11institutetext: European Southern Observatory, Karl-Schwarzschild-Str. 2,
85748 Garching-bei-München, Germany
11email<EMAIL_ADDRESS>22institutetext: Max-Planck-Institut für
Astrophysik, Karl-Schwarzschild-Str. 1, 85748 Garching, Germany
33institutetext: Department of Physics and Astronomy, University of Bologna,
via Gobetti 93/2, I-40129 Bologna, Italy 44institutetext: Instituto de
Investigacion Multidisciplinar en Ciencia y Tecnología, Universidad de La
Serena, Raul Bitrán 1305, La Serena, Chile 55institutetext: Departamento de
Astronomía, Universidad de La Serena, Av. Juan Cisternas 1200 Norte, La
Serena, Chile 66institutetext: Department of Physics and Astronomy, University
of Victoria, Victoria, BC, V8P 5C2, Canada

# Revisiting the tension between fast bars & the $\Lambda$CDM paradigm

F. Fragkoudi 1122<EMAIL_ADDRESS>R. J. J. Grand, 22 R. Pakmor2 V.
Springel2 S. D. M. White2 F. Marinacci3 F. A. Gomez4,5
J. F. Navarro6

(Received xx xx; Accepted xx xx)

The pattern speed with which galactic bars rotate is intimately linked to the
amount of dark matter in the inner regions of their host galaxies. In
particular, dark matter haloes act to slow down bars via torques exerted
through dynamical friction. Observational studies of barred galaxies tend to
find that bars rotate fast, while hydrodynamical cosmological simulations of
galaxy formation and evolution in the $\Lambda$CDM framework have previously
found that bars slow down excessively. This has led to a growing tension
between fast bars and the $\Lambda$CDM cosmological paradigm. In this study we
revisit this issue, using the Auriga suite of high resolution, magneto-
hydrodynamical cosmological zoom-in simulations of galaxy formation and
evolution in the $\Lambda$CDM framework, finding that bars remain fast down to
$z=0$. In Auriga, bars form in galaxies that have higher stellar-to-dark
matter ratios and are more baryon-dominated than in previous cosmological
simulations; this suggests that in order for bars to remain fast, massive
spiral galaxies must lie above the commonly used abundance matching relation.
While this reduces the aforementioned tension between the rotation speed of
bars and $\Lambda$CDM, it accentuates the recently reported discrepancy
between the dynamically inferred stellar-to-dark matter ratios of massive
spirals and those inferred from abundance matching. Our results highlight the
potential of using bar dynamics to constrain models of galaxy formation and
evolution.

###### Key Words.:

giant planet formation – $\kappa$-mechanism – stability of gas spheres

##  1 Introduction

Bars are common structures in spiral galaxies in the local Universe (Eskridge
et al., 2000; Menéndez-Delmestre et al., 2007) and are able to redistribute
angular momentum (Lynden-Bell & Kalnajs, 1972) from the inner regions of the
disc to the outer disc and dark matter halo. The distribution function of the
disc and halo determine the amount of material which is available to ‘emit’
and ‘absorb’ this angular momentum (Athanassoula, 2003). The density
distribution of dark matter in the inner regions of haloes is therefore one of
the main factors driving the formation and evolution of bars themselves
(Debattista & Sellwood, 2000), while other important factors include the
velocity dispersion of the disc and halo, the central mass concentration and
the gas fraction in the disc (Ostriker & Peebles, 1973; Combes & Sanders,
1981; Athanassoula et al., 2013).

The inner regions of stellar discs lose angular momentum via the bar, by
trapping stars onto elongated orbits – and therefore growing in mass and
length over time (Athanassoula, 2003) – and through dynamical friction with
the dark matter halo (Tremaine & Weinberg, 1984b; Weinberg, 1985). Dynamical
friction can act to slow down the angular rotation speed of the bar, commonly
referred to as the bar pattern speed, $\Omega_{\rm p}$, without increasing the
length of the bar, $R_{\rm bar}$ (Debattista & Sellwood, 2000). When bars slow
down, the corotation radius, $R_{\rm CR}$ – the radius at which stars on near
circular orbits move around the galaxy with the same angular frequency as the
bar – increases. As a result, the corotation radius normalised by the bar
length, $\mathcal{R}=R_{\rm CR}/R_{\rm bar}$, can shed light on how much
dynamical friction the halo exerts on the bar, and therefore on the amount of
dark matter in the inner galaxy. As is common in the literature, in what
follows we will refer to bars with $\mathcal{R}<1.4$ and $\mathcal{R}>1.4$ as
“fast” and “slow” respectively (Debattista & Sellwood, 2000).

Numerous observational efforts have been made to measure the pattern speed of
bars, finding that bars tend to rotate fast, i.e. have $\mathcal{R}<$1.4
(Corsini, 2011; Aguerri et al., 2015; Guo et al., 2019). This suggests that
dark matter haloes do not exert much dynamical friction on bars, and are
therefore likely subdominant in the central regions of galaxies. However,
uncertainties in obtaining the mass-to-light ratio (M/L) of stellar discs make
determining the baryon-to-dark matter ratio in the inner regions of galaxies
difficult. Dynamical studies of massive spiral galaxies in the local Universe
tend to find that these are baryon-dominated in their central regions (Weiner
et al., 2001; Kranz et al., 2003; Bovy & Rix, 2013; Fragkoudi et al., 2017).
This is also found by studies which use the less uncertain M/L obtained from
stellar population models in the near-infrared (Lelli et al., 2016). On the
other hand, studies such as the DiskMass Survey point to contradictory
results, suggesting that dark matter haloes dominate in the central regions of
spiral galaxies (Bershady et al., 2011). Therefore, whether or not spiral
galaxies are baryon-dominated in their central regions, is still under debate.

On the other hand, advances in numerical and physical implementations have led
to drastic improvements in hydrodynamical cosmological simulations, which are
now able to routinely form spiral galaxies with extended discs (Agertz et al.,
2011; Vogelsberger et al., 2014; Schaye et al., 2015; Grand et al., 2017).
However, the relative contribution of baryonic and dark matter in the inner
regions of galaxies in $\Lambda$CDM is an ongoing topic of debate, as baryonic
processes play an important role in reshaping the inner profiles of dark
matter haloes and in setting the baryon-to-dark matter ratio in the central
regions (Pontzen & Governato, 2012; Lovell et al., 2018). Studying the
dynamics of barred galaxies in cosmological simulations provides a powerful
tool for constraining the relative amount of baryons and dark matter in the
inner regions.

There have been a handful of studies which have explored the properties of
bars in the full $\Lambda$CDM cosmological context to date (e.g. Kraljic et
al. 2012; Scannapieco & Athanassoula 2012; Zana et al. 2018; Rosas-Guevara et
al. 2020; Blázquez-Calero et al. 2020). However, few of these have explored
their pattern speeds, and only two explore this in a large sample of galaxies
(Algorry et al., 2017; Peschken & Łokas, 2019), using the EAGLE (Schaye et
al., 2015) and Illustris (Vogelsberger et al., 2014) simulations. These
studies found that by $z=0$, bars have slowed down excessively, making them
incompatible with observations. This has resulted in a growing tension between
$\Lambda$CDM cosmological simulations and fast bars, raising the question:
_are fast bars incompatible with the $\Lambda$CDM framework?_

Here we revisit the question of the slow-down of bars in the $\Lambda$CDM
context, using the Auriga simulations, a suite of state-of-the-art high-
resolution magneto-hydrodynamical cosmological zoom-in simulations of the
formation of Milky Way-mass galaxies. In Section 2 we show that bars in Auriga
remain fast until $z=0$, compatible with observations. In Section 3 we compare
our results to previous findings in the literature, and discuss some of the
differences in the simulations that can give rise to the different dynamical
behaviour. In Section 4 we discuss some of the implications of our findings
and conclude.

##  2 Fast-rotating bars in the Auriga Simulations

The Auriga simulations (Grand et al., 2017) are a suite of 30 magneto-
hydrodynamical cosmological zoom simulations of haloes with masses in the
range of $1\times 10^{12}-2\times 10^{12}M_{\odot}$. By $z=0$, the simulations
form star-forming disc 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 structures such as bars and bulges which have properties
compatible with those of observed galaxies (Blázquez-Calero et al., 2020;
Fragkoudi et al., 2020; Gargiulo et al., 2019). For this study our barred
sample includes all the Auriga galaxies with bar strength $>0.2.$ at $z=0$,
where bar strength is defined as the maximum amplitude of the $m=2$ Fourier
mode of the surface density. We exclude the galaxies which are undergoing an
interaction at $z=0$. For more details on the sample of Auriga simulations and
how the bar strength is derived, we refer the reader to Appendix A & B.

Figure 1: Bar corotation radius vs bar length for Auriga galaxies: _Top
panel:_ Corotation radius as a function of bar length for the barred Auriga
galaxies, denoted by the symbols in the figure legend. The colour-coding
corresponds to measurements at different redshifts, while for the galaxies for
which we use the TW method (Au-TW) we estimate the pattern speed at $z=0$.
These are compared to data from Corsini (2011) and Aguerri et al. (2015) (grey
points); the error bars indicate the 1$\sigma$ uncertainties. The lower and
upper dashed grey lines indicate $\mathcal{R}=1$ and $\mathcal{R}=1.4$
respectively. Bars in Auriga are fast across all redshifts. _Bottom panel:_ We
show the mean $\mathcal{R}$ for the aforementioned observations, for barred
galaxies in Auriga, and for the EAGLE and Illustris barred galaxies from
Algorry et al. (2017) and Peschken et al. (2019) at $z=0$. The error bars
indicate the 2$\sigma$ error on the mean. The dashed line indicates
$\mathcal{R}=1.4$, below which bars are considered fast. Bars in the Auriga
simulations are fast, compatible with observations, while bars in EAGLE and
Illustris are excessively slow at $z=0$.

As mentioned above, while the absolute value of the pattern speed of bars can
shed light on the exchange of angular momentum in the galaxy, the parameter
used to determine the bar slow-down due to dynamical friction is $\mathcal{R}$
= $R_{\rm CR}$/$R_{\rm bar}$ due to the fact that dynamical friction can act
to slow down the bar without a corresponding increase in its length.
Calculating the length of bars is non-trivial as it oscillates due to the
coupling of different structures in the disc, such as the bar with the spiral
arms (e.g. Petersen et al. 2019; Hilmi et al. 2020). For the estimate of
$R_{\rm bar}$ we employ a method similar to that used in analysing
observations, i.e. the ellipse fitting method in Erwin (2005). We tested
however that our results on $\mathcal{R}$ do not change if we opt for a
different bar length estimate (see Appendix B for details on the bar length
calculation and various tests). The pattern speed in Auriga is calculated from
the temporal evolution of the $m=2$ Fourier phase of the surface density, or
via the Tremaine-Weinberg method (TW; Tremaine & Weinberg 1984a) for
simulations without high cadence outputs (see Appendix B for more details).

We plot $R_{\rm CR}$ versus $R_{\rm bar}$ in the top panel of Figure 1 for the
barred Auriga galaxies and compare these to observations (Corsini, 2011;
Aguerri et al., 2015), in which the bar pattern speed is obtained using the TW
method. The upper and lower solid lines in Figure 1 correspond to
$\mathcal{R}$ = 1.4 and 1 respectively, which denote the regime in which bars
are considered ‘fast’. For the five Auriga galaxies for which we have high
cadence outputs (see Appendix A) we calculate the corotation radius and bar
length at redshifts $z=0,0.25,0.5,0.75,1$ (provided the bar has already formed
by the corresponding redshift), as denoted by the colour-coding of the
symbols. We find that when bars in the Auriga galaxies are formed, they have
$\mathcal{R}<1.4$, i.e. they are dynamically fast, and that they remain so
throughout their evolution. We note that 25% of the sample have $\mathcal{R}$
slightly higher than 1.4, but are compatible with 1.4 within the errors. We
therefore find that the Auriga bars are on average fast at $z=0$, compatible
with observations.

##  3 Comparison with previous pattern speed estimations from cosmological
simulations

In the bottom panel of Figure 1 we show the mean and 2$\sigma$ error on the
mean of $\mathcal{R}$, for barred galaxies in the Auriga simulations, for the
above-mentioned observations, as well as for the barred galaxies studied
previously in $\Lambda$CDM cosmological simulations (Algorry et al., 2017;
Peschken & Łokas, 2019). As was shown by these latter works, bars in the EAGLE
and Illustris simulations tend to have $\mathcal{R}>2.5$, in tension with
observations. Bars in Auriga have $\mathcal{R}<1.4$, thus demonstrating that
bars can remain dynamically fast in cosmological simulations within the
$\Lambda$CDM paradigm. This therefore lessens, to some extent, the previously
reported tension between observed fast bars and $\Lambda$CDM.

However a natural question to ask is ‘what gives rise to the different
behaviour of $\mathcal{R}$ in the different simulations?’ As mentioned above,
$\mathcal{R}$ depends on the amount of dynamical friction exerted by the halo
on the disc, which in turn depends strongly on the amount of dark matter
present in the central region of the halo, as compared to the disc. The
subgrid physics modelling employed in cosmological simulations can therefore
play a critical role in determining the bar properties, and more crucially,
its slowdown rate, as it sets the disc-to-halo mass ratio in the central
regions of galaxies. In what follows we investigate the effect of numerical
resolution, as well as the properties of the galaxies in the different
simulations that might lead to differences in $\mathcal{R}$.

### Resolution:

Figure 2: (a) Resolution test for $\mathcal{R}$: The value of $\mathcal{R}$
for the two lower resolution runs (Au17-5 and Au18-5) compared to the standard
resolution runs (Au17-4 and Au18-4). For reference we also show the mean and
2$\sigma$ error of $\mathcal{R}$ for the whole bar sample of Auriga, EAGLE and
Illustris.  (b) Disc thickness: The flattening, $\epsilon=1-c/a$, of disc
galaxies in Auriga, Illustris and EAGLE. The average values of the
distributions (vertical dashed lines) are 0.42, 0.36 and 0.46 respectively.
(c) Gas fraction vs radius: The average gas density over stellar density in
the Auriga galaxies as compared to disc galaxies in EAGLE and Illustris. (d)
Gas fraction vs normalised radius: As in panel (c), with the radius normalised
by the average bar length in each of the simulations.

Low numerical resolution has been extensively discussed in the literature as a
parameter which could affect the evolution of bars and the exchange of angular
momentum at resonances, often with different studies reaching contradictory
conclusions (Weinberg, 1998; Weinberg & Katz, 2002; Valenzuela & Klypin, 2003;
Sellwood, 2006, 2008).

In order to test the effect of resolution in our models, and to explore
whether it could be the primary reason for why previous studies using the
EAGLE and Illustris cosmological simulations found large values of
$\mathcal{R}$, we carry out a resolution test. We re-ran two of our Auriga
haloes (Au17 and Au18) with 8$\times$ lower mass resolution, and 2$\times$
lower spatial resolution. This leads to a mass resolution that is closer to –
although still slightly higher than – the EAGLE and Illustris resolution. In
the low resolution Auriga runs the stellar and dark matter particles have a
mass of 4$\times 10^{5}M_{\odot}$ and 3.2$\times 10^{6}M_{\odot}$
respectively, while in EAGLE (Illustris) the stellar mass is 1.8$\times
10^{6}M_{\odot}$ (1.3$\times 10^{6}M_{\odot}$) and the dark matter particle
mass is 9.7$\times 10^{6}M_{\odot}$ (6.3$\times 10^{6}M_{\odot}$). If
resolution is the major contributor to the high values of $\mathcal{R}$ in
previous studies we expect to see a significant effect on $\mathcal{R}$ when
decreasing the resolution to be close to that of EAGLE and Illustris.

In Figure 2a we show the $\mathcal{R}$ values for the level 4 (high
resolution) and level 5 (low resolution) runs for Au17 and Au18. We find that
$\mathcal{R}$ has a slight increase of $\sim$15% in the low resolution runs.
However, the increase is within the 2$\sigma$ error of the mean $\mathcal{R}$
values for the high resolution barred galaxies explored in this study.
Therefore, this small increase in $\mathcal{R}$ for the low resolution Auriga
runs does not appear to be sufficient to explain the much higher values found
in the EAGLE and Illustris simulations111It is also worth noting that high-
resolution zoom simulations in the literature (e.g. Zana et al. 2018), which
study the pattern speed of bars also find $\mathcal{R}>1.4$ at $z=0$.. These
considerations suggest that resolution is likely not the main culprit for the
high $\mathcal{R}$ values found in previous simulations such as EAGLE and
Illustris.

Figure 3: Global and local stellar-to-dark matter ratios in the Auriga
galaxies as compared to EAGLE and Illustris: _Left:_ The mean (lines) and
1$\sigma$ (shaded region) values of the $M_{\star}$ vs $M_{\rm h}$ of Auriga,
EAGLE and Illustris disc galaxies at $z=0$ compared to the abundance matching
relation from Moster et al. (2018). _Middle:_ The efficiency with which gas is
turned into stars, i.e. $f_{\star}=\frac{M_{\rm\star}}{f_{b}M_{\rm h}}$ as a
function of stellar mass for the Auriga, EAGLE and Illustris galaxies,
compared to the relation from abundance matching. The mean and 1$\sigma$
values for disc (barred) galaxies are shown with the circles/thick lines
(squares/thin lines) and shaded region respectively. For comparison we also
show the high mass disc galaxies from the SPARC sample explored in Marasco et
al. (2020), where the error bars denote 16th-84th percentile uncertainties.
_Right:_ The local baryon dominance, $V_{\star}/V_{\rm tot}$ as a function of
radius, of Auriga, EAGLE and Illustris disc galaxies in the mass range
$3\times 10^{10}<M_{\star}/M_{\odot}<10^{11}$.

### Disc thickness:

The thickness of discs – which is related to their vertical velocity
dispersion – has been reported in the literature as playing a role in the slow
down of bars (Misiriotis & Athanassoula 2000; Klypin et al. 2009). While the
exact mechanism for this is still unclear, it has been suggested that bars are
slower in thicker discs (Klypin et al. 2009). In Figure 2b we investigate the
differences in disc thickness between Auriga, Illustris and EAGLE, by
exploring the flattening of their mass distributions, defined as
$\epsilon=1-c/a$, where $c$ ($a$) corresponds to the smallest (largest) axis,
i.e. $\epsilon=0$ corresponds to a spheroidal distribution. We obtain the axes
of the ellipsoid by calculating the reduced inertia mass tensor of the
galaxies (for more details see Appendix C). We find that the disc galaxies in
Auriga, Illustris and EAGLE have similar distributions of flattening, with
Illustris having a slightly lower average value (0.36) than Auriga (0.42) and
EAGLE (0.46). While a slight difference in disc thickness might be
contributing to the slow down of bars in Illustris, the flattening of discs in
Auriga and EAGLE is very similar, and it is therefore unlikely that disc
thickness alone can explain the bar slowdown.

### Gas fraction:

The fraction of gas within discs, and in particular within the bar radius, can
play a role in both the formation and strength of bars. It has also been
suggested that gas can play a role in maintaing fast bars (Athanassoula,
2014), although the topic has given rise to considerable debate (e.g. Bournaud
et al. 2005; Berentzen et al. 2007; Athanassoula 2014; Sellwood & Debattista
2014 and references therein). In the two lower panels of Figure 2 we explore
the gas content of disc galaxies in Auriga, EAGLE and Illustris. Auriga and
Illustris have similar gas fractions within 10 kpc, while EAGLE has lower gas
fractions, which could possibly contribute to the slower bars in EAGLE as
compared to Auriga. However, when the radius is normalised by the average bar
length in each simulation, we find that, within the bar radius, Auriga
galaxies have similar gas fractions to EAGLE, and lower gas fractions than
Illustris. Therefore, while the gas fraction might contribute to the
differences in pattern speed between the different simulations, it is likely
not the main culprit for allowing Auriga to maintain fast bars.

### Baryon-dominance

We therefore turn our attention to the relative contribution of the stellar
component compared to dark matter in the Auriga simulations. We explore this
in Figure 3 and compare Auriga to the EAGLE and Illustris cosmological
simulations. In the left panel we plot the average and 1$\sigma$ relation for
the stellar mass as a function of the dark matter halo mass ($M_{\rm\star}$ vs
$M_{\rm h}$) for the Auriga galaxies (red), as well as for the Illustris and
EAGLE discs (green and purple respectively). We compare these to the commonly
used abundance matching relation (Moster et al. 2018; dashed blue line), which
is obtained by matching the observed stellar mass function to the halo mass
function in cosmological simulations. Auriga galaxies are offset from the
relation, and lie above both the EAGLE and Illustris discs, i.e. they are
globally more baryon-dominated.

In the middle panel of Figure 3 we plot the efficiency with which – given a
universal baryon-to-dark matter ratio $f_{\rm b}=\Omega_{\rm b}/\Omega_{\rm
c}$ – galaxies convert their gas into stars, i.e.
$f_{\rm\star}=M_{\rm\star}/(f_{\rm b}M_{\rm h})$, as a function of stellar
mass. This is shown for the Auriga, EAGLE and Illustris simulations, for both
the entire disc sample (circles) and for the barred galaxies (squares). The
Auriga galaxies are offset from the relation predicted by abundance matching –
by more than 2$\sigma$ at $M_{\star}=9\times 10^{10}M_{\odot}$. They also lie
above both the EAGLE and Illustris galaxies, and thus have higher global
stellar-to-dark matter ratios for a given stellar mass at $z=0$. Recent work
by Posti et al. (2019) explored galaxies in the SPARC sample (Lelli et al.,
2016), finding that massive spirals lie above the relation for $f_{\rm\star}$
derived from abundance matching, which implies that massive spirals are overly
efficient at converting gas to stars (the so-called ‘failed feedback
problem’). A subsequent study by Marasco et al. (2020) found that massive
spirals in SPARC are more baryon-dominated than spiral galaxies in
cosmological simulations such as EAGLE and IllustrisTNG100. Interestingly, the
Auriga galaxies follow a similar trend as the high mass spiral galaxies in the
SPARC sample, which are denoted by the grey symbols.

While the $M_{\star}$ – $M_{\rm h}$ relation tells us about the global ratio
of baryons-to-dark matter, the local distribution of baryons-to-dark matter is
more relevant for the dynamics of disc galaxies and for the bar instability.
We show this in the right panel of Figure 3, where we plot $V_{\star}/V_{\rm
tot}$, i.e. the ratio of the stellar component to the total rotation curve, as
a function of radius for disc galaxies in Auriga, Illustris and EAGLE (in the
mass range $3\times 10^{10}<M_{\star}/M_{\odot}<10^{11}$). This reveals how
‘maximal’ or baryon-dominated a galaxy is222Historically, disc maximality has
been calculated at a given radius, such as at 2.2 disc scalelengths., which in
turn will partially determine whether the disc is able to form a bar that
remains fast. We find that the Auriga barred galaxies are overall more baryon-
dominated than EAGLE and Illustris at all radii. In Figure 7 we explore this
trend also at redshift $z=0.5$ and find that Auriga galaxies are already more
baryon-dominated at higher redshifts.

What causes Auriga to be more baryon-dominated than the EAGLE and Illustris
simulations is the complex interplay of the subgrid implementations of the
baryonic physics in the simulations. A full exploration of these is well
beyond the scope of this letter, but here we outline a few differences that
are likely important, and refer the reader to the papers describing the
simulations for more details (Schaye et al., 2015; Vogelsberger et al., 2014;
Grand et al., 2017). In terms of the stellar feedback: the wind model in
Auriga has significant differences from its predecessor, Illustris, in both
its parametrization and its implementation, e.g. the winds are isotropic in
Auriga vs bipolar in Illustris. This might lead to the winds being more
effective at higher redshifts in Auriga where galaxies are irregular. On the
other hand, EAGLE employs a thermal feedback prescription which is more
‘bursty’, heating particles stochastically to sufficiently high temperatures
to avoid catastrophic cooling, which is likely more effective at removing
baryons from the central regions of galaxies. In terms of the AGN feedback:the
prescription in Auriga is ‘smoother’ than that in Illustris and EAGLE. In
Auriga the ‘bubble’ radio-mode feedback provides a more gentle and distributed
heating of the circum-galactic medium than in Illustris, in which a smaller
number of very energetic bubbles were able to blow out all the gas from the
halo; in EAGLE there is only one mode of AGN feedback in which thermal energy
is injected stochastically by heating particles around the black hole to very
high temperatures. Furthermore, Auriga includes magnetic fields, in contrast
to both EAGLE and Illustris. The differences in these physical modelling
assumptions, which combine non-linearly, give rise to discs which are more
massive in Auriga.

##  4 Conclusions

To summarise, we find that bars in the Auriga cosmological simulations are
fast, with $\mathcal{R}<1.4$, in agreement with observations, and contrary to
the results from previous cosmological simulations. The considerations of the
previous section provide us with a likely explanation for why bars in other
cosmological simulations tend to slow down so dramatically by $z=0$: their
host galaxies are embedded in dominant dark matter haloes – both locally and
globally – i.e. they have a higher stellar-to-dark matter ratio, and thus
their bars suffer more dynamical friction. Disc galaxies in the Auriga
simulations, on the other hand, are more baryon-dominated, and thus are able
to remain fast for over half the age of the Universe. This alleviates the
previously reported tension between fast bars and the $\Lambda$CDM
cosmological paradigm. However, our findings suggest that in order to
reproduce the dynamics of barred galaxies, massive spirals must be baryon-
dominated, and should lie above the abundance matching relation. This is
consistent with the recently reported findings of Posti et al. (2019) and
Marasco et al. (2020), which highlight the tension between the stellar-to-dark
matter ratio of high mass spirals obtained dynamically and via abundance
matching.

Our results imply that, for cosmological simulations of galaxy formation and
evolution to reproduce the dynamical properties of observed barred spirals,
they need to build up more stellar mass in their discs than that predicted by
abundance matching. This suggests that care is needed to not make the stellar
and AGN feedback prescriptions overly efficient, as this can lead to lower
stellar-to-dark matter ratios in high-mass spirals, which will prevent them
from maintaining fast bars. As bars are present in a significant fraction
($\geq$ 50%) of the population of spiral galaxies in the local Universe
(Eskridge et al., 2000; Menéndez-Delmestre et al., 2007), this effect is
likely to considerably increase the expected scatter around the abundance
matching relation, particularly for spirals at the high-mass end, where bars
are more numerous (Masters et al., 2012; Gavazzi et al., 2015). Whether these
results have repercussions on the expected value of the abundance matching
relation or simply on its scatter will depend on the relation followed by the
global population of galaxies at this mass range, including early type
galaxies. In terms of the implications for lower mass galaxies, there is still
considerable debate in the observational community with regards to the
frequency of bars in low mass spirals (Nair & Abraham, 2010; Erwin, 2018), and
as such, an exploration of the properties of lower mass barred spirals is
timely. Our results highlight the importance of taking into account the
dynamics of barred spiral galaxies when constraining models of galaxy
formation and evolution.

###### Acknowledgements.

We thank the anonymous referee for a constructive report. FF thanks L. Posti,
B. Moster and C. Damiani for useful discussions and N. Peschken for providing
the halo IDs of barred galaxies in Illustris. FM acknowledges support through
the Program “Rita Levi Montalcini” of the Italian MUR.

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##  Appendix A Barred galaxies in the Auriga simulations

The Auriga simulations (Grand et al. 2017) are a suite of 30 magneto-
hydrodynamical cosmological zoom simulations of haloes with masses in the
range of $1\times 10^{12}-2\times 10^{12}M_{\odot}$ which run from redshift
$z=127$ to $z=0$ with cosmological parameters $\Omega_{m}=0.307$,
$\Omega_{b}=0.048$, and a Hubble constant of $H_{0}=67.77$ km$\,\rm
s^{-1}\,\rm Mpc^{-1}$ (Planck Collaboration et al. 2014); unless otherwise
stated we use these parameters throughout the paper. The simulations are
performed with the magneto-hydrodynamic code AREPO (Springel 2010; Pakmor et
al. 2016), with a comprehensive galaxy formation model (Vogelsberger et al.
2013; Marinacci et al. 2014; Grand et al. 2017) and form star-forming disc
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 form structures
such as bars and boxy/peanuts which have properties compatible with those of
observed bars (Blázquez-Calero et al. 2020; Fragkoudi et al. 2020) and mainly
consist of so-called pseudo-bulges (Gargiulo et al. 2019). For more details we
refer the reader to the aforementioned papers and references therein.

In this study our ‘barred sample’ includes all the Auriga galaxies with bar
strength $A_{2}>0.2.$ at $z=0$. This excludes five Auriga galaxies which are
undergoing an interaction at $z=0$ (Au1, Au11, Au20, Au25 and Au30). For these
galaxies the pattern speed cannot be reliably measured using the Tremaine-
Weinberg method, for which the continuity equation must hold. We therefore
have 16 Auriga galaxies in our barred sample. Five of these haloes (Au9, Au17,
Au18, Au26 and Au28) are re-runs of the original Auriga haloes with higher
cadence outputs (i.e. every 10 Myr), in order to be able to determine the
pattern speed from the temporal evolution of the bar and to test our
implementation of the Tremaine-Weinberg method (due to the high computational
cost we do do not re-run all the Auriga galaxies). For the re-runs, the
initial conditions of the haloes and the physics implementations are the same
as those of the original Auriga haloes. We refer the reader to the Appendix
where we describe in detail how we obtain the bar strength, length and pattern
speed of our simulated galaxies.

##  Appendix B Bar strength, length & corotation radius

Figure 4: Bar properties as a function of time: _Top panel:_ Bar strength for
the five Auriga galaxies for which we have high cadence outputs as a function
of lookback time. The horizontal dashed line indicates $A_{2}=0.2$ above which
we consider the bar to have formed. _Bottom panel:_ Bar pattern speed for
these galaxies, once the bar is formed, as a function of lookback time.

### Bar Strength

We define the bar strength as the maximum amplitude of the normalised $m=2$
Fourier mode of the surface density. We select the stellar particles in the
disc (i.e. with $|z|<1\,\rm kpc$) in annuli of width 0.5 kpc to calculate,

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

where $\rm m_{i}$ is the mass of particle $i$, $R$ is the cylindrical radius,
$N$ is the total number of particles in that annulus and $\theta$ is the
azimuthal angle. To obtain a single value for the bar strength at each time-
step, we take the maximum value of the normalised $m=2$ component,

$A_{2}=\mathrm{max}\left(\frac{\sqrt{a_{2}^{2}(R)+b_{2}^{2}(R)}}{a_{0}(R)}\right)$
(3)

in radius.

We also carry out visual inspections of the snapshots in order to ensure that
the $m=2$ mode is due to the bar, and not some other spurious short-lived
effect. For example, at high redshifts – when mergers are frequent – short-
lived $m=2$ modes can appear due to the centre of mass of the merging system
not being at the centre of the disc (e.g. in the final stages of a merger).
This would therefore lead our algorithm to detect a spurious and short-lived
$m=2$ mode, which is however not related to a bar mode in the disc. However we
note that such spurious and short-lived spikes in $m=2$ are essentially only
found at high redshifts (see e.g. the large peak in $m=2$ for Au26 at $\sim$9
Gyr in Figure 4).

Figure 5: Bar strength, pattern speed and length variation for Au18 with time:
_Top panel:_ We show the pattern speed as estimated from the phase of the
$m=2$ Fourier mode (righ axis – blue), the $m=2$ Fourier mode (bar strength)
of the surface density, as well as the $m=1$ Fourier mode (dashed red line),
as a function of lookback time. _Middle panel:_ Various measures of bar length
as a function of lookback time (see text). _Bottom panel:_ Variation of
$\mathcal{R}$ as a function of time using different estimates for the bar
length.

### Pattern Speed

For the five galaxies for which we have high cadence outputs the bar pattern
speed is obtained by examining the temporal evolution of the bar phase, i.e.
by calculating the $m=2$ phase from the aforementioned Fourier decompositions
in each snapshot and as a function of radius as,

$\theta(R)=0.5\arctan\left(\frac{b_{2}(R)}{a_{2}(R)}\right).$ (4)

The bar pattern speed $\Omega_{\rm p}$ is then calculated as an annular
average within the bar radius as,

$\Omega_{\rm p}=\frac{\Delta\langle\theta\rangle}{\Delta t}.$ (5)

The uncertainty on this value mainly derives from the oscillations in pattern
speed at different snapshots (see for example the bottom panel of Figure 4),
which are typically around $\pm$3-5km/s/kpc. We explored the uncertainty this
implies in terms of the corotation radius and found that this translates to
$\sim\pm$0.5 kpc.

Changes in the pattern speed as a function of time can occur due to coupling
with spiral arms or odd modes such as $m=1$. To check whether $m=1$ can have a
significant effect on our estimates of the pattern speed, in the top panel of
Figure 5 we show the variation of the $m=2$ and $m=1$ modes as well as the
pattern speed as a function of lookback time. We find that at higher
redshifts, when gas accretion and minor mergers are more frequent, $m=1$ is
non-negligible, and this induces some noise in the pattern speed of the bar.
However, by low redshifts the contribution from $m=1$ is negligible, of the
order of a few percent, and as such does not seem to affect our estimates of
the bar pattern speed – especially as we exclude from our sample the galaxies
which are undergoing interactions at $z=0$.

In Figure 4 we show the evolution of the bar strength and pattern speed as a
function of lookback time for five of the Auriga galaxies for which we have
high cadence outputs. In the top panel of the Figure we show the evolution of
the bar strength. Bars form at a range of lookback times in our simulations,
with some forming at $z>1$ (e.g. Au17 and Au18), or as late as 1 Gyr ago
(Au28). Bars in Auriga are long-lived structures, i.e. once the bar has formed
it does not dissolve. In the bottom panel of Figure 4 we show the bar pattern
speed, $\Omega_{\rm p}$, for these five barred galaxies as a function of
lookback time. Most of the bars have pattern speeds which remain roughly
constant or are slightly decreasing (e.g. Au26 after $t_{\rm lookback}\sim
2\,\rm Gyr$). There is also the curious case of Au17, in which the bar pattern
speed increases between formation time and until $t_{\rm lookback}\sim 3\,\rm
Gyr$ due to a resonant interaction with another nearby massive system (a more
detailed exploration of this system will be presented elsewhere). The galaxy
with the largest decrease in $\Omega_{\rm p}$, Au26, shows a prototypical
example of bar growth in an isolated disc, i.e. while the galaxy evolves in
isolation the bar grows gradually stronger over time, while $\Omega_{\rm p}$
decreases. In a number of the other cases shown here, bar formation is
triggered after a significant merger, with the disc losing a large fraction of
its angular momentum by torques induced during the merger (Fragkoudi et al.
2020), with the bar subsequently not growing much stronger or slower.

For the rest of the Auriga barred sample for which we do not have high cadence
outputs we employ the commonly-used Tremaine-Weinberg method (Tremaine &
Weinberg 1984a), which uses the continuity equation to estimate the bar
pattern speed from a single snapshot at $z=0$. We tested our method on the
five simulations for which we have high cadence outputs – and therefore a
reliable estimate for the bar pattern speed – and found that we can typically
recover the pattern speed to within $\pm$5km/s/kpc. This translates to an
error on the corotation radius of $\sim\pm$1 kpc. These are the uncertainties
employed in Figure 1.

Figure 6: Estimating the bar length and corotation radius: _Left panel:_ The
bar length is estimated by fitting ellipses to iso-density contours. The
vertical dashed lines denote the estimated bar length. _Middle panel:_
Ellipticity and position angle as a function of semi-major axis of the
ellipses. The bar length is the minimum of two values: the radius of the first
minimum after the ellipticity maximum (dot-dashed orange line) or the radius
at which there is the first extremum in position angle outside the bar (dashed
line). _Right panel:_ The bar corotation radius (marked by the vertical dashed
blue line) is the radius at which stars in the disc have the same angular
frequency (solid black line) as the bar pattern speed (horizontal dashed blue
line). The dashed and dot-dashed lines denote the $\Omega-\kappa/2$ and
$\Omega+\kappa/2$ curves for the galaxy.

### Bar Length

There is no standard way for estimating the bar length and various methods
have been used in observations and simulations – e.g. visual estimation,
isophote ellipse fitting, measurements based on Fourier analysis etc. – each
with its own advantages and disadvantages (for a discussion of the various
methods see Erwin 2005; Petersen et al. 2019; Hilmi et al. 2020). Here we
calculate the bar length using a method commonly used in observations (such as
the observations presented in Figure 1), i.e. we fit ellipses to isodensity
contours of face-on images of the galaxies.

We follow a prescription similar to the one used in Erwin (2005), in which the
bar length is defined as the minimum of two ellipse fitting measures: the
radius of the first minimum after the maximum of ellipticity, or the radius at
which the position angle has an extremum outside the bar. An example of this
is shown in the left and middle panels of Figure 6. If there is no extremum in
position angle then we use instead the radius at which the position angle
changes by 10 degrees. For this technique, we do not implement an automated
method, but inspect the outputs on a snapshot by snapshot basis. Therefore
this method is not efficient at obtaining bar lengths for a large number of
snapshots, but it does give a rather accurate bar length estimate. Other
techniques which are more conducive to being automatised exist, such as
defining the bar length as the radius at which $A_{2}$ drops below a certain
percentage of its maximum value, or the radius at which there is a certain
change in the phase $\theta$ of the $m=2$ component. These methods however can
lead to an over- or under-estimation of the bar length, due to the complex
interplay between various modes in the disc; for example, when the spiral arms
and bar align this can lead to an overestimate of the bar length (see e.g.
Hilmi et al. 2020 for a detailed exploration of various estimates of the bar
length and the effects of bar-spiral coupling on these estimates), or an
underestimation of the bar length can occur when a secondary mode (such as an
inner bar) forms with a different pattern speed.

To explore whether the value of $\mathcal{R}$ we obtain for the Auriga
galaxies in this work would be significantly altered if we used a different
measure of the bar length, we explore, in the middle and bottom panels of
Figure 5, different bar length estimates. We employ two commonly-used methods
from the literature, i.e. when the $m=2$ Fourier mode drops below 60% of the
maximum (orange), and when the phase of the $m=2$ Fourier mode changes by more
than $\Delta\theta_{m=2}$=7.5 degrees (blue). Employing different values for
the $A_{2}$ threshold or $\Delta\theta_{m=2}$ does not qualitatively affect
these results. The estimates we use based on the ellipse fitting method (and
which is used in the main part of the paper) are shown with black crosses. In
the middle panel of Figure 5 we see that our ellipse-fitting method for
estimating the bar length coincides with the lower values obtained from the
other two methods, in agreement with what is estimated as closer to the ‘true’
bar length in Hilmi et al. (2020). The bar length we use in the main part of
the paper therefore gives values on the low side of that obtained from the
fluctuations in bar length due to bar-spiral coupling. This in turn leads to
higher values of $\mathcal{R}$, as is shown in the bottom panel of Figure 5,
i.e. for most of the temporal range our method gives consistent or even
slightly higher values of $\mathcal{R}$ than the other two methods. Thus, we
see that using another method to estimate the bar length would not change the
results of our study by any significant amount, i.e. we would not find values
of $\mathcal{R}>1.4$, thus changing our conclusion on whether Auriga bars are
fast or slow. We note that at $z\sim 0$ the algorithm which obtains the bar
length from the difference in the $m=2$ phase ($\Delta\theta_{m=2}$) gives
shorter bar estimates than the one we obtain from our ellipse-fitting method
and the maximum of $A_{2}$ method. This is due to the growth of an ‘inner
bar’-like structure at late times in the galaxy, which has a different pattern
speed than the ‘main’ bar of the galaxy. This leads to abrupt phase changes at
low radii. However, upon visual inspection it is clear that the primary bar is
longer.

### Corotation Radius

To obtain the corotation radius of Auriga barred galaxies we first need to
calculate the circular velocity of the galaxy $V_{\rm
c}^{2}(r)=r\frac{\partial\Phi}{\partial r}$. We make the simplifying
assumption of spherical symmetry, which reduces the circular velocity to
$V_{\rm c}=\sqrt{GM(r)/r}$ where $M(r)$ is the enclosed mass within a sphere
of a radius $r$. This simplification will tend to under-predict the true
circular velocity, which will be higher for flatter systems. We tested this by
assuming a Miyamoto-Nagai disc (Binney & Tremaine 2008) with realistic
flattening of the disc, for which we can calculate $V_{\rm c}$ analytically.
We then calculated the approximate value of $V_{\rm c}$ using the spherical
symmetry assumption. We find that the error in $V_{\rm c}$ is of the order of
$\sim 5-10\%$ depending on the flattening of the disc – with the error
increasing for thinner discs. For galaxies in our sample which also have a
spherical component due the dark matter halo this error will be smaller,
roughly of the order of 5%, which is acceptable for the purposes of this
study. The angular frequency of stars is then obtained according to
$\Omega=V_{\rm c}/r$ and the epicyclic frequency is calculated as
$\kappa=\sqrt{R\frac{d\Omega^{2}}{dR}+4\Omega^{2}}$ (Binney & Tremaine 2008).
The corotation radius is the radius at which stars on nearly circular orbits
in the disc have the same angular frequency as the bar. We therefore calculate
this as the intersection between the $\Omega$ curve with the bar pattern speed
$\Omega_{\rm p}$ (right panel of Figure 6).

We note that for the $\mathcal{R}$ values of barred galaxies in EAGLE and
Illustris shown in the bottom panel of Figure 1, we use the values reported in
Algorry et al. (2017) and Peschken & Łokas (2019) respectively. For the
Illustris $\mathcal{R}$, we divide the value reported in Peschken & Łokas
(2019) by 2, since – as stated by the authors – they used a proxy for the bar
length which underestimates the true bar length by a factor of $\sim$2.

##  Appendix C Estimating the galaxy flattening

Galaxies with thinner disks (i.e. with a smaller scale-height to scale length
ratio) have been found to host fast bars, compared to models with thicker
discs which appear to be slower (Misiriotis & Athanassoula 2000; Klypin et al.
2009). We therefore explore whether disc thickness could be the main culprit
for making bars in Auriga faster as compared to Illustris or EAGLE. We
calculate the thickness of the discs in Auriga, and compare to publicly
available values obtained for Illustris and EAGLE (see Genel et al. 2015 and
Thob et al. 2019). For Illustris and EAGLE we take only discs galaxies in the
mass range $3\times 10^{10}<M_{\star}/M_{\odot}<10^{11}$, i.e. the mass range
covered by the Auriga galaxies.

To calculate the disc thickness in Auriga we employ the same method as that in
Illustris and EAGLE, i.e. we model the galaxies as three dimensional
ellipsoids, and calculate their reduced inertial mass tensor,

$I_{ij}=\frac{1}{M}\sum_{n}m_{n}\frac{r_{n,i}r_{n,j}}{r^{2}_{n}}$ (6)

where $i,j=\\{0,1,2\\}$ denote the three principal axes, $r_{n,i}$ is the
$i$th component of the coordinate vector of particle $n$, $m_{n}$ is the
particle’s mass and $M$ is the total mass of the galaxy within the radius we
consider. To calculate the reduced mass tensor we use only stellar particles
inside 2$\times r_{50}$, where $r_{50}$ is the half-mass radius of the galaxy.
The axis lengths of the galaxy are then defined as the square root of the
eigenvalues $\lambda_{i}$ of this matrix. We then compare the flatness of the
discs in the different simulations, defined as $\epsilon=1-c/a$, where $c$
denotes the smallest axis, and $a$ the largest axis of the ellipsoid. The
results of this comparison can be seen in Figure 2b.

##  Appendix D The abundance matching relation, $M_{\rm\star}$ and $M_{\rm h}$
and maximality

In Figures 3 and 7 we use the abundance matching relation from Moster et al.
(2018), employing the values for “All centrals” from their Table 8 and using
their derived relation for the scatter in equation 25 (Moster et al. 2018).
The values of $M_{\rm\star}$ and $M_{\rm h}$ for Auriga, Illustris and EAGLE
used in the aforementioned figures are obtained as follows: For Auriga, we
calculate $M_{\rm\star}$ by summing the mass of stellar particles inside 10%
of the virial radius of the halo. $M_{\rm h}$ is the total mass of dark matter
particles within the virial radius of the halo. For Illustris, we extract the
$M_{\rm\star}$ within 20 kpc and the total dark matter mass inside the virial
radius $M_{\rm h}$ for disc galaxies using the publicly available Illustris
data (Nelson et al. 2015). We use the definition of disc galaxies employed by
Peschken & Łokas (2019), i.e. selecting galaxies which have more than 20% of
their stellar mass with circularity parameter above 0.7 and with flattened
distributions, i.e. which have a flatness ratio (defined by the ratio
eigenvalues of the stellar mass tensor) $<0.7$. For the values extracted for
the barred Illustris galaxies, we take as barred Illustris galaxies those
studied in Peschken & Łokas (2019), in their ”total bar” sample (N. Peschken
kindly provided us with these galaxy IDs) where barred galaxies are defined as
those disc galaxies with $A_{2}>0.15$ with some additional criteria (see
Section 2.2 of their paper for more details). For EAGLE, we use the publicly
available data from McAlpine et al. (2016) and where relevant the subsequent
particle data release (The EAGLE team 2017) to obtain $M_{\star}$ (stellar
mass within 20 kpc) and $M_{\rm h}$ (dark matter mass within the virial
radius) values for disc galaxies in EAGLE, which we define as having stellar
velocity rotation-to-dispersion ratios (_RotToDispRatio_), and the stellar
disc-to-total ratio from counter rotation (_DiscToTotal_) larger than 1.7 and
0.7, respectively. For the barred galaxies, we use the IDs of the galaxies
identified as barred in Algorry et al. (2017).

To explore the local baryon dominance of barred galaxies in Figure 3, we
follow the definition of disc maximality used in previous works, i.e. we
employ the ratio of the circular velocity due to the stellar component as
compared to the total circular velocity. We obtain the stellar and total
circular velocities as described in the previous section.

Figure 7: Stellar to dark matter ratios at higher redshifts: _Top:_
$f_{\star}$ vs. $M_{\star}$ for Auriga, EAGLE and Illustris disc galaxies at
$z=0.5$. The dotted blue line denotes the relation determined using the Moster
et al. (2018) abundance matching relation and the shaded region denotes the
1$\sigma$ scatter around the relation. _Bottom_ $V_{\star}/V_{\rm tot}$ for
the galaxies shown in the top panel. Auriga is locally more baryon-dominated
than EAGLE and Illustris already at higher redshifts.