# On the Decline of Star Formation during the Evolution of Galaxies

Adelheid Teklu  Universitäts-Sternwarte, Fakultät für Physik, Ludwig-
Maximilians-Universität München, Scheinerstr.1, 81679 München, Germany
Excellence Cluster Origins, Boltzmannstr. 2, 85748 Garching, Germany  Rolf-
Peter Kudritzki  Universitäts-Sternwarte, Fakultät für Physik, Ludwig-
Maximilians-Universität München, Scheinerstr.1, 81679 München, Germany
Institute for Astronomy, University of Hawaii at Manoa, 2680 Woodlawn Drive,
Honolulu, HI 96822, USA  Klaus Dolag  Universitäts-Sternwarte, Fakultät für
Physik, Ludwig-Maximilians-Universität München, Scheinerstr.1, 81679 München,
Germany  Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741
Garching, Germany  Rhea-Silvia Remus  Universitäts-Sternwarte, Fakultät für
Physik, Ludwig-Maximilians-Universität München, Scheinerstr.1, 81679 München,
Germany  Lucas Kimmig  Universitäts-Sternwarte, Fakultät für Physik, Ludwig-
Maximilians-Universität München, Scheinerstr.1, 81679 München, Germany

###### Abstract

Cosmological simulations predict that during the evolution of galaxies, the
specific star formation rate continuously decreases. In a previous study we
showed that generally this is not caused by the galaxies running out of cold
gas but rather a decrease in the fraction of gas capable of forming stars. To
investigate the origin of this behavior, we use disk galaxies selected from
the cosmological hydrodynamical simulation Magneticum Pathfinder and follow
their evolution in time. We find that the mean density of the cold gas regions
decreases with time. This is caused by the fact that during the evolution of
the galaxies, the star-forming regions move to larger galactic radii, where
the gas density is lower. This supports the idea of inside-out growth of disk
galaxies.

galaxies: evolution, gas masses, star formation

##  1 Introduction

The formation and evolution of galaxies is a complex interplay between
different physical effects. One of the most important processes involved is
the formation of stars from the galaxies’ cold gas reservoir. In a simplified
picture, the galaxies accrete cold gas from the cosmic web, which settles into
disks by redistributing angular momentum due to its non-collisionless nature,
where then stars are formed. During their evolution, star-forming galaxies
move along the so-called main sequence, with an increase in stellar mass due
to star formation accompanied by a growth in the total amount of gas and thus
an increase in star formation rate. However, at a given stellar mass, this
star formation rate is generally found to be larger at higher redshifts, with
an overall decrease in star formation rate towards lower redshifts (e.g.,
Santini et al. 2017; Pearson et al. 2018), which leads to a slower growth of
the stellar mass. When the star formation is shut down (due to several
different possible quenching mechanisms), the galaxies fall below the main
sequence.

As the stars form out of molecular gas, understanding the cosmic evolution of
the molecular gas mass density is of uttermost importance given that the
amount of molecular gas is a natural limiting factor for the amount of
possible star formation. Theoretical studies (see e.g. Obreschkow et al. 2009;
Obreschkow & Rawlings 2009; Lagos et al. 2011; Popping et al. 2014) have
predicted that from redshift $z\approx 2$ to $z=0$ the cosmic density of the
molecular gas decreases. These predictions were confirmed by several
observational studies (see e.g. Walter et al. 2014; Decarli et al. 2016;
Scoville et al. 2017; Decarli et al. 2019; Riechers et al. 2019; Lenkić et al.
2020; Tacconi et al. 2020). However, measuring the molecular gas masses from
observations has its caveats: In most cases, the molecular gas can only be
observed through tracers, with the most prominent tracer being CO.
Unfortunately, the conversion factor from CO to $H_{2}$ is still a matter of
debate (see e.g. Bolatto et al. 2013; Somerville & Davé 2015). Furthermore,
Shetty et al. (2014) discuss that CO might not be a direct tracer for the star
formation, as it traces the total molecular gas instead of the dense molecular
gas (Gao & Solomon 2004), which is the factual source of star formation.
Additionally, observational studies found that there is a component of the
molecular gas that is actually not star-forming but residing in a thick
diffuse disk (Caldú-Primo et al. 2013; Pety et al. 2013). Therefore, simply
assuming the molecular gas mass as tracer of star formation or vice versa may
be misleading, and thus connecting the available reservoir in molecular gas to
the actual star formation properties is an important yet not well understood
issue.

Many modern simulations have shown that with varying the physical parameters,
e.g. the modelling of the feedback mechanisms (e.g. Davé et al. 2020),
resolution (e.g. Crain et al. 2017) or the star formation recipes (e.g.
Valentini et al. 2019), the simulations can be fine-tuned to match the
observational results. This is very helpful for understanding the interplay
between the different quantities, but also comes with the caveat that a given
property can be obtained through changing different model assumptions while
obtaining the same result. Furthermore, comparing simulations directly to
observations is not always straight forward. On the one hand, an issue arises
when calculating the fractions of $H_{2}$, as these can vary strongly
depending on the used prescription (see e.g. Lagos et al. 2015; Valentini et
al. 2022). On the other hand, it is important to take selection effects into
account when models are compared to observations, as highlighted by Popping et
al. (2019). Additionally, in theoretical works it is commonly assumed that the
molecular gas mass is the equivalent of the star-forming gas. This is not only
in disagreement with observations as discussed above, but such a non-star-
forming $H_{2}$ component has also been found in simulations (Lagos et al.
2015). Therefore, comparisons between simulations and observations need to
take such issues into account to ensure a meaningful comparison. Nevertheless,
to understand the complicated interplay between the available gas reservoir of
a galaxy and its resulting star formation is one of the key problems to be
deciphered in current galaxy formation studies.

The molecular gas content of a galaxy is thought to be fed from the overall
(cold) gas content of a galaxy, and for star forming disk galaxies this
reservoir of cold yet not molecular gas in the form of HI is known to be
rather extended, in some cases twice the size of the stellar disk, even
exhibiting low amount of star formation without showing large molecular clouds
(e.g. Gil de Paz et al. 2005; Thilker et al. 2005). The amount of HI gas
around galaxies is known to be larger for galaxies of smaller stellar masses,
and flattens to values of about $M_{\mathrm{HI}}\approx
10^{10}-10^{11}M_{\odot}$ for galaxies of stellar masses above
$M_{\mathrm{*}}\approx 10^{10}M_{\odot}$ in the local universe (e.g., Maddox
et al. 2015). However, exploring the gas reservoir of galaxies at higher
redshifts is increasingly difficult due to the weakness of the HI 21 cm line,
which is the only direct tracer for the HI content (e.g. Chowdhury et al.
2022). However, when comparing to simulations, this total gas content is
usually what is the most direct to measure from simulations.

In a previous study by Kudritzki et al. (2021, hereafter K21), we compared
look-back models with the hydrodynamical cosmological Magneticum Pathfinder
simulations as well as observations, connecting stellar mass, gas mass, and
star formation rates in star forming galaxies through cosmic time. We found
that during the evolution of galaxies on the main sequence, their star
formation rates decline towards low redshifts, not because the galaxies are
running out of cold gas, but because the fraction of the star-forming gas
declines. This is in excellent agreement with the observed properties as
discusses above, however, the reasons for this behaviour are still unclear. In
this follow-up study we investigate the physical processes responsible for
this decline.

More precisely, we shed light on the process that leads to the decrease of the
fraction of the star-forming gas compared to the total cold gas content. For
this we use the Magneticum Pathfinder simulations, which are described in
Section 2. We then show star-forming relations in Section 3 and analyze gas
properties in Section 4. In Section 5 we compare to observations and conclude
our findings in Section 6.

##  2 The Magneticum Pathfinder Simulations

The Magneticum111www.magneticum.org Pathfinder simulations are a set of fully
hydrodynamical cosmological simulations of different box-volumes and
resolutions. They follow the formation and evolution of cosmological
structures through cosmic time, accounting for the complex physical processes
which shape the first building blocks of galaxies into the galaxies seen
today. For details on the simulations see Hirschmann et al. (2014) and Teklu
et al. (2015). A WMAP-7 $\Lambda\mathrm{CDM}$ cosmology (Komatsu et al. 2011)
is adopted with $h=0.704$, $\Omega_{m}=0.272$, $\Omega_{b}=0.0451$,
$\Omega_{\lambda}=0.728$, $\sigma_{8}=0.809$, and an initial slope of the
power spectrum of $n_{s}=0.963$.

###  2.1 Implementation of Physical Processes

Star formation and galactic winds are treated in the same way as described by
Springel & Hernquist (2003). In this multiphase model for star formation, the
inter-stellar medium (ISM) is treated as a two-phase medium, where clouds of
cold gas form from cooling of hot gas and are embedded in the hot gas phase
assuming pressure equilibrium whenever gas particles are above a given
threshold density. The density of each gas particle is calculated using a
weighted sum over particle neighbors, where the weight decreases with
increasing distance. The hot gas within the multiphase model is heated by
supernovae (SNe) and can evaporate the cold clouds. Around 10% of massive
stars is assumed to explode as SNe II. The released energy by SNe II
($10^{51}$ erg) is modeled to trigger galactic winds with a mass loading rate
being proportional to the star formation rate (SFR) to obtain a resulting wind
velocity of $v_{\mathrm{wind}}=350$km/s. Our simulations also include a
detailed model of chemical evolution according to Tornatore et al. (2007).

Metals are produced by SNe II, by SNe Ia, and by intermediate- and low-mass
stars in the asymptotic giant branch (AGB). Metal radiative cooling rates are
implemented according to Wiersma et al. (2009). Metals and energy are released
by stars of different mass by integrating the evolution of the stellar
population (for details see Dolag et al. 2017), properly accounting for mass-
dependent lifetimes using a lifetime function according to Padovani &
Matteucci (1993), the metallicity-dependent stellar yields by Woosley & Weaver
(1995) for SNe II, the yields by van den Hoek & Groenewegen (1997) for AGB
stars, and the yields by Thielemann et al. (2003) for SNeIa. Stars of
different mass are initially distributed according to a Chabrier initial mass
function (Chabrier 2003).

Our simulations also include a prescription for black hole (BH) growth and
feedback from active galactic nuclei (AGN) based on the model presented by
Springel et al. (2005) and Di Matteo et al. (2005), including the same
modifications as Fabjan et al. (2010a) and some minor changes. The accretion
onto BHs and the associated feedback adopt a sub-resolution model (for further
details see Hirschmann et al. 2014; Steinborn et al. 2015; Teklu et al. 2015).
BHs are represented by collisionless “sink particles” that can grow in mass by
accreting gas from their environments, or by merging with other BHs. They are
seeded in galaxies with stellar masses above $\approx 4\cdot
10^{9}h^{-1}M_{\odot}$ with an initial mass of $\approx
10^{5}h^{-1}M_{\odot}$. The gas accretion rate $\dot{M}_{\bullet}$ is
estimated using the Bondi-Hoyle-Lyttleton approximation (Hoyle & Lyttleton
1939; Bondi & Hoyle 1944; Bondi 1952):

$\dot{M}_{\bullet}=\frac{4\pi
G^{2}M_{\bullet}^{2}f_{\mathrm{boost}}\rho}{(c_{s}^{2}+v^{2})^{3/2}},$ (1)

where $\rho$ and $c_{s}$ are the density and the sound speed of the
surrounding (ISM) gas, respectively, $f_{\mathrm{boost}}$ is a boost factor
for the density, which typically is set to $100$ and $v$ is the velocity of
the BH relative to the surrounding gas. The BH accretion is always limited to
the Eddington rate (maximum possible accretion for balance between inward-
directed gravitational force and outward-directed radiation pressure):
$\dot{M}_{\bullet}=\min(\dot{M}_{\bullet},\dot{M}_{\mathrm{edd}})$. Note that
the detailed accretion flows onto the BHs are unresolved, and thus we can only
capture BH growth due to the larger-scale gas distribution, which is resolved.
Once the accretion rate is computed for each BH particle, the mass
continuously grows. To model the loss of this gas from the gas particles, a
stochastic criterion is used to select the surrounding gas particles to be
removed. Unlike in the model described by Springel et al. (2005), in which a
selected gas particle contributes with all its mass, we included the
possibility for a gas particle to lose only a slice of its mass, which
corresponds to 1/4 of its original mass. In this way, each gas particle can
contribute with up to four ‘generations’ of BH accretion events, thus
providing a more continuous description of the accretion process.

Figure 1: r-Band mock images of an example disk galaxy from the high-mass bin
sample, at 10 different redshift bins from $z=0$ (upper left) to $z=2$ (lower
right). First and third row show the face-on view, while second and fourth row
show the edge-on views. Mock images were made using the code by Martin et al.
(2022).  Figure 2: The evolution of the Kennicutt-Schmidt relation for the
galaxies in the highest mass bin. The black triangles represent the mean star
formation rate density and mean gas surface density within the gas surface
density bins, for the star-forming gas. The small data points represent mean
values within radial bins of single galaxies, note that one galaxy can have up
to 16 data points. Blue diamonds are the cold gas including star-forming gas
and turquoise triangles are only star-forming gas.

We assume that a fraction $\epsilon_{\mathrm{f}}$ of the radiated energy is
thermally coupled to the surrounding gas so that
$\dot{E}_{\mathrm{f}}=\epsilon_{\mathrm{r}}\epsilon_{\mathrm{f}}\dot{M}_{\bullet}c^{2}$
is the rate of the energy feedback; $\epsilon_{\mathrm{f}}$ is a free
parameter and typically set to $0.1$ (see discussion by Steinborn et al.
2015). The energy is distributed kernel weighted to the surrounding gas
particles in an SPH-like manner. Additionally, we incorporated the feedback
prescription according to Fabjan et al. (2010b): we account for a transition
from a quasar- to a radio-mode feedback (see also Sijacki et al. 2007)
whenever the accretion rate falls below an Eddington ratio of
$f_{\mathrm{edd}}:=\dot{M}_{\bullet}/\dot{M}_{\mathrm{edd}}<10^{-2}$. During
the radio-mode feedback we assume a 4 times larger feedback efficiency than in
the quasar mode. This way, we want to account for massive BHs, which are
radiatively inefficient (having low accretion rates), but which are efficient
in heating the ICM by inflating hot bubbles in correspondence to the
termination of AGN jets. Note that we also, in contrast to Springel et al.
(2005), modify the mass growth of the BH by taking into account the feedback,
e.g., $\Delta M_{\bullet}=(1-\eta_{r})\dot{M}_{\bullet}\Delta t$.
Additionally, we introduced some technical modifications of the original
implementation (for details see Hirschmann et al. 2014).

###  2.2 Galaxy Sample Selection

For our study we use the Magneticum Box4/uhr simulation, which has a box
volume of (48 $h^{-3}$Mpc)3 with initially $2\times 576^{3}$ (dark matter and
gas) particles. The particle masses are $m_{\mathrm{DM}}=3.6\times
10^{7}h^{-1}M_{\odot}$ and $m_{\mathrm{gas}}=7.3\times 10^{6}h^{-1}M_{\odot}$,
respectively, and each gas particle can spawn up to four stellar particles
(i.e. the stellar particle mass is approximately 1/4th of the gas particle
mass), with a softening length of
$\epsilon_{\mathrm{DM}}=\epsilon_{\mathrm{gas}}=1.4h^{-1}$ kpc and
$\epsilon_{*}=0.7h^{-1}$ kpc.

The analysis of internal evolution of the gas within starforming galaxies
requires a high enough resolution. To this end, we select the highest
resolution volume simulation, Box4 uhr, which allows for the study of a broad
range of stellar masses down to halos of log10(M∗/M⊙) = 9.26 which are still
resolved with 1000 stellar particles. Previous studies have found galactic
properties to agree well with observations (e.g., Teklu et al. 2015; Schulze
et al. 2018; Remus & Forbes 2022). While the smaller simulation volume may
result in missing some rare or particularly unique objects, our focus is on
the global trend of galactic evolution for which this volume is sufficient.

For identifying disk galaxies we use their position in the stellar
mass–angular momentum ($M_{*}-j_{*}$) plane, quantified by the b-value

$b=\mathrm{log_{10}}\left(\frac{j}{\mathrm{kpc~{}km/s}}\right)-\frac{2}{3}\log_{10}\left(\frac{M_{*}}{M_{\odot}}\right),$
(2)

where $j$ is the specific angular momentum of the galaxy stellar component
(see especially Teklu et al. 2017, but also Teklu et al. 2015 and Schulze et
al. 2018 for more details). The connection between morphology and the position
on the $M_{*}-j_{*}$ plane was first noticed by Fall (1983) and was revisited
by Romanowsky & Fall (2012), who proposed scaling relations for disc and
elliptical galaxies, which would correspond to large and small b-values,
respectively. At redshift $z=0.07$, galaxies with $b\geq-4.35$ are considered
disks. We select all disk galaxies at $z=0.07$ with $\log_{10}M_{*}\geq
9M_{\odot}$. Note that at higher redshift these galaxies could also have been
intermediates, spheroids or even passive.

This leaves us with a sample of 621 galaxies selected at $z\approx 0$, and for
each of these galaxies the evolution is traced back to redshift $z=4.2$. The
galaxies are divided into 7 bins according to their stellar mass at $z\approx
0$, where the mean stellar mass of the galaxies in the bins are
$\log_{10}M_{*}/M_{\odot}$ = 9.28, 9.65, 10.01, 10.16, 10.34, 10.60, and
11.31, respectively, and the number of galaxies in the according mass bin is
107, 186, 124, 93, 61, 36 and 14. The selected mass bins are an arbitrary
choice to represent the range from lower to higher galaxy stellar masses.

Furthermore, we adopt an additional selection criterion for each snap shot
based on the SFR law by Pearson et al. (2018, their Appendix C) with some
modifications, namely with a correction factor as a function of redshift and a
modiﬁcation of the power law, which describes the dependence of stellar mass
(for details see Kudritzki et al. 2021). We use the threshold of 0.8 dex below
our adopted star formation law and include only galaxies above this threshold.

Fig. 1 depicts r-band mock images of an example disk galaxy from the high mass
bin of the Magneticum galaxy sample, followed back in time to show its
evolution. For each redshift, the galaxy is shown face-on (upper rows) and
edge-on (lower rows). As can clearly be seen, it has a large disk both in
stellar component as well as in the gas component (see Fig. 11 in the Appendix
for the gas component) at $z=0$. At higher redshifts, the disk is already
present albeit several small merger events had occurred since $z=2$, although
as expected the radial extend of the disk is significantly smaller.

In the following, we will study the gas components of the selected disk
galaxies in detail, especially the properties of the gas particles. When
referring to gas particles that are currently in the process of forming stars,
we name those star-forming gas. These particles have densities above
$10^{7.2}M_{\odot}\mathrm{kpc}^{-3}$. If not specified otherwise, the cold gas
component includes gas particles with temperatures below $10^{5}K$ and star-
forming gas. The hot gas component consists of gas particles which have
temperatures above that temperature threshold and are non-star-forming.

##  3 Star Formation in Magneticum

As described previously, star formation is treated within a sub-resolution
model (Springel & Hernquist 2003), where gas above a certain density threshold
is treated as a two-phase medium. Above this threshold, cold, star-forming
clouds form from cooling of hot gas and are embedded in the hot gas phase
assuming pressure equilibrium plus a stellar component. This sub-resolution
model describes the unresolved ISM by solving a set of connected differential
equations for mass and energy flows with a closure condition described in
detail by Springel & Hernquist (2003, and references therein). Within this
sub-resolution model, every gas element in the simulation has an associated
star formation rate following from the solution of this set of differential
equations. This allows then to map in detail the star formation within the
simulated galaxies.

Figure 3: The evolution of the mean temperature-density diagram for the gas of
the galaxies in the highest mass bin. The intensity of the red color encodes
the fraction of gas resolution elements in a grid cell. The dashed vertical
line represents the star formation threshold, and the horizontal line the
temperature cut for cold gas. For examples of different mass bins, see Fig. 13
in the Appendix. Note that we chose to show only six representative timesteps
in this figure different than in other figures depicting time evolution in
this study, as in these mean stacked evolution plots the overall trend is
important to see and individual smaller timesteps in this case do not differ
strongly but rather demonstrate a smooth trend as can clearly be seen from the
six timesteps shown here. Also note that the color bar changes according to
the maximum fraction of the shown timestep.

First and foremost, the question is how the gas reservoir of a given galaxy
evolves with redshift with respect to its global star formation properties.
Thus, in Fig. 2 we show the mean gas surface density against the mean star
formation rate surface density, i.e. the Kennicutt-Schmidt (KS) relation, of
the cold gas (including star-forming gas, dark blue diamonds) and only the
star-forming gas (turquoise triangles), for the galaxies in the highest mass
bin as an example. Each data point represents the mean value of one galaxy in
a radial bin out to $40\mathrm{kpc}$ with 16 equally spaced bins à
$2.5\mathrm{kpc}$. Black triangles show the average star formation rate
density and average gas surface density within gas surface density bins, for
the star-forming gas. The star-forming gas of the galaxies, on average,
follows the evolution according to the KS-law (Kennicutt 1998, solid lines)
within its uncertainties (dashed lines), at all redshifts up to $z\approx
3.5$. We find that at high redshifts the galaxies exhibit higher densities
both in the cold gas but also in star formation rate, and that these densities
decrease towards low redshifts. The cold gas falls off the relation toward
lower SFR-densities as in these densities the mean values are dominated by
cold non-star-forming gas. This is in agreement with previous observational
studies, for example Bigiel et al. (2008) found that the relationship of the
total gas surface density and the SFR surface density varies within and
between the galaxies, and that there is almost no correlation between the
surface density of the HI gas, which is generally associated with the cold
non-star-forming gas, and the SFR surface density. Interestingly, this
deviation of the cold gas from the KS-law is relatively constant over time.
This points out the importance of using only the star-forming gas instead of
the cold gas, when investigating and comparing the gas properties to
observations. Note that, as the star-forming gas is a subsample of the cold
gas, the values for the SFR surface density are the same for each radial bin,
while the gas surface density is equal or lower for the star-forming gas
compared to the cold gas. In Fig. 12 in the Appendix we show the KS-relation
for all selected galaxies at $z=0.06$ in order to demonstrate that this
relation is valid for all mass bins.

In Fig. 3 (see also Fig. 13 in the Appendix) we show the evolution of the
stacked temperature–density phase diagrams for all gas of galaxies in the
highest mass bin, where the intensity of the red color mirrors the fraction of
the resolution elements of gas in the grid cell. The dotted vertical line
represents the density threshold of 7.2 $M_{\odot}kpc^{-3}$, above which the
star formation sets in, and the dotted horizontal line shows the temperature
cut made to distinguish between cold and hot gas. At high redshift most of the
gas resolution elements are cold and dense and thus have a high star formation
activity. Towards lower redshift there is still a large amount of cold gas,
which however becomes less dense and thus does not reach the star-forming
threshold anymore. At the lowest redshift we can see the build-up of a hot and
diffuse gas component, i.e. gas with high temperature and low density, which
corresponds to gas that has been heated, for example by AGN feedback.

Figure 4: The evolution of the temperature-density diagram for one example
galaxy, where each data point represents the physical properties within a
resolution element. Red is gas without any star-forming component, star-
forming gas ranges from turquoise and blue tones to dark blue (with increasing
activity). The dashed vertical line represents the star formation threshold,
and the horizontal line the temperature cut for cold gas.

As in the stacked phase diagram possibly interesting signatures could get
lost, we show in Fig. 4 the evolution of the phase diagram for one example
galaxy from the highest mass bin, the same as shown in Fig. 1. Each data point
represents one gas resolution element of the galaxy within $5R_{1/2}$. The
colors indicate their star-forming activity, where red is non-star-forming,
while turquoise, the different blue tones and dark blue are star-forming (with
increasing activity). At high redshifts most of the gas is cold and dense, and
forms stars. This changes with decreasing redshift, where the amount of cold
gas which is not dense enough to reach the SF threshold increases and star
formation decreases, respectively. Still, at low redshift there is a large
amount of cold gas. Note that gas which is assumed to be in the wind phase, is
prevented from forming stars and therefore some gas even when above the SF
threshold does not form stars.

##  4 Evolution of Cold and Star-Forming Gas

In the following, we will study different quantities of the cold and star-
forming gas, such as the mass, the density and the radial profiles in order to
shed light on the origin of the decrease of the star-forming gas fraction.

Figure 5: The colors and lines encode the mean values of the galaxies in the
different stellar mass bins, with mean $log_{10}M_{*}/M_{\mathrm{sol}}$ values
of 9.3, 9.66, 9.95, 10.14, 10.33, 10.59 and 11.15 (from light blue
representing the lowest mass bin to dark blue showing the most massive bin).
Top: The cold gas mass increases with time for the three lowest mass bins,
whereas for the higher mass bins it remains relatively constant after the
first steep increase. Middle: The fraction of the star-forming gas mass
compared to the cold gas content decreases continuously, where there is no
difference between the mass bins except for the two lowest ones. Bottom:
Initially, the mass of the star-forming gas increases for the high mass bins
and then decreases, while it decreases for the low mass bins.

Fig. 5 shows the time evolution of the cold gas mass (top panel), the fraction
of the star-forming gas compared to the cold gas (middle panel), and the star-
forming gas mass (bottom panel). For this, we calculate the mean values of the
galaxies in the different stellar mass bins, which are encoded by different
shades of blue with dark blue being the highest mass bin, the next lower mass
bins becoming lighter, and light blue being the lowest mass bin. As we focus
on the global trends and not those of individual galaxies, and as it would be
confusing to show the lines of the errors or the scatter, in Fig. 14 in the
Appendix we show the standard deviation exemplary. The mean cold gas mass
continuously increases with time for the three lowest mass bins, whereas for
the higher mass bins it remains relatively constant after the first rapid
increase. We can clearly see that the galaxies do not run out of cold gas.
Instead, as already shown in K21, the fraction of the star-forming gas
decreases with evolving time independently of the stellar mass, as aside from
the two lowest stellar mass bins the curves generally lie on top of each
other. Interestingly, at early times the fraction of the star-forming gas is
very similar for all mass bins; however, the evolution then differs, as the
mass of the star-forming gas increases and then decreases for the galaxies in
the high mass bins, while for those in the two lowest mass bins it decreases
from the beginning. This highlights that it is not the lack of cold gas for
galaxies in any of the stellar mass bins but the capability of the gas to form
stars, which leads to a decrease of the star formation. We note that, given
the smooth trends found between the various mass bins in mean star-forming and
cold gas content, we expect that a variation of the mass bin borders would
result in little change of the findings presented here. However, the lowest
mass bin is to be taken with caution as these galaxies are very small at all
redshifts.

Figure 6: The mean logarithmic mass density of cold gas as function of look-
back time, where the different mass bins are color-coded as in Fig. 5. The
black dashed line represents the threshold for star formation.  Figure 7: The
fraction of cold gas resolution elements with a given density for the highest
mass bin, where the different colors and lines represent the different look-
back times. The dashed line represents the threshold for star formation.
Towards lower look-back times the peak shifts from high to low density, lying
below the SF threshold at later times.

Figure 8: Left: The radial profile of the mean logarithmic mass density of
cold gas for the highest mass bin, where the different colors and lines
represent the different look-back times and the dashed line represents the
threshold for star formation. Right: The radial profile of the mean
logarithmic surface density of cold gas.

As the stars form out of cold gas which condenses to dense clouds, we next
study the density of the cold gas in Fig. 6. Here, we use the (mass) density
of the cold gas, which is calculated using a weighting function around the
resolution elements and in this way roughly expresses the local densities,
i.e. these gas resolution elements represent large gas regions in which stars
can form. This is in contrast to averaging over parts of the galaxy or the
whole galaxy, as e.g. done for the cosmic gas density. In Fig. 15 of the
Appendix we show that the “global” density
$=M_{\mathrm{coldgas}}/(4/3\cdot\pi\cdot(5\cdot R_{1/2})^{3})$ results in
different curves. In Fig. 6 we find that with evolving time the mean density
of the cold gas decreases for all galaxies. It then drops below the threshold
for the star formation (black dashed line), which happens earlier for low mass
galaxies and later for high mass galaxies.

In the following we want to explore why this mean density of the cold gas
decreases with time. For this, we analyze the galaxies in the highest mass bin
in more detail. Fig. 7 shows the average fraction of cold gas in $5\cdot
R_{1/2}$ within a certain density bin. Here, the different lines and colors
encode the look-back times and the dashed line the density threshold for star
formation. As clearly can be seen, at high redshift most of the cold gas has a
density above the star formation threshold. Towards lower redshifts the
distribution becomes broader and the peak is shifting to lower density, below
the star formation threshold. So while the total amount of cold gas at low
redshifts is still rather large, its density clearly is too low for most of it
to continue star formation. This raises the question what is causing this
behavior.

To understand this, we take a closer look at the radial distribution of the
cold and star-forming gas, as this aspect has so far been neglected in the
above quantities. The left panel of Fig. 8 shows the radial profile of the
average gas density in a certain radial bin. The different lines and colors
represent the different look-back times and the dashed line represents the
density threshold for star formation. Overall, the mean density decreases with
evolving time. Another general, redshift independent trend is that the density
is higher in the center and decreases towards larger radii. This can be
explained by the hydrostatic pressure equilibrium, where the gas experiences
the gravitational force and thus pressure and density are higher in the
center, where most of the galaxy’s mass resides. We note that in the center of
the galaxies, especially at small look-back times, some of the galaxies have a
ring of cold gas and thus it is possible that only a small number of galaxies
contributes to the data points in the very central regions. However, this plot
does not account for the number of gas resolution elements nor the mass of the
gas that is found in these radial bins. Therefore, on the right panel of Fig.
8 we show the surface density of the cold gas, as this quantity takes the
spatial distribution better into account.

Interestingly, we clearly see that the maximum of the surface density is
moving to larger radii towards low redshifts. Note that for all radial bins we
only consider galaxies with some amount of cold gas to calculate the mean
surface density. As many galaxies in our sample at the lowest redshifts have
little or no cold gas at their centers given that many of them have rings of
cold gas as mentioned already above, the mean values for the two centralmost
radial bins are dominated by the few galaxies which do, resulting in a
spurious peak from the low number statistics for these two bins. The clear
peak in cold gas density moving towards larger radii with lower redshifts,
however, is not affected by low number statistics and is a real result of
physical processes.

Figure 9: The mean number of cold gas (top), star-forming (middle) resolution
elements and the fraction of star-forming to cold gas resolution elements
(bottom) in each radial bin for the galaxies in the highest mass bin, where
the different colors and lines represent the different look-back times.

Fig. 9 shows the mean number of gas resolution elements in a certain radial
bin, with the top panel showing the cold gas, the middle panel showing the
star-forming gas and the bottom panel the fraction of star-forming to cold gas
(including star-forming gas). We can clearly see that at large look-back times
the majority of the gas is in the center and that the peak moves further
outside with evolving time. The distribution seems to become broader, which
reflects that the total number of cold gas resolution elements does not
decrease. However, the peak of the star-forming gas becomes noticeably
smaller, albeit it also moves to larger radii. The fraction shown in the
bottom panel displays that the star-forming gas is becoming less over time as
the cold gas regions move to larger radii. To further illustrate that the
star-forming regions are moving outside with decreasing look-back times, in
Fig. 11 in the Appendix we show the evolution of one example galaxy projected
to the $xy$\- and $xz$-plane.

##  5 Comparison with Observations

For the comparison with observations we use the star-forming gas as a proxy
for the H2 gas. However, this has to be taken with some caution, as e.g. Lagos
et al. (2015) have found that in the EAGLE simulations this approximation is
poor for H2 content above a certain mass and that the H2 masses are depending
on the resolution. As a proxy for the atomic hydrdogen we use the cold gas
excluding the star-forming gas. We calculate the gas and stellar masses within
a radius of 40 kpc in order to cover a similar size compared to the
observational results by Chowdhury et al. (2022), who analyzed cubes of 90 kpc
length for their samples of star-forming galaxies to investigate the HI
contents at high $z$.

Figure 10: The redshift evolution of the mean mass of the cold gas without SF
gas divided by the mean mass of the SF gas only. The observational data (red
squares) for $z\approx 1$ and $z\approx 1.3$ were presented in Chowdhury et
al. (2022) and the estimated observational value for $M_{\mathrm{atomic}}$ and
$M_{\mathrm{molecular}}$ at $z\approx 0$ were taken from Catinella et al.
(2018) and Saintonge et al. (2017), respectively. The colors encode the lowest
(light blue) and highest (dark blue) as well as the mass bin corresponding
approximately to the mass the observations covered (blue dash-dotted line).

Fig. 10 shows the mean mass of the cold gas without star-forming gas divided
by the mean mass of the star-forming gas, and the observed mean mass of atomic
gas divided by the mean molecular gas mass, respectively. For better
visibility, we only show three mass bins, namely the highest mass bin (dark
blue), the lowest mass bin (light blue) and the mass bin of about
$10^{10}M_{\odot}$ (blue dash-dotted line) roughly corresponding to the mean
stellar mass used in the observations. The fractions decrease with increasing
redshift for the galaxies in our simulation. They agree well with the
observations for $z\approx 0$ and $z\approx 1$. On the other hand, we find
some disagreement with the Chowdhury et al. (2022) value at $z\approx 1.3$,
which indicates an increasing fraction. We note, however, that Chowdhury et
al. (2022) discuss potential uncertainties for the mass of molecular gas,
which would lower their fraction to 2.5.

Our results are in agreement with the trend presented by Obreschkow & Rawlings
(2009), and Power et al. (2010), who analyzed predictions of semi-analytic
models by Baugh et al. (2005); Bower et al. (2006); De Lucia & Blaizot (2007);
Font et al. (2008), where the fraction of the molecular compared to atomic
hydrogen is increasing towards higher redshifts. Similarly, Lagos et al.
(2011) using a galaxy formation model find that the fraction of H2 compared to
HI increases to a peak at $z\approx 3.5$, then H2 dominates until $z\approx 5$
and above this redshift HI is the dominant component of the ISM.

##  6 Summary and Discussion

We have analyzed a sample of galaxies in the cosmological hydrodynamical
simulation Magneticum Pathfinder. The galaxies were selected at $z=0.1$ such
that they are disk galaxies according to the $b$-value. These galaxies were
put into mass bins and were followed backwards in time. In this way, we
calculate average properties for different mass bins and study the average
evolution of today’s galaxies that are similar in mass at present instead of
averaging over a population at different points in time. Our aim was to shed
light on the question why the fraction of star-forming gas is decreasing,
while there is still much cold gas in these galaxies at low redshift (see also
Kudritzki et al. 2021). Our analysis has shown the following:

  * •

The density distribution of cold gas clouds is shifting from higher to lower
densities with evolving time. As a result, the number of clouds reaching the
threshold for star formation is decreasing.

  * •

The reason for this behavior is a shift of the maximum of the number
distribution of clouds to larger galactic radii indicating an inside-out
growth of disk galaxies, in agreement with observations (see e.g. González
Delgado et al. 2015; Goddard et al. 2017). Since the gas clouds are in
pressure equilibrium with the galaxy’s gravitational potential, their density
is smaller at larger radii and a larger fraction of their distribution falls
below the star formation threshold.

While the average density of the cold gas in the whole galaxy as well as seen
in the outer parts of the galaxies is below the star formation threshold
(especially towards low redshifts), individual regions of cold gas still reach
densities high enough to form stars. These regions are shifting towards larger
radii with evolving time.

In summary, the fraction of the star-forming gas is simply becoming smaller as
the gas clouds move further out to larger radii during the evolution of the
galaxy, where the gas density is smaller.

Our findings show the need to better consider not just the total galactic gas
content, but also where it is radially distributed. The inside-out growth of
disk galaxies provides a natural mechanism for a soft quenching, not for lack
of cold gas material but rather due to lack of pressure. Processes which clump
together this gas may then result in reignition of the star formation.

We thank Andi Burkert and Tadziu Hoffmann for useful discussions. We thank our
referee for engaged and very constructive comments. This work has been
supported by the Munich Excellence Cluster Origins funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy EXC-2094 390783311\. KD and LK acknowledge support by the
COMPLEX project from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation program grant agreement
ERC-2019-AdG 882679. The Magneticum Pathfinder simulations were performed at
the Leibniz-Rechenzentrum with CPU time assigned to the Project “pr86re”. We
are especially grateful for the support by M. Petkova through the
Computational Center for Particle and Astrophysics (C2PAP).

##  Appendix A Spatial Distribution of Gas and Stars

Fig. 11 shows the evolution of an example galaxy in the highest mass bin. For
twelve redshifts we plot the spatial distribution of stars (yellow), hot gas
(red), cold gas (blue) and star-forming gas (turquoise) in face-on projection
(upper panels, odd rows) and edge-on projection (lower panels, even rows). The
circle indicates $5\cdot R_{1/2}$. This figure illustrates that at early times
the galaxy is small and compact and that the star formation takes place in the
inner part. With evolving time the gas is pushed outwards and the stars form
at larger radii.

Figure 11: The evolution of the spatial distribution of stars (yellow), hot
gas (red), cold gas (blue) and star-forming gas (turquoise) for one example
galaxy. The circle indicates $5\cdot R_{1/2}$.

##  Appendix B Kennicutt-Schmidt Relation for All Disk Galaxies at $z=0$

Fig. 12 shows the Kennicutt-Schmidt relation for all galaxies fullfilling the
selection criteria at redshift $z=0$. The colors encode the mass bins of the
galaxies, where light colors represent the low mass galaxies and dark blue the
high mass galaxies. We note that we omitted the innermost radial bin due to
poor resolution. As discussed above, our results are in agreement with
previous observational studies. However, we only consider late-type galaxies
with a certain amount of cold and star-forming gas. So with our sample of
galaxies, we cannot predict whether the low surface brightness galaxies in our
simulation fall below the KS relation as found in observational studies
(Bigiel et al. 2008; Nagesh et al. 2023).

Figure 12: The KS relation for all selected star-forming disk galaxies.

##  Appendix C Temperature vs. Density

In Fig. 3 we have shown the evolution of the stacked temperature–density phase
diagrams for the gas resolution elements of galaxies in the highest mass bin.
Here in Fig. 13, we show it for two other mass bins, namely the lowest and one
intermediate. Again, the intensity of the red color mirrors the fraction of
gas in the grid cell. Similar to the galaxies in the highest mass bin, at high
redshift most of the gas is cold and dense, with a high star formation
activity. Towards lower redshift there is still a large amount of cold gas,
however, less dense, and not reaching the star-forming threshold, any more. In
contrast to the galaxies in the highest mass bin, at the lowest redshift there
is no hot and diffuse gas. This reflects the fact that the galaxies in the
lowest mass bin do not have AGNs, which could heat and dissipate the gas.

Figure 13: The evolution of the mean temperature vs. density diagram for the
galaxies in the lowest (top panels) and intermediate (bottom panels) mass bins
of mean values of $log_{10}M_{*}/M_{\odot}=9.28$ and $10.16$, respectively.
The intensity of the red color encodes the fraction of gas resolution elements
in a grid cell. The vertical dashed line represents the star formation
threshold, while the horizontal line shows the temperature cut made for cold
gas.

##  Appendix D Example for Standard Deviations

Fig. 14 shows the same as Fig. 5 but only three mass bins, namely the mass
bins of mean values of $log_{10}M_{*}/M_{\odot}=9.28$, $10.16$, and 11.15,
respectively. In order to indicate the significance of our results we show the
lines for $\pm\sigma$. We report the values for the minimum and maximum
standard deviation over all redshifts separately for the three different mass
bins in Table 1.

Figure 14: The same as Fig. 5 but only the lowest, intermediate and highest mass bins are shown in order to illustrate the standard deviation.  Mass bin $logM_{*}/M_{\odot}$ | Min($\sigma(logM_{coldgas}/M_{\odot})$) | Max($\sigma(logM_{coldgas}/M_{\odot})$)
---|---|---
9.28 | 0.1146 | 0.1803
10.16 | 0.1595 | 0.2776
11.15 | 0.1441 | 0.4478
Mass bin $logM_{*}/M_{\odot}$ | Min($\sigma(M_{SF}/M_{coldgas})$) | Max($\sigma(M_{SF/}M_{coldgas})$)
9.28 | 0.0471 | 0.1890
10.16 | 0.0986 | 0.1577
11.15 | 0.0456 | 0.1631
Mass bin $logM_{*}/M_{\odot}$ | Min($\sigma(logM_{SF}/M_{\odot})$) | Max($\sigma(logM_{SF}/M_{\odot})$)
9.28 | 0.1555 | 0.3615
10.16 | 0.1687 | 0.3536
11.15 | 0.1487 | 0.4944
Table 1: The values of the minimum and maximum standard deviations for the
quantities of the three mass bins shown in Fig. 14.

##  Appendix E Global Cold Gas Density

In Section 4 we discuss the evolution of the densities of individual gas
regions. Here we intent to demonstrate that this is different from the
evolution of the ”global” density of cold gas in a galaxy. For that purpose we
calculate the density of the gas mass in the whole galaxy. This is done by
using the whole gas mass within a sphere and divide by the volume of this
sphere, i.e. $=M_{\mathrm{coldgas}}/(4/3\cdot\pi\cdot(5\cdot R_{1/2})^{3})$.
The comparison with observations should be used with caution. If a fixed
aperture is used for different types or sizes of galaxies, small galaxies can
show a low density while large galaxies show a higher density. Due to our
selection of the gas inside a sphere which is scaling with the stellar half
mass radius within the virial radius we mostly avoid this issue. In Fig. 15 we
can see that it makes a difference compared to the ”local” gas density (see
Fig. 6). The density all over the galaxy is lower than that averaged over
individual gas regions. This quantity shows implicitly how the mass grows
compared to the size. For lower mass galaxies the size increases less compared
to the gas mass than for higher mass galaxies. The most interesting aspect
here is that this global density does not tell us if the individual gas
regions are dense enough to form stars. i.e. the global density is high for
low mass galaxies (light blue) while the local density is lower for them than
for the high mass galaxies (dark blue).

Figure 15: The “global” density of the cold gas over time (i.e. the cold gas
mass within a sphere divided by the sphere’s volume), colored by the final
stellar mass as in Fig. 5.

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