# On bursty star formation during cosmological reionization - how does it
influence the baryon mass content of dark matter halos?

Anand Menon  International Centre for Radio Astronomy Research, The University
of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009,
Australia  Chris Power  International Centre for Radio Astronomy Research, The
University of Western Australia, 35 Stirling Highway, Crawley, Western
Australia 6009, Australia  [

###### Abstract

The baryon mass content (i.e. stellar and gas mass) of dark matter halos in
the early Universe depends on both global factors - e.g. ionising ultraviolet
(UV) radiation background - and local factors - e.g. star formation efficiency
and assembly history. We use a lightweight semi-analytical model to
investigate how both local and global factors impact the halo baryon mass
content at redshifts of $z\geq 5$. Our model incorporates a time delay between
when stars form and when they produce feedback of $0\leq t^{d}/\text{Myr}\leq
30$, which can drive bursts of star formation, and a mass and redshift
dependent UV background, which captures the influence of cosmological
reionization on gas accretion onto halos. We use statistically representative
halo assembly histories and assume that the cosmological gas accretion rate is
proportional to the halo mass accretion rate. Delayed ($t^{d}$¿0) feedback
leads to oscillations in gas mass with cosmic time, behaviour that cannot be
captured with instantaneous feedback ($t^{d}$=0). Highly efficient star
formation drives stronger oscillations, while strong feedback impacts when
oscillations occur; in contrast, inefficient star formation and weak feedback
produce similar long-term behaviour to that observed in instantaneous feedback
models. If the delayed feedback timescale is too long, a halo retains its gas
reservoir but the feedback suppresses star formation. Our model predicts that
lower mass systems (halo masses $m_{\text{h}}\leq 10^{7}\text{M}_{\odot}$) at
$z\leq 10$ should be strongly gas deficient ($m_{\text{g}}\rightarrow 0$),
whereas higher mass systems retain their gas reservoirs because they are
sufficiently massive to continue accreting gas through cosmological
reionization. Interestingly, in higher mass halos, the median
$m_{\star}/(m_{\star}+m_{\text{g}})\simeq 0.01-0.05$, but is a factor of 3-5
smaller when feedback is delayed. Our model does not include seed supermassive
black hole feedback, which is necessary to explain massive quenched galaxies
in the early Universe.

###### keywords:

galaxies: formation, galaxies: high-redshift, cosmology: theory, cosmology:
dark ages, reionization, first stars, methods: numerical

ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D)
Anand<EMAIL_ADDRESS>
##  1 Introduction

Recent observations with the JWST have revealed a potential overabundance of
massive galaxies ($M_{\star}\sim 10^{10.5-11}{\rm M}_{\odot}$) in the high
redshift Universe ($z\gtrsim 7$) when compared to the predictions of
theoretical models of galaxy formation (Finkelstein2023, 15, 24, e.g.). This
has prompted questions about the validity of the standard cosmological model
and the early growth of dark matter halos (e.g. boylan-kolchin2023 (5), but
see KraghJespersen2024 (22)), as well as questions about our understanding of
the interplay between gas reservoirs, star formation, and feedback at early
times (Dekel2023, 8, 4, e.g.). Whether or not there is an actual overabundance
of massive galaxies when compared to theoretical model predictions
(Finkelstein2023_2, 16, see discussion in, e.g., ), the questions these
observations have prompted highlight the fundamental time limit imposed by the
age of the Universe at early cosmic times when assessing galaxy properties,
given the rate at which the highest mass dark matter halos can grow and
consequently the stellar masses of the galaxies that they contain.

A similar time limit exists at lower halo masses. It has been understood for
some time that the presence of photo-ionizing backgrounds should inhibit the
formation of galaxies in low mass dark matter halos (Efstathiou1992, 10, 47,
e.g.). For example, the presence of an ultraviolet (UV) background will photo-
ionize warm diffuse gas in the inter-galactic medium (IGM), preventing its
accretion onto low mass dark matter halos while also preventing gas already
present in halos from cooling(Efstathiou1992, 10, e.g.). Such a UV background
emerged during cosmological reionization driven by high mass stars
(Barkana2001, 1, 50, e.g.), leading to the suppression of galaxy formation in
lower mass halos (Gnedin2000, 19, 51, 33, 23, e.g.). Therefore, for lower mass
halos to host galaxies, we might argue that there is a time limit imposed by
the requirement that they form their stars prior to cosmological reionization,
at $z\gtrsim 6$.

In this paper, we investigate the consequences of the time limit imposed by
cosmological reionization for the baryon content - the stellar mass,
$m_{\ast}$, and gas mass, $m_{\text{g}}$ \- of dark matter halos at high
redshifts. In particular, we examine the interplay between the UV background-
driven suppression of gas accretion onto lower dark matter halos, stellar
feedback driven by star formation in these halos, and their baryon mass
content - $m_{\ast}$ and $m_{\text{g}}$. Galaxies embedded within low-mass
halos will experience episodes of supernova-driven feedback that deplete their
reservoir of star forming gas via powerful winds (Dekel1986, 7, 11, e.g.), and
prevent further stellar mass assembly and gas accretion. It’s interesting to
note that the main sequence lifetimes of the high mass stars ($M\geq 8{\rm
M}_{\odot}$) that result in supernovae are relatively short - of order
$10^{7}$ yrs - which, at $z\gtrsim 6$, is an appreciable fraction -
approximately 10% - of a dark matter halo’s dynamical time. This means that a
common assumption of galaxy formation models - that the time between star
formation and supernova feedback is short and be treated as instantaneous -
should break down in the high redshift Universe. Instead, feedback is delayed,
leading to episodic or bursty star formation (Scalo1986, 40, 14, 35, 17, 36,
41, 46, e.g.).

FaucherGiguere2018 (14) modelled such bursty star formation and showed that it
can arise when the effective equilibrium between the self-gravity of gas in a
galaxy and stellar feedback following star formation cannot be maintained.
This should be commonplace in galaxies at high redshifts and in lower mass
galaxies across cosmic time. Indeed, there is observational evidence for
bursty star formation in galaxies at high redshifts (Faisst2019, 13, 28, 44,
e.g.). Faisst2019 (13) concluded from a statistical sample of $z\\!\sim$4.5
galaxies that the significant scatter in UV and H$\alpha$ luminosities and
star formation rates implied episodes of bursty star formation during the
prior $50$ Myr. Drawing on JWST/NIRSpec data, Strait2023 (44) found star
formation is in the process of being shut down in a $z=5.2$ galaxy, while
Looser2023 (28) found star formation has been quenched in a $z=7.3$ galaxy;
both are consistent with galaxy formation model predictions that incorporate
bursty star formation Dome2023 (9). Similarly, there is observational evidence
for bursty star formation in dwarf galaxies (Weisz2012, 48, 12, e.g.), based
on the distributions of UV and Hα that probe their star formation histories.
Such bursty star formation arises naturally in hydrodynamical galaxy formation
simulations that can track resolved star formation and feedback (Hopkins2023,
20, e.g.), especially in lower mass galaxies (Onorbe2015, 34, 30, 42, e.g.)
and during the epoch probed by JWST (Pallottini2023A&A, 36, 41, 46, e.g.).

Using a lightweight semi-analytical model that we have written, we examine how
the combination of UV suppression of gas accretion from cosmological scales
and bursty star formation driven by delayed feedback influence the baryon
content of dark matter halos at high redshifts. We model the assembly
histories of these halos using Monte Carlo merger trees that have been
calibrated against cosmological $N$-body simulations (Parkinson2008, 37, cf.)
and therefore capture plausible variations in their growth over cosmic time.
We assume that cosmological gas accretion tracks halo growth, modulated by the
presence of the UV background, following the approach of Kravtsov2022 (23). We
parameterise star formation and feedback efficiencies, allowing for delayed
feedback following the approach of Furlanetto2022 (17). In this way, we can
assess how local factors (star formation efficiency, onset and strength of
stellar feedback, variation in dark matter halo assembly history) and global
factors (onset of a UV ionizing background and the suppression of cosmological
gas accretion) influence the baryon content of halos at $z\geq 5$.

In the following sections, we describe the main features of our model (§ 2);
we present our results for the baryon content of halos as a function of cosmic
time and halo mass (§ 3.1), showing how they are sensitive to the assumptions
of our model (instantaneous versus delayed feedback, absence or presence of UV
background), model parameters (e.g. star formation efficiency, delayed
feedback time interval), and variations in dark matter halo assembly history;
and we summarise our key findings and their implications for our understanding
of galaxy formation and evolution in the low-mass halo regime (§ 4). We use
the cosmological parameter values $\Omega_{\text{b}}=0.0484$,
$\Omega_{\text{m}}=0.308$, $\Omega_{\Lambda}=0.692$, $h=0.678$,
$\sigma_{8}=0.815$, and $n_{\text{s}}=0.968$, which are consistent with the
results obtained by the Planck2018 (38).

##  2 Theoretical Model

We have written a lightweight semi-analytical model111This python code can be
made available by contacting the corresponding author. for studying the growth
of gas and stellar mass in dark matter halos, which is motivated by the
philosophy of equilibrium models set out in Dave2012 (6). We track the mass
assembly of dark matter halos and the corresponding cosmological accretion of
gas from the IGM; the accumulation of a cold gas reservoir at the centre of
the halo; star formation; and the expulsion of gas via stellar feedback
(supernovae-driven winds). In particular, we follow Furlanetto2022 (17) and
introduce a delay between when stars form and when they produce feedback, and
we adopt a time-dependent suppression of cosmological gas accretion that
reflects the growth of a UV ionizing background, following Kravtsov2022 (23).

Note that we do not account for the growth of the seeds of super-massive black
holes and the feedback they produce. We expect this to be important in more
massive galaxies - star formation will be preferred at the expense of black
hole growth in lower mass galaxies because black hole growth is limited to
occur on a Salpeter timescale (Nayakshin2009MNRAS, 32, 3, see, e.g.). This
difference in timescales on which stellar and black hole driven feedback acts
(Power2011, 39, e.g.) should give rise to more complex bursty star formation
histories in more massive galaxies; we will investigate this in a future
paper.

###  2.1 Evolutionary Equations

We track the time rate of change of gas ($\dot{m}_{\text{g}}$), star formation
($\dot{m}_{\star}$), and supernova-driven wind mass loss ($\dot{m}_{w}$) at
time $t$ via a set of coupled differential equations:

$\displaystyle\dot{m}_{\text{g}}(t)$
$\displaystyle=\dot{m}_{\text{c,g}}(t)-\dot{m}_{\star}(t)-\dot{m}_{\text{w}}(t),$
(1) $\displaystyle\dot{m}_{\star}(t)$
$\displaystyle=\epsilon_{\text{sf}}\frac{m_{\text{g}}(t)}{\tau_{\text{sf}}},$
(2) $\displaystyle\dot{m}_{\text{w}}(t)$
$\displaystyle=\eta_{\text{fb}}\dot{m}_{\star}(t-t^{\text{d}}).$ (3)

It is straightforward to convert between $t$ and redshift $z$ for our adopted
cosmology, which is important for evaluating the contribution of the UV
background. The various parameters are now described in the following
subsections.

####  2.1.1 Growth of the Gas Reservoir

Equation 1 tracks the gas reservoir in a galaxy, accounting for the accretion
rate of gas from cosmological scales ($\dot{m}_{\text{c,g}}$), gas that is
converted to stars via $\dot{m}_{\star}$, and gas that is lost via winds
($\dot{m}_{w}$). As is commonly assumed (White1991, 49, e.g.),
$\dot{m}_{\text{c,g}}$ tracks the accretion rate of the dark matter halo that
hosts the galaxy, $\dot{m}_{\text{h}}$;

$\dot{m}_{\text{c,g}}=\epsilon_{\text{in}}\quantity(\frac{\Omega_{\text{b}}}{\Omega_{\text{m}}})\dot{m}_{\text{h}}.$
(4)

Here the quantities $\Omega_{\text{b}}$ and $\Omega_{\text{m}}$ refer to the
total baryon and matter densities of the Universe, respectively, and the
assumption is that the halo accretes a baryon mass equivalent to the cosmic
fraction ($\Omega_{\text{b}}/\Omega_{\text{m}}$) per unit dark matter
accreted. We also use a pre-factor $0\leq\epsilon_{\text{in}}\leq 1$, such
that $\epsilon_{\text{in}}=1$ corresponds to the accretion of the cosmic
baryon fraction and $\epsilon_{\text{in}}=0$ corresponds to complete
suppression of baryon accretion. We discuss this further in the context of UV
suppression of cosmological accretion below.

####  2.1.2 Growth of Stellar Mass

Equation 2,

$\dot{m}_{\star}=\epsilon_{\text{sf}}\frac{m_{g}}{\tau_{\text{sf}}},$

links the star formation rate to the mass of the gas reservoir,
$m_{\text{g}}$, a star formation timescale, $\tau_{\text{sf}}$, and an
efficiency factor, $\epsilon_{\text{sf}}$. We assume that $\tau_{\text{sf}}$
is proportional to the dynamical time,

$\tau_{\text{sf}}\propto t_{\text{dyn}}=\frac{R}{\sigma}$ (5)

where $R$ and $\sigma$ are the characteristic radius and velocity dispersion
of the system respectively. If we adopt the dynamical time of the halo as
characterising this timescale, we expect that

$t_{\text{dyn}}=\left(\frac{4}{\Delta_{\text{vir}}}\right)^{1/2}\frac{1}{H(t)}\simeq\frac{0.15}{H(t)}$
(6)

where $\Delta_{\text{vir}}\simeq 178$ is the virial overdensity of the halo
and

$H(t)=H_{\text{0}}\sqrt{\Omega_{\text{m}}(1+z(t))^{3}+\Omega_{\Lambda}}.$ (7)

is the Hubble parameter at cosmic time, $t\equiv~{}t(z)$. We assume that

$\tau_{\text{sf}}=0.15\frac{f_{\text{sf}}}{H(z)}$ (8)

and expect that $f_{\text{sf}}\gtrsim 1$. We take $\epsilon_{\text{sf}}=0.015$
as our fiducial value, following Furlanetto2022 (17) and motivated by
Murray2011 (31) and Leroy2017 (27), but we allow this parameter to vary
between $\epsilon_{\text{sf}}=0.0015$ (low efficiency) and
$\epsilon_{\text{sf}}=0.1$ (high efficiency) to gauge its influence on our
results.

####  2.1.3 Supernovae-Driven Wind Mass Loss

Equation 3,

$\dot{m}_{\text{w}}=\eta_{\textbf{fb}}\dot{m}_{\star}(t-t^{\text{d}}),$

links the rate of mass loss driven by supernovae to $\dot{m}_{\star}$ at some
time $t-t^{\text{d}}$, where $t^{\text{d}}$ is the delayed feedback timescale.
If $t^{\text{d}}=0$, then this mass loss rate is driven by the instantaneous
star formation rate; however, if $t^{\text{d}}>0$, then it depends on star
formation at some earlier time and corresponds to delayed feedback. We adopt
$t^{\text{d}}$=0.015 Gyr (15 Myr) as our fiducial value, but have explored
values of $0\leq t^{\text{d}}\leq 0.03$ Gyr.

We follow Furlanetto2017 (18) in defining our choice of feedback efficiency
parameter; they model momentum-regulated supernova feedback and assume that
supernovae accelerate the wind to the escape velocity of the halo. This gives,

$\eta_{\textbf{fb}}=\epsilon_{\text{fb}}\pi_{\text{p}}\quantity(\frac{10^{11.5}M_{\odot}}{m_{\text{h}}})^{1/3}\quantity(\frac{9}{1+z})^{1/2},$
(9)

where $\epsilon_{\text{fb}}$ parameterises feedback efficiency and corresponds
to the momentum injected by the supernovae driving the wind, and
$\pi_{\text{p}}$ is the total amount of momentum from each supernova. We
follow Furlanetto2022 (17) in adopting $\epsilon_{\text{fb}}=5$ as our
fiducial value, but we also consider values in the range
$2\leq\epsilon_{\text{fb}}\leq 7$. Following Furlanetto2017 (18), we assume
$\pi_{\text{p}}=2\times 10^{33}$ g cm $\text{s}^{-2}$.

####  2.1.4 UV Suppression of Cosmological Gas Accretion

We follow the approach of Kravtsov2022 (23) by noting that the gas mass within
a dark matter halo may be a function of both mass and redshift (Okamoto2008,
33, e.g.),

$m_{\text{g}}(z)=f_{\text{b}}(m_{\text{h}},z)m_{\text{h}}(z).$ (10)

The quantity $f_{\text{b}}(m_{\text{h}},z)$ is related to
$\epsilon_{\text{in}}$ in Equation 1, as we make clear below. Following
Gnedin2000 (19, 33), we write,

$f_{\text{b}}(m_{\text{h}},z)=\frac{\Omega_{\text{b}}}{\Omega_{\text{m}}}s(\mu_{\text{c}},\omega),$
(11)

where $\Omega_{\text{b}}/\Omega_{\text{m}}$ is the cosmic baryon fraction and
$s(x,y)$ is defined as,

$s(x,y)=\quantity[1+(2^{\frac{y}{3}}-1)x^{-y}]^{-\frac{3}{y}};$ (12)

$\mu_{\text{c}}=m_{\text{h}}/M_{\text{c}}(z)$, where $M_{\text{c}}(z)$ is a
characteristic mass below which $s(\mu_{\text{c}},\omega)\rightarrow 0$.
Following Kravtsov2022 (23), we adopt

$M_{\text{c}}=1.69\times
10^{10}\frac{\text{exp}(-0.63z)}{1+\text{exp}([z/\beta]^{\gamma})}M_{\odot},$
(13)

where $\gamma=15$ and $\beta$ is given by,

$\beta=z_{\text{rei}}\quantity\Bigg[\text{ln}\quantity\Big(1.82\crossproduct
10^{3}\text{exp}(-0.63z_{\text{rei}})-1)]^{-1/\gamma};$ (14)

here $z_{\text{rei}}$ is the redshift of reionization. We adopt as our
fiducial value $z_{\text{rei}}=7$ but have checked the sensitivity of our
results to this choice.

If we consider the time rate of change of Equations 10 and 11, we see that

$\displaystyle\dot{m}_{\text{g}}(t)=\dot{f}_{\text{b}}m_{\text{h}}+f_{\text{b}}\dot{m}_{\text{h}}$
$\displaystyle=\left(f_{\text{b}}+\dot{f}_{\text{b}}\frac{m_{\text{h}}}{\dot{m}_{\text{h}}}\right)\dot{m}_{\text{h}}$
$\displaystyle=\left(s+\dot{s}\frac{m_{\text{h}}}{\dot{m}_{\text{h}}}\right)\left(\frac{\Omega_{\text{b}}}{\Omega_{\text{m}}}\right)\dot{m}_{\text{h}},$

which is equivalent to Equation 4, and so we can evaluate
$\epsilon_{\text{in}}$, given the form of Equation 12.222We note that
Kravtsov2022 (23) explored the general case in which gas accretion is
suppressed by the presence of a UV background ($\epsilon_{\text{UV}}$), the
presence of a hot halo ($\epsilon_{\text{hot}}$), and preventive feedback
($\epsilon_{\text{prev}}$), such that
$\epsilon_{\text{in}}=\epsilon_{\text{UV}}\epsilon_{\text{hot}}\epsilon_{\text{prev}}$.
We argue that suppression by hot halos and by preventive feedback can be
neglected at high redshifts (i.e.
$\epsilon_{\text{hot}}\epsilon_{\text{prev}}=1$, and so we consider
$\epsilon_{\text{in}}=\epsilon_{\text{UV}}$. We write,

$\displaystyle\epsilon_{\text{in}}=$
$\displaystyle\text{max}\bigg{(}0,s\quantity(\mu_{\text{c}},\omega)\bigg{[}(1+X)-$
$\displaystyle
2\epsilon(z,\gamma)\frac{M_{\text{h}}}{\dot{M}_{\text{h}}}X(1+z)H(z)\bigg{]}\bigg{)},$
(15)

where

$\displaystyle\epsilon(z,\gamma)$
$\displaystyle=\frac{0.63}{1+e^{(z/\beta)^{\gamma}}}+\frac{\gamma
z^{\gamma-1}}{\beta^{\gamma}}\frac{e^{(z/\beta)^{\gamma}}}{(1+e^{(z/\beta)^{\gamma}})^{2}},$
$\displaystyle X$
$\displaystyle=\frac{3c_{\omega}M_{\omega}}{1+c_{\omega}M_{\omega}},\quad
c_{\omega}=2^{\omega/3}-1,$ $\displaystyle M_{\omega}$
$\displaystyle=\quantity(\frac{M_{\text{c}}(z)}{M_{\text{h}}})^{\omega},\quad\omega=2.$

This has the effect of suppressing gas accretion at halo masses below the
threshold mass $M_{\text{c}}(z)$ at $z<z_{\text{rei}}$, with $M_{\text{c}}(z)$
increasing with decreasing $z$ (i.e. accretion is suppressed onto
progressively higher mass halos). For redshifts $z>z_{\text{rei}}$, there is
no suppression of gas accretion. We refer the interested reader to Figure 1 of
Kravtsov2022 (23) for an illustration of this behaviour.

###  2.2 Modelling Dark Matter Halo Growth

We have used the algorithm of Parkinson2008 (37) to generate Monte Carlo dark
matter halo merger trees based on Extended Press-Schechter theory (Bond1991,
2, 25, e.g.). This has been calibrated to provide predicted halo masses as a
function of cosmic time that are consistent with merger trees derived from the
Millennium Simulation (Springel2005, 43, cf.). For each of the 11 halo mass
bins equally spaced (linearly) in the interval $10^{6}\text{M}_{\odot}\leq
m_{\text{h}}\leq 10^{11}\text{M}_{\odot}$, we sample 100 realisations
following mass assembly histories in the redshift range $5\leq z\leq 25$
equally spaced in the logarithm of the expansion factor, $a=1/(1+z)$. This
provides us with $m_{\text{h}}$ as a function of $z$; we calculate
$\dot{m}_{\text{h}}$ as a function of $t(z)$ by constructing a smooth spline
interpolant, which we differentiate numerically.

Figure 1: Mass assembly history of a sample of 100 dark matter halos with a
halo mass of $m_{\text{h}}=10^{8}\text{M}_{\odot}$ at $z$=5 generated using
the Parkinson2008 (37) Monte Carlo merger tree algorithm. The upper and lower
panels show $m_{\text{h}}$ (in $\text{M}_{\odot}$) and $\dot{m}_{\text{h}}$
(in $\text{M}_{\odot}/{\text{Gyr}}$) against cosmic time (in Gyrs; lower
horizontal axis) and redshift (upper horizontal axis); red solid and dotted
curves indicate the median, 10${}^{\text{th}}$, and 90${}^{\text{th}}$
percentiles of the distributions at a given time.

In Figure 1, we show the mass assembly histories of a sample of dark matter
halos that have masses of $m_{\text{h}}=10^{8}\text{M}_{\odot}$ at $z=5$. Each
individual halo is represented by a light grey curve; the median,
10${}^{\text{th}}$, and 90${}^{\text{th}}$ percentiles are indicated by the
red solid and dotted curves. The upper panel shows the variation of
$m_{\text{h}}$ with cosmic time (lower horizontal axis) and redshift (upper
horizontal axis), starting at $z$=25, while the lower panel shows the
corresponding values of $\dot{m}_{\text{h}}$. This shows that our Monte Carlo
merger trees produce a diversity of assembly histories and that halos that
have the same value of $m_{\text{h}}$ at $z$=5 can differ by up to an order of
magnitude in $m_{\text{h}}$ and $\dot{m}_{\text{h}}$ at earlier times. This
allows us to explore how such variation in $\dot{m}_{\text{h}}$ and the
corresponding influence on $\dot{m}_{\text{c,g}}$ and $\dot{m}_{\star}$ affect
the halos’ baryon mass content.

##  3 Results

###  3.1 Bursty star formation & UV suppression of accretion

Figure 2: Baryon mass assembly history of an example halo with
$m_{\text{h}}=10^{7}\text{M}_{\odot}$ at $z$=5. The upper and middle panels
show $m_{\text{g}}$ and $m_{\star}$ (in $\text{M}_{\odot}$) against cosmic
time (in Gyrs); solid (dotted) curves correspond to instantaneous star
formation and feedback without (with) UV suppression of accretion
(instantaneous + No UV, instantaneous + UV), while dashed (dot-dashed) curves
correspond to bursty star formation from delayed feedback without (with) UV
suppression of accretion (delayed + No UV, delayed + UV). For comparison, we
show also the growth of $m_{\text{h}}$ with cosmic time in the lower panel.

We begin by considering the growth of gas mass (upper panel), stellar mass
(middle panel), and halo mass (lower panel; for comparison) in Figure 2,
focusing on an example halo with $m_{\text{h}}=10^{7}\text{M}_{\odot}$ at
$z=5$, drawn from the sample of 100 halos in this mass bin. We choose this
halo mass bin because it marks the transition between low-mass halos that can
be efficiently quenched via delayed feedback alone and high-mass halos that
continue to accrete and form stars after the onset of UV suppression; we show
examples of low- and high-mass systems’ assembly histories in A. To illustrate
the effects of bursty star formation and UV suppression on the growth of gas
and stellar mass as a function of cosmic time, we consider the cases of
instantaneous star formation and feedback and no UV suppression of accretion
(hereafter instantaneous + No UV; solid curves); instantaneous star formation
feedback and UV suppression of gas accretion with $z_{\text{rei}}=7$
(hereafter instantaneous + UV; dotted curves); bursty star formation from
delayed feedback and no UV suppression of accretion (hereafter delayed + No
UV; dashed curves); and bursty star formation and feedback and UV suppression
of accretion with $z_{\text{rei}}=7$ (hereafter delayed + UV; dot-dashed
curves). We adopt our fiducial parameters of $\epsilon_{\text{sf}}=0.015$ and
$\epsilon_{\text{fb}}=5$.

In the instantaneous + No UV case, we find that both $m_{\text{g}}$ and
$m_{\star}$ show an initial steep rise at early times ($\leq 0.5$ Gyr), and
continue to grow but at a slower at later times (up to 2 Gyr). There are
noticeable increases in $m_{\text{g}}$ at approximately 0.5 and 0.6 Gyr; these
correlate with increases in $m_{\text{h}}$ evident in the lower panel, which
translate into increases in $m_{\text{g}}$ given our assumption that
$\dot{m}_{\text{c,g}}$ tracks $\dot{m}_{\text{h}}$. In contrast, for
instantaneous + UV, we find that $m_{\text{g}}$ and $m_{\star}$ show a similar
initial steep rise at early times but the effect of the UV background shutting
off cosmological accretion is evident - $m_{\text{g}}$ peaks after 0.75 Gyr
and then declines with increasing time, while $m_{\star}$ plateaus. Because
$m_{\text{g}}$ is linked to $\dot{m}_{\text{h}}$ and the magnitude of UV
suppression is both halo mass and redshift dependent, we see that
$m_{\text{g}}$ shows an initial sharp decline but plateaus after 1.5 Gyr. For
the delayed + No UV case, we see the characteristic strong oscillations in
$m_{\text{g}}$ \- and to a lesser extent in $m_{\star}$ \- evident during the
initial 0.5 Gyr, but as the halo grows, these oscillations damp away, and both
quantities are indistinguishable from the case with instantaneous star
formation and feedback. Similarly, for the delayed + UV case, we see these
oscillations in $m_{\text{g}}$ and $m_{\star}$ repeated.

###  3.2 Halo mass assembly history

Figure 3: Influence of halo mass assembly history: Baryon mass assembly
history of a sample of 100 halos with $m_{\text{h}}=10^{7}\text{M}_{\odot}$ at
$z$=5. The upper and lower panels show the median values of $m_{\text{g}}$ and
$m_{\star}$ (in $\text{M}_{\odot}$) against cosmic time (in Gyrs); dotted
curves correspond to the instantaneous + UV case, while dot-dashed curves
correspond to the delayed + UV case. The coloured bands indicate the range of
the 10${}^{\text{th}}$ and 90${}^{\text{th}}$ percentiles.

In Figure 3, we show the variation in $m_{\text{g}}$ and $m_{\star}$ with
cosmic time for a sample of 100 halos, each with
$m_{\text{h}}=10^{7}\text{M}_{\odot}$ at $z=5$. We now focus on models with
the UV suppression of accretion of gas from the IGM. The dotted (dot-dashed)
curves correspond to the median values of $m_{\text{g}}$ (upper panel) and
$m_{\star}$ (lower) for the instantaneous + UV (delayed +UV) case, while the
coloured bands indicate the range of the 10${}^{\text{th}}$ and
90${}^{\text{th}}$ percentiles. This shows how variations in the assembly
history of the underlying dark matter halo, whose growth rate
$\dot{m}_{\text{h}}$ governs the growth rate of the gas mass and consequently
the stellar mass. The variations introduced by the diversity of halo assembly
histories correspond to approximately $0.5-1$ dex in both $m_{\text{g}}$ and
$m_{\star}$. The variations in $m_{\text{g}}$ are larger for the delayed + UV
case, as we might expect, while we we see that the variations in $m_{\star}$
are similar in magnitude for both the instantaneous + UV and delayed + UV
cases ($\sim 0.5$ dex at early times, $\sim 1$ dex at later times).

###  3.3 Efficiency of star formation, $\epsilon_{\text{sf}}$, and feedback,
$\epsilon_{\text{fb}}$

Figure 4: Influence of star formation efficiency: Baryon mass assembly history
of a sample of 100 halos with $m_{\text{h}}=10^{7}\text{M}_{\odot}$ at $z$=5.
The upper and lower panels show the median values of $m_{\text{g}}$ and
$m_{\star}$ (in $\text{M}_{\odot}$) against cosmic time (in Gyrs) for
$\epsilon_{\text{sf}}$=0.0015 (inefficient) and $\epsilon_{\text{sf}}$=0.1
(efficient) respectively. Dotted curves correspond to the instantaneous + UV
case, while dot-dashed curves correspond to the delayed + UV case. The
coloured bands indicate the range of the 10${}^{\text{th}}$ and
90${}^{\text{th}}$ percentiles. The greyed bands and curves correspond to the
counterpart cases with the fiducial value of $\epsilon_{\text{sf}}$=0.015.

We now examine how our choices of star formation and feedback efficiency
affect the growth of gas mass (upper panel) and stellar mass (lower panel) in
Figures 4 and 5. As in Figure 3, we focus on a sample of 100 halos with
$m_{\text{h}}=10^{7}\text{M}_{\odot}$ at $z=5$, but for cases of low and high
star formation and feedback efficiencies, $\epsilon_{\text{sf}}$ and
$\epsilon_{\text{fb}}$, respectively. We keep $z_{\text{rei}}=7$ fixed and
look at the cases of low (high) efficiency in the upper (lower) panels, with
reference curves corresponding to our fiducial values plotted in light grey
($\epsilon_{\text{sf}}$=0.015 and $\epsilon_{\text{fb}}$=5).

As before, we focus on the cases of instantaneous + UV (dotted curves) and
delayed + UV (dot-dashed curves). By reducing (increasing) the star formation
efficiency, we expect a reduction in the number of stars to form over a fixed
interval to be lower (higher) than in the fiducial case; however, the coupling
of star formation to feedback means that a higher star formation efficiency
should boost the strength of the feedback, which, in the case of delayed
feedback, should lead to stronger oscillations in the ${m}_{\text{g}}$ and
${m}_{\star}$ (Furlanetto2022, 17, see, e.g.,). We anticipate that stronger
(weaker) feedback efficiency should impact $\dot{m}_{\text{c,g}}$ and
consequently $m_{\text{g}}$, but it’s likely the that the difference with
respect to the fiducial run will be more pronounced for weaker feedback - it
may evacuate the halo of gas more quickly, but the accretion rate onto the
halo is unchanged.

In Figure 4, we fix feedback efficiency at $\epsilon_{\text{fb}}$=5 and look
at star formation efficiencies of $\epsilon_{\text{sf}}$=0.0015 (low
efficiency, upper panel) and $\epsilon_{\text{sf}}$=0.1 (high efficiency,
lower panel). As expected, in the case of higher star formation efficiency, we
see evidence for enhanced star formation efficiency driving enhanced feedback.
Consider the instantaneous + UV case. At a given time, we see that
$m_{\text{g}}$ is smaller by approximately 0.5 dex compared to the fiducial
run up to the point at which the effect of UV suppression kicks in, following
which $m_{\text{g}}$ rapidly declines as its converted to stars; the evolution
of $m_{\star}$ is relatively unchanged from the fiducial case. In contrast,
the delayed + UV case shows some interesting changes compared to the fiducial
case; the initial evolution of $m_{\text{g}}$ is similar but from
approximately 0.2 Gyr it experiences 3 longer periods in which
$m_{\text{g}}\rightarrow 0$, while we also note that the typical system forms
more stars at a given time, and $m_{\star}$ is $\sim 1$ dex higher after 1.2
Gyr. For the low star formation efficiency case, we see that the differences
between the two models are dramatically reduced. In particular, the strong
oscillations apparent in the fiducial delayed + UV case are very weak - only a
dip in $m_{\text{g}}$ is evident at 0.2 Gyr - and the evolution in
$m_{\text{g}}$ and $m_{\star}$ are very similar, with the delayed + UV case
having higher values of $m_{\text{g}}$ by $\sim 0.1$ dex and lower values of
$m_{\star}$ by $\sim 0.1$ dex, when compared to the instantaneous + UV case.

Figure 5: Influence of feedback efficiency: Baryon mass assembly history of a
sample of 100 halos with $m_{\text{h}}=10^{7}\text{M}_{\odot}$ at $z$=5. The
upper and lower panels show the median values of $m_{\text{g}}$ and
$m_{\star}$ (in $\text{M}_{\odot}$) against cosmic time (in Gyrs) for
$\epsilon_{\text{fb}}$=2 (inefficient) and $\epsilon_{\text{fb}}$=7
(efficient) respectively. Dotted curves correspond to the instantaneous + UV
case, while dot-dashed curves correspond to the delayed + UV case. The
coloured bands indicate the range of the 10${}^{\text{th}}$ and
90${}^{\text{th}}$ percentiles. The greyed bands and curves correspond to the
counterpart cases with the fiducial value of $\epsilon_{\text{fb}}$=5.

In Figure 5, we fix star formation efficiency at $\epsilon_{\text{sf}}$=0.015
and look at feedback efficiencies of $\epsilon_{\text{fb}}$=2 (low efficiency,
upper panel) and $\epsilon_{\text{fb}}$=7 (high efficiency, lower panel). As
anticipated, stronger feedback acts to accelerate when the median halo loses
its gas mass (i.e. $m_{\text{g}}\rightarrow 0$), especially following the
onset of UV suppression, and the spread in times when this occurs is larger
than in the fiducial model (0.3 Gyr compared to 0.2 Gyr in the fiducial case);
however, it has little effect on the evolution of the stellar mass. We find
that weaker feedback has a dramatic effect on the evolution of $m_{\text{g}}$
in the delayed +UV model; the initial oscillations seen in the fiducial case
are present, but after 0.3 Gyrs $m_{\text{g}}$ grows up to the onset of UV
suppression, before showing a slow decline; the total gas mass in this model
is larger than in the instantaneous feedback case after the initial
oscillations have damped away. The behaviour of the instantaneous +UV models
in both the strong and weak feedback cases track that in the fiducial case,
with the key difference being in the rate of decline of $m_{\text{g}}$ after
the onset of UV suppression - the halo retains more (less) gas mass at a given
time when feedback is strong (weak).

###  3.4 Delayed Feedback Timescale, $t^{\text{d}}$

In Figure 6 we assess how our choice of the delayed feedback timescale,
$t^{\text{d}}$, influences the growth of $m_{\text{g}}$ and $m_{\star}$ with
cosmic time. We focus on a sample of 100 halos with
$m_{\text{h}}=10^{7}\text{M}_{\odot}$ at $z=5$, with $z_{\text{rei}}=7$ fixed
and our fiducial values of star formation and feedback efficiency of
$\epsilon_{\text{sf}}$=0.015 and $\epsilon_{\text{fb}}$=5. According to our
formulation, the instantaneous star formation and feedback model corresponds
to $t^{\text{d}}$=0, while our fiducial delayed feedback model assumes
$t^{\text{d}}$=0.015 Gyr; we consider two further cases, with
$t^{\text{d}}$=0.0075 Gyr (upper panel) and 0.03 Gyr (lower panel).

Figure 6: Influence of delayed feedback timescale: Baryon mass assembly
history of a sample of 100 halos with $m_{\text{h}}=10^{7}\text{M}_{\odot}$ at
$z$=5. The upper and lower panels show the median values of $m_{\text{g}}$ and
$m_{\star}$ (in $\text{M}_{\odot}$) against cosmic time (in Gyrs) for
$t^{\text{d}}$=0.0075 Gyr and $t^{\text{d}}$=0.03 Gyr respectively. Dotted
curves correspond to the instantaneous + UV case, while dot-dashed curves
correspond to the delayed + UV case. The coloured bands indicate the range of
the 10${}^{\text{th}}$ and 90${}^{\text{th}}$ percentiles. The greyed bands
and curves correspond to the counterpart cases with the fiducial value of
$t^{\text{d}}$=0.015 Gyr.

Compared to our fiducial model with $t^{\text{d}}$=0.015 Gyr, we see that a
shorter (longer) time delay increases (decreases) the number of initial
oscillations in both $m_{\text{g}}$ and $m_{\star}$, and the peak median
masses are similar. There are two noticeable differences - in the lengths of
the intervals when $m_{\text{g}}\rightarrow 0$ at early times, with
$t^{\text{d}}$=0.03 Gyr experiencing longer intervals when $m_{\text{g}}\simeq
0$; and in the evolution of $m_{\text{g}}$ after the onset of UV suppression
of gas accretion, such that $m_{\text{g}}$ plateaus close to its peak value in
the model with the long time delay through to 1.2 Gyr, whereas
$m_{\text{g}}\rightarrow 0$ after the onset of UV suppression as in the
fiducial case, albeit with a delay of $\sim 0.1$ Gyr. These trends highlight
the complex interplay between gas accretion, star formation, and delayed
feedback - especially the manner in which gas mass is retained at later times
when $t^{\text{d}}$=0.03 Gyr.

###  3.5 Redshift of Reionization, $z_{\text{rei}}$

In Figure 7 we assess how our choice of the redshift of reionization,
$z_{\text{rei}}$, influences the growth of $m_{\text{g}}$ and $m_{\star}$ with
cosmic time. Again we focus on a sample of 100 halos with
$m_{\text{h}}=10^{7}\text{M}_{\odot}$ at $z=5$, with our fiducial values of
star formation and feedback efficiency of $\epsilon_{\text{sf}}$=0.015,
$\epsilon_{\text{fb}}$=5 fixed. We look at the case of $z_{\text{rei}}$=10,
which corresponds to the time when the Universes was approximately 60% of its
age at $z$=7 (our fiducial redshift of reionization). The trends are
qualitatively similar to those we see when $z_{\text{rei}}$=7; the peak values
of $m_{\text{g}}$ and $m_{\star}$ are lower, and the rate at which
$m_{\text{g}}\rightarrow 0$ is more rapid in the case of instantaneous
feedback with UV suppression, as we would expect because of the shorter
dynamical times in these systems.

Figure 7: Influence of redshift of reionization: Baryon mass assembly history
of a sample of 100 halos with $m_{\text{h}}=10^{7}\text{M}_{\odot}$ at $z$=5.
We show the median values of $m_{\text{g}}$ and $m_{\star}$ (in
$\text{M}_{\odot}$) against cosmic time (in Gyrs) for $z_{\text{rei}}$=10.
Dotted curves correspond to the instantaneous + UV case, while dot-dashed
curves correspond to the delayed + UV case. The coloured bands indicate the
range of the 10${}^{\text{th}}$ and 90${}^{\text{th}}$ percentiles. The greyed
bands and curves correspond to the counterpart cases with the fiducial value
of $z_{\text{rei}}$=7.

###  3.6 Baryon Mass Content - Variation with Halo Mass

We have explored how local factors - star formation and feedback efficiencies,
burstiness of star formation, variation in dark matter halo assembly histories
- and global factors - the onset of UV suppression arising from cosmological
reionization - influence the time evolution of the gas and stellar masses of
halos. We focused on a halo mass scale - $m_{\text{h}}=10^{7}\text{M}_{\odot}$
at $z=5$ \- that we argued is transitional, being sufficiently massive enough
not to experience early truncation of gas accretion and star formation, but
not overly massive that it’s unaffected by delayed feedback and UV suppression
of gas accretion from the IGM. We have this information for halos spanning a
mass range $10^{6}\text{M}_{\odot}\leq m_{\text{h}}\leq
10^{11}\text{M}_{\odot}$ at that epoch, as we show in A.

In Figure 8 we quantify how $m_{\text{g}}$ and $m_{\star}$ vary with
$m_{\text{h}}$, looking at three epochs - $z$=10, 7, and 5 (from right to
left). Because we are interested in relative trends, we plot the quantity
$m_{\star}/(m_{\star}+m_{\text{g}})$ against $m_{\text{h}}$ at that epoch. As
the system becomes gas depleted, we expect that
$m_{\star}/(m_{\star}+m_{\text{g}})\rightarrow 1$ because
$m_{\text{g}}\rightarrow 0$, which we expect at lower halo masses and should
move to a higher mass scale following the onset of the UV suppression of gas
accretion from the IGM.

At $z$=10, we see that this ratio shows little variation across the mass
range, with the median $m_{\star}/(m_{\star}+m_{\text{g}})\simeq 0.05$ in the
instantaneous + UV case, compared to $m_{\star}/(m_{\star}+m_{\text{g}})\simeq
0.03$ in the delayed + UV case. There is a weak trend for
$m_{\star}/(m_{\star}+m_{\text{g}})$ to decrease (increase) with increasing
mass in the instantaneous (delayed) + UV case. Interestingly, we see an
enhancement $m_{\star}/(m_{\star}+m_{\text{g}})$ in the delayed + UV case in
the halo mass range $10^{7}\text{M}_{\odot}\leq m_{\text{h}}\leq
10^{8}\text{M}_{\odot}$ \- this reflects the transitional nature of this halo
mass range and how halo mass assembly histories interact with burstiness in
star formation and delayed feedback to produce a range in gas mass.

By $z$=7, we now see a marked variation across the mass range. At higher
masses, the median $m_{\star}/(m_{\star}+m_{\text{g}})\simeq 0.05$ in the
instantaneous + UV case, unchanged from $z$=10; in contrast,
$m_{\star}/(m_{\star}+m_{\text{g}})\simeq 0.02$ in the delayed + UV case,
which has is lower than at $z$=10. The ratio increases smoothly towards lower
masses in the instantaneous + UV case, whereas the ratio is flat down to
$m_{\text{h}}\simeq 10^{7}\text{M}_{\odot}$ before sharply turning up to
$m_{\star}/(m_{\star}+m_{\text{g}})\simeq 1$. At $z$=5, the marked variation
with mass has strengthened. At higher masses, the median
$m_{\star}/(m_{\star}+m_{\text{g}})\simeq 0.05$ in the instantaneous + UV case
is unchanged from $z$=10, whereas $m_{\star}/(m_{\star}+m_{\text{g}})\simeq
0.01$ in the delayed + UV case, which is marginally lower than at $z$=7. We
now see a gradual rise in $m_{\star}/(m_{\star}+m_{\text{g}})$ towards lower
$m_{\text{h}}$ becoming evident at $m_{\text{h}}\simeq
10^{10}\text{M}_{\odot}$; by $m_{\star}/(m_{\star}+m_{\text{g}})$=1 by
$m_{\text{h}}\simeq 10^{7}\text{M}_{\odot}$ in both the instantaneous + UV and
delayed + UV cases.

At the three epochs we have considered, we see a general trend for
$m_{\star}/(m_{\star}+m_{\text{g}})$ to be approximately a factor of $3-5$
larger in the instantaneous + UV case compared to the delayed + UV case.
Depending on epoch and mass scale, this can mean both a relative enhancement
in stellar mass or a relative deficit in gas mass in the instantaneous + UV
model. However, the trend is for the lower mass galaxies to have higher
stellar masses in the delayed + UV case, while - in the context of this
particular physical model - higher mass galaxies can retain their gas
reservoirs. Physically, we would expect seed super-massive black hole growth
and associated feedback to reduce gas mass in these high mass galaxies. We
will revisit this in a future paper.

Figure 8: Baryon mass assembly history of a
$m_{\text{h}}=10^{7}\text{M}_{\odot}$ at $z$=5. The upper and lower panels
show $m_{\text{g}}$ and $m_{\star}$ (in $\text{M}_{\odot}$) against cosmic
time (in Gyrs); solid (dotted) curves correspond to instantaneous star
formation and feedback without (with) UV suppression of accretion, while
dashed (dot-dashed) curves correspond to bursty star formation without (with)
UV suppression of accretion.

##  4 Conclusions

We have investigated how the baryon mass content (i.e. the stellar and gas
mass) of dark matter halos in the early Universe ($z\geq 5$) is affected by
the complex interplay between bursty star formation - arising from delayed
feedback - and the UV suppression of gas accretion from the IGM driven by
cosmological reionization. Using a lightweight semi-analytical model that we
have written, we have assessed how global factors - the emergence of an
ionising ultraviolet (UV) radiation background - and local factors - star
formation and feedback efficiency, time delayed feedback, and variations in
dark matter halo assembly histories - influence the evolution of baryon mass
content as a function of cosmic time ($z\geq 5$) and dark matter halo mass
($10^{6}\text{M}_{\odot}\leq m_{\text{h}}\leq 10^{11}\text{M}_{\odot}$).

Using the assembly histories of baryons in a sample of halos with
$m_{\text{h}}=10^{7}\text{M}_{\odot}$ at $z$=5 to illustrate the relative
importance of physical processes, we investigated how star formation
efficiency, the strength of feedback, and the delayed feedback timescale
influenced the stellar and gas masses over cosmic time. As shown by previous
studies, delayed feedback leads to oscillations in gas mass, evident more
subtly in the stellar mass, as a function of cosmic time. If star formation is
inefficient, gas and stellar mass growth histories are similar regardless of
whether or not feedback is instantaneous or delayed; in contrast, highly
efficient star formation can drive strong oscillations, as we would anticipate
given that this population produces feedback via supernovae winds. Weaker
delayed feedback cannot suppress the oscillatory behaviour in lower mass
systems and these systems can retain their gas for a more extended period,
whereas the main effect of stronger feedback is to drive
$m_{\text{g}}\rightarrow 0$ more quickly. Longer delayed feedback timescales
reduce the number of oscillations in $m_{\text{g}}$ and can result in more gas
being retained at later times, whereas an earlier redshift of reionization has
the simple effect of nudging the evolutionary trends in $m_{\text{g}}$ and
$m_{\star}$ (i.e. $m_{\text{g}}\rightarrow 0)$, plateauing of $m_{\star}$ to
the correspondingly earlier time.

If we consider the variation of baryon mass content - which we parameterise by
the ratio $m_{\star}/(m_{\star}+m_{\text{g}})$ \- with halo mass, we find at
earlier times that the median trend is for the ratio to be relatively flat and
small in both the instantaneous and delayed feedback cases - of order
$0.01-0.1$ \- but with $m_{\star}/(m_{\star}+m_{\text{g}})$ to be
approximately a factor of $3-5$ larger in the instantaneous case. At later
times, we find that the ratio increases with decreasing halo mass such that
$m_{\star}/(m_{\star}+m_{\text{g}})\rightarrow 1$ in the lowest mass systems.
Higher mass systems retain their gas reservoirs in both cases considered
because they are sufficiently massive to continue accreting gas from the IGM
in a way that less massive systems cannot during cosmological reionization.

The relative gas richness of massive systems is at odds with both observations
and theoretical expectations (Moster2018, 29, 24, e.g.), but we would expect
feedback from super-massive black holes in these massive galaxies to expel gas
and quench star formation. As noted in § 2, we do not model seed supermassive
black growth or its associated feedback; we expect this to be important
(Power2011, 39, e.g.) in an interesting, mass scale dependent manner
(Nayakshin2009MNRAS, 32, cf.) and we shall explore it in a future paper.

We have not explicitly considered the influence of metallicity in this paper,
although we have investigated straightforward extensions to our model to do
this (Kravtsov2022, 23, e.g.). We expect that feedback efficiency should
depend on metallicity (Jecmen2023, 21, 45, e.g.). For example, the ability of
massive stars to drive stellar winds depends on the interaction cross-section
of energetic photons with gas in stellar atmospheres, which depends on the
presence of heavier elements (Lamers1999, 26, cf.). This will introduce
another mass-scale dependence, reflecting the number of generations of stars
and metal enrichment events, and we shall explore it in future work.

##  5 Acknowledgement

AM and CP thank the anonymous referee for their very positive report. CP
acknowledges the support of the ARC Centre of Excellence for All Sky
Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013.

##  Appendix A Evolution of Baryon Mass Content at Low and High Halo Masses

In Figure 9, we show the variation in $m_{\text{g}}$ and $m_{\star}$ with
cosmic time for a sample of 100 halos, each with
$m_{\text{h}}=10^{7}\text{M}_{\odot}$ at $z=5$. The dotted (dot-dashed) curves
correspond to the median values of $m_{\text{g}}$ (upper panel) and
$m_{\star}$ (lower) for instantaneous (bursty) star formation and feedback
with UV suppression of accretion, while the coloured bands indicate the range
of the 10${}^{\text{th}}$ and 90${}^{\text{th}}$ percentiles. This shows how
variations in the assembly history of the underlying dark matter halo, whose
growth rate $\dot{m}_{\text{h}}$ governs the growth rate of the gas mass and
consequently the stellar mass. Interestingly we see a large variation in gas
mass at late times for the lower mass system - although the median
$m_{\text{g}}=0$, there is a large number of systems with
$m_{\text{g}}=10-100\text{M}_{\odot}$.

Figure 9: Baryon mass assembly history of a sample of 100 halos with
$m_{\text{h}}=10^{6}\text{M}_{\odot}$ (upper panel) and
$m_{\text{h}}=10^{11}\text{M}_{\odot}$ (lower panel) at $z$=5. The upper and
lower panels show the median values of $m_{\text{g}}$ and $m_{\star}$ (in
$\text{M}_{\odot}$) against cosmic time (in Gyrs); dotted curves correspond to
instantaneous star formation and feedback with UV suppression of accretion,
while dot-dashed curves correspond to bursty star formation with UV
suppression of accretion. The coloured bands indicate the range of the
10${}^{\text{th}}$ and 90${}^{\text{th}}$ percentiles.

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Machine learning can be employed to automate the entire misinformation
detection pipeline, or to assist in specific stages of it. Figure fig:2
identifies the three central stages of the process, and provides a generalized
view of the target objective at each of them. While specific design choices
may vary across systems (e.g., choosing if check-worthiness is a multi-class
label prediction problem or a regression problem), this general formulation
enables us to situate the different types of injustices that may emerge at
distinct functional stages of these systems.

Studying, anticipating or mitigating algorithmic harms requires the
identification of stakeholder(s) affected by incorrect algorithmic
predictions, together with the notion(s) of justice relevant to a stakeholder
when a certain type of error is made. For example, some errors might result in
undue scrutiny of speech or disproportionate harms stemming from the spread of
misinformation, depending on whether the algorithm is over- or under-
scrutinizing the content relevant to a particular group. In this section, we
discuss the potential harms associated with different algorithms in automated
misinformation detection, grounded on the taxonomy introduced in Section
sec:tax. For each algorithmic stage in the pipeline, we discuss: (1) a
generalized framework/objective for the machine learning task; (2) the
relevant sense of injustice to different stakeholders when affected by errors
in algorithmic decisions, and (3) an overview of potential sources of these
injustices.

[width=]Taxonomy_of_Harm.pdf

Figure : A generic framework, used in conjunction with Figure LABEL:fig:1, to
examine issues of justice regarding algorithmic decision-making in the
automated misinformation detection pipeline.

##  Automated Claim Detection

###  The Machine Learning Task

The first task in the automated misinformation detection pipeline is
identifying “check-worthy” claims. According to claimbuster, check-worthy
claims are claims for which the “general public” would be interested in
assessing their validity. For fact-checking organizations, these would be the
claims (out of broader possible universe of claims) that they choose to
evaluate. Snopes, for instance, says this about how they decide what to fact-
check: “We have long observed the principle that we write about whatever items
the greatest number of readers are asking about or searching for at any given
time, without any partisan considerations” [snopes_2021]. It is worth noting
that the motivations for identifying a subset of check-worthy claims may vary
depending on the context and system design. Most commonly, check-worthiness is
a prioritization mechanism to allocate the scarce resource that is human fact-
checkers. Given the scalability of algorithmic fact-checking, such scarcity of
fact-checking resources could be altered, but check-worthiness may remain as a
relevant mechanism to screen content that should be fact-checked, conditioned
on the nature of the claim. Several model architectures have been proposed for
detecting check-worthiness of textual data (e.g. [checkthat,
automated_claim_detection, claimbuster, twitter_metadata,
checkworthy_multitask]). claimbuster, for instance, gather numerous expert
crowd-sourced labels for political speech sentences as either _Non-Factual
Sentence_ (opinion, belief), _Unimportant Factual Sentence_ (facts that are
not relevant to general pubic), and _Check-Worthy Factual Sentence_ (facts
that are relevant to general public). They then extract sentence sentiment,
named entities, and tf-idf features from the vocabulary and optimize a support
vector machine classifier predicting check-worthiness. As common in
misinformation detection, this system relies on training data from a very
specific, and likely not generalizable, corpus: political speeches.
CheckThat!, an annual misinformation prediction conference, has provided a
dataset that contains 629 labeled English tweets pertaining to COVID-19 as a
benchmark dataset [checkthat]. The researchers gathered survey responses to
five questions that attempted to capture the check-worthiness of each tweet.
Participants in the machine learning competition were then asked to aggregate
these labels and develop a model to predict check-worthiness for novel tweets.
Researchers investigating COVID-19 misinformation adopted a very similar
approach to checkthat for crowd-sourcing labels to detect claims made on
social media, except instead of aggregating the label, they implemented a
multi-task learning approach, in which the correlated survey questions used to
determine check-worthiness were modelled jointly [checkworthy_multitask]. Some
researchers have also framed the task as rumor detection, and have found
success using metadata on social media (author information, rate of decay of
engagement, reactions, and replies) jointly with semantic representations of
the content to spot early propagation of rumors that should be fact-checked
[twitter_metadata]. In general, we frame the machine learning task as trying
to recover the function:

$\overline{Y}_{c}=f_{1}(X,X_{meta})$ (1)

Where $\overline{Y}_{c}$ is the predicted check-worthiness label, score or
vector of labels (binary, multi-class, or a tuple in the context of multi-task
learning) for a given claim, $X$ is a semantic representation of the claim,
and $X_{meta}$ is a vector of metadata associated with the claim (likes,
reposts, etc). If a piece of content is predicted to be check-worthy, then it
will pass to further stages in the misinformation detection pipeline, where
evidence will be gathered in order to produce a veracity score or label. The
check-worthiness algorithm is concerned with reducing the scale of information
for which evidence retrieval is necessary.

###  Issues of Justice

Under-scrutiny of the claims reported by, of interest to, or about a
particular group can lead to a distributive injustice for sources of evidence,
seekers and subjects of information, as a key moderator of the informational
good (fact-checking) is not being distributed evenly amongst groups. Seekers
of information may be harmed because they receive a comparatively lower
standard of information quality, allowing them to more directly feel the harms
of misinformation than others. Recall the example from Section LABEL:sec:rel,
where Facebook’s seemingly ad-hoc content moderation policy led to claims of
bias from conservative sources of information–a claim of participatory
injustice in our framework. In response, Facebook changed the standards for
evaluating misinformation from conservative sources of information, reducing
scrutiny of some of these sources. By changing the standard for misinformation
detection for specific groups (in this case, conservatives), harms fall upon
(conservative) seekers of information, who are now more likely to be exposed
to misinformation. Under-scrutiny errors at the claim detection stage may also
affect subjects of information, who could be harmed if false claims were made
against a community or an individual that could damage their reputation or
fuel stigma and stereotypes and these claims were not prioritized to be fact-
checked. Thus, failing to assess these claims could result in a recognitional
injustice, by enabling inaccurate representations of members of society.
Finally, in settings where user reports are an input to check-worthiness
assessment, a disproportionate lack of attention to claims flagged by a
community could constitute an epistemic injustice, which may additionally
result in some of the injustices listed above for the same individuals and/or
communities, when their role in relation to a claim is multi-faceted. While
the type of injustice stemming from under-scrutiny will be different for
different stakeholders, they all correspond to false negative errors. Thus,
assessing _equal opportunity_ hardt2016equality as an algorithmic fairness
metric is of relevance, where the sensitive attribute category will vary
conditioned on the stakeholder. Over-scrutiny may also constitute harms,
although these will depend on the interaction with other portions of the
system. By itself, marking something to be verified does not necessarily harm
sources, seekers or subjects, especially if we assume perfect accuracy in the
fact-checking stage. However, that is an unrealistic assumption, and it is
necessary to consider fairness under composition [dwork2018fairness]. In
particular, it has been shown in the algorithmic fairness literature that
making the same type of errors, e.g. false negatives, when the output of one
algorithm determines the input of the following one will result in compounding
imbalances, which may correspond to compounding injustices [de2019bias]. Thus,
monitoring _false positive rates_ associated with predicted check-worthiness
is especially pressing in conjunction with the errors at other stages of the
pipeline.

###  Sources of Harms

Check-worthiness is an ambiguous task, and those designing a system must first
decide how to define what being check-worthy means, and establish a
quantitative approximation to it. This problem formulation stage may already
induce bias. For example, check-worthiness may be defined as what interests
the “general public” [claimbuster], or what “the greatest number of readers
are asking about” [snopes_2021]. Clearly, a simple focus on majorities
disregards minority communities along multiple axes. Misinformation targeting
specific minorities, e.g. an immigrant community, will by definition not be
considered check-worthy unless it also affects other groups, constituting a
distributive injustice for this group of seekers of information. Similarly,
when these communities participate in the ecosystem as sources of evidence, by
reporting or searching for certain items, their input will be neglected unless
it matches the interest of majority groups, resulting in an epistemic
injustice. Thus, the harms associated with false negative errors, or under-
scrutiny of claims, may be disproportionately concentrated in certain groups
as a result of measurement bias in the training labels jacobs2021measurement.
Measurement bias–when an inappropriate proxy is used to measure a construct
[machine_learning_bias_general]–may result from a choice of metric, e.g. the
most frequently searched items, or from ambiguity in the formulation of survey
questions used to collect check-worthiness labels. For instance, questions
inquiring whether something is of interest to the “general public” may elicit
very subjective responses regarding who the “general public” is, and may
downplay claims that are highly relevant (with potentially high stakes) to a
particular minority group. This issue may be exacerbated for subjects of
information if there is a cultural disconnect between those evaluating the
claim (the sources of evidence) and the subject of information. If there is a
cultural disconnect, those evaluating the content may fail to appropriately
understand it as well as its implications or implied sentiment, as these may
all vary contextually across cultures. For example, social stereotypes affect
how people understand language directed toward minority groups, leading to
higher false negative assessments when labelers have negative stereotypes
about the subject of information [davani2021hate]. One direction for future
research would be to evaluate the impact of cultural alignment between the
subject of (mis)information and the source of evidence to see if there are
significant differences between a culturally-aligned and culturally-misaligned
subject-source pairs. If one were to find differences between the subject-
source pairs, this might imply that culturally-misaligned annotators provide
lower signal (or high bias) labels.