# Bar formation and evolution in the cosmological context: Inputs from the
Auriga simulations

Francesca Fragkoudi1, Robert J. J. Grand2, Rüdiger Pakmor3, Facundo Gómez4,5
<EMAIL_ADDRESS>Federico Marinacci6, Volker Springel3

1Institute for Computational Cosmology, Department of Physics, Durham
University, South Road, Durham DH1 3LE, UK
2Astrophysics Research Institute, Liverpool John Moores University, 146
Brownlow Hill, Liverpool, L3 5RF, UK
3Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85748
Garching, Germany
4Instituto de Investigación Multidisciplinar en Ciencia y Tecnología,
Universidad de La Serena, Raúl Bitrán 1305, La Serena, Chile
5Departamento de Astronomía, Universidad de La Serena, Av. Juan Cisternas 1200
Norte, La Serena, Chile
6Department of Physics & Astronomy “Augusto Righi”, University of Bologna, via
Gobetti 93/2, 40129 Bologna, Italy

(Accepted -. Received -; in original form -)

###### Abstract

Galactic bars drive the internal evolution of spiral galaxies, while their
formation is tightly coupled to the properties of their host galaxy and dark
matter halo. To explore what drives bar formation in the cosmological context
and how these structures evolve throughout cosmic history, we use the Auriga
suite of magneto-hydrodynamical cosmological zoom-in simulations. We find that
bars are robust and long-lived structures, and we recover a decreasing bar
fraction with increasing redshift which plateaus around $\sim 20\%$ at $z\sim
3$. We find that bars which form at low and intermediate redshifts grow longer
with time, while bars that form at high redshifts are born ‘saturated’ in
length, likely due to their merger-induced formation pathway. This leads to a
larger bar-to-disc size ratio at high redshifts as compared to the local
Universe. We subsequently examine the multi-dimensional parameter space
thought to drive bar formation. We find that barred galaxies tend to have
lower Toomre $Q$ values at the time of their formation, while we do not find a
difference in the gas fraction of barred and unbarred populations when
controlling for stellar mass. Barred galaxies tend to be more baryon-dominated
at all redshifts, assembling their stellar mass earlier, while galaxies that
are baryon-dominated but that do not host a bar, have a higher ex-situ bulge
fraction. We explore the implications of the baryon-dominance of barred
galaxies on the Tully-Fisher relation, finding an offset from the unbarred
relation; confirming this in observations would serve as additional evidence
for dark matter, as this behaviour is not readily explained in modified
gravity scenarios.

###### keywords:

Galaxy: formation - Galaxy: evolution - galaxies: kinematics and dynamics -
methods: numerical

††pubyear: 2024††pagerange: Bar formation and evolution in the cosmological
context: Inputs from the Auriga simulations–C

##  1 Introduction

The question of why some galaxies form a stellar bar and others do not, has
puzzled astronomers for over a century now. Since the early studies on the
morphology of ‘extra-galactic nebulae’, Curtis (1918) noted that some spiral
galaxies have _“a band of matter extending diametrically across the nucleus
and inner parts of the spiral”_ , which he dubbed the ‘$\phi$-type spirals’,
due to their rough similarity to the Greek letter $\phi$. Hubble later
classified galaxies into the now commonly used ‘Hubble Sequence’, separating
spiral galaxies into what he called the ‘normal’ spirals, and the ‘barred’
spirals (Hubble, 1926, 1936). We now know that such barred spirals are in fact
as common, if not more so, than unbarred spirals in the local Universe, with
some studies finding bar fractions as high as 70%, depending on the wavelength
and method used to identify them (Eskridge et al., 2000; Menéndez-Delmestre et
al., 2007; Aguerri et al., 2009; Gadotti, 2009; Masters et al., 2011; Erwin,
2018).

While the fraction of barred galaxies is high in the local Universe, it is
generally found to decrease towards higher redshifts (Sheth et al. 2008;
Melvin et al. 2014; Simmons et al. 2014, but see also Jogee et al. 2004).
Studies using data from the Hubble Space Telescope find that bars are present
in about $\sim 10\%$ of disc galaxies up to $z\sim 2$ (e.g. Simmons et al.
2014), while more recent studies using data from the James Webb Space
Telescope (JWST) find multiple bars at $z>1$ (e.g. Guo et al. 2023; Costantin
et al. 2023) and a higher bar fraction than previously reported: $\sim$20% at
$z\sim 1-2$ and $\sim$15% between $z\sim 2-3$ (Le Conte et al., 2024). Bars
have also recently been identified out to redshifts of $z\sim 4$ (Smail et
al., 2023; Tsukui et al., 2024; Amvrosiadis et al., 2024). At the same time,
studies exploring stellar populations in local galaxies find evidence of old
bars in the local Universe (Gadotti et al. 2015; de Sá-Freitas et al. 2023; de
Sà-Freitas et al. in prep.), suggesting that these structures are robust and
long-lived. The decreasing fraction of bars at higher redshifts, hints at that
the tight link between bar formation and the evolution of spiral galaxies,
while their presence in the early Universe suggests that spiral galaxies might
be more dynamically mature at these early times than previously thought (see
also e.g. Rizzo et al. 2020; Lelli et al. 2021).

Bars affect their host galaxy in a variety of ways, and their formation marks
a period in which internal evolutionary processes begin to dominate (e.g.
Kormendy & Kennicutt 2004). As they evolve, bars are found to grow longer with
time (e.g. Athanassoula & Misiriotis 2002; Ghosh & Di Matteo 2024), due to
their ability to exchange angular momentum in the galaxy (Lynden-Bell &
Kalnajs, 1972), i.e. by emitting angular momentum from material at resonance
in the inner disc, which is then absorbed by resonant material in the outer
disc and the spheroidal halo (Sellwood, 1980; Athanassoula, 2002). Bars affect
the galaxy’s evolution by funnelling gas within the bar to the central-most
regions of the galaxy, where it can form nuclear structures such as nuclear
discs and rings (e.g. Athanassoula 1992; Knapen et al. 2002; Comerón et al.
2010; Ellison et al. 2011; Fragkoudi et al. 2016; Sormani et al. 2018; de
Lorenzo-Cáceres et al. 2019; Méndez-Abreu et al. 2019; Leaman et al. 2019;
Gadotti et al. 2019; Bittner et al. 2020). They also re-shape the inner
regions of their host galaxies via the formation of a vertically extended
bulge, referred to as an X-shaped or boxy/peanut (b/p) bulge, which forms due
to vertical instabilities in the bar, is supported by resonant orbits (Combes
et al., 1990; Raha et al., 1991; Pfenniger & Friedli, 1991; Patsis et al.,
2002; Martinez-Valpuesta et al., 2006; Quillen et al., 2014; Portail et al.,
2015; Fragkoudi et al., 2015; Sellwood & Gerhard, 2020; Li et al., 2023), and
which redistributes stars and gas in the inner regions (e.g. Ness et al. 2013;
Di Matteo et al. 2015; Fragkoudi et al. 2017b, c, 2018; Debattista et al.
2020).

There are a number of properties which are known to be important in
determining whether a bar can form or not, and which determine how it
subsequently evolves. For example, an axisymmetric stellar disc will be
unstable to local instabilities if it is too ‘cold’, i.e. if it does not have
any _random motions_ (Toomre, 1964). The early work of Hohl (1971) explored
the long-term stability of stellar discs using N-body simulations, to
establish that even when a disc has enough velocity dispersion to prevent
axisymmetric instabilities (i.e. when the Toomre stability criterion $Q>1$),
it can still be unstable to global bar-forming modes. Subsequent work has
shown that further increasing the velocity dispersion of a disc can delay bar
formation, and lead to an overall weaker bar, but might not be enough to
completely stabilise the disc against bar formation (e.g. Athanassoula &
Sellwood 1986; Combes et al. 1990).

Another fundamental parameter which sets whether a galaxy will form a bar, is
its disc-dominance, i.e. how much the stellar disc dominates over the dark
matter halo in the inner regions. In this work we will interchangeably use
either ‘disc-dominance’ or ‘baryon-dominance’ to refer to this, as both are
used in the literature. Ostriker & Peebles (1973) showed that embedding an
otherwise unstable disc within a massive halo, can stabilise it against bar
formation; the existence of a large number of unbarred galaxies in the
Universe therefore provided an early indication of the existence of such
massive halos surrounding spiral galaxies. The disc-dominance of a galaxy is
usually defined as the disc’s contribution to the circular velocity curve at a
given characteristic radius (e.g. at 2.2 disc scale-lengths Sackett 1997; see
also Efstathiou et al. 1982) – and as such depends on the enclosed mass within
that radius. Later works further expanded on the interaction between bars and
halos, such as the seminal paper of Athanassoula (2002), which showed that
while dark matter halos can stabilise the disc against bar formation for an
initial period of time, the existence of a _live_ dark matter halo in fact
helps bars grow longer and stronger later in their evolution. This occurs
because a ‘live’ halo is able to absorb the angular momentum emitted by the
disc via the bar, whereas in the case of a disc embedded within a static halo,
only the outer disc is available to act as a ‘absorber’ of angular momentum.
This ‘emission’ and ’absorption’ of angular momentum in the galaxy is driven
by the bar, i.e. the inner regions of the disc emit angular momentum – which
leads to a longer and stronger bar – while the outer disc and halo absorb the
angular momentum, leading to a more extended disc and to an increase in net
rotation in the halo. It is important to note that the velocity dispersion of
the halo – not just its mass – play a role in the angular momentum transfer
(Athanassoula, 2003), which further complicates matters.

While the dark matter halo is the most massive ‘spheroid’ in the galaxy, a
_spheroidal bulge component_ in the central regions will also contribute to
stabilising the disc against bar formation (Ostriker & Peebles, 1973), while
also acting as an absorber of angular momentum from the bar (e.g. Saha et al.
2012). The resonant processes driving this redistribution of angular momentum
are similar to those taking place between the bar and the halo; the absorption
of angular momentum by the bulge/spheroid reduces the angular momentum of the
disc and can lead to the spin-up and flattening of an initially non-rotating
spheroidal bulge (e.g. Saha et al. 2012, 2016).

Finally, the _gas fraction_ of galaxies has long been thought to play a role
in the formation and subsequent evolution of bars. The works of Bournaud &
Combes (2002) and Bournaud et al. (2005), which explored bar formation in
N-body and hydrodynamics simulations, suggested that bars can be weakened or
even completely destroyed in the presence of gas. They attributed this to two
mechanisms; first, the formation of a central mass concentration (CMC) due to
bar-induced gas inflows, which weakens the bar, and second, the transport of
angular momentum from the gas to the bar itself. Berentzen et al. (2007) later
suggested that the weakening of bars in discs with higher gas fractions is not
due to the angular momentum transport between these two components, but rather
due to the differences in bar buckling induced by the CMC. CMCs that make up a
few percent of the mass of the stellar disc, are thought to contribute to bar
weakening and even destruction111It is worth noting that the masses and
compactness of the CMCs required to completely destroy bars are not readily
found in nature (see for example the discussion in Athanassoula et al. 2005).,
and the more concentrated and massive the CMC, the larger this effect (Hasan
et al., 1993; Norman et al., 1996; Athanassoula et al., 2005). Athanassoula et
al. (2013) explored the formation of bars within galaxies with varying gas
fractions and halos of different triaxiality. They found that galaxies that
are more gas-rich stay axisymmetric for longer, and that once bars start to
grow, they do so at a slower rate. Therefore, while gas appears to play a role
in bar formation, it is unclear whether this occurs ‘directly’ due to the
exchange of angular momentum between the gas and the bar itself, or whether
gas has an ‘indirect’ effect on bar formation, for example through the
formation of a CMC which forms when gas is pushed to the centre by the bar.

The aforementioned studies have largely been carried out using isolated N-body
simulations. These lend themselves to exploring the internal evolution of
galaxies, due to the higher resolution they offer, while they also enable one
to isolate specific physical properties to study. However, it is becoming
increasingly clear that the mechanisms responsible for bar formation and
evolution cannot be decoupled from the cosmological context, since processes
such as mergers (e.g. Athanassoula et al. 2017), interactions (e.g. Łokas
2019), external gas accretion (e.g. Bournaud & Combes 2002), the physics of
star formation and various feedback processes (e.g. Irodotou et al. 2022), as
well as the assembly and properties of the dark matter halos themselves (e.g.
Collier et al. 2018), all play a role in the formation and evolution of bars.
Recent advances in terms of numerical resolution and modeling of the physics
of baryons have made it possible to begin to use cosmological simulations to
explore the internal dynamical evolution of galaxies both in ‘zoom-in
simulations’ (e.g. Scannapieco & Athanassoula 2012; Kraljic et al. 2012; Buck
et al. 2018; Gargiulo et al. 2019; Fragkoudi et al. 2020, 2021; Merrow et al.
2023) as well as in ‘large box’ cosmological simulations (e.g. Algorry et al.
2017; Peschken & Łokas 2019; Rosas-Guevara et al. 2020; Roshan et al. 2021;
Frankel et al. 2022; Gargiulo et al. 2022; Izquierdo-Villalba et al. 2022;
López et al. 2024). It is worth noting that all these simulations, which use
different prescriptions for modelling the baryonic physics of galaxy formation
and evolution, often lead to substantially different predictions for the
fraction of bars (e.g. finding very few bars at low redshifts $z\sim 0.25$;
Reddish et al. 2022) as well as for their intrinsic properties (e.g. weak and
transient bars; Ansar et al. 2023). As the galaxy formation (or ‘baryon’
physics) models become more advanced, we can use the properties of bars to
constrain these models, since bars act as a litmus test of whether the
dynamics of observed discs are being adequately modelled in cosmological
simulations.

In this work we explore the properties of barred galaxies and how these evolve
over cosmic history in the Auriga suite of magneto-hydrodynamical cosmological
zoom-in simulations, to provide predictions for the properties of bars in
Auriga at both low and high redshifts. We then explore the multi-dimensional
parameter space of bar formation within the cosmological context, by examining
the aforementioned parameters thought to be important for bar formation. The
paper is structured as follows: in Section 2, we briefly describe the Auriga
cosmological simulations, as well as the analysis tools used in this study. In
Section 3 we present some of the global properties of barred galaxies in
Auriga, and show how the length of bars evolves with redshift. In Section 4 we
explore the different parameters that affect bar formation, and how these
contribute to whether a galaxy will be barred or not in the cosmological
context. In Section 5 we discuss some of the implications of our findings on
the Tully Fisher relation, on bars at high redshift and on the multi-
dimensional parameter space of bar formation. In Section 6 we conclude and
summarise our results.

Figure 1: Face-on RGB stellar light projections of the barred galaxies in the
Auriga simulations. The panels have a size of 50 kpc$\times$50 kpc.  Figure 1:
Continued

##  2 The Auriga Simulations

Figure 2: _Top:_ Distribution of barred (red), unbarred (blue) and
barred+interacting (dashed brown) galaxies in our sample in stellar mass.
_Middle:_ Fraction of barred galaxies as a function of stellar mass. The
shaded area is the uncertainty on the fractions assuming binomial statistics,
given by $\sigma=\sqrt{f_{\rm bar}(1-f_{\rm bar})/N}$, where $N$ is the number
of galaxies. _Bottom:_ Bar strength versus stellar mass for the 40 galaxies in
the Auriga suite of simulations. The horizontal dashed line indicates a bar
strength of $A_{2}$=0.25, above which we consider the galaxies to be barred.
Squares indicate barred galaxies, blue circles indicate unbarred galaxies, and
crosses indicate barred galaxies which are undergoing an interaction at $z\sim
0$ (see the text for more details). Figure 3: The $m=2$ Fourier mode of the
surface density as a function of lookback time for galaxies which are barred
at $z=0$, colour-coded by the formation time of the bar (dark purple: bar
forms at larger lookback time, light yellow: bar forms close to $z=0$). The
galaxies that end up barred at $z=0$ develop their bars at various times in
the simulation, some forming early on and staying barred, while others have
bars that develop only at late times.

Figure 4: Statistical properties of the entire sample of 40 Auriga galaxies.
_Left:_ Bar fraction as a function of redshift (top axis) and lookback time
(bottom axis) for the suite of Auriga simulations, compared to observations
from Eskridge et al. (2000), Sheth et al. (2008) and Le Conte et al. (2024).
_Right:_ Bar fraction as a function of redshift for galaxies in two stellar
masses bins, as indicated in the legend. The right axis and grey line indicate
the fraction of galaxies in the low mass bin. We note that in these plots,
time is plotted in the opposite direction from the other plots in the paper,
in order to match the convention used in observational studies. We find a
decreasing bar fraction with redshift in the Auriga simulations, and that more
massive galaxies host more bars at all the times we are able to explore.

Figure 5: _Top:_ Bar strength at $z=0$ versus the age of the bar. The colour-
coding gives the bar strength (used in subsequent plots), and the size of the
points indicates the length of the bar at $z=0$. We indicate the Pearson
coefficient $r$ and the p-value in the figure. _Bottom:_ Bar formation time as
a function of the time at which the galaxy becomes baryon-dominated. The
dashed line indicates is 1-to-1 line.  Figure 6: Bar formation time as a
function of various parameters: _Left:_ Bar formation time as a function of
the stellar-to-dark matter ratio inside 5 kpc at $z=0$. The numbers indicate
the Pearson coefficient $r$ and the p-value. _Middle:_ Bar formation time as a
function of the stellar mass of the galaxy at $z=0$. _Right:_ Bar formation
time as a function of the stellar mass of the galaxy at the time of bar
formation. There is a mild correlation between the bar formation time and the
stellar-to-dark matter ratio at $z=0$, while there is no correlation between
bar formation time and stellar mass at $z=0$ (i.e no obvious signs of mass
downsizing within this mass range). There is a trend of decreasing stellar
mass for earlier bar formation time, as galaxies at higher redshifts have
lower stellar masses. Figure 7: Bar length as a function of time for galaxies
that form bars between four different time-frames, as indicated by the figure
inset. The solid lines show the average values of the bar length and the
shaded region shows the $1\sigma$ dispersion. We see that bars that form at
high redshifts tend to already form long, while bars forming at low redshifts
form short and grow longer with time. The inset compares the time of
significant merging events to the time of bar formation for the oldest bars,
with the dashed line indicating the 1-to-1 line. The oldest bars in our sample
seem to form around the time of a significant merger event. Figure 8: Bar
length as a function of the formation time of the bar. _Left:_ The bar length
at the time of bar formation, as a function of bar formation time. _Middle:_
The bar length at $z=0$ for these same galaxies, as a function of the bar
formation time. _Right:_ The relative change in the bar length as a function
of bar formation time. The crosses indicate the average value for a given age
bin, and the shaded region shows the 1$\sigma$ spread. Bars that form at high
redshifts do not change their lengths much because they form long already,
while the youngest bars have had limited amount of time to grow their bar
length. Figure 9: The bar length-to-disc size ratio as a function of the
formation time of the bar. _Left:_ The ratio at the time of bar formation;
_Middle:_ The ratio at $z=0$; _Right:_ The change in the ratio from bar
formation time to $z=0$. We see that the ratio of bar length-to-disc size is
larger at the time of bar formation for early forming bars, and decreases at
$z=0$ due to the increasing disc size.

We explore the formation and evolution of bars in the Auriga simulations which
are a suite of cosmological magneto-hydrodynamical zoom-in simulations of
galaxies which span a range of halo masses at $z=0$ between $M_{\rm
200}=0.5\times 10^{12}-2\times 10^{12}\rm{M}_{\rm\odot}$ (Grand et al., 2017,
2019). The simulations are run from redshift $z=127$ to $z=0$ with
cosmological parameters $\Omega_{\rm m}=0.307$, $\Omega_{\rm b}=0.048$ and
$\Omega_{\rm\Lambda}=0.693$, and a Hubble constant of $H_{\rm 0}=100h\,\rm
km\,s^{-1}\,\rm Mpc^{-1}$,where $h=0.6777$ (Planck Collaboration et al.,
2014). The simulations are performed with the magnetohydrodynamic code AREPO
(Springel, 2010; Pakmor et al., 2016), with a comprehensive galaxy formation
model (see Vogelsberger et al. 2013; Marinacci et al. 2014; Grand et al. 2017,
for more details) which includes primordial and metal line cooling, a
prescription for a uniform background ultraviolet field for reionization
(completed at $z=6$), a subgrid model for star formation, stellar evolution
and feedback, magnetic fields, as well as black hole seeding, accretion and
feedback. The dark matter particles have a mass of $\sim 4\times
10^{5}\rm{M}_{\odot}$ and the stars and gas have a mass resolution $\sim
5\times 10^{4}\rm{M}_{\odot}$. The physical softening of collisionless
particles grows with time and corresponds to a fixed comoving softening length
of 500 $h^{-1}$ pc, while the maximum physical softening allowed is 369 pc
(see Power et al. 2003 for a criterion on how to choose softening parameters).
The physical softening for the gas cells is scaled by the gas cell radius with
a minimum limit of the softening equal to that of the collisionless particles.

Star formation and stellar feedback are modelled as follows: if a given gas
cell is eligible for star formation, it is converted (according to the
Chabrier 2003 initial mass function) either into a star particle – in which
case it represents a single stellar population of a given mass, age and
metallicity – or into a site for SNII feedback, i.e. a ‘wind’ particle. Wind
particles are launched in a random direction with a velocity that scales with
the 1-D local dark matter velocity dispersion (see Grand et al. 2017 for more
details). The wind metal content is determined by the initial metallicity of
the gas cell from which the wind particle originated, i.e. it is ‘loaded’ with
$\eta=0.6$ of the total metals of the parent gas cell. For the stellar
particles, the mass loss and metal enrichment from SNIa and AGB stars is
modelled and the mass and metals are distributed among nearby gas cells with a
top-hat kernel. We track a total of nine elements: H, He, C, O, N, Ne, Mg, Si,
and Fe.

By $z=0$, the simulations form disc-dominated star-forming galaxies with flat
rotation curves that reproduce a range of observed scaling relations such as
the Tully-Fisher relation (Grand et al., 2017) and the size-mass relation of
HI gas discs (Marinacci et al., 2017). They also form instabilities in the
discs such as bars and boxy/peanut bulges (Fragkoudi et al., 2020), which have
structural properties (Blázquez-Calero et al., 2019) and pattern speeds
(Fragkoudi et al., 2021) in agreement with those of observed bars, and mainly
consist of so-called pseudo-bulges (Gargiulo et al., 2019), reproducing what
is found for disc galaxies in the local Universe (e.g. Kormendy & Kennicutt
2004; Gadotti 2009).

We note that the full simulation output data for the Auriga simulations has
been made publicly available – see Grand et al. (2024) for more details.
Various data products on barred galaxies in Auriga, collected in this paper,
are now also available with the public data release222The data can be found at
https://wwwmpa.mpa-garching.mpg.de/auriga/.

###  2.1 Barred galaxy sample

In this work, we focus on the galaxies which at $z=0$ have halo masses in the
range of $M_{\rm 200}$ = $0.5\times 10^{12}-2\times 10^{12}\rm{M}_{\odot}$.
This includes the 30 original Auriga halos presented in Grand et al. (2017)
(Au1-Au30), as well as the 10 slightly lower mass galaxies, first presented in
Grand et al. (2019) (Au31-Au40). We exclude Au34 from all the analysis that
follows as it has a low resolution dark matter particle within its virial
radius.

In what follows we focus most of the analysis on the barred galaxies in the
sample, which can be seen in Fig. 1. We determine whether a galaxy is barred
by examining the bar strength, $A_{2}$, which we define as the $m=2$ Fourier
mode of the stellar surface density at $z=0$ (see the next subsection for more
details on how the bar strength is obtained). Our ‘barred’ sample includes all
the galaxies which at $z=0$ have $A_{2}\geq 0.25$. This results in 26 out of
the 39 halos explored here. The other 13 galaxies are classified as the
unbarred sample. For parts of the analysis that follow, we further label the
galaxies that undergo an interaction at $z\sim 0$ with a companion. These are
labeled as ‘interacting’ galaxies and are identified by visually inspecting
for disturbances in the stellar distribution of the galaxies within a lookback
time of $t_{\rm lb}=500$ Myr. These samples are summarised in Table 1 (see
column ‘Classification’) as well as in Fig. 2, where we plot the bar strength
$A_{2}$ as a function of stellar mass (bottom panel), the bar fraction versus
stellar mass (middle panel) and the mass distribution of the three subsamples
(unbarred – blue, barred without interacting – red, and barred including
interacting – dashed brown).

###  2.2 Analysis

Here we describe how we obtain the bar-related parameters for the galaxies in
our sample, such as the bar strength, length and bar formation time.

Bar strength, $A_{2}$: To obtain the bar strength we first re-align the
galaxy, such that the angular momentum vector of the stellar disc is parallel
to the $z$-axis. We then select the stellar particles close to the disc,
within a height of $|z|<0.8\,\rm{kpc}$ above and below the plane. The Fourier
modes of the surface density are given by,

$a_{\rm{m}}(R)=\sum^{N}_{i=1}m_{i}\cos(\rm{m}\theta_{i}),\,\,\,\,\,\,\,\,\,\,m=0,1,2,...,$
(1)

$b_{\rm{m}}(R)=\sum^{N}_{i=1}m_{i}\sin({\rm{m}}\theta_{i}),\,\,\,\,\,\,\,\,\,\,{\rm{m}}=0,1,2,...$
(2)

where m denotes the Fourier mode, $m{{}_{i}}$ is the mass of particle $i$, $R$
is the cylindrical radius, $N$ is the total number of particles at that radius
and $\theta_{i}$ is the azimuthal angle. In order to obtain a single value for
the bar strength we take the maximum value of the relative $m=2$ component
within the inner 10 kpc as,

$A_{2}=\rm
max\left(\frac{\sqrt{\left(a^{2}_{2}(R)+b^{2}_{2}(R)\right)}}{a_{0}(R)}\right).$
(3)

We note that for estimating the bar fraction in the analysis that follows, we
also visually inspect the galaxies, to ensure that if $A_{2}\geq 0.25$, this
is not due, for example, to a transient off-centering of the disc due to a
merger, which can occur frequently at high redshifts.

Bar formation time, $t_{\rm bf}$: In what follows we define the bar formation
time $t_{\rm bf}$ as the first lookback time at which $A_{2}$ becomes, and
subsequently remains, larger than 0.25. In Fig. 3 we show the bar strength
$A_{2}$ as a function of lookback time, for the barred galaxies in our sample,
which we colour-code according to their bar formation formation time.

Bar length, $R_{\rm bar}$: There are various methods for estimating the length
of the bar, from ellipse-fitting methods used widely in observational studies,
to methods using the $m=2$ Fourier mode of the surface density, which are more
commonly employed in numerical studies (see e.g. Erwin 2005; Gadotti 2011;
Hilmi et al. 2020; Fragkoudi et al. 2021; Petersen et al. 2023). Here we use a
method which is well-suited for estimating the bar length in an automated way
for simulated galaxies, across a large number of snapshots, i.e. we use the
radius at which $A_{2}$ falls below 70% of its maximum value. We note that we
have tested various methods for obtaining the bar length, such as a definition
based on the constancy of the phase of the $m=2$ Fourier mode. We found this
latter method to be more prone to noisy bar length estimates. This lead us to
employ the former method, as it is more robust (see also Figure B2 of
Fragkoudi et al. 2021).

Disc size, $R_{\rm d}$: We define the size of the disc for the galaxies in our
sample as the radius at which the stellar surface density drops below
1$\rm{M}_{\odot}/\rm pc^{2}$.

Time at which the galaxy becomes baryon-dominated, $t_{\rm gal,bd}$: We define
the time at which the galaxy becomes baryon-dominated, $t_{\rm gal,bd}$, as
the time at which the stellar mass begins to contribute at least 80% of the
total mass within 5 kpc. In what follows, when referring to the stellar mass
of a galaxy, we use the mass within 10% of the virial radius of the galaxy,
unless otherwise noted.

##  3 Bar properties & evolution

We start by exploring some statistical and global properties of bars in our
sample (Section 3.1), such as the fraction of bars across stellar mass and
redshift, the formation time of bars and their longevity. We then look into
one of the most important bar properties, the bar length, its relation to the
size of the disc, and how these evolve in time (Section 3.2).

###  3.1 Global Properties of Barred Galaxies

Halo |  $A_{2}(z=0)$ |  $t_{\rm bf}$ |  $R_{\rm b}(z=0)$ |  $R_{\rm b}(z=z_{\rm bf})$ |  $R_{\rm d}(z=0)$ |  $R_{\rm d}(z=z_{\rm bf})$ |  $M_{\rm\star}(z=0)$ |  $M_{\rm\star}(z=z_{\rm bf})$ |  Classification
---|---|---|---|---|---|---|---|---|---
|  |  [Gyr] |  [kpc] |  [kpc] |  [kpc] |  [kpc] |  [$10^{10}\rm{M}_{\odot}$] |  [$10^{10}\rm{M}_{\odot}$] |
Au1 |  0.55 |  3.97 |  4.00 |  3.16 |  18.38 |  8.38 |  2.74 |  1.63 |  bar, int
Au2 |  0.43 |  7.29 |  5.49 |  6.82 |  41.88 |  20.04 |  7.04 |  3.55 |  bar
Au3 |  0.11 |  |  |  |  32.12 |  |  7.75 |  |  unbarred
Au4 |  0.10 |  |  |  |  20.38 |  |  7.09 |  |  unbarred
Au5 |  0.35 |  2.17 |  3.84 |  2.62 |  20.62 |  18.04 |  6.72 |  6.02 |  bar
Au6 |  0.20 |  |  |  |  27.62 |  |  4.75 |  |  unbarred
Au7 |  0.29 |  1.66 |  3.77 |  1.92 |  22.38 |  20.88 |  4.87 |  3.68 |  bar
Au8 |  0.18 |  |  |  |  27.88 |  |  2.99 |  |  unbarred
Au9 |  0.56 |  6.50 |  5.92 |  1.97 |  19.38 |  11.79 |  6.10 |  3.76 |  bar
Au10 |  0.60 |  6.17 |  4.88 |  4.85 |  14.88 |  10.29 |  5.94 |  2.85 |  bar
Au11 |  0.52 |  4.91 |  4.95 |  2.68 |  20.38 |  15.21 |  5.56 |  2.66 |  bar, int
Au12 |  0.38 |  2.67 |  3.05 |  2.51 |  18.88 |  18.38 |  6.01 |  4.39 |  bar
Au13 |  0.38 |  3.45 |  1.73 |  1.65 |  15.62 |  15.04 |  6.19 |  4.38 |  bar
Au14 |  0.37 |  1.47 |  4.37 |  2.74 |  22.38 |  19.71 |  10.4 |  8.98 |  bar
Au15 |  0.10 |  |  |  |  23.12 |  |  3.93 |  |  unbarred
Au16 |  0.10 |  |  |  |  36.88 |  |  5.41 |  |  unbarred
Au17 |  0.55 |  10.37 |  6.37 |  5.24 |  19.12 |  11.54 |  7.61 |  1.68 |  bar
Au18 |  0.38 |  9.16 |  5.33 |  3.81 |  20.88 |  12.38 |  8.04 |  2.35 |  bar
Au19 |  0.08 |  |  |  |  25.88 |  |  5.32 |  |  unbarred
Au20 |  0.49 |  6.68 |  6.88 |  2.80 |  28.38 |  15.88 |  4.63 |  2.01 |  bar, int
Au21 |  0.22 |  |  |  |  24.38 |  |  7.72 |  |  unbarred
Au22 |  0.43 |  8.09 |  5.37 |  5.88 |  13.62 |  11.04 |  6.02 |  2.70 |  bar
Au23 |  0.42 |  5.06 |  8.46 |  3.64 |  26.12 |  16.96 |  9.02 |  6.27 |  bar
Au24 |  0.63 |  8.53 |  4.76 |  5.00 |  31.62 |  13.21 |  6.55 |  2.65 |  bar
Au25 |  0.28 |  0.34 |  1.77 |  2.67 |  26.12 |  25.12 |  3.14 |  3.08 |  bar, int
Au26 |  0.46 |  5.38 |  4.53 |  3.80 |  19.12 |  15.46 |  11.0 |  6.55 |  bar
Au27 |  0.42 |  2.97 |  4.96 |  3.15 |  24.62 |  20.62 |  9.61 |  8.21 |  bar
Au28 |  0.51 |  2.79 |  6.32 |  4.37 |  18.12 |  15.88 |  10.4 |  8.80 |  bar
Au29 |  0.19 |  |  |  |  18.62 |  |  9.03 |  |  unbarred
Au30 |  0.48 |  0.81 |  4.60 |  4.11 |  18.62 |  19.38 |  4.25 |  3.86 |  bar, int
Au31 |  0.13 |  |  |  |  19.88 |  |  1.80 |  |  unbarred, int
Au32 |  0.54 |  3.15 |  4.70 |  3.27 |  18.88 |  21.12 |  2.12 |  1.24 |  bar, int
Au33 |  0.35 |  1.14 |  4.70 |  6.34 |  23.12 |  19.21 |  4.50 |  3.16 |  bar, int
Au35 |  0.16 |  |  |  |  19.38 |  |  1.94 |  |  unbarred, int
Au36 |  0.22 |  |  |  |  20.88 |  |  3.04 |  |  unbarred, int
Au37 |  0.13 |  |  |  |  25.38 |  |  2.81 |  |  unbarred
Au38 |  0.40 |  3.45 |  5.88 |  3.40 |  19.12 |  18.12 |  5.05 |  3.69 |  bar
Au39 |  0.52 |  5.22 |  6.52 |  4.20 |  21.12 |  13.04 |  2.68 |  1.87 |  bar
Au40 |  0.60 |  7.12 |  5.07 |  5.47 |  21.88 |  11.46 |  3.46 |  1.74 |  bar, int
Table 1:  Properties of barred galaxies: (1) Halo name; (2) $A_{2}$ the bar
strength at $z=0$ for the 39 galaxies in this sample; (3) the lookback time of
bar formation ($t_{\rm bf}$) in Gyr; (4) The length of the bar at $z=0$, (5)
the length of the bar at the time of bar formation, (6) the size of the disc
at $z=0$, (7) the size of the disc at the time of bar formation, (8) the
stellar mass of the galaxy at $z=0$, (9) the stellar mass of the galaxy at the
time of bar formation, (10) classification of the galaxy as barred, unbarred
and whether it falls within our definition of interacting at $z=0$ (int; see
text for more details).

Bar Formation Time: In Fig. 3 we show the $m=2$ Fourier mode for the barred
galaxies in our sample, as a function of time, colour-coded according to their
bar formation time. We see that bars have a wide range of formation times (see
also Table 1), with the oldest bar in our sample forming at a lookback time of
$t_{\rm lb}=10.4$ Gyr and the youngest bar forming at $t_{\rm lb}=0.3\,$Gyr.
It is worth emphasizing that, from $z\sim 1-2$, bars in Auriga are long-lived
structures, i.e. they do not dissolve and are not recurring. This is evident
in Fig. 3 where we see that once a bar forms, i.e. once the bar strength is
above $A_{2}>0.25$, it remains above 0.25 and does not go through cyclical
periods of destruction and reformation. The single exception to this rule is
Au36, whose bar is severely weakened by an interaction with another galaxy,
resulting in it being defined as unbarred by $z=0$ (see Fig. 18; we will
explore this system in more detail elsewhere).

Bar fraction: In Fig. 4 we explore the fraction of bars as a function of
redshift (and lookback time) in our sample, comparing this to observations
from Eskridge et al. (2000), Sheth et al. (2008) and Le Conte et al. (2024).
We find that $60\%$ of the galaxies in our sample at $z=0$ are barred333This
is slightly lower than the fraction found in Fragkoudi et al. (2020), where we
only included the 30 high mass Auriga halos, while here we use $A_{2}>0.25$ to
identify bars, as opposed to 0.2 which was employed in our previous study.,
with the fraction of bars decreasing for higher redshifts and reaching about
$\sim 20\%$ at $z\sim 3$, in agreement with the aforementioned observations.
Beyond $z\sim 1$ there appears to be a slight increase in the bar fraction in
our simulations (although not that this is within the uncertainties),
indicating that some structures identified as bars at high redshifts might be
transient.

The right panel of Fig. 4 shows the bar fraction as a function of lookback
time in two stellar mass bins, with masses greater than, and smaller than
$M_{\star}=5\times 10^{10}\rm{M}_{\odot}$. The right $y$-axis and grey line
indicate the fraction of galaxies in the low mass bin. We find that the bar
fraction is higher in the high mass bin at all lookback times at which these
galaxies exist in our sample. This emphasizes that at all times, bars are more
common in massive galaxies. These findings seem to be compatible with the mass
downsizing hypothesis, i.e. that more massive galaxies form bars first (see
for example Sheth et al. 2008), but as we show below, this does not
necessarily imply a straightforward correlation between the stellar mass of
the galaxy and bar formation time.

Mass Downsizing and Correlations with $t_{\rm bf}$: In the top panel of Fig. 5
we show the bar strength at $z=0$ as a function of the bar formation time. We
find a weak correlation between the bar strength at $z=0$ and the age (or
formation time) of the bar, indicating that older bars tend to be slightly
stronger. In the bottom panel of the Figure we show the bar formation time as
a function of the time at which the galaxy becomes baryon dominated (see
Section 2.2 for details). We see that there is a correlation between these
quantities, suggesting that galaxies form a bar when the galaxy passes a
threshold of baryon-dominance (and see also Section 4.1).

In Figure 6 we explore whether the time of bar formation is correlated to the
stellar mass of the galaxy at $z=0$, as one might naively expect in a mass
downsizing scenario. In the middle panel we show the formation time of the bar
as a function of the stellar mass of the galaxy at $z=0$, in which we do not
find a correlation. Nor do we find a correlation between bar formation time
and the stellar mass of the galaxy at the time of bar formation, as is shown
in the right panel of Figure 6 – indeed we find a weak anti-correlation
between these quantities. This can be understood since bars that form at
higher redshifts will form in galaxies that have not yet fully assembled their
stellar mass. On the other hand, we see that the formation time of the bar
correlates (albeit weakly) with how much the stellar component dominates over
the dark matter component within 5 kpc at $z=0$, as can be seen in the left
panel of Figure 6. This indicates that at $z=0$, we can relate how old a
galaxy’s bar is, with how baryon-dominated that galaxy is. It is worth noting
the scatter in this relation, which increases at older ages, likely due to
effects related to mergers which, as we will see below, can also act as
triggers of bar formation.

###  3.2 Evolution of Bar Properties

In this section we explore the evolution of bar properties in Auriga as a
function of time, focusing in particular on the length of bars and the ratio
of bar length to disc size.

In Figure 7 we show the evolution of bar length as a function of time. In
particular, we group together bars according to their formation time, i.e. we
select bars that form between lookback times of $t_{\rm lb}=0-2.7$ Gyr ago
(dark purple), $t_{\rm lb}=2.8-4.9$ Gyr (dark blue), $t_{\rm lb}=5-7.1$ Gyr
(dark green) and $t_{\rm lb}=7.2-11$ Gyr (light green), as indicated by the
figure inset. These bins are chosen such that they split the sample into 4
groups with approximately the same number of galaxies. We find that bars that
form at low redshifts, tend to form shorter bars which subsequently grow
longer over time. On the other hand, bars that form at higher redshifts tend
to already be quite long when they form, and their length does not evolve much
over the galaxy’s lifetime. This is likely due to the fact that galaxies that
form bars at high-$z$ are triggered by a merger, and therefore form long and
“saturated”, i.e. they are not able to grow by exchanging more angular
momentum. Indications of this can be seen in the inset of Fig. 7, where we
show the time of one of the three most massive mergers in the galaxy, versus
the time of bar formation. We see that the galaxies that form bars at high
redshift all do so around the time of one of their most massive mergers.

We quantify these trends further in Fig. 8, where we plot the bar length of
galaxies as a function of the formation time of the bar, in the left panel at
the time of bar formation, and in the middle panel at $z=0$. In the right
panel we show the relative change in the bar length from the time of bar
formation until $z=0$ for each galaxy. This quantifies the bar growth for the
various galaxies, reinforcing the fact that the oldest bars undergo less
growth – as they already form long – while the bars that undergo the most
growth are those of intermediate ages. The youngest bars also grow in time,
but as they have less time to evolve than the intermediate age bars, their
relative growth is limited.

Given that galaxies tend to have smaller discs at higher redshifts (e.g. van
der Wel et al. 2014; Ormerod et al. 2024), it might be somewhat surprising
that bars that form early on in the Universe in our simulations have longer
bars. This would imply that the ratio of bar length-to-disc size changes as a
function of redshift. We explore the ratio of bar length to disc size in Fig.
9; we focus here on the non-interacting barred galaxies, since interacting
galaxies have disturbed discs and as such the concept of the disc size becomes
ambiguous. In the left panel of the figure we show the ratio of bar length-to-
disc size _at the time of bar formation_ for each of the galaxies, as a
function of bar formation time. We find a trend in which this ratio is higher
for galaxies whose bars form at higher redshifts. We therefore see that when
bars form at high redshifts they extend over a larger range of the disc. In
the middle panel we show this ratio at $z=0$, and in the right panel we show
the change in the ratio from the time of bar formation until $z=0$. In the
middle panel it is clear that the bar length-to-disc size relation flattens
out by $z=0$. By examining the middle and right panels of the figure, we see
that the ratio decreases for the older bars, due to the fact that the disc
size increases for these galaxies, while the bar length doesn’t increase
significantly (as shown in Figs. 7 and 8). On the other hand, for the younger
bars the ratio increases at $z=0$, which is due to the increase in the bar
length for intermediate and young bars over time. Therefore, when observing
bars at higher redshift, we might expect to find a higher bar-to-disc size
ratio than in the local Universe.

##  4 Drivers of bar formation

In this section we explore the various factors that are thought to be
important for the formation of bars – namely the baryon-dominance (4.1),
accreted bulge component (4.2), gas fraction (4.3) and the Toomre Q parameter
(4.4) – examining the role these play in bar formation within the cosmological
context.

###  4.1 Baryon dominance

Figure 10: _Left:_ The loci of galaxies in our sample in the stellar mass -
halo mass plane at $z=0$. We indicate whether the galaxy is unbarred (blue
circle), barred (red square) or barred and undergoing an interaction at $z=0$
(cross). The abundance matching relation from Moster et al. (2013) is
indicated with the blue line and shaded region. _Top Right:_ The distance of
galaxies from the abundance matching relation for barred (red), unbarred
(blue) and barred+interacting (dashed brown) galaxies. _Bottom Right:_ The
baryon-dominance as a function of radius of the barred (without interacting
galaxies; red) and unbarred (blue) galaxies in our sample, as indicated by
$V_{\star}$/$V_{\rm{tot}}$. We compare these to the Milky Way model from Bovy
& Rix (2013) (black). While barred galaxies are on average more baryon-
dominated than unbarred galaxies in Auriga, both are less baryon-dominated
than the MW14 model for the Milky Way.  Figure 11: Build-up of stellar and
dark matter mass for barred and unbarred galaxies. _Left:_ The ratio of
stellar over dark matter mass within 5 kpc for barred (red) and unbarred
(blue) galaxies in the Auriga suite of simulations. On average, for barred
galaxies, the stellar mass dominates over the dark matter mass within the
central 5 kpc by $z\sim 0$, which is in contrast to unbarred galaxies which
tend to be more dark matter dominated over all cosmic times. _Right:_ The
build-up of stellar (solid) and dark matter (dashed) mass within 5 kpc for
barred (red) and unbarred (blue) galaxies. Barred galaxies assemble their
stellar mass earlier on and form on average 3 Gyr before their unbarred
counterpart. Figure 12: Average fraction of accreted stars in the ‘bulge’, at
$z=0$, for the barred (without and with interacting galaxies, left and middle)
and unbarred (right) galaxies in our sample. Unbarred galaxies have on average
$\sim 2\times$ more ex-situ stars in their bulge than the barred sample
(without interacting galaxies) and 60% more ex-situ stars in the bulge than
the total barred sample. Figure 13: The relation between the $M_{\star}-M_{\rm
h}$ plane and the fraction of ex-situ stars in the central 4 kpc. _Left:_ The
$M_{\star}-M_{\rm h}$ plane colour-coded by the accreted fraction of stars in
the bulge. _Right:_ The fraction of accreted stars in the bulge as a function
of the offset from the abundance matching relation for the unbarred galaxies
(top) and the barred galaxies (bottom). We see that galaxies that are more
offset from the abundance matching relation tend to have higher ex-situ
fractions, indicating that the ex-situ bulge contributes to suppressing bar
formation in baryon-dominated systems. Figure 14: The effect of gas fraction
on bar formation: _First row:_ At $z=0$, the gas fraction for each halo as a
function of stellar mass (left) and as a function of the baryon-dominance of
the galaxy (right). The vertical dashed lines indicate the range in which
galaxies are considered for the averages shown in the panel below. _Second
row:_ The average gas fraction for various samples: all galaxies, galaxies
with the $M_{\star}$ and $\delta M_{\star}$ cuts shown above, separating into
all the bars, bars excluding interacting galaxies and unbarred galaxies.
_Third row:_ The gas fraction for all barred and unbarred galaxies in our
sample across time. _Fourth row:_ As above, but for galaxies within the mass
range shown in the top left panel. If we consider galaxies in a given mass
bin, the gas fractions are the same for barred and unbarred galaxies.  Figure
15:  The Toomre Q parameter for the galaxies in our sample. _Top:_ Toomre Q
inside stellar $r_{50}$ at the time of bar formation, for barred (red squares)
and unbarred (blue circles) galaxies, as a function of the stellar mass of the
galaxy at that time (note that for the unbarred galaxies we take these values
at the time corresponding to the average bar formation time – see text).
Unbarred galaxies tend to have higher Toomre Q than barred galaxies at the
time of bar formation for all stellar masses. _Bottom:_ The average Toomre Q
inside $r_{50}$ for barred (red) and unbarred (blue) galaxies in our sample as
a function of time.

It is well established that the ratio of the stellar disc to the dark matter
halo mass within a spiral galaxy is a determining factor in how unstable the
galaxy will be to the formation of a bar (e.g. Ostriker & Peebles 1973; Combes
et al. 1990; Athanassoula 2002; Fujii et al. 2018). Here we explore how the
baryon-dominance of a galaxy affects the formation of a bar in the Auriga
cosmological simulations.

As discussed in Section 3.1, in the bottom panel of Fig. 5 we compare the bar
formation time to the time at which the galaxy has at least 80% of the total
mass within 5 kpc dominated by the stellar component (which we define as the
time at which the galaxy becomes baryon-dominated). We find that there is a
correlation – albeit with significant scatter – between the time at which the
bar forms and the time at which the galaxy becomes baryon-dominated,
suggesting that once a disc becomes baryon-dominated it will tend to form a
bar.

To further explore the role baryon-dominance plays on bar formation in the
cosmological context, in Fig 10, we show the stellar mass against the halo
mass for the galaxies in our sample, which are labeled according to whether
they are barred, unbarred, or have a bar but are also interacting at $z\sim
0$. We also plot the abundance matching relation from Moster et al. (2013) for
comparison. We find that barred galaxies tend to be displaced from the
commonly used abundance matching relation, as compared to unbarred galaxies,
which lie closer to the relation. We quantify this offset from the abundance
matching relation in the top right panel of Fig. 10: we show the vertical
displacement of the galaxy from the abundance matching relation (i.e. the
over- or under-abundances in stellar mass as compared to the abundance
matching relation, for a given halo mass), normalised by the $1\sigma$ scatter
of the abundance matching relation of Moster et al. (2013). The barred galaxy
sample is shown in red, the unbarred sample in blue, and the barred sample
including the interacting galaxies with the dashed brown line. We see that
barred galaxies tend to be more than 2$\sigma$ offset from the abundance
matching relation, compared to unbarred galaxies which lie closer to the
abundance matching relation. The vertical arrows indicate the median value for
each population. It is worth noting, that this ’offset’ from the abundance
matching relation correlates to the baryon-dominance $f_{\rm bd}$ of the
galaxy, as can be seen in Fig. 19. The bottom right panel of Fig. 10 shows the
$V_{\rm\star}$/$V_{\rm tot}$ of the barred and unbarred galaxies in our
sample, and in black for the Milky Way (as determined by Bovy & Rix 2013). We
see that barred galaxies have higher $V_{\star}$/$V_{tot}$ than unbarred
galaxies at all radii. However, this is not as high as that for the Milky Way,
suggesting that Milky Way-like galaxies in cosmological simulations might
still be less baryon-dominated than massive spiral galaxies in the local
Universe. This fact, i.e. that cosmological simulations are not as baryon-
dominated as suggested by dynamical models of massive spiral galaxies in the
local Universe (e.g. Kranz et al. 2003; Weiner et al. 2001; Fragkoudi et al.
2017a), is commonly found in the literature (Lovell et al., 2018; Posti et
al., 2019; Posti & Fall, 2021; Fragkoudi et al., 2021).

Fig. 11 explores the ratio of stellar-to-dark matter within 5 kpc ($f_{\rm
bd}$) as a function of time. In the left panel of the figure we see that
already at earlier lookback times, the barred galaxies are more baryon
dominated than the unbarred sample. In the right panel we show the growth of
the stellar (solid) and dark matter (dashed) mass of the barred (red) and
unbarred (blue) galaxies. We see that barred galaxies assemble their stellar
mass more rapidly than unbarred galaxies (see also Rosas-Guevara et al. 2020),
while the dark matter mass is assembled similarly when comparing the barred
and unbarred galaxies. We quantify this by exploring when the samples assemble
50% of their stellar mass, and find that barred galaxies assemble their
stellar mass on average $\sim$3 Gyr earlier than unbarred galaxies. Barred
galaxies therefore tend to grow their stellar component more rapidly than
unbarred galaxies, likely forming an early massive stellar disc, which is
conducive to bar formation.

###  4.2 The ex-situ bulge

We now explore how the fraction of ex-situ (accreted) stars in the central
regions of galaxies affects the formation of bars. The fraction of accreted
stars in the inner regions can be thought of as a proxy for a merger-built hot
(classical) component, since these accreted stars will be brought in via
mergers. For example, Gargiulo et al. (2022) showed that simulated galaxies
whose bulge has a high Sérsic index will also tend to have a larger fraction
of ex-situ. As hot spheroids (either stellar or dark matter) have been shown
to delay the formation of bars (e.g. Ostriker & Peebles 1973; Athanassoula
2002), one might expect such an ex-situ component to have an impact on bar
formation in cosmological simulations. We use the fraction of ex-situ stars
rather than a kinematic characterisation of the bulge because commonly-
employed kinematic estimates (such as the circularity parameter Abadi et al.
2003) are affected by the presence of a bar (which induces large non-circular
motions in the orbits of stars in the central region). This makes it
challenging to separate the contribution of the spheroid from the bar. In what
follows we consider the accreted fraction within the inner 5 kpc, but we note
that our results are qualitatively similar when considering different radii.

We start by comparing in Fig. 12 the fraction of accreted stars in the bulge,
for the barred and unbarred galaxies in our sample (see also Fragkoudi et al.
2020). We compare unbarred galaxies to barred galaxies (excluding interacting
galaxies at $z=0$) and find that unbarred galaxies have $\times$2 higher
fraction of accreted stars; this supports the previous theoretical results
that have found that massive classical bulges can impede bar formation. The
difference in the accreted bulge fraction between barred and unbarred galaxies
becomes somewhat less prominent if we include the interacting galaxies in the
barred sample (see middle column of Fig. 12), however the unbarred galaxies
still have $\sim 50\%$ higher accreted fraction in the bulge as compared to
the total barred sample. Similar results were found previously in Fragkoudi et
al. (2020) for the Auriga galaxies and in Gargiulo et al. (2022) for the
IllustrisTNG50 simulations, who found that galaxies with a low fraction of ex-
situ stars in their bulge are more likely to host a long-lived bar.

We further investigate whether the accreted fraction can prevent galaxies from
forming bars by exploring how this relates to the baryon-dominance of
galaxies. As discussed in the previous section, galaxies that are more baryon-
dominated are more likely to form a bar. However, there are some galaxies in
our sample which _are_ baryon-dominated (and offset from abundance matching
relation – see Fig. 10), which have not formed a bar by $z=0$, indicating that
something may be preventing them from forming a bar. In Fig. 13 we explore how
the offset from the abundance matching relation depends on the accreted
fraction of stars for unbarred galaxies. In the left panel we plot the
$M_{\star}-M_{\rm h}$, colour-coded by the faction of accreted stars. We see
that galaxies that are baryon-dominated, i.e. are offset from the abundance
matching relation, which do not host a bar, also tend to have a higher
fraction of accreted stars. This is shown explicitly in the top right panel of
Fig. 13, where we plot the fraction of accreted stars versus the offset from
the abundance matching relation for the unbarred galaxies in our sample. While
there isn’t a strong correlation between the two parameters, we find that
galaxies which are more offset from the abundance matching relation tend to
have a higher fraction of accreted stars. This seems to suggest that having a
higher fraction of accreted stars could suppress the formation of the bar,
even when the galaxy is baryon-dominated. On the other hand, there is no
obvious trend for the barred galaxies with accreted fraction (see bottom right
panel of Fig. 13), and as already mentioned, these galaxies tend to have lower
accreted fractions.

###  4.3 Gas Fraction

In Fig. 14 we show the fraction of gas in barred and unbarred galaxies at
$z=0$ within a radius of $r=5$ kpc, in order to explore the effect it has on
bar formation in cosmological simulations. We focus on the gas fraction within
the inner regions of the disc where the bar is, as that is where it will have
the largest effect on the bar dynamics. In the top left column, we show the
gas fraction as a function of stellar mass. In the top right column we show
the gas fraction as a function of the offset from the abundance matching
relation (which as discussed previously, correlates with the baryon-
dominance). We see that there is an overall decrease in the gas fraction for
higher stellar mass galaxies, while a similar decrease in gas fraction can be
seen as a function of baryon-dominance, with both barred and unbarred galaxies
occupying similar regions in these planes. Interestingly, in the top panel, we
see that the barred galaxies that have lower masses (and high gas fractions)
are also the ones that appear to have had a recent interaction.

In the middle left panel of the figure we show the average gas fraction in the
barred sample, with and without interacting galaxies (square and triangle
points respectively) and unbarred galaxies (circle), in the leftmost point
(labelled ‘All halos’). We find that when considering all galaxies in our
sample, unbarred galaxies tend to have higher gas fractions (by about $\sim
13\%$) than barred galaxies. This is accentuated (to $\sim$60% higher gas
fraction) when comparing unbarred galaxies to barred galaxies which are not
interacting. This might lead one to conclude that the gas fraction is higher
for unbarred galaxies; however, as there is a dependence of bar fraction on
stellar mass – see e.g. in Fig. 2 – we repeat this measurement by factoring
out the mass dependence. That is to say, since there is an anti-correlation
between gas fraction and stellar mass (i.e. low mass galaxies tend to have
higher gas fractions), it is unclear whether the average higher gas fractions
of unbarred galaxies could be due simply to the fact that low mass galaxies in
our sample tend to be unbarred. To factor out the stellar mass dependence, we
apply a mass cut as indicated by the vertical dashed lines in the top left
panel, and recalculate the average gas fraction for our three samples in that
mass range; this is shown in the middle left panel (labelled ‘$M_{\rm\star}$
cut’). We also explore the average gas fraction for a given baryon-dominance
cut (labelled ‘$\delta M_{\star}$ cut’), with the restricted range shown with
the vertical lines in the top right panel. We find that both when restricting
our sample to a smaller range in stellar mass or baryon-dominance, the average
gas fractions of barred and unbarred galaxies are very similar, differing by
less than 3%.

In the two bottom panels of Fig. 14 we show the gas fraction within 5 kpc for
barred and unbarred galaxies as a function of time. In the second to last
panel, we see that unbarred galaxies tend to have slightly higher gas
fractions across time. However, in the bottom panel we see that, if we
restrict the sample to galaxies within the same mass bin (as in the top left
panel of Fig. 14), we find similar gas fractions for both barred and unbarred
galaxies across all cosmic times.

We therefore conclude that, gas fraction does not seem to be a driving factor
of bar formation in and of itself, since if we restrict our samples to a given
stellar mass, or indeed a given baryon-dominance, barred galaxies do not tend
to be more gas rich. Rather the difference in average gas fractions in the
barred and unbarred sample over all stellar masses, seems to be a consequence
of the dependence of gas fraction on stellar mass, which sets how baryon-
dominated the galaxies are.

###  4.4 The Toomre Q parameter

A parameter which is known to play an important role in the formation of bars
is the in-plane radial velocity dispersion of the stellar component (e.g.
Combes & Sanders 1981; Athanassoula & Sellwood 1986), which affects how cold
the disc is, and therefore its stability. This has often been explored in the
literature by examining the dependence of bar properties on the Toomre $Q$
parameter (Toomre, 1964),

$Q=\frac{\sigma_{r}\kappa}{3.36G\Sigma_{\star}}$ (4)

where $\sigma_{r}$ denotes the radial velocity dispersion of the stellar disc,
$\kappa$ the epicyclic frequency, $G$ Newton’s gravitational constant and
$\Sigma_{\star}$ is the surface density of the stellar disc.

In the top panel of Fig. 15 we show the average Toomre $Q$ parameter within
the stellar half mass radius of the barred and unbarred galaxies in our sample
at the time of bar formation, as a function of stellar mass (for the unbarred
galaxies, we calculate this parameter at 4.6 Gyr, which corresponds to the
average time of bar formation for the galaxies in our sample). We see that the
unbarred galaxies have a higher Toomre $Q$ parameter than the barred galaxies.
This occurs for all stellar masses.

In the bottom panel of Fig. 15 we plot the average and 1$\sigma$ dispersion
for $Q$ within the half-mass radius for barred (red) and unbarred (blue)
galaxies, as a function of time. We see that unbarred galaxies tend to have
higher $Q$ across all times, although the difference between the two samples
decreases at lower redshifts. This is likely due to the fact that bars
increase the radial velocity dispersion in the inner galaxy, and as such drive
up the $Q$ parameter.

We therefore find that the in-plane radial velocity dispersion – and therefore
the stability of the disc – does play a role in determining whether a galaxy
will be barred or unbarred in cosmological simulations. As the galaxy’s
formation history will determine the radial velocity dispersion within the
galaxy (both in the innermost regions and in the disc), the relation between
the galaxy’s formation history and Toomre Q is worth exploring in future work.

##  5 Discussion

Figure 16: The stellar Tully-Fisher relation for galaxies in Auriga barred
galaxies (excluding interacting) and unbarred galaxies. _Left:_ The grey dot-
dashed line shows the relation given by fitting all galaxies together, while
the shaded region indicates the intrinsic vertical scatter $\sigma_{y}$.
_Right:_ The red (blue) dashed line shows the relation when fitting only the
barred (unbarred) galaxies. The shaded regions indicate the uncertainties from
boostrapping (see text).We find that the TF relation for barred galaxies is
offset from that of the unbarred galaxies. Figure 17: The multi-dimensional
parameter space of bar formation: dark matter halo mass $M_{\rm DM}$, stellar
mass $M_{\star}$, baryon-dominance $f_{\rm bd}$, fraction of accreted stars in
the central regions (bulge) $f_{\rm acc}$, the Toomre parameter within the
half-mass radius $Q_{r_{50}}$ at the time of bar formation and the gas
fraction at $z=0$ $f_{\rm gas}$. The red (blue) points indicate the barred
(unbarred) sample, including interacting galaxies. We see that the barred-
unbarred populations separate out for the baryon dominance, accreted bulge
fraction and Toomre $Q$ parameters.

In what follows we discuss the implicatios of our results in terms of the
baryon-dominance of galaxies on the Tully-Fisher (TF) relation (Section 5.1)
and what our findings about the evolution of bars imply about bars at high
redshift (Section 5.2).

###  5.1 Implications for the Tully-Fisher relation

The Tully-Fisher relation (TF; Tully & Fisher 1977) is an empirical relation
between the rotation velocity of galaxies – often the radius at which the
rotation curve flattens out – versus their luminosity. This relation has low
intrinsic scatter, making it a powerful tool for determining distances to
galaxies, while it has also proved useful for testing galaxy formation models
in $\Lambda$CDM (Mo et al., 1998) and theories of modified gravity, such as
Modified Newtonian Dynamics, or MOND (see e.g. Milgrom 1983; McGaugh 2012). In
Fig. 16 we explore the consequences of our findings in Section 4.1 – i.e. the
fact that barred galaxies tend to be more baryon-dominated than unbarred
galaxies – on the stellar TF relation (STFR; in Appendix Bar formation and
evolution in the cosmological context: Inputs from the Auriga simulations we
show results for the Baryonic Tully Fisher relation). We obtain the circular
velocity of the galaxies in our sample at 30 kpc – which corresponds roughly
to the flat part of the rotation curve for the galaxies we are exploring, i.e.
$V_{\rm flat}$. We then fit the relation $\log{M_{\star}}=m\log{V_{\rm
flat}}+y_{0}$ to our sample of barred galaxies (excluding interracting
galaxies) and unbarred galaxies. In the left panel of Fig. 16 we show a least
squares fit to this sample (grey dashed curve). The vertical scatter of the
relation in our sample is $\sigma_{y}=0.05$dex, as indicated with the grey
shaded region. In the right panel of the Figure we fit separately the barred
(red) and unbarred (blue) samples. For these fits, we fix the slope of the
relation to be that obtained for the total sample and we perform bootstrapping
to assess the uncertainties in the fits. The 95% percentile uncertainty in the
intercept, as obtained form our fits, is shown with the shaded regions. We
find an offset in the STFR of the barred galaxies as compared to that of the
unbarred galaxies, of the order of $\Delta y_{0}=0.05$dex. This offset is
comparable to the intrinsic scatter of the entire sample, suggesting that,
while challenging, it could in principle be observed.

We now turn our attention to what observational studies have found regarding
the barred and unbarred TF relation. There have been few of these carried out
in the literature, often finding antithetical results. For example, the study
of Sakai et al. (2000) found that the TFR of barred galaxies is offset from
the unbarred one, suggesting that barred galaxies have maximal discs. The
subsequent study of Courteau et al. (2003) explored this in two separate
samples, the Shellflow and SCII surveys. Courteau et al. (2003) found a
difference in the TFR for barred and unbarred galaxies in the Shellflow
survey, while they did not find any offset in the SCII sample. Due to the
marginally larger number of galaxies in the latter sample, the authors
concluded that there is no difference between the barred and unbarred TFR. We
note that these studies did not have resolved HI rotation curves, which would
therefore increase the scatter in the relation (e.g. Ponomareva et al. 2017).
Therefore, the question of whether the barred TFR is offset from the unbarred
one in observational samples still remains, and would be worth exploring with
more recent and larger samples.

It is worth emphasizing, that should such a trend be confirmed in the observed
barred and unbarred TFR, this would be a strong indication for the existence
of dark matter, since such an offset cannot be naturally explained in a
modified gravity framework, such as in Modified Newtonian Dynamics (MOND;
Milgrom 1983; McGaugh 2012). In a MONDian framework, the $V_{\rm flat}$ would
be exactly proportional to the enclosed mass, and as such the relation should
not have any intrinsic scatter (see e.g. McGaugh et al. 2000). Finding an
offset between barred and unbarred galaxies would be indicative of intrinsic
scatter in the relation, which is more naturally explained in a framework with
dark matter, i.e. where barred galaxies are more baryon-dominated in the inner
regions, with the stellar component of the disc dominating over the dark
matter halo.

###  5.2 Bars at high-$z$

A number of observational studies have explored the fraction of barred
galaxies between redshifts of $z=0-1$, with most studies finding a declining
bar fraction as a function of redshift (Abraham et al., 1999; Sheth et al.,
2008; Melvin et al., 2014; Simmons et al., 2014), although some studies find
constant bar fractions (specifically for strong bars) (Jogee et al., 2004).
Studies using data from the Hubble Space Telescope (HST) have found that by
$z=1$, the bar fraction drops to $\sim 10\%$ (e.g. Simmons et al. 2014).
Recent studies using JWST data have been revealing barred galaxies at $z>1$
(e.g. Guo et al. 2023), and shedding new light on the bar fraction at these
redshifts. For example, a recent study by Le Conte et al. (2024) – covering
the redshift range $z=1-3$ with data from the Cosmic Evolution Early Release
Science Survey (CEERS; Finkelstein et al. 2023) – found that compared to
previous HST results, the fraction of bars is higher by a factor of $\sim 2$
when considering the higher sensitivity JWST data, finding a bar fraction of
$20\%$ at $z\sim 1-2$. At $z\sim 2-3$ the bar fraction drops to $15\%$. As
shown in Fig. 4, these trends are reproduced in the Auriga simulations, with a
decreasing fraction of barred galaxies as a function of redshift. It is
important to note that our sample covers a small mass range at $z=0$, and
therefore our results do not make predictions for the behaviour of lower mass
galaxies.

Cosmological simulations with different galaxy formation models do not all
agree on the bar fractiono at low and high redshfits. For example, some
studies find a dearth of bars at high redshifts, which is exacerbated at low
redshifts (e.g. Reddish et al. 2022). Other studies of bar fractions using
‘big box’ cosmological simulations (e.g. Peschken & Łokas 2019; Rosas-Guevara
et al. 2022; Gargiulo et al. 2022) find a flat, or rising bar fraction from
$z=0$ to $z\sim 1$, in contrast to most observational studies of bar fractions
which find a decreasing bar fraction. One possible explanation for this
mismatch (as mentioned in the latter studies) is that observational studies
are not able to find short bars at higher redshifts, due to resolution limits.
However, another possible explanation for this mismatch could be that the
galaxy formation models employed affect the bar fraction across redshift.
Whether the bar fraction remains flat/rising or decreases towards higher
redshifts will therefore be an important point to settle in upcoming
observational studies with higher spatial resolution and sensitivity, for
example by using data from facilities such as JWST.

In the mass range probed by our simulations, we find that bars that form at
higher redshifts tend to form longer than their local Universe counterparts.
This occurs because bars at high redshifts tend to be triggered by
interactions and/or mergers, which might cause the bars to form more
‘saturated’, i.e. longer. As shown in Fig. 9 this results in these galaxies
having a higher ratio of bar length relative to the disc size, as compared to
local galaxies. Therefore – for stellar masses in the range probed by our
sample – we might expect to find that observed high redshift bars (e.g. with
JWST) are relatively long, and that the bar length-to-disc size ratio is
larger than that observed in the local Universe.

###  5.3 The multi-parameter space of bar formation

In this work, we have explored various galaxy properties which are thought to
affect bar formation, in order to gain a better understanding of the role
these parameters play when taken into account within the global cosmological
context. This can help pave the way towards establishing the multi-parameter
space that sets whether a galaxy will form a bar or not within the
cosmological setting, which could have a number of useful applications (e.g.
for determining whether a galaxy will form a bar in semi-analytic cosmological
simulations).

We attempt to summarise our findings in Fig. 17, where we show the
distribution of barred and unbarred galaxies in different combinations of the
parameter space we have explored in the previous sections. We find that, as
discussed, the main parameters that separate out the barred-unbarred
population are mainly the baryon-dominance $f_{\rm bd}$, the fraction of
accreted stars within the inner region (bulge) of the galaxy $f_{\rm acc}$ and
the Toomre $Q$ parameter of the inner galaxy $Q_{r_{50}}$, at the time of bar
formation, although we note that there is significant overlap between in which
galaxy will be barred or unbarred in either one of these properties
separately. However, for some of properties, for example the gas fraction
$f_{\rm gas}$ or the dark matter halo mass $M_{\rm DM}$, the barred and
unbarred samples are completely overlapping. In future, this multi-dimensional
parameter space can be used to build a semi-empirical model for bar formation
within the cosmological context.

##  6 Summary & Conclusions

In this work we study the formation and evolution of barred galaxies in the
cosmological context, using the Auriga suite of cosmological zoom-in
simulations, focusing on galaxies around the mass of the Milky Way (i.e. with
halo masses between $0.5\times 10^{12}-2\times 10^{12}\rm M_{\odot}$ at
$z=0$). We start by exploring the properties of barred galaxies and how these
evolve over cosmic history. We then focus our attention on the question ‘which
galaxies form bars?’ – i.e. we examine the various galaxy properties which are
thought to play a role in the formation of bars. This enables us to determine
the importance of these various properties within the cosmological context, in
order to begin piecing together the multi-dimensional parameter space that
drives bar formation. We summarise our results below.

In terms of the global properties of barred galaxies and on the evolution of
bars, we find that:

  * •

In Auriga, the fraction of barred galaxies decreases up to $z\sim 1$, and then
remains $\sim 20\%$ out to $z\sim 3$, consistent with the observed bar
fraction found from recent JWST studies (Fig. 4). The fraction of barred
galaxies is higher in high mass galaxies across the redshifts explored.

  * •

Below $z\sim 2$, bars in the Auriga simulations are robust, long-lived
structures. They have a wide range of formation times, with the oldest bar in
our sample forming at lookback time of $t_{\rm lb}=10.37$ Gyr and the youngest
forming at $t_{\rm lb}=0.34$ Gyr (e.g. Fig. 3).

  * •

At $z=0$, older bars tend to also be stronger, although with significant
scatter in the relation (Fig. 5).

  * •

The bar formation time correlates with the time at which the galaxy becomes
baryon-dominated, as well as with the baryon-dominance of the galaxy at $z=0$,
but does not correlate with the host galaxy’s stellar mass $M_{\rm\star}$
(e.g. Fig. 5 and Fig 6).

  * •

Bars that form at higher redshifts in our simulations tend to be longer than
those that form in the low redshift Universe (Fig. 8). We find that this is
due to the fact that bars at high redshifts are formed just after a
significant merger, which leads them to form with a saturated length (Fig. 7).

  * •

Barred galaxies at high redshifts have larger bar-to-disc size ratios, due to
the fact that bars forming in the early Universe are longer and have smaller
size stellar discs (Fig. 9).

When exploring the physical properties that are important for determining
whether a galaxy will form a bar or not, we find that:

  * •

Baryon-dominance is confirmed to play a prominent role in determining whether
a galaxy will form a bar or not, with barred galaxies being on average more
baryon-dominated from high redshifts, down to $z=0$. This leads to barred
galaxies being offset from commonly-used abundance matching relations (e.g.
Moster et al. 2013) in the $M_{\rm\star}-M_{\rm h}$ plane (Figs. 10 & 11).

  * •

Galaxies that form bars assemble their stellar mass more rapidly (on average
$\sim 3$ Gyr earlier) than unbarred galaxies (Fig. 11).

  * •

Galaxies that are offset from the abundance matching relation – i.e. which are
baryon-dominated – but which do not have a bar at $z=0$, tend to have a higher
fraction of accreted material in their bulge (Fig. 13). This suggests that
galaxies with a larger ex-situ bulge are more stable against bar formation. On
average, barred galaxies have a lower ex-situ bulge than unbarred galaxies at
$z=0$ (Fig. 12).

  * •

Unbarred galaxies tend to have a larger Toomre Q parameter, and therefore more
stable discs, both at high redshifts (i.e. around the time of bar formation)
and at $z=0$ (Fig. 15).

  * •

On average, unbarred galaxies have higher gas fractions than barred galaxies.
However, when controlling for stellar mass, the difference between barred and
unbarred galaxies in terms of gas fraction disappears (Fig. 14). This suggests
that differences in gas fraction are a secondary effect which is due to
differences in the stellar mass (coupled to the fact that bars are more common
in high mass galaxies in our sample). For a given stellar mass and/or baryon-
dominance, barred and unbarred galaxies have similar gas fractions. We
therefore do not find that gas fraction plays a major role in itself, in
whether or not a galaxy will be barred.

  * •

We summarise our results on the multi-dimensional parameter space that drives
bar formation in Fig. 17, where we see that the barred and unbarred samples
separate out best in baryon dominance ($f_{\rm bd}$), accreted bulge fraction
($f_{\rm acc}$) and Toomre $Q$ parameter.

The fact that barred galaxies tend to be more baryon-dominated has important
consequences for the loci of barred and unbarred galaxies in the Tully-Fisher
(TF) relation (see Fig. 16). Namely, barred galaxies are offset from unbarred
galaxies in the TF relation (i.e. for the same $V_{\rm flat}$, barred galaxies
have a larger $M_{\star}$ or $M_{\rm baryonic}$). Confirming this offset for
barred galaxies in the observed TF relation would have important implications
for the existence of dark matter, since such an offset cannot be readily
explained in a modified gravity (e.g. MOND) framework.

## Acknowledgements

The authors thank Marie Martig, Simon White, Federico Lelli and Kyle Oman for
useful discussions, and David Campbell and Adrian Jenkins for generating the
initial conditions and selecting the sample of the Auriga galaxies. FF is
supported by a UKRI Future Leaders Fellowship (grant no. MR/X033740/1). RG
acknowledges support from an STFC Ernest Rutherford Fellowship (ST/W003643/1).
FAG acknowledges support from ANID FONDECYT Regular 1211370, by the ANID BASAL
project FB210003 and acknowledges funding from the Max Planck Society through
a “Partner Group” grant. FM acknowledges support from the European Union -
NextGenerationEU, in the framework of the HPC project – “National Centre for
HPC, Big Data and Quantum Computing” (PNRR - M4C2 - I1.4 - CN00000013 – CUP
J33C22001170001). This work was supported by STFC [ST/X001075/1]. Part of the
simulations of this paper used: the SuperMUC system at the Leibniz Computing
Centre (www.lrz.de), Garching, under the project PR85JE of the Gauss Centre
for Supercomputing e.V. (www.gauss-centre.eu); the Freya computer cluster at
the Max Planck Institute for Astrophysics; and the DiRAC@Durham facility
managed by the Institute for Computational Cosmology on behalf of the STFC
DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital
funding via STFC capital grants ST/K00042X/1, ST/P002293/1, ST/R002371/1 and
ST/S002502/1, Durham University and STFC operations grant ST/R000832/1. DiRAC
is part of the National e-Infrastructure.

## Data Availability

All the data used, as well as parts of the auxiliary data produced for this
paper, are publicly available to download at https://wwwmpa.mpa-
garching.mpg.de/auriga/data.html

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##  Appendix A The evolution of unbarred galaxies

Figure 18: The $m=2$ Fourier mode of the surface density as a function of
lookback time for galaxies which are not classified as barred at $z=0$.

In Fig. 18 we show the evolution of the bar strength $A_{2}$ as a function of
redshift for the galaxies which do not contain a bar at $z=0$ in our sample,
in order to explore if any of these galaxies have at some point hosted a bar.
We examine in detail the galaxies in this sample that have peaks or spikes in
$A_{2}$, to explore whether these are indeed due to the presence of a bar. We
find that in all but one of these galaxies, the peaks in $A_{2}$ are not due
to the presence of a stellar bar, but are transient effects due to
interactions and/or mergers. The exception to this rule is Au36 which does
indeed form a bar at a $t_{\rm lb}\sim 4$ Gyr, which is subsequently gradually
weakened by an interaction. This leads to the dissolution of the bar in this
galaxy, which appears as an unbarred (or very weakly barred) at $z=0$
according to our definition (which requires $A_{2}>0.25$). We therefore see
that in the Auriga simulations bars are long-lived structures, with only one
case of a bar that is severely weakend after its formation.

##  Appendix B Relation between different estimates of baryon-dominance

Figure 19: The relation between the ratio of stars to dark matter within 5 kpc
($f_{\rm bd}$) versus the offset from the abundance matching relation ($\delta
M_{\rm\star}$). The two quantities are correlated. Figure 20: Rotation curve
of barred (left) and unbarred (right) galaxies in the Auriga simulation. We
show the mean and $1\sigma$ dispersion for the total (solid), stars (dashed)
and dark matter (dot-dashed). The contribution to circular velocity from the
gas is not shown here but is included in $V_{\rm{tot}}$. The MW14 model is
shown in black. The radius at which the dark matter begins to dominate over
the stars is shown as vertical arrows on the bottom $x$-axis for the Auriga
galaxies (red and blue) and the MW14 model (black).

In Fig. 19 we show the relation between the ratio of stellar to dark matter
within 5 kpc ($f_{\rm bd}$) versus the offset from the abundance matching
relation of Moster et al. (2013) as defined in Section 4.1 by $\delta
M_{\rm\star,AM}$. We see that the two are correlated. The baryon-dominance
$f_{\rm bd}$ of the barred galaxies can also be seen in Fig. 20, where the
circular velocity curves for the barred sample (left) and the unbarred sample
(right) are plotted. Solid lines give the average values for the two samples
and the shaded regions the 1$\sigma$ dispersion. The thin black lines in both
panels correspond to the MW14 model of Bovy & Rix (2013). We see that the
barred population is more baryon-dominated, i.e. the stellar component
dominates in the inner regions, and they are overall more massive (higher
circular velocity). The vertical lines give the point at which the stellar and
dark matter components cross each other, which for the MW14 model occurs
around $\sim 13.5$ kpc. For the Auriga barred population it occurs around 8
kpc, while for the unbarred population it occurs at smaller radii (as expected
for more dark matter dominated systems) at around 4 kpc. We see that even
though barred galaxies in Auriga are more baryon-dominated than unbarred ones,
they are not as baryon-dominated as dynamical models for the Milky Way
predict.

##  Appendix C The Baryonic Tully Fisher Relation

Figure 21: The Baryonic Tully Fisher relation for galaxies in Auriga. _Left:_
The grey dot-dashed line shows the relation given by fitting all galaxies
together, while the shaded region indicates the intrinsic scatter
$\sigma_{y}$. _Right:_ The red (blue) dashed line shows the relation when
fitting only the barred (unbarred) galaxies. The shaded regions indicate the
uncertainties from boostrapping (see text). The scatter is slightly reduced
for the BTFR as compared to the STFR and the barred and unbarred BTFR
relations are less separated.

Here we extend our discussion in Section 5.1 to the baryonic Tully Fisher
Relation. Studies have shown that using the total baryonic mass of galaxies,
i.e. combining both the gaseous and stellar component, results in even less
intrinsic scatter in the relation, which is referred to as the Baryonic Tully
Fisher relation (BTFR; e.g. McGaugh et al. 2000). The BTFR has less than
0.05dex in orthogonal intrinsic scatter and $\sim 0.075$dex in vertical
scatter (Lelli et al., 2019).

In the left panel of Fig. 21 we explore the BTFR for barred (excluding
interacting) and unbarred galaxies in our sample. To obtain the baryon mass we
add the gas mass of the galaxy – which we obtain by summing all the gas within
0.1$r_{\rm 200}$ and $z<$2 kpc – to the stellar mass. In this mass regime the
gas fractions are quite low (below 10% for most galaxies) and as such, the
results are similar to those found for the STFR, i.e. the barred galaxies are
on average slightly offset from the unbarred galaxies, although the offset is
less pronounced. We also find that the slope is decreased for the BTFR, i.e.
we find a slope of $\sim 3.1$, as opposed to the STFR where we found a slope
of $\sim 3.6$ (see 5.1). We note that it is not uncommon to find shallower
slopes for samples that focus only on the high mass regime (see for example
Oman et al. 2016).