the cooling of the gas becomes dominated
by collisions with dust grains, metal-poor gas remains optically thin to
higher densities and, therefore, is able to cool more quickly, leading to
lower gas temperatures and, potentially, more fragmentation than metal-rich
gas. Bate (2014) demonstrated this by performing simple spherically-symmetric
calculations of collapsing 1-M⊙ Bonnor-Ebert spheres with radiative transfer
and varying opacities, showing that gas temperatures reduce by factors of 5–7
at densities ranging from $10^{-13}$ to $10^{-9}$ g cm-3 when the opacity is
reduced from 3 to 0.01 times the opacity of solar-metallicity gas (see Fig. 23
of Bate, 2014).

These opposing effects of metallicity (i.e. typically hotter low-density gas,
but cooler high-density gas) leads to a more complicated dependence of the
characteristic mass on redshift and metallicity than that postulated by Larson
(1998).

Very recently, Sharda & Krumholz (2022) have investigated analytically how the
characteristic stellar mass may vary with metallicity, decreasing from a
comparatively high value at low metallicity ($Z\mathrel{\hbox{\hbox
to0.0pt{\lower 2.36806pt\hbox{$\sim$}\hss} \kern-3.00003pt\raise
1.72218pt\hbox{$<$}}}10^{-4}~{}{\rm Z}_{\odot}$) to a much lower value once
dust dominates the cooling ($Z\mathrel{\hbox{\hbox to0.0pt{\lower
2.36806pt\hbox{$\sim$}\hss} \kern-3.00003pt\raise
1.72218pt\hbox{$>$}}}10^{-2}~{}{\rm Z}_{\odot}$). They argue that this may
help to explain the bottom-heavy IMFs apparently observed in massive
elliptical galaxies (see Section 4.3.1 below). However, they do not consider
at all how a change in redshift (i.e. the level of CMBR) may also alter the
characteristic mass and its metallicity dependence. The results from this
paper show that both redshift and metallicity need to be taken into account.

In another recent paper, Guszejnov et al. (2022) studied how environment,
initial conditions, and feedback may alter the stellar IMF. Among other
results, they find that the characteristic stellar mass varies with
metallicity and with the level of the ISRF. They find that with a typical
Galactic ISRF, lower metallicity tends to raise the gas temperature and this
in turn produces an increase characteristic stellar mass. This result is at
odds with the results of Bate (2019) who find that under present-day
conditions the characteristic stellar mass is quite independent of the
metallicity (ranging from 1/100 to 3 times the solar value). This may be due
to the limited resolution of the Guszejnov et al. (2022) study – as mentioned
above, Bate (2019) found that at low metallicity the effects of intermediate-
density gas being hotter were offset by enhanced cooling at high densities
that led to enhanced small-scale fragmentation. On the other hand, Guszejnov
et al. (2022) found that subjecting solar-metallicity clouds to a stronger
ISRF increased the characteristic stellar mass, which is in qualitative
agreement with the results we present here.

###  4.2 Extrapolation of the results to different redshifts

Fundamentally, the numerical results from this paper and Bate (2019) point to
an IMF that is relatively independent of metallicity (ranging from
$z=0.01-3~{}{\rm Z}_{\odot}$) for present-day star formation ($z=0$), to one
where the characteristic stellar mass increases with metallicity at high
redshift (assuming all other parameters remain the same). The main cause of
this different qualitative behaviour is that for present-day star formation
increasing the metallicity results in higher extinction of the ISRF and colder
temperatures deep within star-forming clumps. By contrast, at redshift $z=5$
the CMBR component of the ISRF that is able to penetrate into the densest
regions of molecular cloud cores is hotter. There is essentially no difference
in the range of temperatures of high-density gas ($n_{\rm
H}\mathrel{\hbox{\hbox to0.0pt{\lower 2.36806pt\hbox{$\sim$}\hss}
\kern-3.00003pt\raise 1.72218pt\hbox{$>$}}}10^{8}$ cm-3) at low metallicity
($Z=0.01~{}{\rm Z}_{\odot}$) between the $z=0$ and $z=5$ calculations. In the
$z=0$ calculations the vast majority of the gas is much warmer than the
temperature of the CMBR at $z=5$ ($T_{\rm CMBR}=(2.73~{}{\rm K})(1+z)=16$ K),
ranging between $T\approx 10-200$ K for densities $n_{\rm
H}\mathrel{\hbox{\hbox to0.0pt{\lower 2.36806pt\hbox{$\sim$}\hss}
\kern-3.00003pt\raise 1.72218pt\hbox{$>$}}}10^{8}-10^{11}$ cm-3. Therefore,
the hotter CMBR at $z=5$ has little effect and so the low-metallicity IMFs at
$z=0$ and $z=5$ are almost identical. However, at high metallicity (e.g.
$Z={\rm Z}_{\odot}$), for $z=0$ a substantial fraction of the high-density gas
has temperatures below 15 K and temperatures drop as low as $T\approx 6$ K.
Therefore, when otherwise identical calculations are carried out using the
hotter CMBR at $z=5$ high-density gas is kept substantially hotter and
fragmentation is reduced. This leads to fewer protostars being formed and
higher characteristic stellar masses because the protostars that do form are
able to accrete more mass.

Based on these results, we can interpolate to determine how the IMF is likely
to vary with metallicity and redshift between $z=0$ and $z=5$, and extrapolate
for $z>5$. Since the CMBR temperature scales as $T_{\rm CMBR}=(2.73~{}{\rm
K})(1+z)$ we expect the IMF to be essentially identical to the $z=0$ cases up
to $z=1$, since the CMBR temperature is lower than the minimum gas
temperatures found in the $z=0$ calculations ($T_{\min}\approx 5$ K even for
$Z=3~{}{\rm Z}_{\odot}$). There would probably also be no significant change
for $z=2$ either, because although a small amount of gas may otherwise be
colder than $T_{\rm CMBR}$ it would be a negligible amount. We base this on
the fact that the minimum gas temperature for high-density gas is raised by
about a factor of two between $z=0$ and $z=5$ for $Z=0.1~{}{\rm Z}_{\odot}$
but this results in a very small increase in the characteristic stellar mass
(Table 1). For $z=3$, we would expect a very slight increase in the
characteristic stellar mass (i.e. from $\approx 0.15$ to $\approx 0.20$ M⊙ in
terms of the mean of the logarithm of the stellar masses) for
$Z\mathrel{\hbox{\hbox to0.0pt{\lower 2.36806pt\hbox{$\sim$}\hss}
\kern-3.00003pt\raise 1.72218pt\hbox{$>$}}}3~{}{\rm Z}_{\odot}$, similar to
that which is seen between $Z=0.01~{}{\rm Z}_{\odot}$ and $Z=0.1~{}{\rm
Z}_{\odot}$ at $z=5$ because the minimum high-density gas temperature would be
raised from $T\approx 5$ K to $T\approx 11$ K. By redshift $z=4$ this
magnitude of shift in the characteristic stellar mass should be apparent for
solar metallicity, while for $Z=3~{}{\rm Z}_{\odot}$ the characteristic mass
is likely to have doubled to $\approx 0.3$ M⊙).

Beyond $z=5$ we would expect the characteristic stellar masses of the IMFs for
all metallicities $Z>0.01~{}{\rm Z}_{\odot}$ to increase with redshift. As
with the $z=5$ calculations, the greatest effect would initially be found in
the highest metallicity clouds. However, in the limit that $T_{\rm CMBR}$
becomes very large, say $\mathrel{\hbox{\hbox to0.0pt{\lower
2.36806pt\hbox{$\sim$}\hss} \kern-3.00003pt\raise 1.72218pt\hbox{$>$}}}50$ K
which is roughly the median temperature of the high density ($n_{\rm
H}\mathrel{\hbox{\hbox to0.0pt{\lower 2.36806pt\hbox{$\sim$}\hss}
\kern-3.00003pt\raise 1.72218pt\hbox{$>$}}}10^{8}-10^{11}$ cm-3) gas in all
the of the calculations (Fig. 6), then all the IMFs are likely become similar
(and all with characteristic stellar masses significantly larger than the
Galactic value). Since the Jeans mass scales $\propto T^{3/2}\rho^{-1/2}$ it
would be reasonable to expect that the characteristic mass (for clouds similar
to those modelled here) would increase $\propto(1+z)^{3/2}$ in the limit of
high redshift.

Finally, a few remarks of caution. First, all the calculations discussed here
are performed in the regime where dust cooling dominates the cooling of high-
density molecular gas. We have intentionally limited our studies to
metallicities $Z\geq 0.01~{}{\rm Z}_{\odot}$ for this reason. In the limit of
no metals, i.e., Population III stars, gas cooling is very different (e.g.
Bromm et al., 1999; Abel et al., 2000). To model the very-low-metallicity
regime (e.g. $Z=10^{-4}-10^{-2}~{}{\rm Z}_{\odot}$) accurately probably
requires both metal line cooling of the gas and dust cooling to be modelled
accurately (e.g. Sharda & Krumholz, 2022). Second, to date we have performed
such calculations using only a single type of initial condition – relatively
dense initial clouds ($n_{\rm H}\approx 6\times 10^{4}$ cm-3) with decaying
turbulence and an initial root-mean-square Mach number of ${\cal M}=11.2$ at
15 K. Jones & Bate (2018b) studied how the IMF may vary with cloud density for
present-day star formation and found that the characteristic stellar mass
varied relatively weakly with cloud density $\propto\rho^{-0.2}$. However,
other effects may come into play that we do not include, such as the effects
of feedback from protostellar outfows (e.g. Mathew & Federrath, 2021; Tanvir,
Krumholz & Federrath, 2022). All in all, there is still a great deal of work
to be done in mapping out how the IMF may vary over a wide range of initial
conditions.

###  4.3 Observational evidence for a varying IMF

####  4.3.1 Galactic stellar populations

Although there is little direct evidence for variation of the IMF (Bastian,
Covey & Meyer, 2010), there are a number of observations that provide indirect
evidence in favour of variable IMFs. Larson (1998) lists several points
including: no metal-free stars have ever been found (implying that few, if
any, of the first stars had masses $M\mathrel{\hbox{\hbox to0.0pt{\lower
2.36806pt\hbox{$\sim$}\hss} \kern-3.00003pt\raise
1.72218pt\hbox{$<$}}}0.8~{}{\rm M}_{\odot}$); the ‘G-dwarf problem’ in which
the numbers of metal-poor stars in the solar neighbourhood are deficient
compared to the predictions of simple chemical models (van den Bergh, 1962;
Schmidt, 1963); the large total mass of heavy elements in large clusters of
galaxies (Elbaz et al., 1995; Loewenstein & Mushotzky, 1996; Gibson, 1997;
Gibson & Matteucci, 1997); and the observed increases of the metallicities
(Faber 1973; Worthey, Faber & Gonzalez 1992; Vazdekis et al. 1996, 1997;
Trager et al. 2000; Gallazzi et al. 2006; Graves, Faber & Schiavon 2009a, b;
Kuntschner et al. 2010; McDermid et al. 2015) and mass-to-light ratios
(Guzman, Lucey & Bower 1993; Jorgensen, Franx & Kjaergaard 1996; Burstein et
al. 1997; Thomas et al. 2011; Cappellari et al. 2012, 2013; Dutton, Mendel &
Simard 2012;,Wegner et al. 2012; Dutton et al. 2013; Tortora, Romanowsky &
Napolitano 2013; McDermid et al. 2014; Davis & McDermid 2017) of the central
regions of early-type galaxies with increasing central velocity dispersion
(i.e. mass). Recently, gravitational lensing has also been used to measure the
total projected mass within the Einstein radius to obtain mass-to-light ratios
(Ferreras, Saha & Williams 2005, 2008; Auger et al. 2010; Ferreras et al.
2010; Treu et al. 2010; Barnabè et al. 2011; Spiniello et al. 2015; Newman et
al. 2017). Although the central regions of early-type galaxies are observed to
have metallicities and mass-to-light ratios that increase with the central
mass, observations of radial gradients that the outskirts of such galaxies are
generally composed of metal-poor stars (e.g. Greene et al., 2015).

Historically, these observations and, in particular the high values of
[$\alpha$/Fe] observed in massive early-type galaxies, have been used argue
for a top-heavy IMF (e.g. Larson, 1998). Gunawardhana et al. (2011) and
Weidner et al. (2013) have proposed that the IMF becomes more top-heavy with
increasing star-formation rate. However, Larson (1998) argued instead that
many of these observations can potentially be explained by IMFs that have the
same (Salpeter-type) slope for high-mass stars, but a varying (characteristic)
mass below which the IMF flattens or turns over. For example, mass-to-light
ratios of old stellar populations (e.g. those in elliptical galaxies) depend
sensitively on the behaviour of the IMF in the vicinity of a solar-mass, since
stars above $\approx 0.8$ M⊙ have evolved into dark stellar remnants, while
the light is generated by stars below this mass. In general, mass-to-light
ratios can be increased either by an excess of very low-mass stars (i.e.
bottom-heavy), or an excess of stellar remnants (i.e. top heavy).

In principle, spectral analysis of stellar populations can be used to
constrain directly the fraction of low-mass stars in early-type galaxies
(Spinrad 1962; Cohen 1978; Faber & French 1980; Carter, Visvanathan & Pickles
1986; Hardy & Couture 1988; Delisle & Hardy 1992; Cenarro et al. 2003; Falcón-
Barroso et al. 2003; van Dokkum & Conroy 2010, 2011; Conroy & van Dokkum
2012a, b; Smith, Lucey & Carter 2012; Spiniello et al. 2012, 2014; Ferreras et
al. 2013; Ferreras et al. 2015; Ferreras, La Barbera & Vazdekis 2015; La
Barbera et al. 2013, 2017; Martín-Navarro et al. 2015; van Dokkum et al.
2017). Recent spectra analysis of the centres of early-type galaxies show a
substantial population of low-mass stars (e.g. van Dokkum & Conroy, 2010) and
point to a bottom-heavy stellar population, although not in all early-type
galaxies (Smith & Lucey 2013; Smith, Lucey & Conroy 2015; Leier et al. 2016).
Rosani et al. (2018) find that need for a bottom-heavy IMF does not depend on
the environment of the early-type galaxy (i.e. galaxy hierarchy or host halo
mass). To produce both the high metallicity and enhanced [$\alpha$/Fe] of
early-type galaxies and a bottom-heavy IMF, a time-dependent IMF may be
required that changes from top- to bottom-heavy during the early formation
process (Vazdekis et al., 1997; Weidner et al., 2013; Ferreras et al., 2015).

Recently, Guszejnov, Hopkins & Graus (2019) examined whether the near
universal Milky Way IMF could be reconciled with proposed extragalactic IMF
variations. They found that it was very difficult to match the proposed
extragalactic IMF variations without simultaneously predicting too much
variation of the IMF in extreme Galactic environments.

Using the calculations presented in this paper, we cannot comment on
variations in the slope of the IMF at the high-mass end since the calculations
are only large enough to produce low- and intermediate-mass stars. However, we
can use them to predict the dependence of the characteristic stellar mass on
both metallicity and redshift (Section 4.2). Unfortunately, it is not
immediately clear how the predictions of this paper can be used to explain the
above observations. A characteristic stellar mass that increases for
decreasing redshift and increasing metallicity would fit well with the picture
presented by Larson (1998) of a deficit of low-mass stars being able to
explain many observations. However, it does not appear to fit well with the
recent spectral analysis results that point to an abundance of low-mass stars
and a bottom-heavy IMF. It may be that several different processes are
contributing to the unusual stellar populations in the centres of early-type
galaxies (e.g. high levels of turbulence or intense radiation fields from
starbursts), rather than only changes due to the varying CMBR. For example,
the recent calculations of Tanvir et al. (2022) show that at very high
molecular cloud densities, feedback from protostellar outflows may result in
an IMF with a lower characteristic stellar mass (and a narrower overall mass
range) than the Galactic IMF. Alternately, the results obtained in this paper
do provide a mechanism for metal-rich gas to produce time-dependent IMFs that
change from top-heavy to a standard IMF as redshift decreases (similar to that
postulated by Vazdekis et al., 1997; Weidner et al., 2013; Ferreras et al.,
2015).

####  4.3.2 Globular clusters

Globular clusters are much more simple objects than the central regions of
galaxies. They (largely) are thought to have formed from single star formation
events, and they are not thought to contain significant dark matter. When
interpreting their stellar mass functions care does needs to be taken with the
effects of dynamical evolution – the lowest mass stars are lost
preferentially. But if this can be taken into account, they can in principle
be used to study the form of the low-mass ($M\mathrel{\hbox{\hbox
to0.0pt{\lower 2.36806pt\hbox{$\sim$}\hss} \kern-3.00003pt\raise
1.72218pt\hbox{$<$}}}0.8$ M⊙) IMF at high redshift.

In particular, a study of old globular clusters in M31 has found that more
metal-rich globular clusters become increasingly bottom-light (i.e. have a
deficit of low-mass stars). Strader, Caldwell & Seth (2011) studied more than
a hundred globular clusters in M31, the vast majority of which were very old
(i.e. formed at high redshift). They divided their sample into several
metallicity bins. Their most metal-rich sample had metallicities
[Fe/H]$~{}>-0.4$. Crucially, Strader et al. (2011) were able to detect that
those clusters that were thought to be more dynamically-evolved were more
bottom-light as expected due to dynamical evolution. However, due to their
large sample, they were also able to show that even accounting for the effects
of dynamical evolution, globular clusters that were more metal-rich were
increasingly bottom-light. There are questions over this result, however.
Shanahan & Gieles (2015) claim that the masses of the M31 globular clusters
were systematically underestimated and that the bias is more important at high
metallicities.

Nevertheless, four of the M31 metal-rich globular clusters were also studied
using the spectral analysis method of Conroy & van Dokkum (2012a) by Conroy &
van Dokkum (2012b) to determine the distributions of low-mass stars. In
agreement with Strader et al. (2011) they found that they had a significant
deficit of low-mass stars relative to a Galactic Kroupa IMF.

This observed trend of the stellar populations of more metal-rich globular
clusters being more bottom-light is the opposite of the trends seen in
spectral studies of the centres of early-type galaxies. However, it is exactly
the sort of trend that is expected from the numerical results presented in
this paper. Assuming that the old globular clusters were formed at redshifts
of $z\mathrel{\hbox{\hbox to0.0pt{\lower 2.36806pt\hbox{$\sim$}\hss}
\kern-3.00003pt\raise 1.72218pt\hbox{$>$}}}5$, the results presented here can
be used to predict that the characteristic stellar masses of old globular
clusters should indeed have been higher for greater metallicity. Similarly,
the most metal-rich old globular clusters should have been born with deficits
of low-mass stars compared to the present-day Galactic IMF.

##  5 Conclusions

We have presented results from three radiation hydrodynamical simulations of
star cluster formation that are subjected to the cosmic microwave background
radiation at redshift $z=5$. The three calculations have identical initial
conditions except for their metallicity which ranges from 1/100 to 1 times the
solar value. The calculations resolve the opacity limit for fragmentation,
protoplanetary discs (radii $\mathrel{\hbox{\hbox to0.0pt{\lower
2.36806pt\hbox{$\sim$}\hss} \kern-3.00003pt\raise 1.72218pt\hbox{$>$}}}1$ au),
and multiple stellar systems. The calculations treat gas and dust temperatures
separately and include a thermochemical model of the diffuse ISM that is
important for setting the temperatures at low densities. We compare the
results presented in this paper with the results of present-day ($z=0$) star
cluster formation calculations already published by Bate (2019).

We draw the following conclusions:

  1. 1.

At redshift $z=5$ we find that the stellar mass functions produced by the
calculations are become increasingly bottom-light with increasing metallicity.
In other words, the characteristic (median) stellar mass increases with
increasing metallicity. This behaviour is qualitatively different to that
found for present-day ($z=0$) star formation in which the stellar mass
function is found to be independent of metallicity (in the range
$Z=0.01-3~{}{\rm Z}_{\odot}$). At redshift $z=5$, the distribution of stellar
masses at low metallicity ($Z=0.01~{}{\rm Z}_{\odot}$) is similar to the
parametrisation of the observed Galactic IMF given by Chabrier (2005). But at
$z=5$ with solar metallicity the characteristic stellar mass is approximately
3 times higher.

  2. 2.

Stellar multiplicity strongly increases with primary mass, similar to observed
Galactic stellar systems. But whereas at $z=0$ the multiplicity for a given
primary mass is found to be similar for all metallicities ($Z=0.01-3~{}{\rm
Z}_{\odot}$), at $z=5$ the multiplicity of M-dwarfs is found to decrease with
increasing metallicity, such that their multiplicity is a factor of two lower
at $Z={\rm Z}_{\odot}$ compared to $Z=0.01~{}{\rm Z}_{\odot}$.

  3. 3.

At both $z=0$ and $z=5$ there is an anti-correlation between metallicity and
the frequencies of close (semi-major axes $a<10$ au) bound protostellar pairs
with primary masses $M_{1}=0.1-1.5~{}{\rm M}_{\odot}$. However, the trend is
not as strong at $z=5$ as at $z=0$, and the frequencies of close pairs are
lower at $z=5$ than at $z=0$.

  4. 4.

We also find that close (semi-major axes $a<10$ au) bound protostellar pairs
have greater fractions of unequal-mass systems with lower metallicity at both
$z=0$ and $z=5$.

  5. 5.

The above differences between star formation at $z=5$ and $z=0$ are a result
of metal-rich gas being unable to cool to as lower temperatures at $z=5$ as at
$z=0$ due to the hotter cosmic microwave background radiation. This inhibits
fragmentation at high densities and high metallicity, meaning that the
protostars that do form tend to accrete more gas (i.e. they have a greater
characteristic stellar mass) and they have lower companion frequencies.

  6. 6.

Comparing our results to observational evidence for a varying IMF, we find
that the observed trend of old globular clusters in M31 being more bottom-
light with increasing metallicity (Strader et al., 2011; Conroy & van Dokkum,
2012b) is consistent with the dependencies of the IMF on redshift and
metallicity that we find from the numerical simulations. The results may also
help to explain some other observations that point to the need for varying
IMFs, although they do not provide an explanation for bottom-heavy IMFs that
are apparently observed in the centres of many massive elliptical galaxies.

The calculations discussed in this paper begin with idealised initial
conditions (i.e. uniform-density, spherical clouds initialised with a
particular ‘turbulent’ velocity field). This is done to allow careful
comparison between calculations to determine the effects that varying the CMBR
and/or metallicity have. More realistic initial conditions (e.g. self-
consistent turbulence and different cloud structures, such as turbulence
driven by external supernovae; Seifried et al. 2018) or different initial
clouds (e.g. mean cloud densities; Jones & Bate 2018b; Tanvir et al. 2022) may
lead to some what different stellar properties. Similarly, these calculations
do not include magnetic fields or protostellar outflows. Nevertheless, we
expect the general trends for how stellar properties depend on redshift and
metallicity that have been identified in this paper to be similar even if more
complex initial conditions were employed or additional physical processes were
to be included.

## Acknowledgements

MRB thanks Charlie Conroy for reading and commenting on a draft version of the
manuscript.

This work was supported by the European Research Council under the European
Commission’s Seventh Framework Programme (FP7/2007-2013 Grant Agreement No.
339248). The calculations discussed in this paper were performed on the
University of Exeter Supercomputer, Isca, and on the DiRAC Complexity system,
operated by the University of Leicester IT Services, which forms part of the
STFC DiRAC HPC Facility (www.dirac.ac.uk). The latter equipment is funded by
BIS National E-Infrastructure capital grant ST/K000373/1 and STFC DiRAC
Operations grant ST/K0003259/1. DiRAC is part of the National
E-Infrastructure. This research was supported in part by the National Science
Foundation under Grant No. NSF PHY-1748958. Some of the figures were produced
using SPLASH (Price, 2007), an SPH visualization tool publicly available at
http://users.monash.edu.au/$\sim$dprice/splash.

## Data availability

Data that can be used to produce Table 2 and Figs. 8 to 15, and to calculate
many of the values in Table 1 are provided as Additional Supporting
Information (see below). Input and output files from the three calculations
that were performed for this paper are available from the University of
Exeter’s Open Research Exeter (ORE) repository (Bate, 2022). This dataset
includes the initial conditions and input files for the three SPH
calculations, the SPH dump files that were used to create Figs. 3–7, and the
output files that were used to produce Fig. 2 and Tables 1, 3, and 4.

## Supporting Information

Additional Supporting Information may be found in the online version of this
article:

Data files for protostars. We provide text files of Tables 3 and 4 that give
the properties of the protostars and multiple systems for each of the three
calculations. These files contain the data necessary to construct Figs. 8 to
15, and to produce Table 2. Their file names are of the format
Table3_Stars_z5_MetalX.txt and Table4_Multiples_z5_MetalX.txt where ‘X’ gives
the metallicity (‘001’ for 0.01, ‘01’ for 0.1, or ‘1’).

Animations. We provide animations of the evolution of the column density, and
the gas, dust, and radiation temperatures for each of the three calculations
(i.e. 12 animations). Sink particles are represented by white circles. These
animations are provided to support the snapshots provided in Figs. 3 to 5.
Their file names are of the format Bate2022_z5_MetalX_Y.txt where ‘X’ gives
the metallicity and ‘Y’ gives the variable (‘Density’, ‘Tgas’, or ‘Tdust’ or
‘Trad’).

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