# A Value-added COSMOS2020 Catalog of Physical Properties: Constraining
Temperature-dependent Initial Mass Function

Vadim Rusakov Cosmic Dawn Center (DAWN)  Niels Bohr Institute, University of
Copenhagen, Lyngbyvej 2, DK-2100 Copenhagen Ø  Charles L. Steinhardt Cosmic
Dawn Center (DAWN)  Niels Bohr Institute, University of Copenhagen, Lyngbyvej
2, DK-2100 Copenhagen Ø  Albert Sneppen Cosmic Dawn Center (DAWN)  Niels Bohr
Institute, University of Copenhagen, Lyngbyvej 2, DK-2100 Copenhagen Ø

###### Abstract

This work presents and releases a catalog of new photometrically-derived
physical properties for the $\sim 10^{5}$ most well-measured galaxies in the
COSMOS field on the sky. Using a recently developed technique, spectral energy
distributions are modeled assuming a stellar initial mass function (IMF) that
depends on the temperature of gas in star-forming regions. The method is
applied to the largest current sample of high-quality panchromatic photometry,
the COSMOS2020 catalog, that allows for testing this assumption. It is found
that the galaxies exhibit a continuum of IMF, and gas temperatures, most of
which are bottom-lighter than measured in the Milky Way. As a consequence, the
stellar masses and star formation rates of most galaxies here are found to be
lower than those measured by traditional techniques in the COSMOS2020 catalog
by factors of $\sim 1.6-3.5$ and $2.5-70.0$, respectively, with the change
being the strongest for the most active galaxies. The resulting physical
properties provide new insights into variation of the IMF-derived gas
temperature along the star-forming main sequence and at quiescence, produce a
sharp and coherent picture of downsizing, as seen from the stellar mass
functions, and hint at a possible high-temperature and high-density stage of
early galactic evolution.

Catalogs(205) — Initial mass function(796) — Galaxy masses(607) — Galaxy
properties(615) — High-redshift galaxies(734) — Stellar mass functions(1612) —
Galaxy evolution(594)

††software: EAZY (Brammer et al., 2008; Brammer, 2021), astropy (Astropy
Collaboration et al., 2013, 2018), matplotlib (Hunter, 2007), numpy (Harris et
al., 2020), scipy (Virtanen et al., 2020).

##  1 Introduction

Most galaxies outside of the Local Group are studied by fitting photometric
templates with known physical properties. These templates rely on strong
assumptions about the formation and evolution of stellar populations (Conroy,
2013). The problem is complicated, as most of the starlight is integrated and
comes from the high-mass stars that compose a small fraction of its total
stellar mass. Typically in models, the stellar population is produced by some
form of a Galactic stellar initial mass function (IMF; such as Salpeter, 1955;
Miller & Scalo, 1979; Kroupa, 2001; Chabrier, 2003) combined with a specific
star formation history and dust extinction. Although the IMF is constrained
fairly well in the locality of the Milky Way (MW), where it is possible to
resolve individual stars, the same task is impossible for distant enough
galaxies. Therefore, as this assumption fixes a certain stellar budget for
galaxies, it is crucial that it holds so that the physical properties are
estimated accurately. Otherwise, it can break the results and the
interpretation of different stages of galaxy evolution.

Several observations have hinted that models of integrated starlight
calibrated against stellar populations in the local universe may sometimes
fail at increasingly high redshift. Large galaxy surveys (Hildebrandt et al.,
2009; Caputi et al., 2015; Finkelstein et al., 2015; Steinhardt et al., 2016)
have reported that dark matter halos inferred from stellar masses at high
redshift become too massive to fit cosmological predictions. Furthermore,
recent observations of galaxies at $z>8$ (Atek et al., 2023; Labbé et al.,
2023; Naidu et al., 2022, among others) have revealed that by drawing from the
same stellar budget they require more baryons than there are produced at that
time (Boylan-Kolchin, 2023). At the same time, the observed luminosity
functions have been under-predicted in some of the semi-analytical simulations
at $z>13$ by a factor of 30111 It is worth mentioning that other studies have
not reported the tension with interpreted or simulated galaxy properties
(Adams et al., 2023; McCaffrey et al., 2023). Others have indicated that it
can be resolved by changing assumptions in galaxy models (Yung et al., 2023).
Therefore, the apparent nature of the tension reinforces the fact that it is
likely a test of the galaxy models and not the cosmology..

On the other hand, low-redshift observations sensitive to low-mass stars hint
at underestimated stellar mass in early-type galaxies. For example,
measurements of stellar kinematics, absorption lines or various line indices
find excess in mass-to-light ratio in local elliptical galaxies relative to
the Milky Way IMF (Cappellari et al., 2012; Conroy & van Dokkum, 2012; van
Dokkum & Conroy, 2012; Martín-Navarro et al., 2015; van Dokkum et al., 2017a).
Other observations of resolved stellar populations in the MW or local dwarfs
demonstrate similar IMF results in old stellar structures (Geha et al., 2013;
Hallakoun & Maoz, 2021).

Together, these pieces of evidence independently suggest that the mass-to-
light ratio in galaxy models appears to be off in the edge cases with
drastically different star-forming conditions. It is too high for active
galaxies at high redshift and too conservative for passive ones at low
redshift. At the same time the galaxy populations with intermediate properties
likely represent the bulk of the distribution.

In the high redshift cases where the tension with the cosmology arises,
Steinhardt et al. (2022); Harikane et al. (2023) suggested that a bottom-
lighter (or top-heavier) IMF can account for the over-estimation of stellar
mass inferred from observations. Jermyn et al. (2018); Steinhardt et al.
(2022) proposed that the bottom-light IMF at high redshift is likely driven by
the cosmic microwave background (CMB) temperature. Similarly, Yung et al.
(2023) proposed that the simulations can match the higher galaxy number
densities with a top-heavier IMF. It was also possible to resolve the tension
by more properly accounting for cosmic variance, uncertainty in the stellar
mass and accounting for possible additional gas for star formation (Chen et
al., 2023).

Consequently, one could also argue for a more bottom-heavy IMF to solve the
tension on the other redshift end (in the local universe) to account for the
reported under-estimation of stellar mass in the early-type galaxies. This
would be in agreement with the IMF measurements of Cappellari et al. (2012);
Conroy & van Dokkum (2012); van Dokkum & Conroy (2012); Martín-Navarro et al.
(2015); van Dokkum et al. (2017a). Therefore, if the shape of the IMF varies
with average conditions of star formation in different galaxies, there is
likely to be a continuum of stellar mass functions.

In theory, changes in temperature, density or metallicity of the star-forming
material must lead to a corresponding change in the relevant mass scales of
the stellar population (Jeans, 1902; Low & Lynden-Bell, 1976; Larson, 1998,
2005) and modify the shape of the IMF (Bate & Bonnell, 2005; Jappsen et al.,
2005; Krumholz, 2011; Hopkins, 2012; Jermyn et al., 2018). Although, the star-
forming gas conditions have been difficult to determine from line ratios at
high redshift, the temperature of cold dust serves as a strong indicator of
conditions of the interstellar medium (ISM). A possible link between the
thermal state of the gas and dust in the main-sequence galaxies has been shown
by comparing mass and luminosity-weighted dust temperatures with excitation
temperature from [C i] line ratio (Valentino et al., 2020).

Empirically, measurements of dust temperature hint that the state of the ISM
(and possibly the star-forming gas) changes with redshift and position with
respect to the star-forming main sequence (SFMS; Magnelli et al., 2014;
Lamperti et al., 2019). The latter studies show that the temperature of the
cold component of dust measured from the far-IR (FIR) continuum increases from
$\sim 20$ to 40 K for galaxies depending on the offset from the SFMS. Besides,
the average dust temperature increases with redshift (Schreiber et al., 2018;
Cortzen et al., 2020), which is similar to the reported increase in the
intensity of the radiation field at higher redshift (Béthermin et al., 2015;
Magdis et al., 2017). If the molecular clouds exhibit similar behavior, this
evidence builds on to suggest that the IMF should change accordingly (Kroupa,
2001; Bastian et al., 2010).

Moreover, Gunawardhana et al. (2011); Nanayakkara et al. (2017) show that the
excessive equivalent widths of the $H\alpha$ line are likely due to a top-
heavier IMF at $z\sim 0.35$ and $z\sim 2$. As this excess correlates with the
star formation rate, they suggest that the effect increases with redshift for
star-forming galaxies. In addition, a more direct probe of the IMF in Zhang et
al. (2018) – ${}^{13}{\rm C}/^{18}{\rm O}$ transitions – reveals a top-heavier
IMF in several starbursts at $2<z<3$.

The recently introduced technique in Sneppen et al. (2022) was used to
constrain the IMF in some of the most well-measured galaxies. However, it has
been challenging to test the idea due to rare photometric band coverage and
relatively low quality of available measurements. Therefore, most samples are
prone to the strong covariances between various model parameters. As such, the
lack of emission from the bulk of stellar mass leads to degeneracies between
the IMF, the law of dust attenuation or the star formation history. However,
Sneppen et al. (2022) showed that improved measurements in the recent data
sets (such as COSMOS2015; Laigle et al., 2016) can provide more statistical
power to relax the assumption of the local IMF and break the existing
degeneracies to some extent.

This work employs the fitting technique to produce a catalog of new physical
properties for galaxies in the COSMOS field, available for use by the
community. It is expected that new higher quality observations released in the
COSMOS2020 catalog (Weaver et al., 2022a) can reveal fainter galaxies across a
range of redshifts, as well as refine the observations of the previous
catalogs. With these data, it is possible to improve on the estimates of
stellar mass ($M_{\star}$) and star formation rate (SFR), by starting with
the, possibly, more physically plausible set of assumptions for the stellar
IMF. In addition, this update provides the IMF property theoretically related
to the temperature of the star-forming gas (Jermyn et al., 2018) and argues
for its importance for galaxy evolution. The work here shows that the measured
change in stellar mass and star formation rate and the associated increase in
the inferred gas temperature at lower redshifts ($0<z<2$) reinforces the
importance of the proper treatment of stellar populations particularly at high
redshift, where the cosmological predictions can be tested in the most direct
way.

The data used in this work are described in § 2. Next, § 3 outlines the
procedure of SED fitting, which includes the description of the temperature-
dependent IMF and the associated templates. Then, § 4 details the quality
choices made to produce the final data set and demonstrates the best-fit
physical properties. The following subsections demonstrate some well-known
relationships with the new properties and describe new insights driven by the
best-fit IMF. Finally, the discussion in § 5 reiterates some of the most
critical approximations made when interpreting observations in application to
galaxy evolution. This work assumes a Flat Lambda Cold Dark Matter cosmology
($H_{0}=70$ km s-1 Mpc-1, $\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$).

##  2 Data catalog

Previously, work to constrain the IMF using this technique was done with the
COSMOS2015 catalog (Laigle et al., 2016) in Sneppen et al. (2022), which when
published was the largest existing multi-band photometric data set. COSMOS2020
(Weaver et al., 2022a) is an updated catalog with more detections and deeper
and more precise photometry than the previous versions in the COSMOS field
(Capak et al., 2007; Scoville et al., 2007; Ilbert et al., 2009, 2013; Laigle
et al., 2016). The most significant advantages arise from deeper infrared
bands and from including bluer bands in its detection image. The latter is
expected to result in a higher number of small blue galaxies, which is
particularly useful for this work, where compact blue galaxies are of special
interest, as discussed in § 4.8.

There are two primary versions of the COSMOS2020 catalog, CLASSIC and FARMER.
This work makes use of the CLASSIC version, which is preferred for the
brightest galaxies. In contrast to the modeled fluxes in the FARMER version,
it derives aperture-based photometry from PSF-homogenized images similarly to
the previous catalogs. Despite the differences in approach, the precision of
measurements and the fraction of outliers in terms of photometric redshifts
are similar between them on average. For the brightest galaxies CLASSIC
performs marginally better both in precision and number of redshift outliers
using the standard photometric fitting procedure. The FARMER catalog is
analysed in a similar experiment with a different photometric fitting code,
LePhare (Arnouts et al., 1999; Ilbert et al., 2006) in Rusakov et al. (2023;
in prep.).

The photometric bands were selected as per the suggestions in Weaver et al.
(2022a). Several filters were discarded as too shallow and thus not providing
additional constraints: Spitzer/IRAC Channels 3 and 4; Subaru Suprime-Cam (SC)
broad bands, $NB711$, $NB816$, $NB118$; and GALEX FUV and NUV. The final list
of photometric bands included 25 filters: Canada France Hawaii Telescope
(CFHT) $u$, $u^{*}$; five Subaru/Hyper Suprime-Cam (HSC) bands $grizy$; four
UltraVISTA DR4 bands $YJHK_{S}$; twelve Subaru/Suprime-Cam (SC) bands $IB427$,
$IB464$, $IA484$, $IB505$, $IA527$, $IB574$, $IA624$, $IA679$, $IB709$,
$IA738$, $IA767$, $IB827$; and Spitzer/IRAC channels 1, 2. The work makes use
of the deeper fluxes derived with the 2-arcsecond apertures. The photometry
was corrected for aperture sizes and the Milky Way extinction based on the
offsets provided in the catalog.

To obtain the most accurate and well-measured objects in the COSMOS2020
catalog, several selections were made. Of the 1,720,700 sources, approximately
437,000 were kept by removing the objects that were flagged by SExtractor,
including sources biased by the bright neighbors, at the boundaries of the
images, saturated detections or instances that failed at execution. These
sources were selected in “ultradeep” stripes of the UltraVISTA photometry,
which reduced the survey area from $\sim 3.4$ to 0.9 deg2 and decreases the
volume completeness of the final catalog. Further selections of objects were
made after solving for SED templates with the best-fit IMF, as some of the
poor measurements were found to be most prone to degeneracies in the solutions
and ended up at the edges of the IMF temperature range. This quality cut is
described in § 4.1.

##  3 SED Fitting with EAZY

The algorithm for fitting photometric SEDs with EAZY (Brammer et al., 2008;
Brammer, 2021) is based on a linear combination of SED templates from a model
set. This procedure significantly reduces the computation time, as the best-
fit linear combination can be determined via matrix inversion rather than via
extended grid sampling.

However, if the set of basis templates is not carefully constructed, the best-
fit linear combination may produce an nonphysical shortcut solution. This is
possible as almost all of the resulting solutions are represented by a linear
mixture of young and old stellar populations with varying amounts of dust.
Even if the problem is well-determined, statistically, the ultraviolet and
infrared components of observed SEDs can be reconstructed with different
combinations of dust, star formation history and the shape of the IMF. Thus,
for example, a quiescent galaxy at may be fitted with an SED of a dusty star-
forming galaxy at lower redshift, unless the strong emission lines can be
sampled by a fortunate coincidence of some narrow bands.

Therefore, it is crucial to construct a set of model templates to match the
prior expectations of physical properties of galaxies in the targeted redshift
regime. The templates are typically made to span a limited and physically-
motivated space of such parameters, as stellar metallicity, age and dust
extinction. On the other hand, the spanned space is also made substantially
large to sample the properties as much as possible and avoid forcing most of
the inferred galaxy properties to look the same. The balance between asserting
a narrow physical domain of templates and still obtaining a representative fit
is achieved by ensuring that the template galaxies efficiently span the color
space (Brammer et al., 2008). This work builds on the templates provided for
COSMOS2020 at the time of publication222Standard COSMOS2020 templates made
with a Chabrier IMF are available at: https://github.com/gbrammer/eazy-
photoz/..

###  3.1 Temperature-dependent IMF

It is likely that different phases of gas in the giant molecular clouds result
in IMF variability across individual galaxies at different physical locations
and at different times of their evolution. Various dependencies of the IMF on
the gas phase have been proposed (Hopkins, 2012; Krumholz, 2011; Jappsen et
al., 2005; Bate & Bonnell, 2005), where some of the simplest approaches use
the scaling of the IMF mass with the temperature only (Jermyn et al., 2018).
The mass scale of cloud fragmentation in the Kroupa IMF (Kroupa, 2001) is made
proportional to the square of the temperature:

$\frac{dN}{dm}(T)\propto\left\\{\begin{array}[]{ll}m^{-0.3},&m<0.08M_{\odot}f(T)\\\
m^{-1.3},&0.08M_{\odot}f(T)<m<0.50M_{\odot}f(T)\\\
m^{-2.3},&0.50M_{\odot}f(T)<m\end{array}\right.,$ (1)

where $f(T)=(T/T_{0})^{2}$ is the scaling factor and $T_{0}=20$ K is the
reference temperature set to approximately the temperature of the molecular
clouds in the Milky Way (Schnee et al., 2008). The scaling of the power-law
break-points is tied to the minimum fragmentation mass of a gas cloud, i.e.
the low-mass end of the IMF (Jermyn et al., 2018). Therefore, throughout this
paper we refer to the relative IMF changes as bottom-light or bottom-heavy,
although in practice the bottom-light(heavy) and top-heavy(light) shapes are
degenerate here, as the power-law slopes are fixed.

Although it is unclear precisely how the complex interplay of various
astrophysical mechanisms in molecular clouds sets their exact temperature,
perhaps, a minimum at $z>7$ is set by the CMB, where it exceeds the local
temperature of 20 K. At lower redshifts, where the temperature exceeds the
microwave background, as for the $\sim 20$ K in star-formirng regions of the
Milky Way, it could be regulated by stellar radiation in regular main-sequence
galaxies or by cosmic rays in stellar explosions (Papadopoulos, 2010).

Although theoretically, the IMF is parameterised using the gas temperature, it
is unclear whether the best-fit IMF constrains this quantity or is degenerate
with other model SED inputs. For example, it was argued by Steinhardt et al.
(2022a) that the SED templates with a parametrised IMF are likely sensitive to
either the sharp decline in the star formation history or just the IMF as the
first order effect. There is less confidence that either of these can directly
translate to the temperature of the molecular gas. However, the star-forming
main sequence is found to correlate with the IMF similarly to the temperature
of dust with redshift in Magnelli et al. (2014), which lends more credence to
the IMF-temperature interpretation. Moreover, the connection of the IMF
temperature to galaxy morphologies in Steinhardt et al. (2023) further
reinforces the physical origin of this property.

Photometric templates used in this work were constructed to have as similar
properties as possible to the standard COSMOS2020 templates333The standard
templates preserved the records of their parameters, except for the star
formation history. Therefore it was adjusted to produce as similar templates
here as possible., with the sole exception of changing the IMF. The standard
17 templates span stellar populations with ages from 0.1 to 7 Gyr, dust
extinction ranging $\sim 0-2$ magnitudes, constant star formation histories
with specific star formation rate $sSFR\sim 2\times 10^{-8}$ yr-1 and the
attenuation curve from Kriek & Conroy (2013) (Table 1). Building upon this
17-template basis, a series of templates were made with the varying IMF shapes
that corresponded to molecular gas temperatures ranging from 10 to 60 K in
steps of 1 K. They were produced using Flexible Stellar Population Synthesis
(FSPS) code (Conroy et al., 2009; Conroy & Gunn, 2010) and made available
publicly444The templates with a range of varying IMFs are made publicly
available at https://github.com/vvrus/sed-templates..

Table 1: Properties of the stellar populations in the 17-template basis of standard templates used with COSMOS2020. The columns show: ID of a template; age of a stellar population; fraction of a solar metallicity $log(Z/Z_{\odot})$; and dust extinction in the V band, $A_{V}$. The first 7 templates are produced using Padova isochrones with $Z_{\odot}=0.019$ and for the last 10 MIST isochrones that assume $Z_{\odot}=0.0142$ were used.  ID | Age (Gyr) | $log(Z/Z_{\odot})$ | $A_{V}$
---|---|---|---
1 | 0.1 | 0.0 | 0.005
2 | 0.1 | 0.0 | 0.5
3 | 0.1 | 0.0 | 0.005
4 | 0.1 | 0.0 | 0.5
5 | 0.1 | 0.0 | 1.0
6 | 0.1 | 0.0 | 2.0
7 | 0.1 | 0.0 | 3.0
8 | 0.31 | 0.0 | 0.005
9 | 0.31 | 0.0 | 1.0
10 | 0.31 | 0.0 | 2.0
11 | 0.62 | 0.0 | 0.005
12 | 0.62 | 0.0 | 1.0
13 | 0.62 | 0.0 | 2.5
14 | 1.76 | 0.0 | 0.005
15 | 1.76 | 0.0 | 1.0
16 | 7.18 | 0.0 | 0.005
17 | 0.91 | 0.0 | 0.005

##  4 Results

Generally, parameters inferred from large photometric surveys are analyzed
from the perspective of population statistics, rather than studying individual
objects. In part, this is because the fits for any individual galaxy are often
not well constrained, with star formation rates particularly poorly-
determined. This is even more relevant when the SED fitting is performed with
the addition of new parameters, which increases random uncertainty and
possibly causes additional degeneracies or strong covariances between
parameters. Therefore, it is necessary to consider quality cuts to maximize
the statistical confidence of the fit results and identify potential outliers
arising from the existing degeneracies. The relevant selections and validation
of photometric redshifts are described below. Additionally, the subsections
here show some of the well-known relationships, as well as the new ones driven
by the change of the IMF shape.

###  4.1 Quality Cuts

At first, the results of the SED fitting were used to discard poor samples.
The data here was cut based on photometric coverage to ensure that constraints
on the IMF were as strong as possible. Then, the samples where no solutions
could be found or they were poorly defined were discarded.

First, it is crucial to have a sample observed over as broad of a set of bands
as possible in order to be able to constrain the solution parameters. Thus,
the redshift range was reduced to include objects in the range $0<z<2$, as too
many photometric bands drop out at $z>2$.

It was demonstrated in Sneppen et al. (2022), using mock SED observations,
that the IMF parameter can be safely recovered only for the objects with the
highest S/N ratios and at the IMF temperature above $\sim 20-25$ K. Most of
the objects in the COSMOS2020 catalog lie near the detection limit and thus
have relatively low-quality flux measurements, which results in weaker
constraining power for the given number of parameters and, ultimately,
inferred physical properties that cannot be trusted. For example, some
solutions were often found to be erroneously fit by the most bottom-heavy or
bottom-light IMFs or, alternatively, by the IMFs at edges of the sampled
$T_{IMF}$ range. As such, some faint quiescent galaxies with high mass-to-
light ratio and weak constraints in the UV can be sometimes fitted with the
most bottom-heavy IMF or a bottom-light IMF and high dust extinction.

Therefore, Sneppen et al. (2022) and Steinhardt et al. (2022a) defined the
quality cut $S/N>10$ in the Suprime-Cam $V$ band or UltraVISTA $K$ band, at
which the number of such objects best fitted at the boundaries of the
temperature range dropped significantly to $\sim 10-15\%$. Here, instead most
of the objects at the temperature boundaries are discarded as having poorly-
defined solutions in their chi-squared landscapes $\chi^{2}(T_{IMF})$ at
$10<T_{IMF}<60$ K with steps of 1 K. Specifically, the solutions with
$\chi^{2}_{max}-\chi^{2}_{min}<0.1$, $FWHM(T_{IMF})<4$ K were discarded. If
several solution temperatures were identified, the ones with the largest area
$\sum_{T_{IMF}}{\chi^{2}(T_{IMF})}$ were selected. As a result, $\sim 250,000$
samples out of $\sim 437,000$ (the 437,000 objects were selected as described
in § 2) passed with best-fit temperatures, out of which only 60% of objects
had $S/N<10$. The fraction of any $S/N$ at the margins of the temperature
range did not exceed 7%. On the other hand, the discarded sample was the
noisiest, with 90% of its objects having $S/N<10$ in $K_{S}$ flux.

By cutting objects based on the quality of their $\chi^{2}$ peaks, the final
sample comprised 249,494 objects with 205,750 in the range $0<z<2$ where they
were still covered by most of the photometric bands. The sample was reduced
even further to 156,389 objects by flagging the remaining solutions as
outliers. Those were identified as faint objects with too low or too high
mass-to-light ratios and, thus, likely suffering from degenerate template
parameters (more description is included in Appendix A). Alternatively, some
objects discarded as outliers had additional local minima, similar to objects
identified in Sneppen et al. (2022). They could arise due to incorrectly
fitting the highest-mass breakpoint in the Kroupa IMF in galaxies that had
recently lost their most massive stars, but still have the intermediate-mass
population. Lastly, only 72,805 objects in the catalog have uncertainty in
$T_{IMF}$ estimated using the minimum variance bound, as described in § 4.4,
as the rest of them had low significance of chi-square solutions. These best
objects were used to obtain the results in § 4.4–4.8.

###  4.2 Stellar Mass Completeness

Figure 1: Counts of all galaxies in the bins of photometric redshift and
stellar mass. The points show 95% mass completeness limits estimated by
rescaling down to the detection limit in IRAC Channel 1 for star-forming and
quiescent. These limits $\log_{10}{\mathcal{M}}(z)$ were fitted with a
Schechter function and shown as solid lines. The uncertainties are taken as
the median errors in bins of redshift and rescaled mass. The levels for star-
forming galaxies are below the limits of quiescent galaxies by $0.2-0.3$ dex
at $0<z<2$, which reflects the higher $\mathcal{M}/L$ ratio for the passive
population.

The lower limit of stellar mass of a representative sample of galaxies depends
on the survey depth and the mass-to-light ratio of galaxies and evolves with
redshift. Owing to the lack of UV and optical emission per stellar mass,
quiescent galaxies are expected to be less complete than the brighter star-
forming sample at every redshift in the photometry of COSMOS2020. In addition,
it is expected that at $0.0<z<2.0$ the galaxy sample is missing a population
of blue dwarf galaxies, which fall outside of the detection image, and,
possibly, some galaxies at the large stellar mass end at the lowest redshift
due to a smaller volume. The separation of galaxies into star-forming and
quiescent samples was done in the UVJ color space (Williams et al., 2009;
Arnouts et al., 2013).

By using the approach of Pozzetti et al. (2010), the limiting mass was defined
by taking the faint objects that are likely to have complete stellar masses in
bins of redshift and rescaling their masses to the magnitude of the survey
depth. Following the prescriptions in Weaver et al. (2022b), 1% of the
photometric fits with the largest chi-squared were removed from the sample.
Then, the stellar masses $\mathcal{M}$ of the 30% faintest objects in IRAC
Channel 1 magnitudes $m$ were rescaled to the detection limit
$m^{\prime}=26.4\pm 0.1$ (Weaver et al., 2022a) to obtain
$\mathcal{M^{\prime}}$:

$\begin{array}[]{c}\log_{10}{\mathcal{M^{\prime}}}=\log_{10}{\mathcal{M}}+0.4(m-m^{\prime}).\end{array}$
(2)

The 95% completeness level $\log_{10}{\mathcal{M}_{95}}$ was taken as the 95th
percentile of the $\log_{10}{\mathcal{M^{\prime}}}$ distribution and is shown
with dots on Figure 1. The calculation is done separately for star-forming and
quiescent galaxies to account for their different $\mathcal{M}/L$ ratios. The
key assumption made here is that galaxies with $\mathcal{M^{\prime}}$ and
$\mathcal{M}$ have the same relation to luminosity. However, as the IMF
correlations with the physical properties in § 4.4–4.5 imply, $\mathcal{M}/L$
of star-forming galaxies correlates with the stellar mass and the proper
scaling should account for this difference by using
$(\mathcal{M^{\prime}}/L^{\prime})^{\beta_{1}}$ and
$(\mathcal{M}/L)^{\beta_{2}}$, where $\beta$ increases for larger stellar mass
(i.e. $\beta_{1}>\beta_{2}$). By assuming that $\beta_{1}=\beta_{2}$, as it is
done here, the rescaled stellar mass $\mathcal{M^{\prime}}$ and, consequently,
the mass completeness are overestimated, albeit insignificantly, as the
difference between the 30% faintest galaxies in magnitude at the same redshift
should be minor. Therefore, the completeness used here acts as conservative
upper limit. Finally, although the IRAC bands were not included in the
detection image of COSMOS2020, Weaver et al. (2022b) argue that this should
not affect the results significantly.

The completeness levels were calculated in bins of redshift at $0.1<z<3.4$ for
quiescent galaxies and $0.1<z<4.3$ for star-forming galaxies in steps of 0.3.
Finally, the 95% mass completeness levels $\mathcal{M}$ as a function of $z$
were fitted with a Schechter function (Schechter, 1976; shown as solid lines
in Figure 1; best-fit parameters are shown in Table 2):

$\begin{array}[]{ll}\log_{10}{\mathcal{M}(z)}&=(0.4\ln{10})\log_{10}{\mathcal{M}^{\star}}\times\\\
&\times[10^{0.4(z^{\star}-z)}]^{\alpha+1}\exp{(-10^{0.4(z^{\star}-z)})},\end{array}$
(3)

where uncertainties were taken as median $\sigma_{z}$ in the bins of redshift
and bins of stellar mass $\sigma_{\log_{10}{\mathcal{M}}}$ defined as
$\log_{10}{\mathcal{M}_{95}}\pm 0.2\log_{10}{\mathcal{M}}$.

Table 2: Best-fit parameters and their statistical uncertainties for the Schechter function representing the limits of mass completeness for star-forming and quiescent galaxy samples (see Eq. 3). Sample | $\log_{10}{\mathcal{M}^{\star}}$ | $z^{\star}$ | $\alpha$
---|---|---|---
Star-forming | 9.56$\pm$0.53 | -1.63$\pm$0.28 | -1.01$\pm$0.01
Quiescent | 9.10$\pm$0.24 | -1.89$\pm$0.13 | -1.03$\pm$0.01

This work applies two separate 95%-completeness limits to star-forming and
quiescent galaxies estimated in Weaver et al. (2022b) for all relationships,
except the redshift validation in Figure 2 and the change of properties due to
the IMF modification in Figure 4.

###  4.3 Cross-check of Photometric Redshifts

Spectroscopic redshifts have usually been used as a partial validation of
photometric template fits. Here, the same check is applied to the variable-IMF
fits in the presented catalog. Figure 2 demonstrates the correspondence of the
sample with the spectroscopic redshifts from the latest Super-deblended
catalog (Jin et al., in prep.) and standard photometric redshifts based on
Chabrier IMF (at 20 K in the Milky Way) from the COSMOS2020 survey (Weaver et
al., 2022a). The outliers in the one-to-one comparison are defined as points
satisfying the condition $|\Delta z|>0.15(1+z)$ (Hildebrandt et al., 2009).
The fraction of outliers $\eta$ is indicated on the plot panels. It shows that
the spectroscopic sample disagreed in $\sim 3$ % of cases, similar to the
number outliers in the standard catalog, as shown in Figure 2 and in Weaver et
al. (2022a). The precision of the photo-$z$ is estimated by using the
normalized median absolute deviation $\sigma_{NMAD}$ (Brammer et al., 2008),
which is not strongly sensitive to outliers:

$\begin{array}[]{c}\sigma_{NMAD}=1.48\times{\rm median}\left(\frac{|\Delta
z-{\rm median}(\Delta z)|}{1+z_{spec}}\right)\end{array}$ (4)

.

Finally, the bias in the estimates is denoted as $b={\rm
median}(z_{phot}-z_{spec})$. These metrics perform similarly to the standard
measurements reported in the COSMOS2020 catalog. The agreement confirms that
even more significant differences in templates have little effect on the
redshifts reconstructed with the same code (Sneppen et al., 2022). Finally,
the outliers identified in Figure 2, have been discarded from the scientific
results in the rest of the work.

Figure 2: Density diagrams showing the validation of photometric redshifts
$z_{phot}$ obtained with the best-fit IMFs. Top panel: comparison of redshifts
obtained with the Chabrier IMF at 20K (W22) against the spectroscopic
redshifts $z_{spec}$ from (Jin et al. in prep.) (1,773 objects). Middle panel:
redshifts obtained with the Kroupa IMF at the best-fit temperature against the
spectroscopic sample. Right panel: comparison of the photometric redshifts
$z_{phot}$ obtained with the Kroupa IMF at the best-fit temperature and with
the Chabrier IMF at 20 K (189,231 objects). The fraction of higher than
one-$\sigma$ outliers is indicated as $\eta$. Overall, the redshifts
constrained with the new SED templates appear to be consistent with the
standard COSMOS2020 templates and the spectroscopic sample.

###  4.4 IMF Temperatures

Figure 3: The number of objects in bins of IMF temperature and lookback time,
normalized in lookback-time bins $\Delta t_{i}$, for the total (left), star-
forming (middle) and quiescent sample (right). Dashed lines show best-fit
exponentials fit in the space of temperature and redshift $(T,z)$ using
individual non-normalized data. Uncertainties in redshift are based on
$1\sigma$ width of $p(z)$ solutions from EAZY (median binned errors range
between $\sigma_{z}\sim 0.01-0.27$). The errors in IMF temperature are
computed using the likelihood based on chi-squared of the SED solutions, and
are very loose (see text for details; median binned errors range between
$\sigma_{T_{IMF}}\sim 3-12$ K). Most of the objects have IMF temperature above
that of the analogs in the local universe forming a continuum of bottom-
lighter IMFs. On average, the best-fit IMF temperature increases with redshift
as in Sneppen et al. (2022); Steinhardt et al. (2022b). Finally, the
temperature and corresponding IMF behaviour is distinct between active and
passive galaxy populations, where the former is almost exclusively responsible
for the temperature increase in the total sample. This result for star-forming
and quiescent galaxies is consistent with the observed trends in temperature
of cold dust in similar redshift ranges (see text for references).

The main result in this work is that most of the galaxies are unlike the local
analogs and have bottom-lighter IMFs, as traced by the IMF temperature
parameter (Figure 3). These best-fit IMFs indicate the larger relative
abundance of the most massive stellar populations at all redshifts.
Theoretically, this change is expected to correspond to hotter (and,
presumably, denser and metal-poorer) molecular gas.

Possible relationships in the distribution of IMF temperatures $T_{IMF}$ at
different redshifts $z$ are investigated by fitting an exponential function
$T_{IMF}(z)=A\exp{(-z/\tau)}$ (dashed lines in Figure 3). There is not a
proper way in the SED-fitting procedure implemented here for estimating
uncertainties in temperature, so the errors are calculated using the chi-
square $\chi^{2}$ statistic computed by EAZY for best-fit models produced at
each IMF temperature in $10<T_{IMF}<60$ K in steps of 1 K. By assuming that
the temperature estimator is unbiased (at $T_{IMF}\gtrsim 20-25$ K) and using
the Minimum Variance Bound, the uncertainty $\hat{\sigma_{\hat{T}}}$ was
calculated as $\Delta\hat{T}_{-}$ and $\Delta\hat{T}_{+}$ at
$\log{L(\hat{T}\pm\hat{\sigma}_{\hat{T}})}=\log{L_{max}}-1/2$ with the
likelihood as $\log{L}=-2\chi^{2}$. For the solutions that did not have the
sufficient significance, the uncertainty fields in the catalog are left empty.
The best-fit function in the total sample (left panel) shows that the
temperature increases from $\sim 25$ K to 35 K at $0<z<2$, mostly driven by
star-forming galaxies which were selected in UVJ color space (Williams et al.,
2009; Arnouts et al., 2013).

Although there is a large scatter in solutions for IMF temperature in the
whole sample, there is a systematic trend for increasing temperature at higher
redshift. One component of the scatter is added by the fitting procedure and
is expected to be mostly random at temperatures above $\sim 25$ K. Based on
the fits of synthetic photometry, Sneppen et al. (2022) showed that scatter in
the recovered temperature is $\sigma\approx 5-10$ K and the estimator is only
slightly biased at those temperatures for the strongest photometric
detections, while almost completely inaccurate for “cooler” galaxies (see
Figure 13 in their publication). However, it is possible that other
unaccounted systematic effects contribute to the obtained temperature
distribution. For example, one of the critical limitations of the SED
templates employed here is that the temperature, density and metallicity of
the star-forming gas are not causally connected to the expected changes in the
definition (Eq. 1) of the IMF. Therefore, the resulting “temperature” quantity
is likely a combination of effects due to these three quantities.

The rest of the scatter appears to be due to the differences in the mass to
light ratio, $\mathcal{M}/L$, of stellar populations in different galaxies.
For example, the most common and strong distinction exists between active and
quiescent galaxies, where the latter are expected to have bottom-heavier IMFs.
Indeed, the right panel in Figure 3 shows that the passive population is
consistent with the same $\mathcal{M}/L$ and has relatively small scatter.
Given that the quiescent galaxies with very weak UV emission are expected to
be the least sensitive to probing the IMF, this result validates the fitting
procedure. This would be consistent with the measurements of dust temperatures
in quiescent galaxies that appear to have a lower limit at $T_{dust}\sim 21\pm
2$ K in a similar redshift range in Magdis et al. (2021). However, it is
possible that the temperatures of the star-forming gas and dust are set by
different mechanisms in quiescent galaxies and therefore the former does not
have to be similarly bounded at $\sim 20$ K. This possibility cannot be tested
with the IMF estimator here, as it is completely biased in that temperature
regime. On the other hand, the active population spans the full sampled
temperature range at every epoch and is almost entirely consistent with the
temperature increase in the total sample.

By isolating the quiescent galaxies, it appears that the large scatter is
indicative of various star-forming galaxies. While most appear to be on the
main sequence the rest can represent different sub-populations. Among others,
ULIRGS that are thought to result from major mergers of gas-rich galaxies or
close interactions (from observations Sanders et al., 1991; Solomon et al.,
1997; from simulations Mihos & Hernquist, 1996), have been shown to exhibit a
constant IMF temperature at $\sim 30$ K up to $z=2$ (Sneppen et al., 2022),
consistently with dust temperature estimates in Béthermin et al. (2015). Among
galaxies efficiently forming new stars are “green peas” (Cardamone et al.,
2009) that have small stellar mass and compact morphology, little dust
obscuration, and other starbursts that include galaxies with varying physical
properties but appear to have similarly high sSFR due to a recently ignited
burst of star formation (Kennicutt & Evans, 2012). Based on their properties,
these types are likely to occupy the higher temperature tail, above ULIRGS and
typical SFMS galaxies. Indeed, Zhang et al. (2018) showed that starbursts at
$2<z<3$ have a top-heavy IMF by measuring ${}^{13}{\rm C}/^{18}{\rm O}$
transitions. Note, the density in Figure 3 reveals a grid pattern, which is
likely due to fits with degenerate parameters, described in Appendix A, that
could not be removed entirely.

Finally, the large scatter in the IMF shapes here is in agreement with the
broad distribution of equivalent widths of H$\alpha$ at $z\sim 2$ in
Nanayakkara et al. (2017). That work shows that the most likely contribution
to the large widths is the top-heavy IMF. Moreover, the large scatter in the
widths cannot be explained entirely by the starbursts and likely indicates
either stochasticity or some underlying relationships in the shapes of the
IMF.

This behavior of the IMF shape is similar to that found using galaxies in
COSMOS2015 (Sneppen et al., 2022), although it now extends to fainter galaxies
and lower temperatures at the higher redshift and includes galaxies with
higher measurement quality on average. In particular, more data fills the
lookback time $t>8$ Gyr ($z>1.1$) and $T_{IMF}<30$ K in Figure 3. It
represents a faint red sample at the lower tail of sSFR or faint blue
galaxies, which have been observed owing to the depth of the COSMOS2020
surveys.

###  4.5 Physical Properties of Galaxies

If most studied galaxies have systematically different stellar populations
from what was assumed previously, their inferred stellar mass and the rate of
star formation must change correspondingly. Figure 4 shows the changes in
these two properties at $0.2<z<2.0$.

Clearly, most of the objects have properties characteristic of the bottom-
lighter IMFs, or higher temperature of the star-forming gas. As can be
expected, the bias is associated predominantly with the bottom-lighter IMF
solutions for the star-forming galaxies. There, the magnitude of the change
appears to be correlated with the SFR of galaxies, or the strength of UV
emission. The largest offset reaches $\sim 1$ dex in stellar mass and $\sim 2$
dex in SFR, while the median changes are $\sim 0.2$ and $\sim 0.4$ dex,
respectively.

The best-fit properties for the quiescent population at these redshifts do not
appear to be affected significantly by the change. This indicates two
possibilities that: (1) their last stellar populations formed at the same
temperature as in the local universe; (2) photometric IMF variations in the
quiescent galaxies are below the sensitivity limit of our procedure of $T\sim
20$ K. Nevertheless, the most bottom-light quiescent galaxies are still found
in the lowest redshift bins, at $z<0.8$, which suggests that some internal
feedback (stellar radiation or cosmic rays) on top of the CMB may heat the gas
at the time of the last star-forming episode to above 25 K at $z>0.8$.
However, it is also likely that some of coldest faint quiescent galaxies at
$z>0.2$ may still be undetected.

It is also worth noting that the photometric fitting method is not capable of
recovering IMF shapes with IMF temperature $T<20$ K, as show in the mock
simulations in Sneppen et al. (2022). Therefore, the lower bound found here is
characteristic of the method. Although physically this bound coincides with
the temperature background in the MW, it disagrees with the probes of even
bottom-heavier IMFs in elliptical galaxies and other old stellar structures at
low redshift in Conroy & van Dokkum (2012); van Dokkum & Conroy (2012); Geha
et al. (2013); Martín-Navarro et al. (2015); Conroy et al. (2017); van Dokkum
et al. (2017a); Hallakoun & Maoz (2021).

Figure 4: Difference in stellar mass and star formation rate between the
“best-fit IMF sample” $(\mathcal{M}_{T},{\rm SFR}_{T})$ and the standard
properties from COSMOS2020 $(\mathcal{M},{\rm SFR})$ at $0<z<2$. The
properties $(\mathcal{M}_{T},{\rm SFR}_{T})$ are taken at the best-fit IMF
temperature $T$, while $(\mathcal{M},{\rm SFR})$ are based on the local IMF,
analogous to $T\sim 20$ K. The standard COSMOS2020 properties were converted
from Chabrier to Kroupa IMF for a consistent comparison, scaling properties by
1.06 (eg., $\mathcal{M_{K}}=1.06\mathcal{M_{C}}$, using the conversion from
Zahid et al., 2012). The separation of galaxies into star-forming and
quiescent was done in the $UVJ$ color space of the best-fit IMF colors. The
$UVJ$ classification is consistent between two catalogs with 98.2% agreement.
The contours outline 42,545 active and 8,416 passive galaxies. Clearly, as the
best-fit IMF is almost always bottom-lighter than in the Milky Way, the masses
and star formation rates shift to lower values. The median differences of the
total sample correspond to $log(\mathcal{M}_{T}/\mathcal{M})\approx-0.21$
(1/1.6 linear change) and $log(SFR_{T}/SFR)\approx-0.40$ (1/2.5 linear
change). The offset is on average higher for star-forming galaxies with the
highest SFR, while quiescent galaxies experience little to no change in
stellar mass and a similar change in SFR. The contours in both plots are
consistent and extend to the same number levels (outer contours cover 98% of
the respective total numbers).

###  4.6 Star-forming Main Sequence

Figure 5: All galaxies plotted in the space of the star-forming main sequence
in redshift windows from $z=0.2$ to 2.0. The color corresponds to the median
IMF temperature in a bin of stellar mass and SFR ($log(\mathcal{M})$,
$log(SFR)$). Bins with $\geq 10$ samples are filled, bins with $1-9$ samples
are shown as scatter. Typical star-forming galaxies are found on the main
sequence, showing a gradient of IMF temperatures correlated with the sSFR;
quiescent galaxies turn off the sequence and extend to lower SFR at the
highest stellar masses and lowest IMF temperatures; objects with the highest
IMF temperatures at the low stellar mass end exhibit a distinct temperature
relationship from the typical main sequence (the temperature relationship is
shown on Figure 7).

At every redshift, most of the actively star-forming galaxies of each stellar
mass have been found to exhibit a narrow range of SFR, which defines the star-
forming main-sequence (SFMS) (Brinchmann et al., 2004; Noeske et al., 2007;
Peng et al., 2010; Speagle et al., 2014). This relationship holds in both
photometrically and spectroscopically-derived SFR and despite the fact that
galaxies have a range of different star formation histories, supernova rates,
strengths of active galactic nuclei feedback or cosmic environment. However,
it can still be expected that variability in these factors can play role in
changing the distribution of stellar masses in a galaxy. Therefore, one of the
ways of testing the properties of the variable IMF fits is by verifying that
the main sequence still holds.

Figure 5 shows all galaxies, where the star-forming ones still form the SFMS
at each redshift despite the shift in stellar masses and star formation rates
due to changing the Chabrier IMF at 20 K to IMFs at the best-fit temperatures.
Similarly, the work of Steinhardt et al. (2022b) that implemented the same
temperature-dependent IMF, showed that there is a correlation between SFR,
stellar mass at every redshift, consistent with the known relationship. While
the SFMS appears to be the same qualitatively, the exact offset and gradient
of the relation should be sensitive to the shape of the best-fit IMF and
should correlate with it, as suggested by the change of the SFR and stellar
mass in Figure 4.

The best-fit properties here indicate that there is a gradient of IMFs along
the SFMS at every redshift. The IMF temperature increases most rapidly towards
the high SFR and low stellar mass end. The gradient becomes steeper at the
higher redshift, with more extreme cases of the high-temperature galaxies. As
shown in Figure 1 of Steinhardt et al. (2023), the change of IMF shape may be
correlated with the gradient of the temperature of dust continuum emission
shown in Magnelli et al. (2014). This suggests that the “IMF temperature”
parameter may trace the temperature of molecular clouds at star formation, as
argued theoretically (Jermyn et al., 2018). Therefore, with the evidence
presented here and in the previous work it may be compelling to model a
possible relationship or an equilibrium state between temperature of the
molecular clouds and cold dust. If calibrated at low redshift, this
relationship can be used to constrain the IMF for galaxies at high redshift
for which measuring gas temperatures from line ratios is challenging (see §
5.3 for further discussion).

As expected, the quiescent population is found at the highest mass bins for a
given SFR and redshift. The colors in Figure 5 indicate that the quiescent
galaxies are coolest at the lowest redshift, and gradually become warmer at
the higher redshift. This effect demonstrates that either the star-forming gas
used to form the last stellar population is hotter in earlier epochs due to
some feedback mechanisms or that some of the coolest samples at the higher
redshift are below the detection limit of the COSMOS2020 surveys. The last is
more likely, as observations show that the average temperature in massive
quiescent galaxies at $0.2<z<2.0$ is approximately constant at $T_{d}=21\pm 2$
K (Magdis et al., 2021). Finally, the quiescent galaxies are coolest among
other galaxies at every epoch. Thus, they appear to turn off the main sequence
once some critical mass is built up555Evidence in § 4.7 from quiescent galaxy
fraction appears consistent with there being a characteristic mass for
quenching at every redshift, which monotonically decreases towards the current
epoch. which appears to be traced by the IMF temperature.

A population of low-mass and hot galaxies appears as indicated by the IMF
temperature in the highest redshift panels of the SFMS. These galaxies have
distinct combination of sSFR and IMF temperature, as discussed in more detail
in § 4.8. Steinhardt et al. (2023) suggested that they also appear to have
compact morphologies. It has been hypothesized that these galaxies may
represent the early, core-forming stage of galaxy evolution. If that is the
case, this population is only cooling down onto the SFMS and thus can be
expected to have a different $SFR-M_{\star}$ relationship or none, similarly
to quiescent galaxies.

It is unclear whether this sample is separate from the typical star-forming
galaxies on the main sequence and therefore requires investigation.
Previously, it has been argued by Abramson et al. (2014) that the SFMS can be
explained solely by the spatially-resolved evolution of the star-forming disks
which host evolved bulges. If the fundamental SFMS evolution is driven purely
by the physics of star formation, it is possible that the core-forming
galaxies form a similarly straight SFMS. On the other hand, if the typical
main sequence galaxies are driven by a combination of star formation and disk-
morphology effects, the core-forming galaxies may exhibit a characteristically
different main sequence: a power law with a different exponent, normalization
or a different functional form (more details follow on this in § 4.8). Thus,
morphology may present hints for the mechanisms driving the SFMS relation and
other stages of evolution.

Finally, the galaxy sample covers almost 4 orders of magnitude in SFR with
some of the fainter samples detected only in COSMOS2020 and not the previous
COSMOS catalogs (Figure 5). The low-mass galaxies are cut sharply at the
higher redshift due to a combination of the Malmquist bias at the low SFR, the
quality cuts here at the higher SFR (see § 4.1) and mass completeness cuts. At
the same time, the most massive and luminous galaxies are found in the most
distant observations where the sampled cosmic volume is largest.

###  4.7 Stellar Mass Function

Figure 6: The observed fraction of the quiescent galaxies as a function of
stellar mass defined as $log(\Psi_{QS}/[\Psi_{SF}+\Psi_{QS}])$ for a stellar
masses derived with a best-fit IMF (left) and standard stellar masses form the
COSMOS2020 catalog (right). The uncertainties include Poisson error in star-
forming and quiescent galaxy counts. Stellar mass $log(\mathcal{M})$ is
limited according to completeness levels at different redshift for star-
forming and quiescent galaxies (Weaver et al., 2022b). At every redshift, most
massive galaxies complete their evolution first, and lowest mass galaxies take
the longest time to evolve. The evolution from high to low mass end progresses
smoothly with redshift. Overall, this picture is consistent with the effect of
downsizing. Notably, for the properties with the best-fit IMF this effect
appears consistently at all redshift bins – even at $z>1$ – unlike in some of
the previous studies of stellar mass functions (see text for details). These
properties (left) show a systematically higher fraction of quiescent galaxies
in the most massive tail than inferred from the standard property set (right).
This difference comes from the differential change in the stellar mass, where
the star-forming functions are shifted towards lower masses, while the
quiescent ones mostly remain unchanged (this comparison is shown in Appendix
B).

Stellar mass function, as the comoving volume density of star-forming and
quiescent populations at different redshift snapshots, is an observational
tracer of growth of stellar mass at different scales over time (Fontana et
al., 2006; Drory & Alvarez, 2008; Marchesini et al., 2009; Peng et al., 2010;
Pozzetti et al., 2010; Ilbert et al., 2013; Moustakas et al., 2013; Muzzin et
al., 2013; Grazian et al., 2015; Song et al., 2016; Davidzon et al., 2017;
Stefanon et al., 2021; Weaver et al., 2022b). These observations provide
constraints for theoretical models and simulations of galaxy evolution, and
eventually help to study the mechanisms regulating galaxy assembly, growth and
quenching of star formation. Here, the fraction of quiescent to total galaxies
in terms of the mass functions is used to show the growth and the hierarchy of
the quenched population.

The fraction of quiescent galaxies as a function of stellar mass obtained with
the best-fit IMFs is reproduced with the same overall characteristic behaviour
as in most studies that observe the effect of downsizing, reported first in
(Cowie et al., 1996; Juneau et al., 2005; Fontanot et al., 2009). The mass
scale of quenching on Figure 6 indicates that the most massive galaxies are
formed at earlier epochs and evolve faster than the less massive populations
at lower redshift.

According to the best-fit stellar masses obtained with a bottom-lighter
variable IMF (Figure 4), the stellar mass function for star-forming galaxies
is shifted towards the lower stellar mass with respect to the quiescent which
stays unaffected (individual mass functions are shown in Appendix B).
Therefore, the passive galaxy population becomes more abundant for the most
massive galaxies at every redshift. As a result, the evolution of the
quiescent mass scale is even more sharply consistent with mass downsizing in
comparison to standard photometric properties, similarly to results in
Steinhardt et al. (2022a).

The change in the quiescent galaxy fraction with redshift allows placing
constraints on possible mechanisms of quenching. As shown in Steinhardt et al.
(2022a), there appears to be a characteristic mass at every redshift, above
which majority of the galaxies have stopped forming stars. This characteristic
mass monotonically increases with redshift, consistent with downsizing. This
behaviour is consistent with running out of cold molecular gas or some
universal feedback mechanism and ability to cool it or, more broadly, with the
framework of “mass quenching” (Peng et al., 2010). Some possible quenching
mechanisms include morphological quenching, AGN feedback or cosmological
starvation (Man & Belli, 2018).

###  4.8 Galaxy Population Diagram

This section shows a relation between the sSFR, or inverse galaxy growth
timescale, and the temperature of the star-forming gas derived from the IMF
constraints. It appears that main-sequence galaxies experience a decline in
the efficiency of converting their gas into stars towards lower redshift from
molecular gas studies (Magdis et al., 2021) or, more generally, from abundance
matching (Finkelstein et al., 2015). Similarly, there is a monotonic decrease
in sSFR of main-sequence galaxies from photometric properties (Peng et al.,
2010; Speagle et al., 2014). As seen from the SFMS in this work (Figure 5),
the IMF temperature is proportional to sSFR for star-forming galaxies, but the
relationship is likely different or breaks completely for quiescent galaxies.
Should this behaviour break or change characteristically for any galaxies in
our sample, this may signify a different regime of star formation, as argued
in Steinhardt et al. (2023).

Figure 7 shows the relationship between sSFR and the IMF temperature in the
sample at $0.2<z<2.0$. As can be expected, the densest region of the diagram
is populated by galaxies on the main sequence. There IMF temperature and sSFR
appear to be coupled the strongest for these galaxies, albeit spanning a small
range of both properties. Likely, this is consistent with the strong feedback
mechanisms driving these galaxies on the SFMS.

The $sSFR-T_{IMF}$ coupling appears to break in two different ways for the
extreme tails in temperature and mass, which suggests characteristically
distinct galaxy populations. The low temperature end with the most massive
objects is represented by the quiescent population. Finally, the population at
the high temperature has the shortest growth timescales, lowest masses and
thus is likely to be in the state of rapid star formation and cooling onto the
“typical” main sequence. Perhaps these properties signify three distinct modes
of star formation, connected by galaxy evolution tracks first from early rapid
formation, then to the strongly-regulated main sequence and, finally,
quiescence. Possible physical processes defining these modes are discussed in
more detail in Steinhardt et al. (2023).

In addition, Steinhardt et al. (2023) suggest that the thermal evolution on
and off the star-forming main sequence may be independently linked to the
apparent morphological development of galaxies. The connection shows that the
cooling off the main sequence galaxies is possibly associated with the
transition from disk-like structures with a central bulge or a core to a more
extended elliptical structure. The evolution is also apparent in the change of
photometric colors, from predominantly blue disks with red cores to red
ellipticals. Finally, the new, early stage of evolution appears to have blue
compact cores as their morphological counterparts at all redshifts, which
could be progenitors of the future disk-like galaxies with a predominatly
quiescent core, similarly to the morphologically-driven evolution in Abramson
et al. (2014).

Notably, $sSFR=10^{-8}$ yr-1 is a common upper limit set by star formation
rate in SED templates in different fitting codes. Greater sSFR are not allowed
to avoid degeneracy between the star formation history and dust in solutions.
Therefore, the real gradient at the high temperature tail may be steeper than
observed here. The true slope is going to constrain the potential physical
mechanism driving the evolution of this galaxy population.

On the other end, the quiescent population appears to be limited from the
bottom at around $T_{IMF}\approx 20K$, which is either real or limited by the
sensitivity of the photometric procedure to the weak UV signal. The 20 K limit
is in agreement with the local molecular gas measurements in the Milky Way
(Schnee et al., 2008). The Galaxy is likely on the way down in the quiescent
branch at low temperature and at $sSFR\sim 10^{-11}$ yr-1 (Licquia & Newman,
2015) on Figure 7. Therefore, this suggests that the sharp cut at the low
temperature and at least down to the MW sSFR is real. Nevertheless, the
technique employed here cannot recover temperatures lower than $20-25$ K, as
shown using mock galaxies in Sneppen et al. (2022). Therefore, the method here
is not capable of recovering those IMF that have bottom-heavier shapes than in
the MW, as reported for stellar systems with old stars in Conroy & van Dokkum
(2012); van Dokkum & Conroy (2012); Geha et al. (2013); Martín-Navarro et al.
(2015); Conroy et al. (2017); van Dokkum et al. (2017a); Hallakoun & Maoz
(2021).

Figure 7: Diagram of sSFR as a function of IMF temperature $T_{IMF}$ for
several redshift windows in the COSMOS2020 (top row; provided here) and
COSMOS2015 surveys (bottom; from Sneppen et al., 2022). The bins are colored
by median stellar mass $log(\mathcal{M})$. The black contours show the sample
density for reference. Bins with $\geq 10$ samples are filled, while the bins
with $1-9$ samples are shown as scatter. The gray-shaded area shows the range
where the temperatures cannot be recovered. The densest regions are populated
with the typical main-sequence galaxies, which appear to be driven by the
coupling of sSFR and IMF temperature. The two tails with characteristically
different $sSFR-T_{IMF}$ relationships from the typical main sequence appear
to be quiescent (low $T_{IMF}$) and active (high $T_{IMF}$). These distinct
galaxy regions show a smooth gradient of stellar mass, which together with the
declining sSFR and $T_{IMF}$ suggests a clear evolutionary sequence. The
sample shown at the top excludes outliers resulting from spurious SED fits
(see Appendix A). According to the synthetic tests performed in Sneppen et al.
(2022), the best-fit IMF properties cannot be trusted at $T_{IMF}\lesssim 20$
K. Therefore, sub-samples at $T_{IMF}\approx 10-20$ K that are present in most
panels here cannot be trusted.

##  5 Discussion

The analysis here shows that the new properties show a clear picture of mass-
dependent assembly of galaxies in terms of the stellar mass functions and
provide new valuable insights into the star-forming processes along and
“around” the star-forming main sequence. It is shown that the gas temperature
derived from the best-fit IMF adds a new dimension to investigate efficiency
of star formation and possibly dissect galaxy evolution into different stages.
This is supported by the recent work that finds a connection between IMF-
selected galaxies and their morphologies (Steinhardt et al., 2023) and
suggests a possible interpretation in which galaxies evolve from blue compact
cores, through disk formation around the evolved core to elliptical extended
structures.

This section emphasizes a possible bias introduced by assuming a universal IMF
based on the Milky Way, as well as the recently published issues with even
averaging observations of different parts of individual galaxies. It finally
discusses implications for the problem of the “impossibly early galaxies”,
which becomes relevant here by assuming that the observed increase in the
temperature of the various parts of the ISM continues to high redshift, so
that the bias in the estimated mass of the stellar populations becomes
increasingly large.

###  5.1 The Milky Way Bias

Resolved observations in the local universe uncover the chemical and
structural complexity of galaxies that translates into variations in their
star formation processes (Hodge, 1989; Gallart et al., 1996). It has been
found that the cold ISM exists in a range of phases that are sensitive to the
internal processes as well as the environment (Saintonge & Catinella, 2022).
The latter review shows that there are systematic variations in the amount of
molecular gas and the rate at which it is converted into stars across
individual galaxy populations. These differences in the cold phase of the ISM
are expected to produce different stellar populations resulting in different
feedback, which can build up to produce strong differences in stellar masses,
metallicities and other quantities over time. As a result, it is easy to
introduce systematic bias in inferred astrophysics by making strong
assumptions that neglect these differences.

Historically, the complexity in star-forming conditions has been dealt with by
assuming that the IMF derived in the MW is universal (eg., Kroupa, 2001;
Franceschini et al., 2006; Arnouts et al., 2013), although some hints of IMF
variability have become available (Treu et al., 2010; Martín-Navarro et al.,
2015; van Dokkum et al., 2017b). However, galaxy evolution over cosmic time
has been shown to break even under different IMF measurements made only in the
Milky Way (Speagle et al., 2014).

Evidence from cold dust continuum emission and atomic gas measurements
(Valentino et al., 2020) suggests that molecular gas temperatures at high
redshift are unlikely to be the same as in the local universe. The dust
temperature from stacked measurements increases from 20 K at $z=0$, equivalent
of the MW gas temperature, to 40 K at $z=2.3$ (Magnelli et al., 2014) and to
50 K at $z=4$ (Schreiber et al., 2018; Cortzen et al., 2020). This is
consistent with the evidence of increasing intensity of the radiation field
(Béthermin et al., 2015; Magdis et al., 2017) in earlier galaxies. Based on
theoretical work, temperature changes in the star-forming gas should be
sensitive for the mass scale of the stellar population (Jeans, 1902; Low &
Lynden-Bell, 1976; Larson, 1998, 2005) and modify the shape of the IMF (Jermyn
et al., 2018).

Therefore, it is necessary to relax the assumption of the local IMF at least
for non-local galaxies to reflect the expected changes in the star-forming
conditions. Previous studies of H$\alpha$ equivalent width (Gunawardhana et
al., 2011; Nanayakkara et al., 2017; and references therein) indicated that
galaxies at $z\sim 0.35$ and $z\sim 2$ have flatter high-mass IMF slopes,
which likely evolve with redshift. Besides, Zhang et al. (2018) find a more
direct indication of a top-heavy IMF in starburst galaxies at $2<z<3$ from
${}^{13}{\rm C}/^{18}{\rm O}$ lines. Based on the photometric constraints
obtained in this work and previously in Sneppen et al. (2022), the IMF at
$0<z<2$ is systematically different in star-forming galaxies and is likely to
change as the CMB temperature increases at higher redshift and due to a
complex interplay of various feedback processes. Consequently, the stellar
masses of typical star-forming galaxies appear overestimated by up to 1.0 dex
in the most extreme cases (Figure 4). As a result, the interpretation of such
key processes in galaxy evolution as quenching becomes more pronounced - it is
more consistent with the effect of downsizing (see § 4.7). In addition, the
IMF-derived temperature of star-forming gas hints at an early and hot stage of
galaxy evolution (Steinhardt et al., 2023), as well as provides insights into
the star-forming main sequence (see § 4.8).

When it is not possible to constrain an IMF, which is the case for most
current observations outside the COSMOS field, it is instead necessary to
assert a physically motivated IMF, because using the 20 K local IMF can break
the inferred galaxy properties in the different ways addressed in this work in
§ 4.5–4.8. The assumption should be based on the CMB temperature at the
relevant redshift regime666It is worth noting that the IMF-derived temperature
theoretically corresponds to the temperature in the star-forming clouds of the
photometrically-constrained stellar population. Thus, for the quiescent
population this temperature is offset from the observation by the time when
the last stellar population formed. and the latest understanding of the ISM
temperature. While there is no complete theory of the feedback-driven gas
temperature in galaxies, it is possible that the temperature measurements of
dust continuum or [C i] gas line ratio in specific galaxies or galaxy
populations can be used as a proxy (more on this strategy in § 5.3).

Finally, the observations of galaxy populations throughout the cosmic time
suggest that even if there is a most-likely universal IMF, it likely does not
look like the MW IMF. The most typical observed galaxy at the currently
sampled redshifts is on the star-forming main sequence. However, our Galaxy
with $sSFR\sim 10^{-11}$ yr-1 (Licquia & Newman, 2015) is consistent with
being quiescent. Besides, the existing empirical IMFs probe only small
selected regions in the MW, which are not guaranteed to represent the average
conditions in the Galaxy. Thus, if it is possible to fix an IMF in a random
photometric-template fitting experiment, the local IMF is unlikely to be the
preferred option.

###  5.2 Monolithic Galaxy Bias

Assuming that the diversity of gas temperatures and compositions can be solved
for, the other key approximation is that of galaxies as point sources. The
structural extent and large crowding of stars is neglected in spatially
unresolved studies. So it is possible that the integrated light can be biased
towards the brightest stars. In this case, the old stellar populations may be
outshined by luminous young O and B-type stars in star-forming galaxies.
Besides, galaxies can have different morphology, from compact cores to complex
spiral arms with bars and ellipticals, where, depending on relative locations
of different stellar populations, each probably results in a different bias.

Recent studies comparing spatially-resolved and integrated photometric
observations discover relative bias in recovered stellar populations. Giménez-
Arteaga et al. (2023) show that for five compact early galaxies at $5<z<9$
this leads to their masses being underestimated by a factor of up to 1.0 dex.
It is found that the young stars born in the central compact region dominate
the integrated photometry over the older stars. Sorba & Sawicki (2015, 2018)
showed similar results, with up to 80% of the spatially-resolved stellar mass
missing from their integrated analysis at $0.25<z<2.50$.

As most extragalactic observations lack spatial resolution, it is key to
quantify the reported bias and factor it into physical properties of galaxy
ensembles. This effect is likely to be the strongest for the most actively
star-forming galaxies with blue colors and especially young starbursts. It is
also likely that it is easier to “conceal” the stellar mass in the more
compact structures. Therefore, it is necessary to investigate this effect for
dusty star-forming galaxies, galaxies with different sSFR and various
morphologies at different epochs. The empirical correction for stellar mass
has been derived by Sorba & Sawicki (2018) for galaxies at $0.25<z<2.50$ based
on the sSFR. The work reported the strongest bias at increasing redshift and
high sSFR, but has not investigated the connection to galaxy morphology.

The effect of the combination of the temperature-dependent or else physically-
motivated IMF with the integrated photometry bias will perhaps depend on the
bottom-lightness of the IMF and the total galaxy luminosity at which the
photometry is dominated by the youngest O and B-type stars. Calculating the
total systematic effect is necessary to properly compare the observed physical
properties of galaxies with the cosmological predictions.

Finally, the photometric decomposition of main-sequence galaxies into disks
and buldges or central cores in Abramson et al. (2014) demonstrates that
morphological properties of galaxies may be used to further dissect galaxy
evolution into its fundamental constituents. It was shown that the SFMS may be
reconstructed with the star-forming galaxy disks alone. Such interpretation
would be consistent with the picture of galaxy evolution here and in
Steinhardt et al. (2023), where the core-forming galaxies may be progenitors
of the main sequence disks (see § 4.8).

###  5.3 Temperature Proxies for IMF

In this work it was possible to constrain the IMF only for the $\sim 10\%$ of
COSMOS2020 catalog with the widest wavelength coverage and highest S/N,
leaving most of the observations unused. The unused part of the sample
demonstrated strong covariances between IMF and other model parameters.
Therefore, additional information in terms of extending the wavelength
coverage, probing deeper photometry or setting a prior IMF is required to make
use of the available data.

In the framework of finding best-fit model photometry it is common to impose a
magnitude-based prior to zoom in on a physically motivated part of the
parameter space, which would otherwise suffer from statistical degeneracies
(Brammer et al., 2008). Similar approach is commonly performed with the IMF
when the MW prior is used. If there are indications that the ISM changes at
higher redshift can drive the changes in mass scale of stellar populations, a
prior on the IMF should be set correspondingly to the relevant physical
regime.

For setting the IMF, it is necessary to determine the gas temperature in the
relevant redshift regime. It is thought that multiple feedback processes may
be responsible for setting the temperature of the star-forming gas in the
galaxy (Papadopoulos, 2010). However, currently there is no preferred method
for predicting this temperature in a galaxy. Therefore, it is only possible to
determine the minimum CMB temperature with certainty. As it is demonstrated in
Steinhardt et al. (2022), the starting point can be setting the IMF based on
the redshift regime, which corresponds to an approximate CMB temperature band.
Although not ideal, this method should almost certainly be more accurate than
the assumption of the local IMF.

In absence of theoretical models for determining the gas temperatures at
redshift $z>2$, it may be possible to use the temperature of the cold dust
component as a proxy. The possible correlation between excitation temperature
of the gas and cool dust in galaxies on the main sequence (Valentino et al.,
2020) may be used as prior temperature for IMF. Thus, at redshifts $2<z<6$
where the CMB is below the local MW temperature, such proxy for an IMF could
be tested for galaxies on the main sequence. However, as discussed in § 4.8,
different feedback processes are likely responsible for galaxies not on the
main sequence, such as the starbursts or quiescent galaxies. Therefore, it is
not guaranteed that the same temperature correlations can be expected in those
galaxies.

A clear example when it is necessary to use prior temperature knowledge are
early galaxies found with the James Webb Space Telescope. Their redshift
regime ($z>10$) corresponds to the most extreme star-forming conditions
compared to the nearby galaxies, where even the CMB temperature places the
lower temperature limit to over 30 K. Therefore, the local IMF assumption
(equivalent to 20 K) is physically implausible in these galaxies. However, it
is likely not possible to constrain the IMF due to insufficient photometric
band coverage in the early observations. Therefore, it is necessary to ensure
that the IMF shape meets the minimum temperature requirement set by the CMB.
Alternatively, the masses can be overestimated by as much as 0.4-1.0 dex
(Steinhardt et al., 2022).

###  5.4 Impossibly Early Galaxies

In the previous decade, it became possible to test the predictions of the
hierarchical assembly of the early galaxies with the advent of such surveys as
Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS;
Grogin et al., 2011; Koekemoer et al., 2011) and Spitzer Large Area Survey
with Hyper-Suprime-Cam (SPLASH) that allowed observations of sizeable samples
of galaxies at $4<z<8$. Davidzon et al. (2017) and Steinhardt et al. (2016),
among others, reported a sharp problem where the observed stellar mass growth
is more efficient than allowed in the current cosmological model at high
redshift. It was found that by converting the photometrically-derived stellar
masses to halo masses based on local measurements or by matching galaxies
luminosities to the available dark matter halos, the inferred halo masses
exceeded the theoretical predictions by as much as $\sim 0.8$ dex at $z=10$
(Steinhardt et al., 2016). Similarly, the stellar mass functions appeared to
outgrow the halo masses at $z>3.5$ (Davidzon et al., 2017).

More recently, the disagreement between theory and observations has been
extended even to higher redshift of $z>8$ with some early JWST observations
(eg., Labbé et al., 2023). It has been shown that these massive galaxies have
outgrown their dark matter halos several times (Boylan-Kolchin, 2023) making
it critical that either the Lambda Cold Dark Matter model (Planck
Collaboration et al., 2020) is incorrect or the interpretation of observations
has a fault. Further studies have either not found the same disagreement
(Adams et al., 2023; McCaffrey et al., 2023) or have resolved it by reducing
its statistical significance (Chen et al., 2023) and by using a top-heavier
IMF (Harikane et al., 2023; Yung et al., 2023).

It is possible that the increasing disagreement at higher redshift is
propagated by the assumptions of the physics of star formation at low
redshift. In this case, possibly the analysis of stellar populations at
increasingly higher redshift in Davidzon et al. (2017); Labbé et al. (2023)
and other work shows a progressively greater bias due to the bottom-heavy IMF.
As it is argued in § 5.1, the conditions for star formation are expected to
become increasingly different from the local universe in the regimes of higher
CMB temperature and correspondingly higher stellar radiation and cosmic ray
feedback at high redshift. If that is the case, the mass of the stellar
populations in the early galaxies has been overestimated.

Although it is not yet possible to constrain an IMF at high redshift due to
the lack of wide photometric information or a large ensemble of rest-frame UV
and optical galaxy spectra, this work attempts to demonstrate the effect at
lower redshift regime where the survey data are more abundant. It is found
that stellar masses even in this already cosmologically “cold” regime at
$0<z<2$ decrease by up to 0.6 dex with the best-fit IMF, with similar results
shown in Sneppen et al. (2022) in COSMOS2015 galaxies.

With the constraints demonstrated at low redshift, it has been possible to
make rough predictions of the IMF at much higher redshift regimes which can
place the interpreted stellar masses of galaxies back in agreement with the
cosmological predictions. Steinhardt et al. (2022) showed that by assuming
that the temperature of the star-forming gas and respective IMF is set only by
the CMB, the stellar mass estimates at $z>8$ can be placed well within the
cosmologically allowed regime. Although at $z\sim 10$ the CMB temperature
($\sim 30$ K) becomes substantially higher than in the Milky Way ($\sim 20$
K), it is likely that the various feedback processes associated with the star
formation and supernovae can set this temperature even higher leading to much
lower estimates of the total stellar masses.

Finally, several assumptions have been made commonly to produce estimates of
total galaxy masses at all redshifts, including the local IMF, the specific
stellar baryon fraction from abundance matching and the stellar to halo mass
fraction. It is argued here that the first assumption is one of the most
counter-intuitive based on the expected temperature changes in the early
galaxies. If the change in estimated stellar masses from lower redshift IMF
constraints is extrapolated to higher redshift, it can be sufficient to
reconcile tension with the theoretical predictions of halo masses. However,
the fate of the stellar mass crisis will only be clearer after also
incorporating the bias from spatially non-resolved stellar populations
(Giménez-Arteaga et al., 2023). Among other assumptions, the stellar-to-halo
mass conversion is also tied to the local measurements which do not have to
hold at high redshifts and thus also have to be tested.

##  6 Conclusion

In summary, this work presents new physical properties of $\sim 10^{5}$ well-
measured COSMOS2020 galaxies at $0<z<2$ for which it is most likely to
constrain an IMF (Sneppen et al., 2022). It is found that the galaxies are
best fitted with a continuum of IMFs. Most of these IMFs are bottom-lighter
than measured in the Milky Way. As a result, stellar mass and star formation
rate estimates change significantly. The decrease in $M_{\star}$ and $SFR$ of
stellar populations is indicative of the increasing gas temperature in
molecular clouds at higher redshift, by construction of the IMF employed here.
Consequently, this work also derives the IMF parameter as a measure of the IMF
shape and the gas temperature, in theory. Below is a list of the main features
and some of the most interesting general implications of this catalog to
galaxy evolution identified so far.

  * •

Change in $SFR$ and $M_{\star}$ properties. From the perspective of the
temperature-dependent IMF, stellar masses and star formation rates of star-
forming galaxies reported in the standard COSMOS2020 catalog up to $z<2$ are
overestimated by factors of $\sim 1.6-3.5$ and $2.5-70.0$, respectively.

  * •

IMF shape as a probe of gas temperature. The changes in physical properties
correspond to the overall increase in the temperature of a star-forming gas as
a function of redshift inferred form the IMF. The theoretical interpretation
of the IMF shape as the gas temperature probe agrees with the evidence of
increasing dust and gas temperature with redshift (Valentino et al., 2020;
Magdis et al., 2017; Magnelli et al., 2014). The properties of quiescent
galaxies exhibit only a small change towards $z\sim 2$, which is consistent
with the measurements of dust temperature for quiescent galaxies in Magdis et
al. (2021).

  * •

Sharp downsizing effect. The fraction of quiescent galaxies as a function of
stellar mass appears to be consistent with the effect of downsizing (Juneau et
al., 2005; Cowie et al., 1996). The change in stellar masses with the best-fit
IMF is differential, such that the star-forming stellar mass functions are
shifted towards the lower stellar mass, while the masses of the quiescent
galaxies mostly remain unchanged. As a result, there is a larger fraction of
high-stellar mass passive galaxies at every studied redshift, which is more
sharply consistent with the effect of mass downsizing in comparison with the
standard set of properties.

  * •

The $SFR-M_{\star}-T_{IMF}$ relation. The best-fit IMF parameter provides new
insights about gas temperature on the SFMS and around it. There is a clear
positive gradient of IMF temperatures with sSFR on the main sequence. In turn,
quiescent galaxies have the lowest temperature at all redshifts. Additionally,
the galaxies with the highest temperatures and lowest stellar masses appear to
have a relationship with the IMF temperature distinct from that of the typical
star-forming galaxies.

  * •

The $sSFR-T_{IMF}$ relation. Main-sequence galaxies exhibit a strong $sSFR-
T_{IMF}$ coupling. However, there appear to be two groups of galaxies for
which this relation breaks. The first tail at low temperature corresponds to
quiescent galaxies, while the tail at the high temperature is populated with
the lowest-mass active galaxies. Based on the likely compact morphologies
(Steinhardt et al., 2023), it has been proposed that the last group is in the
early phase of galactic evolution that is distinct from typical galaxies on
the SFMS.

This work releases the new catalog and the photometric templates used here for
public access.

We would like to thank John Weaver, Gabriel Brammer, Sune Toft, Kate Gould,
Clara Giménez-Arteaga and members of the COSMOS Collaboration for help
provided with this work and useful discussions. The Cosmic Dawn Center (DAWN)
is funded by the Danish National Research Foundation under grant No. 140. This
research has made use of the SIMBAD database, operated at CDS, Strasbourg,
France. This research has made use of the VizieR catalogue access tool, CDS,
Strasbourg, France (DOI: 10.26093/cds/vizier). The original description of the
VizieR service was published in 2000, A&AS 143, 23. This research has also
made use of NASA Astrophysics Data System Bibliographic Services. This work
made use of the photometric data in the COSMOS2020 catalog which, along with
the standard physical properties, can be found at
https://doi.org/10.26093/cds/vizier.22580011 (catalog
10.26093/cds/vizier.22580011).

##  Appendix A Quality cuts & outliers

The best-fit SED templates of some samples converged around several very
specific parts of the parameter space. They appear in a grid-like pattern
shown in Figure 8 as a vertical stripe at the temperature $47-48$ K and a
horizontal stripe at $-12.5<sSFR<-11.5$ yr-1. In addition, most objects at
18-20 K were part of the outliers as spurious fits with narrow chi-square
solutions. Most of these fits appeared to have the IMF constraint degenerate
with the star formation history. In comparison, the sample of COSMOS2015
objects with variable IMF had a smaller number of apparently similar outliers,
although in a different part of the parameter space (cf. Steinhardt et al.,
2022a).

These samples were removed from the results demonstrated in this work using
the following selection criteria. They appeared to have distinct mass-to-light
ratios $M/L_{V}$ for their specific star formation rates $SFR/M_{\star}$, as
well as unusually narrow width of the chi-squared solutions $\sigma_{T}$.
Thus, the following criteria were used to cut the solutions corresponding to:
horizontal outliers with $log_{10}(SFR/M_{\star})<-11.5$ &
$M/L_{V}<0.9~{}M_{\odot}/L_{\odot}$; vertical outliers with $46<T_{IMF}<49$ &
$M/L_{V}>0.3$. These criteria were not optimal, as some viable solutions were
cut as a result and some outliers remained in the sample.

Given that this catalog cuts the original data after several selections, and
identifies outliers in the best-fit results, it is important to cross-check
the data consistency at every stage. Figure 9 shows the distributions of
$K_{S}$ magnitude after each of the following selections (the sizes of the
data sets are given in brackets):

  * •

Cut 1 (418,000): “ultradeep” stripes of the UltraVISTA photometry. Removing
the sources that were biased by the bright neighbors, appeared at the image
boundaries, were saturated or failed at execution, as flagged by SExtractor.
The selection reduced the survey area from $\sim 3.4$ to $0.9$ deg2.

  * •

Cut 2 (243,774): valid IMF solutions, as defined by chi-squared values in the
sampled temperature range $10<T<60$ K.

  * •

Cut 3 (189,332): removing solutions flagged as “outlier” (appeared as a grid
in Figure 8), based on the mass-to-light ratio cuts described above.

  * •

(Optional) Cut 4 (73,573): this sample includes only the solutions where the
uncertainty could be estimated based on the chi-squared metric using the
minimum variance bound estimator. This sample was used to demonstrate most
results in this work and is optional. The histograms in Figure 9 show that
this cut removes a fraction of faint sources.

The histograms of the $K_{S}$ magnitude do not differ significantly and thus
indicate that there are no major biases introduced by the selections made in
the data. However, the last cut is not advised, as it appears to remove a
fraction of $K_{S}$-faint source.

Figure 8: Diagram of sSFR as a function of temperature for galaxies at
$0.2<z<2.0$. The grid-like pattern with the vertical stripe at the temperature
$47-48$ K and the horizontal stripe at $-12.5<sSFR<-11.5$ yr-1 was flagged as
outliers. They likely originate due to the highest-mass IMF break being
covariant with the star formation history. Figure 9: Distributions showing
$K_{S}$ magnitude number histogram (left) and density (right) of the data set
after each selection or quality cut. The sizes of the data set are indicated
in the brackets. The magnitude observable is used to cross-check that the
selections do not introduce significant bias in the data.

##  Appendix B Changes in Stellar Mass Functions

This appendix shows the differences between the stellar mas functions of
active and passive galaxies computed using the standard stellar masses from
COSMOS2020 and the masses from the value-added catalog with the best-fit IMFs.
These functions are plotted in Figure 10 for star-forming (top row) and
quiescent (bottom row) galaxies separately.

Figure 10: Stellar mass functions for star-forming (top) and quiescent
(bottom) galaxies at different redshift windows in $0.2<z<2.0$. The standard
stellar mass has been converted from Chabrier to Kroupa IMF to match the
properties in the value-addded catalog. Most masses of active galaxies have
been shifted towards lower values as a result of using the best-fit IMF. On
the other hand, the masses of the passive galaxies remain similar to the
properties with the Milky Way IMF. This differential change in stellar mass
results in a systematic change of the quiescent galaxy fraction as a function
of the mass, which implies a sharper effect of mass downsizing (see Figure 6).

##  Appendix C Catalog Table

[b]

Table 3: Description of the column labels in the value-added catalog. Rows
1-3: ID and the sky coordinates are taken from the COSMOS2020 CLASSIC catalog.
Rows 4-55: standard EAZY-generated best-fit output. Rows 56-59: best-fit IMF
temperature (i.e., bottom-lightness of the Kroupa IMF) with its uncertainties
and outlier flags.

Num | Label | Unit | Description
---|---|---|---
1 | ID | — | COSMOS2020 CLASSIC catalogue ID
2 | ALPHA_J2000 | deg | Right ascension (J2000)
3 | DELTA_J2000 | deg | Declination (J2000)
4 | z_phot | — |  EAZY maximum likelihood photo-z
5 | z_phot_chi2 | — | chi-squared at z_phot
6 | z_phot_risk | — | risk parameter (Tanaka et al., 2018) at z_phot
7 | z_min_risk | — | z_phot where risk parameter z_phot_risk is minimised
8 | min_risk | — | risk at z_min_risk
9 | z_raw_chi2 | — | photo-z where z_phot_chi2 is minimised, without priors
10 | raw_chi2 | — | chi-squared at z_raw_chi2
11 | zxxx$\star$ | — | percentiles of photo-z
12 | nusefilt | — | number of filters used for photo-z
13 | lc_min | 0.1nm | minimum effective wavelength of valid filters
14 | lc_max | 0.1nm | maximum effective wavelength of valid filters
15 | restU |  $\mu$Jy | rest-frame U-band flux density
16 | restU_err |  $\mu$Jy | rest-frame U-band flux density uncertainty
17 | restB |  $\mu$Jy | rest-frame B-band flux density
18 | restB_err |  $\mu$Jy | rest-frame B-band flux density uncertainty
19 | restV |  $\mu$Jy | rest-frame V-band flux density
20 | restV_err |  $\mu$Jy | rest-frame V-band flux density uncertainty
21 | restJ |  $\mu$Jy | rest-frame J-band flux density
22 | restJ_err |  $\mu$Jy | rest-frame J-band flux density uncertainty
23 | dL | [Mpc] |  $\log_{10}$ luminosity distance at z_phot
24 | mass | [$M_{\odot}$] | $\log_{10}M_{star}$
25 | sfr | [$M_{\odot}$ yr-1] | $\log_{10}{\rm SFR}$
26 | ssfr | [yr-1] | $\log_{10}{\rm sSFR}$
27 | Lv | [$L_{\odot}$] |  $\log_{10}L$ (V-band)
28 | LIR | [$L_{\odot}$] |  $\log_{10}L$ (8-1000 $\mu$m)
29 | energy_abs | [$L_{\odot}$] |  $\log_{10}E_{\rm absorbed}$ associated with $A_{V}$
30 | Lu | [$L_{\odot}$] |  $\log_{10}L$ (U band)
31 | Lj | [$L_{\odot}$] |  $\log_{10}L$ (J band)
32 | L1400 | [$L_{\odot}$] |  $\log_{10}L$ in 200 $\text{\AA}$-wide top-hat filter (at 1400 $\text{\AA}$)
33 | L2800 | [$L_{\odot}$] |  $\log_{10}L$ in 200 $\text{\AA}$-wide top-hat filter (at 2800 $\text{\AA}$)
34 | LHa | [$L_{\odot}$] |  $\log_{10}L$(H$\alpha$)
35 | LOIII | [$L_{\odot}$] |  $\log_{10}L$(O III)
36 | LHb | [$L_{\odot}$] |  $\log_{10}L$(H$\beta$)
37 | LOII | [$L_{\odot}$] |  $\log_{10}L$(O II)
38 | MLv |  $M_{\odot}$ $L_{\odot}^{-1}$ | mass-to-light ratio (V band)
39 | Av | mag | extinction (V band)
40 | lwAgeV | Gyr | light-weighted age (V band)
41 | mass_pxxx$\star$ | [$M_{\odot}$] | percentiles of $\log_{10}M_{star}$
42 | sfr_pxxx$\star$ | [$M_{\odot}$ yr-1] | percentiles of $\log_{10}{\rm SFR}$
43 | Lv_pxxx$\star$ | [$L_{\odot}$] | percentiles of $\log_{10}L$ (V band)
44 | LIR_pxxx$\star$ | [$L_{\odot}$] | percentiles of $\log_{10}L$ (8-1000 $\mu$m)
45 | energy_abs_pxxx$\star$ | [$L_{\odot}$] | percentiles of $\log_{10}E_{\rm absorbed}$ associated with $A_{V}$
46 | Lu_pxxx$\star$ | [$L_{\odot}$] | percentiles of $\log_{10}L$ (U band)
47 | Lj_pxxx$\star$ | [$L_{\odot}$] | percentiles of $\log_{10}L$ (J band)
48 | L1400_pxxx$\star$ | [$L_{\odot}$] | percentiles of $\log_{10}L$ in 200 $\text{\AA}$-wide top-hat filter (at 1400 $\text{\AA}$)
49 | L2800_pxxx$\star$ | [$L_{\odot}$] | percentiles of $\log_{10}L$ in 200 $\text{\AA}$-wide top-hat filter (at 2800 $\text{\AA}$)
50 | LHa_pxxx$\star$ | [$L_{\odot}$] | percentiles of $\log_{10}L$(H$\alpha$)
51 | LOIII_pxxx$\star$ | [$L_{\odot}$] | percentiles of $\log_{10}L$(O III)
52 | LHb_pxxx$\star$ | [$L_{\odot}$] | percentiles of $\log_{10}L$(H$\beta$)
53 | LOII_pxxx$\star$ | [$L_{\odot}$] | percentiles of $\log_{10}L$(O II)
54 | ssfr_pxxx$\star$ | [yr-1] | percentiles of $\log_{10}{\rm sSFR}$
55 | Av_pxxx$\star$ | mag | percentiles of extinction (V band)
56 | t_imf | K |  $T_{\rm IMF}$, maximum-likelihood temperature of Kroupa IMF (Jermyn et al., 2018)
57 | t_imf_siglo | K |  $T_{\sigma_{-}}$, uncertainty in best-fit temperature of Kroupa IMF
58 | t_imf_sigup | K |  $T_{\sigma_{+}}$, uncertainty in best-fit temperature of Kroupa IMF
59 | outlier | — | bad local solutions that appear as a grid of $T_{\rm IMF}$ and sSFR (see § A)

  * 1

${\rm xxx}$ is one of [025, 160, 500, 840, 975] percentiles, i.e. [2.5%, 16%,
50%, 84%, 97.5%], of a PDF of a quantity.

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