# Non-Universal Stellar Initial Mass Functions: Large Uncertainties in Star
Formation Rates at $z\approx 2-4$ and Other Astrophysical Probes.

Joshua J. Ziegler,1 Thomas D. P. Edwards,2,3,4 Anna M. Suliga,5,6,7 Irene
Tamborra,7 Shunsaku Horiuchi,8,9 Shin’ichiro Ando,2,9 and Katherine
Freese1,3,4
1Department of Physics, Department of Physics, Austin, Texas 78751, USA
2Gravitation Astroparticle Physics Amsterdam (GRAPPA), Institute for
Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics,
University of Amsterdam, Science Park 904, 1090 GL Amsterdam, The Netherlands
3The Oskar Klein Centre, Department of Physics, Stockholm University,
AlbaNova, SE-10691 Stockholm, Sweden
4Nordic Institute for Theoretical Physics (NORDITA), 106 91 Stockholm, Sweden
5Department of Physics, University of California Berkeley, Berkeley,
California 94720, USA
6Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin
53706, USA
7Niels Bohr International Academy and DARK, Niels Bohr Institute, University
of Copenhagen, Blegdamsvej 17, 2100, Copenhagen, Denmark
8Center for Neutrino Physics, Department of Physics, Virginia Tech,
Blacksburg, Virginia 24061, USA
9Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583,
Japan
jjziegler@utexas.eduthomas.edwards@fysik.su.seasuliga@berkeley.edutamborra@nbi.ku.dkhoriuchi@vt.edus.ando<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We explore the assumption, widely used in many astrophysical calculations,
that the stellar initial mass function (IMF) is universal across all galaxies.
By considering both a canonical Salpeter-like IMF and a non-universal IMF, we
are able to compare the effect of different IMFs on multiple observables and
derived quantities in astrophysics. Specifically, we consider a non-universal
IMF which varies as a function of the local star formation rate, and explore
the effects on the star formation rate density (SFRD), the extragalactic
background light, the supernova (both core-collapse and thermonuclear) rates,
and the diffuse supernova neutrino background. Our most interesting result is
that our adopted varying IMF leads to much greater uncertainty on the SFRD at
$z\approx 2-4$ than is usually assumed. Indeed, we find a SFRD (inferred using
observed galaxy luminosity distributions) that is a factor of $\gtrsim 3$
lower than canonical results obtained using a universal Salpeter-like IMF.
Secondly, the non-universal IMF we explore implies a reduction in the
supernova core-collapse rate of a factor of $\sim 2$, compared against a
universal IMF. The other potential tracers are only slightly affected by
changes to the properties of the IMF. We find that currently available data do
not provide a clear preference for universal or non-universal IMF. However,
improvements to measurements of the star formation rate and core-collapse
supernova rate at redshifts $z\gtrsim 2$ may offer the best prospects for
discernment.

###### keywords:

stars: luminosity function, mass function – galaxies: luminosity function,
mass function – stars: formation – supernovae: general – neutrinos – methods:
data analysis

††pubyear: 2022††pagerange: Non-Universal Stellar Initial Mass Functions:
Large Uncertainties in Star Formation Rates at $z\approx 2-4$ and Other
Astrophysical Probes.–11

##  1 Introduction

In order to understand the formation and evolution of stars, an important
quantity is the stellar initial mass function (IMF), the relative numbers of
stars as a function of their mass at the time of their formation. As yet, the
IMF remains only loosely constrained observationally. A common assumption is
that the IMF is universal – the same in all environments and throughout cosmic
time. In this paper, we examine five observables that vary over cosmological
distances and which strongly depend on the high-mass region of the IMF. One of
our goals is to identify the extent to which these observables can be used to
test the assumption of a universal IMF at the high-mass end. In particular, we
study the consequences of non-universal IMFs for various astrophysical
quantities, finding larger uncertainties in the star formation rate and the
core-collapse supernova rate.

The concept of an IMF was introduced by Salpeter (1955), who proposed a single
power law $\frac{dN}{dM}\propto M^{\alpha}\,$ where $N$ is the number of stars
formed with mass $M$; in what is now known as the Salpeter IMF, he took
$\alpha=-2.35$. With the assumption of a single power law, the exponent
$\alpha$ can be measured to within approximately 10% (Baldry & Glazebrook,
2003). Unfortunately, there are fundamental questions about the
parametrization that should be used in describing the IMF. Perhaps most
notably, it was recognized at the end of the 20th century that low mass stars
did not tend to fall on the power-law distribution predicted by Salpeter. This
gave rise to IMF models with low-mass suppressions, such as the broken power
law of Kroupa (2001) and the log-normal distribution of Chabrier (2003).
Recent evidence suggests that the IMF may even have an intrinsic dependence on
the local environment (van Dokkum & Conroy, 2010; Ferré-Mateu et al., 2013;
Cappellari et al., 2012; Ferreras et al., 2012; La Barbera et al., 2019;
Harayama et al., 2008; Gunawardhana et al., 2011). Variations to the low-mass
end of the IMF have been studied extensively in the literature (van Dokkum &
Conroy, 2010; Chabrier, 2003; Geha, 2013). Instead, following recent evidence
(Gunawardhana et al., 2011), we focus on observables sensitive to the high-
mass end of the IMF which may also be non-universal.

Star-forming regions can be distinguished by a variety of properties of the
collapsing gas and dust, including angular momentum, metallicity, density,
temperature, and dust content. The universality of the IMF therefore boils
down to an assumption that all of these properties play little to no role in
the masses of the formed stars. Whether this is theoretically justified
remains unclear. As described in Offner et al. (2014) (and references
therein), perturbations in the density of a star-forming gas cloud can, under
reasonable assumptions, generate a power-law spectrum of core and clump
masses, where cores and clumps refer to gravitationally collapsing gas clouds
that are likely to form at least one star. In contrast to this power-law
distribution, at low masses, turbulence in the star-forming cloud can
naturally produce a spectrum of masses which disfavors lower mass stars
relative to the power-law predictions. In particular, Padoan et al. (1997)
showed that turbulence could give rise to a log-normal mass distribution among
low mass cores/clumps, similar to the IMF described by Chabrier (2003). While
this theoretical explanation would seem to leave very little room for non-
universality in the IMF, the mass function described here is for cores and
clumps, not stars. In relating this mass function to the stellar IMF, numerous
assumptions must be made about the formation of protostars out of collapsing
gas (Offner et al., 2014). The validity of many of these assumptions,
especially in extreme environments, is largely an open question, suggesting
that even within this theoretical framework, there may be room to consider
non-universality without requiring a new paradigm.

The question of whether the IMF is indeed universal has been investigated many
times. For example, despite most observations being consistent with a
universal IMF, authors have regularly suggested a non-universal IMF as a way
to explain other astrophysical tensions (Larson, 1998). Further, over the last
two decades, hints of a tension between universal IMFs and observations have
developed, particularly in early-type elliptical galaxies (van Dokkum &
Conroy, 2010; Ferré-Mateu et al., 2013; Cappellari et al., 2012; Ferreras et
al., 2012; La Barbera et al., 2019) and in environments which experience
extreme properties (Harayama et al., 2008; Gunawardhana et al., 2011).
Theoretical models, such as the Integrated Galaxy-wide IMF (Fontanot et al.,
2017) and the cosmic-ray-regulated star formation discussed in Fontanot et al.
(2018), can offer justifications for some of these observations and pose
additional predictions. On the other hand, due to the inherent difficulty in
measuring the IMF, many authors reject these observational claims, leaving the
question of whether the IMF is indeed universal largely unanswered (see e.g.,
Hopkins (2018), and references within for a recent review of the range of
perspectives).

A large part of the uncertainty in whether the IMF is universal can be traced
to the difficulty in unambiguously measuring it. Locally, where it is possible
to resolve individual stars, one can estimate the IMF by comparing observed
stellar populations to the populations that are predicted to form if different
IMF models are assumed (Calzetti, 2013; Kennicutt & Evans, 2012). While
accurate, this approach can only be used in star-forming regions nearby enough
to resolve individual stars, and requires assumptions about the history of
star formation in that region. On the other hand, for more distant galaxies,
where it is impossible to see inside star-forming regions or where it is
difficult to resolve individual stars, some proxy for the star formation rate
(SFR) must be used. The most common approach is to use the luminosity as a
measure of the rate of star formation (for example Kennicutt (1998)), but this
causes observations of the IMF to depend heavily on the calibration factor
between luminosity and SFR. While it is possible to calculate this calibration
factor numerically, it requires an assumption about the IMF. Unfortunately,
this circular dependence encountered when calculating the IMF of distant
galaxies is rather ubiquitous, making an independent measurement of the IMF
challenging.

While it is difficult to directly measure the IMF, it may be possible to find
indirect ways to probe the effects of a non-universal IMF. Previous works,
such as Fontanot et al. (2017); Fontanot et al. (2018) approach this problem
as well, but use different models of varying IMF and probe different
astrophysical observables than we do here.

In this paper, we examine five observables that vary over cosmological
distances and which depend on the IMF: the star formation rate density (SFRD),
the extragalactic background light (EBL), the core-collapse supernova (CCSN)
rate density, the type Ia supernova (SNIa) rate, and the diffuse supernova
neutrino background (DSNB). For each, we explore how they change when using a
non-universal IMF compared to a universal one, and discuss whether they are
discriminable with current or future data. For simplicity, we focus on the
change induced by a varying IMF and ignore many other uncertainties directly
related to each observable. These additional uncertainties will, in practice,
make it more difficult to observe the IMF induced changes. Our goal is simply
to learn whether astrophysical observations of distant objects could, in
principle, provide indirect evidence for a non-universal IMF.

The rest of this paper is structured as follows. In Sec. 2, we describe the
two IMF models we consider throughout the paper. In Sec. 3, we look at how
these IMF models affect the five quantities described in the previous
paragraph: the star formation rate density (SFRD), the extragalactic
background light (EBL), the core-collapse supernova (CCSN) rate density, the
type Ia supernova (SNIa) rate, and the diffuse supernova neutrino background
(DSNB). Finally, we conclude in Sec. 4.

##  2 Initial Mass Function Models

In a star-forming region, the stellar IMF describes the distribution of masses
with which stars form. A common approach to describing this IMF is through a
probability distribution $\xi(M)$. That is

$\frac{dN}{dM}=\xi(M)N_{\rm tot}\,,$ (1)

where $N$ is the number of stars formed with mass $M$, typically measured in
units of ${\rm M}_{\odot}$, and $N_{\rm tot}$ is the total number of stars
formed. Under this convention, $\xi(M)$ is normalized such that
$\int{\xi(M)dM}=1$, when integrating over all possible stellar masses.

In Salpeter’s seminal work (Salpeter, 1955), the IMF was described as a power
law of the form

$\xi(M)\propto M^{\alpha}\,,$ (2)

where $\alpha=-2.35$ was observed for stars in a mass range $0.4$ to
$1.0\,{\rm M}_{\odot}$. Since then, the range of masses over which the IMF
could be determined has vastly increased, but the practice of describing the
IMF through a power-law slope has remained. However, as IMFs have been
studied, the single straight power law of Salpeter has given way to IMFs with
more features. For example, commonly used IMFs include the piecewise power law
established by Kroupa (2001) (a variant of which was used in for example
Baldry & Glazebrook (2003)) and the Chabrier (2003) IMF which has a log-normal
distribution for stars below approximately $1\,{\rm M}_{\odot}$ and a power
law for stars greater than $1\,{\rm M}_{\odot}$. These two IMFs are plotted in
Fig. 1. As long as a universal IMF is assumed, the high-mass behavior of the
IMF is approximately a power law with $\alpha\approx-2.35$, consistent with a
Salpeter IMF (Baldry & Glazebrook, 2003).111Note that the typical break points
for Kroupa and Chabrier are $0.5\,{\rm M}_{\odot}$ and $1\,{\rm M}_{\odot}$
respectively. Because the log-normal mass function smoothly turns over, they
end up giving a similar distribution of stellar masses.

Figure 1: Initial mass functions: This figure compares the non-universal IMFs
we use in this paper (blue shaded region) to the canonical IMFs used in the
literature: Kroupa (2001) (green dotted) and Chabrier (2003) (red dash-
dotted). We note the similarity of the two canonical cases relative to the
wide range of non-universal IMFs we consider; we will use a proxy we name the
Salpeter-like IMF (purple solid line) as our benchmark for the canonical
cases. Each IMF is normalized so that the integral over mass equals 1. The
Kroupa IMF follows a broken power law, with slope $\alpha=-1.3$ for
$M<0.5\,{\rm M}_{\odot}$ and $\alpha=-2.3$ for $M>0.5\,{\rm M}_{\odot}$. The
Chabrier IMF also behaves differently at low mass versus high mass stars, with
the low mass stars ($M<1\,{\rm M}_{\odot}$) following a log-normal behavior
while high mass stars follow a power law with slope $\alpha=-2.35$. In all
cases we use broken power-law IMFs, defined piecewise with a break at 0.5
${\rm M}_{\odot}$. Non-universal IMF (blue-shaded region): For masses
$M<0.5\,{\rm M}_{\odot},\alpha=-1.3$ and for $M>0.5\,{\rm M}_{\odot},\alpha$
can take values in the range -1.8 to -2.35, shown here in the blue shaded
region. Benchmark universal Salpeter-like IMF (purple solid curve): consists
of a shallow power law like that in the Kroupa IMF at low masses and a
Salpeter $\alpha=-2.35$ IMF at high masses. Throughout this paper, we will use
this case as a benchmark against which to test the effects of allowing the IMF
to vary.

However, while it is broadly accepted that a power-law Salpeter IMF does not
hold true at low masses, questions regarding the range of environments over
which the Salpeter power law is valid for stars with mass greater than
$0.5\,{\rm M}_{\odot}$ remain significantly disputed. One of the most well-
motivated regimes in which deviations from a Salpeter IMF at higher stellar
masses could occur is the set of conditions in which Population III stars
grow. For example, a top-heavy IMF at early times, which favors a higher
average mass for Pop III stars, seems to be preferred by observations (Sharda
& Krumholz, 2021). One possible mechanism that could justify this behavior is
described in Sharda & Krumholz (2021), where a change in metallicity can shift
the peak mass from approximately $0.5\,{\rm M}_{\odot}$ for solar
metallicities to around $50\,{\rm M}_{\odot}$ at metallicities of Pop III
stars. Furthermore, Pop III stars may exhibit energy production mechanisms
inaccessible in Pop I and Pop II stars, as would be the case for Dark Stars
(stars made of hydrogen and helium but powered by dark matter) (Freese et al.,
2016). In that case, it would be quite surprising if the IMF were to be
consistent across all three populations.

In addition, there has been a growing body of evidence that seems to suggest
that while the IMF behaves as a power law at high masses, the slope may depend
substantially on environmental factors. Various authors identify multiple
factors as possible sources of these deviations away from Salpeter. In
addition to metallicity, these include: velocity dispersion (Ferreras et al.,
2012; Ferré-Mateu et al., 2013); radius and surface mass density (La Barbera
et al., 2019); and high turbulence (Chabrier et al., 2014).

In this paper, we focus on a non-universal IMF that varies with the SFR of a
star-forming region. Specifically, we explore a relationship that was
identified in data from the spectroscopic GAMA survey (Driver et al., 2011),
as analyzed in Gunawardhana et al. (2011). The GAMA survey was undertaken by
the Anglo-Australian telescope, which had measured the spectra of 120,000
galaxies at the time Gunawardhana et al. (2011) did their analysis. It has now
taken the spectra of approximately 300,000 galaxies. The analysis in
Gunawardhana et al. (2011) used the emission strength of the H$\alpha$ line as
a proxy to calculate the SFR of a galaxy, and then binned galaxies based on
that SFR. Using a set of simulated galaxies, a power-law IMF was fit to the
observed galaxies in each SFR bin, with the exponent $\alpha$ free to vary.
Using these binned galaxies, they found a clear preference for a non-universal
IMF, and that the variation could be described by the function,
$\alpha_{G}\approx 0.36\log\langle\mathrm{SFR}\rangle-2.6.$ Here, the average
SFR, $\langle\mathrm{SFR}\rangle$, is measured in units of ${\rm
M}_{\odot}\,\mathrm{yr}^{-1}$.

While this expression is the basis of the varying IMF we consider throughout
the rest of this work, it is not in the most convenient form for our purposes.
In particular, the independent quantity is the SFR, which is inferred from the
luminosity of the H$\alpha$ emission line. To calculate the astrophysical
observables discussed below, we will need galaxy luminosity functions up to
high redshifts. Unfortunately, the H$\alpha$ emission line is not the ideal
tracer of these luminosity functions as dust reprocesses most light emitted by
galaxies into the infrared. On the other hand, galaxy surveys (and therefore
galaxy luminosity functions) are more complete and readily available in the
far-infrared (FIR) band (i.e., in the wavelength range 8-1000 $\mu m$) up to
high redshifts. As a result, although using infrared luminosities can
introduce significant uncertainty into the calculated SFRs (Madau & Dickinson,
2014; Wilkins et al., 2019), it is essential for our calculations below.222
Note that Wilkins et al. (2019) found that the precise stellar mass range
considered can alter the FIR and H$\alpha$ calibration factors, although the
alteration is not necessarily the same between the two frequency bands. We
therefore point out that by shifting from H$\alpha$ to FIR, we are introducing
additional error on the overall magnitude of each of the observables discussed
below.  With this in mind, we convert $\langle\mathrm{SFR}\rangle$ to the FIR
luminosity $L_{{\rm FIR}}$, using fixed conversion factors from Kennicutt
(1998).333Specifically, we used the relation $\mathrm{SFR}\,({\rm
M}_{\odot}\,\mathrm{yr}^{-1})=4.5\times
10^{-44}L_{FIR}\,(\mathrm{erg}\,\mathrm{s}^{-1})$(Kennicutt, 1998). We discuss
how this value depends on the IMF in Section 2.1. This mimics the process used
by Kennicutt (1998) in reverse, but implicitly assumes that the SFRs predicted
by both tracers (H$\alpha$ luminosity and FIR luminosity) are consistent.
Using this procedure, we can rewrite the varying IMF expression from Kennicutt
(1998) as

$\alpha_{\mathrm{var},>0.5}\approx 0.36\log\langle L_{{\rm
FIR}}\rangle-6.1\,.$ (3)

Ultimately, the IMF we consider here is empirically based, so we choose to
confine ourselves to the range of IMFs that were observed in the corresponding
data. In particular, the analysis in Gunawardhana et al. (2011) calculated
IMFs ranging from $\alpha\approx-2.35$ to $-1.8$, with some populations of
galaxies having IMFs as steep as $\alpha\approx-2.5$. We limit ourselves to
consider only the range of $\alpha\in[-2.35,-1.8],$ which ensures that low
luminosity galaxies have an IMF with slope $\alpha=-2.35$.444 We note however,
that increasing the range of possible $\alpha$’s does not significantly affect
our results. In particular, we verified that extending the range to
$\alpha\in[-2.5,-1.8]$ has no noticeable effect on all results shown below.
This $\alpha$ range corresponds to enforcing galaxies with a luminosity
$\log\langle L/L_{\odot}\rangle\gtrsim 12$ to have $\alpha=-1.8$, and for
galaxies with luminosity $\log\langle L/L_{\odot}\rangle\lesssim 10.4$ to have
$\alpha=-2.35$. Furthermore, we explore only the effect of varying the IMF
above a mass cutoff of $0.5\,{\rm M}_{\odot}$, which gives comparable low-mass
behavior to the Chabrier and Kroupa IMFs. Below this mass cutoff, we use a
fixed power law

$\alpha_{\mathrm{var},<0.5}=-1.3\,,$ (4)

which matches the low-mass power law of the Kroupa IMF from 0.1 to 0.5 ${\rm
M}_{\odot}$.

We plot the range of IMFs that may appear in this luminosity-dependent varying
IMF in Fig. 1 (blue band). Special attention is given to the IMF which
consists of a shallow power law like that in the Kroupa IMF at low masses and
a Salpeter IMF at high masses. Throughout this paper, we will use this
Salpeter-like IMF (blue line) as a benchmark against which to test the effects
of allowing the IMF to vary. One important fact that is readily seen from Fig.
1, and which has been noted by, for example Hopkins (2018), is that all of the
IMF models we consider are quite similar, with only slight variation between
them. However, despite the smallness of these variations, when the different
IMF models are used to predict the values of observables, especially those
that depend on integration of IMF-dependent quantities, we can see substantial
differences appearing between the predictions made under those IMF models.

###  2.1 Luminosity to SFR Calibration Factor

We are interested in using observables which vary on cosmological scales to
probe the IMF, and on those scales directly measuring the IMF is unrealistic.
Instead, we will be using luminosity as a proxy for star formation, and by
extension the IMF, as described in Eq. (3). Because all of the observables we
consider are related to the rates at which stars form or die, a necessary
factor in their calculation is the calibration factor, which we denote $\chi$,
that relates luminosity to SFR. In general, the calibration factor depends on
the IMF, and because we are considering a non-universal IMF, we must consider
how the $\chi$ will depend on our assumed IMF.

For a Salpeter IMF, multiple calculations of the calibration factor have been
performed. In particular, Kennicutt (1998) found the calibration factor using
three different wavelength ranges. While modeling of factors such as dust has
improved (Kennicutt & Evans, 2012), the values in Kennicutt (1998) are often
useful as a benchmark for illustrative purposes. Throughout this paper, we
focus on the far infrared (FIR) wavelength range, 8-1000 $\mu$m, for which the
value of the calibration factor from Kennicutt (1998), assuming a Salpeter
IMF, is

$\chi_{\mathrm{K98,FIR}}=4.5\times 10^{-44}\,\mathrm{{\rm
M}_{\odot}\,yr^{-1}\,s\,erg^{-1}}\,.$ (5)

While the values obtained in Kennicutt (1998) are derived assuming a Salpeter
IMF, and are reasonably consistent with a Salpeter-like IMF (as shown in Fig.
1), we are interested in IMFs with a range of high-mass behaviors. In order to
calculate the impact that changing the IMF has on the calibration factor, we
use the code Pégase.3 (Fioc & Rocca-Volmerange, 2019), which simulates the
radiation spectrum of a galaxy with a set of user-defined inputs (our
particular scenarios are provided here: Pégase inputs). In each Pégase
simulation, a cloud of gas is converted to stars at a prescribed SFR, and with
a prescribed IMF. The stellar spectrum of each star that is formed is computed
and allowed to evolve according to stellar evolution processes. These stellar
spectra are then summed for the ensemble of stars in the galaxy at each
timestep in the simulation, resulting in a galactic spectrum. That galactic
spectrum is then adjusted to account for the reprocessing of stellar light by
dust. Through this process, we can get a spectrum which may be integrated over
various frequency ranges and used to calculate luminosities in different
frequency bands.

In our simulations, we consider galaxies in the local universe that have a
constant rate of star formation, and with other properties that we allow to
vary. Besides allowing for different IMFs, we look at galaxies with two
different geometries: disky spiral galaxies and spheroidal starburst galaxies,
and with different metallicities and different dust models. In particular, for
metallicity we consider a high-metallicity case with $Z=0.026$ and a low-
metallicity case with $Z=0.013$.555Note that some work has been done to self-
consistently model metallicity evolution together with a varying IMF
(Kobayashi, 2010). For the dust models, we consider the dust models of Zubko
et al. (2004) (ZWD), and Li & Draine (2001); Weingartner & Draine (2001)
(LWD). We summarize the combinations of the models we used in Table 1.

For each of the simulated galaxies, we calculate the luminosity in an
$8$–$1000~{}\mu$m wavelength band at different times from $10$ Myr to $1$ Gyr
after the star formation begins. Figure 2 shows how the calibration factor is
affected considering each of these changes in input parameters. Here it can be
clearly seen that while changing the metallicity and dust model does lead to a
distinguishable difference in the calibration factor, those differences are
small compared to the changes induced by changing the IMF and geometry.

Scenario | $Z$ | Grains File
---|---|---
low Z | 0.013 | ZDA
high Z | 0.026 | ZDA
dust ZDA | 0.0195 | ZDA
dust LWD | 0.0195 | LWD
Table 1: Choices of input parameters in Pégase.3. We use the various scenarios
to estimate the impact of these parameters on the luminosity to SFR
calibration factor. For all other results in the paper, we use the calibration
factors derived with the “low Z” conditions.

Figure 2: Calibration factor vs IMF slope: (Left Panel) Comparison of the
calibration factor $\chi=R_{SF}/L$ for a range of IMF slopes ($\alpha=-2.35$
to $-1.8$) used in our Pégase.3 simulations. We also show the impact of galaxy
morphology by comparing calibration factors assuming spiral galaxies (solid
lines) and spheroidal galaxies (dashed lines). The horizontal black line is
the calibration factor identified in Kennicutt 1998 (Kennicutt, 1998) for an
IMF slope $\alpha=-2.35$ and assuming a spheroidal galaxy. Conventionally, the
calibration factor is reported at 100 Myr following an onset of star
formation, indicated here by a grey dotted vertical line. (Right Panel) The
effect of different dust models and metallicities on the calibration factor,
for both spiral and spheroidal morphologies with an IMF slope $\alpha=-2.35$.
For comparison, all curves in the left panel use the “low Z” scenario.

The convention for reporting a single calibration factor is to take the value
at 100 Myr after the start of star formation. For other tracers of star
formation, particularly ultraviolet luminosity, the calibration factor is
effectively a constant after 100 Myr (Kennicutt, 1998). While that is not the
case for the FIR calibration factor, we adopt the same convention. Under this
definition, we observe that the calibration factor we calculate for a
Salpeter-like IMF and a spheroidal starburst galaxy is within 10% of the
calibration factor determined by Kennicutt (Kennicutt, 1998) for a Salpeter
IMF in a spheroidal galaxy. We do not account for other effects, e.g., stellar
rotation on the calibrations (Horiuchi et al., 2013). Note that for
$\alpha=-2.35$ we compared Pégase with Starburst 99 (Leitherer et al., 1999)
and found similar calibration factors.

##  3 Probes of a Non-universal IMF

Now that we have established our IMF models and the associated calibration
factors, we will explore five astrophysical observables that intrinsically
depend on the IMF. For each, we present the theoretical prediction for both
IMF models and discuss whether current or future data are able to distinguish
between the two.

###  3.1 Star Formation Rate Density

We first explore how the SFR of galaxies could provide constraints on the
nature of the IMF. The SFR is the rate at which gas in a star-forming region
turns into stars, typically measured in $\mathrm{{\rm M}_{\odot}\,yr^{-1}}$.
While an interesting quantity in its own right, we will focus on the related
star formation rate density (SFRD), which measures the star formation rate per
unit volume and typically has units $\mathrm{{\rm
M}_{\odot}\,yr^{-1}\,Mpc^{-3}}$. By looking at the SFRD rather than individual
galaxies’ SFR, we can average over the variance introduced because of
different galactic properties and specifically explore how star formation
depends on redshift. As a result, while both quantities give insight into the
star formation process, the SFRD is more directly tied to the cosmic star
formation history and less dependent on conditions within individual star-
forming regions (Madau & Dickinson, 2014).

While an understanding of the SFR is critical to theories of galactic
evolution, it is challenging to measure directly. In fact, only in local
systems, where stars can be resolved, can the SFR be directly measured
(Calzetti, 2013; Kennicutt & Evans, 2012). Where young stars can be resolved,
namely within the Milky Way and the nearest galaxies, it is possible to count
those young stars and therefore directly estimate the SFR (Chomiuk & Povich,
2011). In systems slightly more distant, where it is possible to resolve stars
but impossible to see young stars shrouded in dust, fitting the galactic
color-magnitude diagram to simulations can provide an accurate measure of the
SFR, among other properties. However, for more distant systems, in which stars
cannot individually be resolved, an indirect measure of the SFR must be used,
typically treating luminosity as a tracer of the SFR.666While the methods
described here are among the most direct ways to estimate the SFR, work has
been done to improve these estimates by combining these methods with
observations that depend indirectly on the SFR. For example, see Wilkins et
al. (2008a); Wilkins et al. (2008b) To convert from luminosity to SFR, one
uses the calibration factor introduced in the previous section. As mentioned
before, unfortunately these calibration factors are determined using
simulations which require assumptions to be made about the IMF. In section
2.1, we describe the impact that allowing the IMF to vary can have on the
calibration factor.

To calculate the SFRD, we need the calibration factor as well as the
luminosity distribution of galaxies. This distribution can be described
through the luminosity function ${dN}/{d\log L}$,777 Note that all equations
below are written for a generic luminosity $L$. For notational simplicity, we
therefore drop the subscript FIR on all luminosities. However, all
calculations are performed within the FIR luminosity range (8-1000$\,\mu$m).
which will generically be a function of redshift $z$. In all of our
calculations, we use the set of luminosity functions calculated from data
collected by the Herschel observatory in multiple complementary surveys
including the PACS Evolutionary Probe (PEP), Herschel Multi-Tiered
Extragalactic Survey (HerMES), and Herschel Great Observatories Origin Deep
Survey (GOODS) (Gruppioni et al., 2013). By including deep, pencil beam
surveys like GOODS, these data include galaxies out to a redshift of $z\sim
4$. Meanwhile, broad, shallow surveys, like those in PEP and HerMES, can help
provide more accurate identification of galaxies’ morphologies, Therefore,
from this combined dataset, Gruppioni et al. (2013) were able to develop
accurate, galaxy-morphology specific luminosity functions, labeled as
“spiral”, “starburst”, and “AGN-SF” for redshifts $z\approx 0-4$.888 Note that
care must be taken in how AGN are considered. The luminosity functions we
consider are derived from the total luminosity of the galaxies, which includes
both AGN and stellar sources. However, when calculating calibration factors,
it is conventional to include only luminosity from stellar sources, not
including AGN. As a result, introducing AGN may decrease the calibration
factors from what we derive in the previous section, but we do not consider
the effects of such a decrease in this work.  Respectively, these describe:
spiral, disky galaxies without extreme star formation; spheroidal galaxies
with intense star formation; and galaxies with a bright active galactic
nucleus (AGN). We further distinguish the AGN category into spiral galaxies
with an AGN and starburst galaxies with anAGN, based on the fraction of each
type presented in Gruppioni et al. (2013). For both spiral galaxies with or
without an AGN, we use the spiral calibration factors from the previous
section. Similarly, for starburst galaxies with or without an AGN, we use the
spheroidal calbiration factors.

In Fig. 3, we show the luminosity density as functions of redshift. In
particular, we show the quantity $\int dL\>{dN}/{d\log L}$ for both spiral and
starburst galaxies, as well as their sum. Assuming a Salpeter-like IMF, the
calibration factor is constant, so this quantity is proportional to the SFRD,
as can be seen clearly below in Eq. (6). We additionally show the luminosity
dependence of this integrated quantity by presenting contributions from three
luminosity ranges.

Figure 3: Luminosity density: We show how the luminosity functions
${dN_{g}}/{d\log L}$ from Gruppioni et al. (2013) depend on redshift. These
luminosity functions give the density of galaxies as a function of luminosity
and redshift. Here we present the integral $\int dL\>{dN_{g}}/{d\log L}$ for
spiral and starburst galaxies. We also show the contribution to these
integrals for the luminosity ranges: $10^{8}$–$10^{10}L_{\odot}$,
$10^{10}$–$10^{12}L_{\odot}$, and $10^{12}$–$10^{14}L_{\odot}$. The luminosity
functions in Gruppioni et al. (2013) are defined by fitting to modified
Schecter functions, with parameters that are defined piecewise on $z$. These
piecewise fits have breaks at the redshifts $z=0.5$ and $z=1.1$, so it is at
these redshifts that we see peaks in observables like the SFRD.

From the calibration factor and luminosity function, we can calculate the SFRD
for a given galaxy morphology $g$ by integrating over the luminosity $L$:

$R_{\rm SF,\,g}=\int\chi(L)L\frac{dN_{g}}{d\log L}d\log L\,.$ (6)

The total observed SFRD is then given by the sum over galaxy types
$\sum_{g}{R_{\rm SF,\,g}}$, where $R_{\rm SF,\,g}$ is the SFRD contribution
from galaxies of type g.999 For simplicity, for the rest of this paper we drop
the subscript g notation. All observable quantities defined below implicitly
sum over galaxy type. In Fig. 4, we compare the SFRD calculated using the
varying IMF and Salpeter-like IMF. We also plot existing estimates of the
SFRD, with data drawn from Magnelli et al. (2011, 2013); Gruppioni et al.
(2013). Note that we only show data that explicitly uses the same FIR range as
we consider for our luminosity functions. In principle, there is a wealth of
other data from different wavelengths to compare to (e.g. Abdollahi et al.
(2018); Madau & Dickinson (2014); Driver et al. (2018); Hopkins & Beacom
(2006); Fardal et al. (2007)). We leave a more careful comparison to these
data sets to future work.

Figure 4: Star formation rate density: We compare the SFRD assuming a
Salpeter-like IMF (green dashed curve) and a varying IMF (blue solid curve) to
observational data as a function of redshift. The data were chosen to match
with the FIR band we explore in this work. To limit clutter in the plot, we
chose a representative subset from the catalogue of data (Madau & Dickinson,
2014). Features in the prediction curves at $z=0.5$, $z=1.1$ arise from non-
smooth features in the luminosity functions we consider, and features at
$z=1.0$, $z=1.2$, and $z=2.0$ arise from a non-smooth division of galaxies
with an AGN into spiral and spheroidal sub-categories. While the observed data
(red colored points) agrees quite well with the SFRD predicted from a
Salpeter-like IMF, the data themselves are derived quantities which assume a
Salpeter, Kroupa, or Chabrier IMF. For this reason, it is unsurprising that
the observed data do not match the SFRD predicted using the varying IMF,
particularly at high redshift. If a varying IMF is assumed when calculating
the SFRD from observations, the results are expected to closely follow the
predictions we make for a varying IMF. For illustration, we perform a
preliminary reanalysis of the (Magnelli et al., 2013) data using the
luminosity functions described therein and the varying IMF we use in this
work, depicted as blue points. This illustration is meant only to show proof
of concept, and a more careful reanalysis should be performed, particularly in
order to estimate uncertainties.

At low redshift ($z\lesssim 0.15$), the SFRD calculated using a varying IMF is
slightly lower than (within about 20% of) the SFRD calculated using a
Salpeter-like IMF and is consistent with observations. The primary reason for
this behavior is that at low redshifts, the dominant contribution to the
luminosity functions comes from spiral galaxies with intrinsically lower
galactic luminosity, which favor an IMF power law of $\alpha\approx-2.35$. At
higher redshifts, the luminosity is dominated by a smaller density of
intrinsically more luminous starburst galaxies, which favor a shallower IMF.
Accordingly, we see that at redshifts $z\gtrsim 2$, the SFRD calculated using
a varying IMF is up to a factor of three lower than the SFRD calculated from a
Salpeter IMF and the reported observational data. However, all of the data
plotted in Fig. 4, which is illustrative of much of the data in the literature
(e.g. Abdollahi et al. (2018); Madau & Dickinson (2014); Driver et al. (2018);
Hopkins & Beacom (2006); Fardal et al. (2007)), assume an IMF with Salpeter-
like behavior at high stellar mass. In particular, in each of the three
observed data sets shown here, the SFRD was calculated while assuming either a
Chabrier IMF, a Kroupa IMF, or a pure Salpeter IMF. It is unsurprising then
that the data so closely matches the Salpeter-like IMF, while disagreeing with
results obtained using any other IMF. We therefore expect that reanalyzing the
SFR observations with a varying IMF would result in an SFRD that is lower than
the existing data and that matches our predictions. We leave this analysis to
future work. Since the effect of a varying IMF is signficant, we also note
that direct observations of the SFR at high redshifts would provide a useful
probe of the IMF variability as a function of galaxy type.

###  3.2 Extragalactic Background Light

In addition to measuring the SFRD directly, we can also look at the
extragalactic background light (EBL) to potentially probe the IMF. The EBL is
the integrated light from all sources in a particular direction. In
particular, it includes light from all galaxies, even those too faint to
resolve, and therefore provides an accurate measure of the luminosity
function. In fact, the total EBL can be calculated as

$L_{\rm EBL,total}=\int L\frac{dn_{g}}{dL}dL=\int\frac{dn_{g}}{d\log L}dL\,.$
(7)

That is, the total EBL is simply an integral of the luminosity function over
galaxy luminosities. Starting from the luminosity functions then, we can
calculate the total EBL without introducing a dependence on the IMF, and so it
is impossible to probe the IMF from the total EBL.

However, while the total EBL may be independent of IMF, the IMF affects the
distribution of stellar masses. Because stars of different masses have
different temperatures, and therefore different spectra, it is possible that
the EBL spectrum may provide a way to probe the IMF. With this in mind, we
followed the procedure outlined in Razzaque et al. (2009) to calculate an
estimate of the EBL flux in the wavelength range 0.1 to 100 $\mathrm{\mu m}$.
Specifically, we calculate the spectrum of a star of given mass $M$ with the
corresponding effective temperature $T(M)$ as a blackbody spectrum
$I_{\nu,BB}(T(M))$. We additionally denote the number of stars of a given mass
inside of a galaxy as $\mathcal{N}(M,L)$. This number therefore intrinsically
depends on the IMF. The EBL at a given frequency can then be calculated as

$I_{{\rm EBL},\nu}=\int\int I_{\nu,BB}(T(M))\mathcal{N}(M,L)dM\frac{dN}{d\log
L}dL\,.$ (8)

Although this technique can give an estimate of the EBL emitted from galaxies,
it does not take into account dust, and because of this offers only limited
insight into the observed EBL. Absorption of starlight by dust causes the
short-wavelength end of the spectrum to be reduced, while re-emission by that
dust causes the long-wavelength end to be increased. While this generic
picture is true in all dust models, exactly how dust affects the shape of the
spectrum depends heavily on the dust model. We leave careful accounting of
these dust effects to future work. Instead, we can look at the narrow range of
frequencies where the effects from both dust absorption and emission on the
EBL are minimized. In particular, we consider the range approximately
$4-7\,\mu$m (Cardelli et al., 1989; Kennedy et al., 2013). As can be seen in
Fig. 5, in this wavelength range, the EBL predicted from both a Salpeter-like
and varying IMF are quite similar, with the varying IMF prediction being very
slightly bluer than the Salpeter-like prediction, and both visually fit
existing data equally well.

Figure 5: Extragalactic Background Light Spectrum: Estimates of the EBL
spectra assuming a varying IMF (blue solid curve), and a universal Salpeter-
like IMF (green dashed curve). These are compared to data from Biteau &
Williams (2015), showing the best fit values of the EBL calculated from
various observed measurements of upper and lower bounds on the EBL spectrum.
The estimates we show do not include processing from dust, so we generally
expect the estimates to overpredict the intensity at short wavelengths, but
begin to match at longer wavelengths. At the longest wavelengths, radiation
from dust, rather than from stellar sources, will dominate the observed
spectrum,and our estimates will underpredict the data. This behavior is seen
in our predictions which, in the wavelength range $3-7\,\mu$m which is least
affected by dust, agrees quite well with data, regardless of IMF considered.

###  3.3 Core-Collapse Supernova Rate

Along with the SFRD and EBL, we can look at the rate at which stars collapse
into supernovae as a way to constrain the IMF. Stars which begin their life
with a mass between 8 and approximately 125 ${\rm M}_{\odot}$ are expected to
end their life as a core-collapse supernova (CCSN) (Heger et al., 2003).
Standard CCSNe occur when their iron or oxygen-neon-magnesium core, produced
through nuclear fusion, exceeds the Chandresekhar mass limit and rapidly
collapses (Couch, 2017; Janka, 2017; Burrows & Vartanyan, 2021). This
facilitates electron capture on nuclei and free protons in the center of the
collapsing core, which makes matter more neutron rich and produces a copious
number of electron neutrinos (Bethe et al., 1979a; Fuller, 1982). Once the
density reaches the nuclear saturation density (approximately $2.6\times
10^{14}~{}\mathrm{g}~{}\mathrm{cm}^{-3}$ (Bethe et al., 1979a)) the strong
nuclear force stops the collapse of the inner core (Bethe et al., 1979b;
Couch, 2017; Bethe et al., 1979a; Branch & Wheeler, 2017). The sudden halt
causes the core to bounce and launches a shock wave, which carries stellar
material away from the core. The shock stalls after losing energy through the
dissociation of heavy nuclei but can be re-energized by neutrinos, which
revive the shock leading to a successful explosion. This is the so-called
neutrino-driven delayed explosion mechanism (Colgate & White, 1966; Bethe &
Wilson, 1985).

Eventually, the electromagnetic radiation emitted by the material ejected in a
successful explosion is observed as a supernova. Depending on the chemical
makeup of the ejected material, CCSNe are classified as Type Ib, Ic, or II
supernovae (Turatto, 2003; Smith, 2014). The supernova can leave behind either
a neutron star or black hole depending on how much energy is imparted to the
shock wave by neutrinos, compared to the gravitational binding energy of the
outer layers of the star. Because some of these types of supernovae will be
more detectable than others and any stars that collapse directly to a black
hole do not produce a supernova, the fraction of core collapses that can be
observed through electromagnetic signatures will depend on the fate of the
collapsing massive star, and that in turn may depend on the IMF.

As with the SFR, we will again focus on the core-collapse supernova rate
density (CCSNRD), which allows us to focus on only the redshift dependence. On
timescales that are long compared to the lifetime of a star massive enough to
undergo core collapse ($\lesssim 10$ Myr), we can assume that when one star
goes supernova another star of equal mass is formed. As a result, we can
calculate the CCSNRD directly from the SFRD of stars with mass greater than
approximately $8\,{\rm M}_{\odot}$. In particular,

$R_{{\rm CCSN}}(z)=\int\chi(L)L\frac{dN}{d\log L}\frac{\int_{8\mathrm{{\rm
M}_{\odot}}}^{M_{\mathrm{max}}}\xi(M)dM}{\int_{M_{\mathrm{min}}}^{M_{\mathrm{max}}}M\xi(M)dM}d\log
L.$ (9)

Here, $M_{\mathrm{min}}$ and $M_{\mathrm{max}}$ refer to the minimum and
maximum masses stars can take in the IMF we consider. In all of our
calculations we use $M_{\mathrm{min}}=0.1\,{{\rm M}_{\odot}}$ and
$M_{\mathrm{max}}=125\,{{\rm M}_{\odot}}$, respectively the approximate lowest
mass at which stars can fuse hydrogen and a somewhat arbitrary high mass
cutoff that does not significantly affect our results. The factor of $M$ in
the denominator is necessary because SFRs measure the total mass of stars that
form, whereas the core-collapse rate measures the number of supernova events.

Figure 6: Core collapse supernova rate density: (Left Panel) Comparison of the
predictions of the rate density of core-collapse supernovae assuming either a
Salpeter-like IMF (green dashed curve) or a varying IMF (blue solid curve)
with observational data as a function of redshift. See Fig. 4 for discussion
of features at $z=0.5$, $z=1.0$, $z=1.1$, $z=1.2$, and $z=2.0$. (Right Panel)
Rate of supernovae in each of the galaxy morphologies we consider. Here, blue
corresponds to spiral galaxies, while green corresponds to spheroidal
starburst galaxies. Again, dashed lines correspond to Salpeter-like IMFs,
while solid lines correspond to the varying IMF.

In Fig. 6, we compare the CCSNRD we predict from a varying IMF to that
predicted from a Salpeter-like IMF and to observed CCSNRD data from
Petrushevska et al. (2016); Strolger et al. (2015); Dahlen et al. (2012);
Mattila et al. (2012). As in the case of SFRs, at low redshift, all three
CCSNRD are consistent. At high redshift, the CCSNRD predicted using the
varying IMF is slightly lower than that predicted using the Salpeter-like IMF,
but the discrepancy between the CCSNRD is significantly smaller than the
related difference between SFRs. This improved agreement is to be expected
because the SFR’s dependence on the IMF partially cancels against the explicit
IMF dependence in Eq. (9), as discussed in Madau & Dickinson (2014).
Ultimately, the existing CCSNe data appears to agree with either IMF model
equally well. However, as new observatories including, the James Webb Space
Telescope (Regős & Vinkó, 2019), the Vera Rubin Observatory (through the LSST
survey) (Ivezić et al., 2019), Euclid (Laureijs et al., 2011), and the Nancy
Grace Roman Telescope (Koekemoer et al., 2019) begin to take data, estimates
of the high redshift CCSNRD will become more precise, potentially favoring one
model over the other. Likewise, gravitational wave detectors may offer another
avenue to probe the supernova rate, and consequently the SFR, by measuring the
binary black hole merger rate (Vitale et al., 2019).

We find that the two IMF models produce an $\mathcal{O}(2)$ difference in the
total CCSN rate at redshifts greater than approximately 1.5. Therefore, at
least a 100% determination of the SN rate for $z>1.5$ will be required to
distinguish these two scenarios. Based on projections of the rates of
supernovae expected to be observed by the Roman Telescope (Rose et al., 2021),
it is not unreasonable to expect observational uncertainties at that level.
Furthermore, a greater difference in the predicted supernova rates under our
two models may appear if changing the IMF changes the fraction of stars that
collapse into black holes versus those that collapse to neutron stars. All
else held equal, a shallower IMF would increase the fraction of stars that
evolve to black holes and decrease the fraction that evolve to neutron stars,
relative to a steeper IMF. As shown in Fig. 7, this is precisely the situation
we expect from the varying versus Salpeter-like IMFs. As a result, we can
expect that a varying IMF would lead to fewer obervable CCSNe than a Salpeter-
like IMF, particularly at high redshifts, all other factors held equal.

Figure 7: Fraction of black hole forming collapses: The fraction of supernovae
that result in black holes, rather than neutron stars. For the Salpeter-like
IMF, approximately $21\%$ of stellar collapses lead to black hole formation
(this fraction is assumed to be constant as a function of the redshift for
simplicity), while for the varying IMF, this fraction depends on redshift,
reaching approximately 35% at $z=3$.

###  3.4 Type Ia Supernova Rate

As with CCSNe, we can hope to probe properties of the IMF by looking at Type
Ia supernovae (SNeIa). Type Ia supernovae occur when mass accretes onto white
dwarfs from a giant star, typically when the two form a binary (Mazzali et
al., 2007). As the mass of the white dwarf approaches the Chandrasekhar mass
(Chandrasekhar, 1931), temperatures and densities within the core of the star
become high enough to initiate nuclear reactions from the abundant carbon and
oxygen. These nuclear reactions produce so much energy that the entire star
becomes unbound. Nuclear decays within the ejected material can then be
observed as a SNeIa.

For our purposes, SNeIa can be seen as a tracer of SFR like the CCSN rate. In
addition, measurements of the SNIa rate do not require one to use a
calibration factor and are therefore relatively independent of the IMF.
However, since SNeIa are sourced by white dwarfs, which form from stars whose
lifetimes are on the order of 1-10 Gyr, they can only occur after enough time
has passed for the white dwarf to accrete sufficient mass. Both the lifetime
of the progenitor star and the period between the formation of a white dwarf
and the occurrence of a SNIa must be accounted for when calculating the SNIa
rate. This accounting is done through a delay-time distribution (DTD)
$F(\tau)$, which measures the probability that a star formed at some time
$t-\tau$ will undergo a supernova at time $t$. In our calculations, we use the
simple approximation that $F(\tau)\propto\tau^{-1}$, justified in Maoz et al.
(2014). The SNIa rate density can therefore be calculated from the convolution

$R_{\rm SNIa}=\int_{0}^{t}R_{\rm SF}(t-\tau)F(\tau)d\tau\,.$ (10)

Here, we convert between redshift and time assuming a flat $\Lambda$CDM
Universe that is dominated by $\Lambda$ and matter, and where
$\Omega_{m,0}=0.308$ (Ade et al., 2016).

In Fig. 8 we compare observed SNIa rates, using data from Strolger et al.
(2020); Perrett et al. (2012); Cappellaro et al. (2015), with predictions of
the supernova rate assuming a Salpeter-like IMF and a varying IMF. To make
this comparison, we fixed the normalization of the DTD to $10^{-3}\,{\rm
M}_{\odot}^{-1}$ in order to match the DTD presented in Maoz et al.
(2014).101010However, as noted in Maoz et al. (2014), measurements of the DTD
in different environments (e.g. dwarf galaxies, galaxies, and galaxy clusters)
can vary by an order of magnitude. Using this normalization, the Salpeter-like
IMF is largely consistent with the data, while the varying IMF leads to a SNIa
rate that is consistently smaller than the observed SNIa rate. However, if we
allow the normalization of the DTD to vary, then increasing it by a factor of
2 produces much closer agreement between the varying IMF and observed SNIa
rates. This change to the normalization is still within the $1\sigma$
confidence range for the DTD. In other words, based on current observations
and their uncertainties, the varying IMF cannot be ruled out, and more precise
measurements of the DTD would be necessary to place meaningful constraints on
either IMF model.

Figure 8: Type Ia supernova rate density: We compare the rate density of type
Ia supernovae, assuming either a Salpeter-like IMF (green dashed) or varying
IMF as described in the text (blue), to a selection of observed data. See Fig.
4 for discussion of features at $z=0.5$, $z=1.0$, $z=1.1$, $z=1.2$, and
$z=2.0$. Predictions of the supernova rate depend on a poorly constrained
delay time distribution, and by increasing the overall normalization of this
DTD by a factor of 2 (still within $1\sigma$ uncertainties), we can increase
the varying IMF result from the blue solid curve to the blue dot-dashed curve.

###  3.5 Diffuse Supernova Neutrino Background

Finally, in addition to directly observing of supernova rates, we can look to
the neutrinos supernovae produce in order to estimate their rate, and
therefore potentially probe the IMF. Despite being relatively rare in any
individual galaxy, supernovae are quite common throughout the Universe.
Combining this with the fact that a single CCSN produces an immense number of
neutrinos (approximately $10^{58}$ (Burrows & Vartanyan, 2021; Mirizzi et al.,
2016)) leads to the emergence of a background of neutrinos that is isotropic
and nearly constant in time. This neutrino flux is commonly named the diffuse
supernova neutrino background (DSNB) (Bisnovatyi-Kogan & Seidov, 1984; Krauss
et al., 1984; Wilson et al., 1986; Beacom, 2010; Lunardini, 2016; Mirizzi et
al., 2016).

Only in recent years have experiments begun to approach the sensitivity
necessary to directly observe the DSNB (Malek et al., 2003; Bays et al., 2012;
Zhang et al., 2015; Abe et al., 2022, 2021; Li et al., 2022). While no signal
has yet been detected, the enrichment of Super-Kamiokande (SK) with gadolinium
(Beacom & Vagins, 2004; Horiuchi et al., 2009) and the future proposed and
planned experiments such as Hyper-Kamiokande (HK), JUNO, Jinping, and THEIA
(An et al., 2016; Beacom et al., 2017; Abe et al., 2018; Sawatzki et al.,
2021; Li et al., 2022) are expected to have enough sensitivity to make a first
detection in the coming years. Once observed, the DSNB will provide a test of
astrophysical observables (Lunardini, 2009; Keehn & Lunardini, 2012; Nakazato,
2013; Nakazato et al., 2015; Priya & Lunardini, 2017; Møller et al., 2018;
Kresse et al., 2021; Singh & Rentala, 2021; Horiuchi et al., 2021; Libanov &
Sharofeev, 2022), neutrino flavor physics (Lunardini & Tamborra, 2012; Tabrizi
& Horiuchi, 2021; Suliga et al., 2022), and physics beyond the Standard Model
(Ando, 2003; Fogli et al., 2004; Goldberg et al., 2006; Baker et al., 2007;
Farzan & Palomares-Ruiz, 2014; Jeong et al., 2018; Creque-Sarbinowski et al.,
2021; de Gouvêa et al., 2020; Das & Sen, 2021; Suliga et al., 2022; de Gouvêa
et al., 2022).

####  3.5.1 Theoretical Models

The calculation of the DSNB flux requires two components: the rate of
supernovae as a function of their progenitor masses and the time-integrated
neutrino energy spectra associated with each supernova. The former can be
calculated as described in Section 3.3, while a calculation of the latter is
sketched here. Following Møller et al. (2018); Ashida & Nakazato (2022), we
consider three possible outcomes of supernovae, depending on the mass of their
stellar progenitor. Stars can either evolve into black holes or low/high mass
neutron stars. A characteristic neutrino spectrum is then associated with each
type of explosion. More details on these spectra can be found in Appendix A.

As discussed in Appendix A, the characteristic neutrino spectra associated
with the different supernovae outcomes can be significantly different. The
DSNB signal is therefore also affected by the fraction of black hole forming
collapses (Lunardini, 2009). Unfortunately, the fraction of stellar collapses
leading to black hole formation is unknown. Recent theoretical work and
observational surveys indicate that this fraction could be approximately
$10\,$–$\,40\%$ of all CCSNe (Byrne & Fraser, 2022; Neustadt et al., 2021;
Kochanek et al., 2008; Lien et al., 2010; Gerke et al., 2015; Horiuchi et al.,
2014; Sukhbold et al., 2016; Ertl et al., 2016; Adams et al., 2017a, b; Davies
& Beasor, 2020). For the Salpeter-like IMF, our DSNB modeling follows one of
the scenarios considered in Møller et al. (2018) where the fraction of
progenitors evolving into black holes is set to $21\%$ and for simplicity it
is assumed to be constant as a function of redshift out to at least $z=5$. To
make this assumption, we implicitly assume that both the stellar masses that
evolve to black holes and the IMF do not change over cosmological history.
However, when we consider the varying IMF, we cannot assume that the IMF is
constant as a function of redshift; in fact, because it depends explicitly on
the star formation rate, which evolves over cosmic history, the varying IMF
cannot remain constant. As a result, we expect to see a substantially larger
fraction of stars evolve to black holes at high redshifts in the non-universal
case than in the Salpeter-like case, as seen in Fig. 7.111111 It should be
noted that this fraction is calculated assuming that the stellar mass is the
only factor which controls whether a supernova leads to a black hole formation
or not (see Appendix A.). It is therefore constant for the universal Salpeter-
like IMF. Other stellar properties, such as metallicity, may lead to changes
in the fraction of supernovae that lead to black hole formation. In addition,
because our main goal is to investigate the impact of the variable IMF on the
DSNB, we do not include neutrino oscillations in our modeling, as they would
change the DSNB in the same way in both cases, i.e., Salpeter-like and
variable IMF.

Making use of the supernova rate and the neutrino spectrum, we can calculate
the DSNB flux $\Phi$ as a function of neutrino energy $E$. Specifically,

$\begin{split}\Phi(E)=c&\int_{0}^{z_{\max}}\frac{\mathrm{d}z}{H(z)}\\\
&\times\int\chi(L)L\frac{dN}{d\log
L}\left[\int_{8M_{\odot}}^{M_{\max}}\frac{\mathrm{d}n}{\mathrm{d}E^{\prime}}\frac{\xi(M)dM}{\overline{M}}\right]d\log
L\,,\end{split}$ (11)

where $\frac{dn}{dE}$121212 Note that although Eq. (11) is true for all
flavors, as mentioned in Appendix A we will focus solely on electron-
antineutrinos since they have the best detection prospects.  is the neutrino
spectrum, $E^{\prime}=E(1+z)$ is the source energy necessary at a redshift of
$z$ to be observed with energy $E$, and
$\overline{M}=\int_{M_{\min}}^{M_{\max}}M\xi(M)dM$ is the average mass of
newly formed stars. The integral $\int\chi(L)L\frac{dN}{d\log L}d\log L$ is
equivalent to $R_{\rm SF}$ if the term in brackets does not depend on
luminosity. While this condition is met for the Salpeter-like IMF, when we
consider a varying IMF, the IMF depends on luminosity, so the factor in
brackets picks up a dependence on luminosity. In addition,
$H(z)=H_{0}(\Omega_{m,0}(1+z)^{3}+(1-\Omega_{m,0}))^{1/2}$ is the Hubble
expansion parameter at $z$, where we assume a flat $\Lambda$CDM universe
dominated by matter and $\Lambda$, with $\Omega_{m,0}=0.308$ and
$H_{0}=67.8\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ (Ade et al., 2016).

Using the CCSNRD calculated from both a Salpeter-like and varying IMF we find
the DSNB fluxes shown in Fig. 9. We emphasize here that we have only
considered a single benchmark for each IMF model, but in reality there are a
number of other uncertainties which would create a band of possible DSNB
realizations. We do not directly consider these uncertainties in the modeling
of the DSNB (i.e., in Fig. 10), although below we account for a systematic
uncertainty when evaluating the discriminability of the two IMF models. For
reference, we also show a hatched region which shows the approximate parameter
space currently ruled out by SK observations, as well as the primary signal
region for SK and future detectors (El Hedri et al., 2021). Perhaps the most
notable result is the similarity between the predictions from using the
Salpeter-like and varying IMFs, especially at energies greater than
approximately 20 MeV. While this will likely make it difficult to distinguish
between the two scenarios once the DSNB is detected, it clearly demonstrates
that the DSNB is robust to significant changes to the IMF.

Figure 9: Diffuse supernova neutrino background flux: (Left Panel) We compare
the predicted DSNB $\bar{\nu}_{e}$ flux, assuming either a Salpeter-like
(green dashed) or varying IMF (blue solid) with the region probed by SK
(hatched) (Abe et al., 2021). The red shaded region indicates the range of
energies that could be observable by SK with the addition of gadolinium. The
different IMFs might have an impact on the observed DSNB flux only at low
energies, and it would be difficult to distinguish this difference against
background neutrino sources. (Right Panel) We show the DSNB broken down into
contributions from different redshift bins for the varying IMF. Because the
emitted neutrinos are redshifted, distant supernovae only contribute
significantly at low energies.

At lower energies (below approximately $20$ MeV), we predict the DSNB flux to
be slightly lower when assuming a varying IMF than when assuming a Salpeter-
like IMF. While the IMFs we consider give similar predictions for the CCSNRD
at low redshift, they begin to disagree at higher redshifts. The difference in
the DSNB flux that we see here ultimately arises from this difference in
supernova rates. This is only noticeable at low neutrino energies because
distant supernovae only contribute significantly to the low energy spectrum.
At higher energies, because of the redshifting of the neutrinos as they
propagate, the source energy $E^{\prime}=E(1+z)$ can be significantly higher
for distant supernovae than for nearby ones. As a result, the contribution to
the DSNB flux from distant supernovae comes from a higher energy portion of
the spectrum, which is exponentially suppressed. This effect is illustrated in
the right panel of Fig. 9 where we show the contribution to the DSNB in
different redshift bins. Therefore, while a varying IMF is likely
indistinguishable from a Salpeter-like IMF at high energies, sufficiently
precise measurements of lower energy neutrinos may allow some degree of
distinction between the two models.

Varying the IMF is not the only source of uncertainty expected to appear in
the low-energy DSNB region (around $20$ MeV). Variations to the star formation
histories, for example, could yield comparable differences between DSNB
predictions (Singh & Rentala, 2021; Kresse et al., 2021), which will be
degenerate with differences due to variations of the IMF. Furthermore,
additional uncertainties in the DSNB flux which may appear, independent of the
choice of IMF, include the unknown fraction of high-mass stars that evolve to
black holes and the neutrino spectra emitted during this evolution (Lunardini,
2009; Horiuchi et al., 2018; Kresse et al., 2021), the precise normalization
of the CCSN rate (Horiuchi et al., 2011; Mathews et al., 2014), the evolution
of neutrino flavors in the dense medium encountered during supernovae (Duan et
al., 2010; Chakraborty et al., 2016; Tamborra & Shalgar, 2021), and any
possible stellar binary interactions (Horiuchi et al., 2021). To partially
account for these uncertainties in the following subsection 3.5.2, we include
a systematic uncertainty of $50\%$ in our calculations of the discriminating
power of the detectors to the Varying IMF.

####  3.5.2 Expected Sensitivities of Super-Kamiokande and Hyper-Kamiokande

Figure 10:  Diffuse supernova neutrino background rate: $\bar{\nu}_{e}$ DSNB
event rates in HK enriched with gadolinium detectors for 10 yrs of data
taking. The sum of the $\bar{\nu}_{e}$ DSNB event rate plus background rate
for the Varying IMF (Salpeter-like) is plotted with solid blue (dashed green)
line. The backgrounds rates are depicted as grey regions, and the error bars
reflect the $\pm 1\sigma$ statistic uncertainties. As discussed in the main
text, while SK cannot distinguish the two investigated IMF scenarios, HK might
present a low significance hint towards a particular scenario. Note that we do
not show other astrophysical uncertainties (e.g., errors on the SFR and
uncertainty in the neutrino flux modeling), but partially account for this
with a systematic uncertainty (see main text for details).

Although no detection of the DSNB has yet been made, the strongest constraints
come from the SK experiment (Bays et al., 2012; Abe et al., 2021). In 2019,
upgrades to SK began which allowed for the introduction of gadolinium into the
SK tank by 2021. The gadolinium doping will make the detection of electron
antineutrinos significantly easier, which will subsequently improve our
ability to detect neutrinos from the DSNB (Beacom & Vagins, 2004). As a
result, it is expected that a positive measurement of the DSNB will be
observed in the near future (Li et al., 2022).

Figure 10 shows the predicted accumulated DSNB flux after 10 years of
operation of HK (3740 kton yr exposure) compared to its respective neutrino
backgrounds, where we assume a concentration of 0.1% GdCl3 in water. The blue
solid (green dashed) line depicts the combined neutrino flux from both the
DSNB and background sources, where we calculate the DSNB flux using a varying
IMF (Salpeter-like IMF). Sources of background neutrinos that we consider
include atmospheric charged-current events, invisible muons, 9Li spallation,
and reactor antineutrinos (Abe et al., 2018, 2021).

We can use a simple $\Delta\chi^{2}$ Pearson test to estimate the detection
prospects of distinguishing between the varying IMF and Salpeter-like IMF. In
both SK (225 kton yr exposure with 0.1% GdCl3) and HK, the two IMFs are not
distinguishable at the $3\sigma$ level, even after 10 years of data
collection. Furthermore, in SK, the two models remain indistinguishable at the
$1\sigma$ level. However, in HK, the varying and Salpeter-like IMFs are
distinguishable at the $1.3\sigma$ level. This marks an upper bound to the
distinguishability, as we have not introduced other sources of uncertainty.
Including a systematic uncertainty for the background of 20% and for the DSNB
models of 50% reduces the difference between the two models to $1.1\sigma$
(note that these additional systematics are not shown in Fig. 10). Once SK or
HK detects the DSNB, it may be possible to generate an improved statistic,
using an unbinned maximum likelihood ratio test with a parameterized family of
models that include the Salpeter-like and varying IMFs, which could allow for
a better ability to distinguish between these two models, assuming all other
uncertainties become negligible in comparison.

##  4 Conclusion and Outlook

In this paper, we explored whether one can find evidence for a non-universal
IMF in five astrophysical observables that arise from integrating over
cosmological scales. Throughout, we have seen that these observables show
small but non-zero differences when assuming the non-universal IMF (described
in Sec. 2) versus a universal Salpeter-like IMF. The differences are typically
too small to distinguish in currently existing data, but as detectors improve,
there are a variety of signals which may offer practical ways to study the
IMF. In particular, we find that studying the star formation rate (SFR) and
the core-collapse rate at high redshifts ($z\gtrsim 0.5$) may offer the
greatest distinguishability between the different IMF models.

At redshifts greater than approximately 0.5, the SFR predicted assuming a
varying IMF is lower than the corresponding prediction from a Salpeter-like
IMF by up to a factor of $\sim 3$ (see Fig. 4). Because observing the SFR
requires a calculation that depends strongly on the assumed IMF, the
predictions, though significantly different, can both still be consistent with
current observations. However, due to the SFR’s prominent role in, for
example, modeling star formation histories and cosmological simulations, it
would be interesting to see whether an indirect test of the SFR could favor
one IMF model over the other. For example, the binary black hole (BBH) merger
rate (Fishbach et al., 2018; van Son et al., 2021) acts as an independent
probe of the SFR. LIGO and Virgo, at design sensitivity, are expected to
observe BBH mergers up to redshifts of around one. On the other hand, 3rd
generation telescopes may reach $z\sim 15$ and thus independently measure the
star formation rate density (SFRD) to a few percent according to Vitale et al.
(2019). We leave a more careful examination of this method to future work.

Similarly to the SFR, above $z\gtrsim 0.5$ the core-collapse rate can differ
by a factor of $\sim 2$ between the two IMF models (see Fig. 6). Current
observations of core-collapse rates are still too poor at high redshifts to
distinguish between these models. However, next-generation telescopes such as
James Webb Space Telescope (Regős & Vinkó, 2019), Roman Space Telescope
(Koekemoer et al., 2019), Vera Rubin Telescope (through the LSST) (Ivezić et
al., 2019), and EUCLID (Laureijs et al., 2011), may provide significantly
better rate estimates at these high redshifts. An additional subtlety here is
that a varying IMF will produce a larger fraction of black hole collapses (see
Fig. 7) which may be less luminous. A careful treatment of the observational
efficiency of any core-collapse rate measurement is therefore required to
distinguish the two IMF models.

The stellar IMF plays a fundamental role throughout astrophysics and many
unsolved questions require an accurate model of the IMF to address. It is
therefore vital that new methods are developed to decipher its dependence on
the local environment. Here we have examined a few indirect methods. Future
work should more carefully examine the most promising of these scenarios with
a meticulous treatment of their associated uncertainties, including studying
whether the impact of different IMFs can be distinguished from potentially
degenerate or partially degenerate sources of uncertainty. Moreover, combining
the different probes presented here may lead to significantly tighter
constraints on possible IMF variations. Through this, we hope that indirect
probes may provide additional evidence towards distinguishing a varying IMF.

## Acknowledgments

We thank Andrew Hopkins for a careful reading of the manuscript and helpful
discussions. T.E. and K.F. acknowledge support by the Vetenskapsrådet (Swedish
Research Council) through contract No. 638-2013-8993 and the Oskar Klein
Centre for Cosmoparticle Physics at Stockholm University. T.E. was supported
by the NWO through the VIDI research program “Probing the Genesis of Dark
Matter” (680-47-5). J.Z. and K.F. are grateful for support from the Jeff &
Gail Kodosky Endowed Chair held by K. F. at the University of Texas. J.Z. and
K.F. acknowledge funding from the U.S. Department of Energy, Office of
Science, Office of High Energy Physics program under Award Number DE-
SC0022021. A.M.S. acknowledges the support form the US National Science
Foundation (Grant No. PHY-2020275). In Copenhagen, this work was supported by
the Villum Foundation (Projects Nos. 13164 and 37358), the Danmarks Frie
Forskningsfonds (Project No. 8049-00038B), the Deutsche Forschungsgemeinschaft
through Sonder- forschungbereich SFB 1258 “Neutrinos and Dark Matter in Astro-
and Particle Physics” (NDM). The work of SH is supported by the US Department
of Energy under the award number DE-SC0020262, NSF Grant numbers AST-1908960
and PHY-1914409, and JSPS KAKENHI Grant Number JP22K03630. SA was supported by
MEXT KAKENHI Grant Numbers, JP20H05850 and JP20H05861. This work was supported
by World Premier International Research Center Initiative (WPI Initiative),
MEXT, Japan. A.M.S. and S.H. would like to thank Kavli Institute for
Theoretical Physics for the hospitality during this work. This research was
supported in part by the National Science Foundation under Grant No. NSF
PHY-1748958.

## References

  * Abdollahi et al. (2018) Abdollahi S., et al., 2018, Science, 362, 1031
  * Abe et al. (2018) Abe K., et al., 2018, ArXiv e-print
  * Abe et al. (2021) Abe K., et al., 2021, Phys. Rev. D, 104, 122002
  * Abe et al. (2022) Abe S., et al., 2022, Astrophys. J., 925, 14
  * Adams et al. (2017a) Adams S., Kochanek C., Gerke J., Stanek K., Dai X., 2017a, Mon. Not. Roy. Astron. Soc., 468, 4968
  * Adams et al. (2017b) Adams S., Kochanek C., Gerke J., Stanek K., 2017b, Mon. Not. Roy. Astron. Soc., 469, 1445
  * Ade et al. (2016) Ade P. A. R., et al., 2016, Astronomy & Astrophysics, 594, A13
  * An et al. (2016) An F., et al., 2016, J. Phys. G, 43, 030401
  * Ando (2003) Ando S., 2003, Phys. Lett. B, 570, 11
  * Ashida & Nakazato (2022) Ashida Y., Nakazato K., 2022, ArXiv e-print
  * Baker et al. (2007) Baker J., Goldberg H., Perez G., Sarcevic I., 2007, Phys. Rev. D, 76, 063004
  * Baldry & Glazebrook (2003) Baldry I. K., Glazebrook K., 2003, Astrophysical Journal, 593, 258
  * Bays et al. (2012) Bays K., et al., 2012, Phys. Rev. D, 85, 052007
  * Beacom (2010) Beacom J. F., 2010, Annual Review of Nuclear and Particle Science, 60, 439–462
  * Beacom & Vagins (2004) Beacom J. F., Vagins M. R., 2004, Phys. Rev. Lett., 93, 171101
  * Beacom et al. (2017) Beacom J. F., et al., 2017, Chin. Phys. C, 41, 023002
  * Bethe & Wilson (1985) Bethe H. A., Wilson J. R., 1985, Astrophysical Journal, 295, 14
  * Bethe et al. (1979a) Bethe H. A., Brown G. E., Applegate J., Lattimer J. M., 1979a, Nucl. Phys. A, 324, 487
  * Bethe et al. (1979b) Bethe H. A., Brown G. E., Applegate J., Lattimer J. M., 1979b, Nuclear Physics A, 324, 487
  * Bisnovatyi-Kogan & Seidov (1984) Bisnovatyi-Kogan G. S., Seidov Z. F., 1984, Annals of the New York Academy of Sciences, 422, 319
  * Biteau & Williams (2015) Biteau J., Williams D. A., 2015, The Astrophysical Journal, 812, 60
  * Branch & Wheeler (2017) Branch D., Wheeler J. C., 2017, Supernova Explosions. Springer
  * Burrows & Vartanyan (2021) Burrows A., Vartanyan D., 2021, Nature, 589, 29
  * Byrne & Fraser (2022) Byrne R., Fraser M., 2022, ArXiv e-print
  * Calzetti (2013) Calzetti D., 2013, Star Formation Rate Indicators. p. 419
  * Cappellari et al. (2012) Cappellari M., et al., 2012, Nature, 484, 485
  * Cappellaro et al. (2015) Cappellaro E., et al., 2015, Astronomy & Astrophysics, 584, A62
  * Cardelli et al. (1989) Cardelli J. A., Clayton G. C., Mathis J. S., 1989, Astrophysical Journal, 345, 245
  * Chabrier (2003) Chabrier G., 2003, Publ. Astron. Soc. Pac., 115, 763
  * Chabrier et al. (2014) Chabrier G., Hennebelle P., Charlot S., 2014, The Astrophysical Journal, 796, 75
  * Chakraborty et al. (2016) Chakraborty S., Hansen R., Izaguirre I., Raffelt G., 2016, Nucl. Phys. B, 908, 366
  * Chandrasekhar (1931) Chandrasekhar S., 1931, Astrophys. J., 74, 81
  * Chomiuk & Povich (2011) Chomiuk L., Povich M. S., 2011, The Astronomical Journal, 142, 197
  * Colgate & White (1966) Colgate S. A., White R. H., 1966, Astrophysical Journal, 143, 626
  * Couch (2017) Couch S. M., 2017, Philosophical Transactions of the Royal Society of London Series A, 375, 20160271
  * Creque-Sarbinowski et al. (2021) Creque-Sarbinowski C., Hyde J., Kamionkowski M., 2021, Phys. Rev. D, 103, 023527
  * Dahlen et al. (2012) Dahlen T., Strolger L.-G., Riess A. G., Mattila S., Kankare E., Mobasher B., 2012, The Astrophysical Journal, 757, 70
  * Das & Sen (2021) Das A., Sen M., 2021, Phys. Rev. D, 104, 075029
  * Davies & Beasor (2020) Davies B., Beasor E., 2020, Mon. Not. Roy. Astron. Soc., 496, L142
  * Driver et al. (2011) Driver S. P., et al., 2011, Mon. Not. Roy. Astron. Soc., 413, 971
  * Driver et al. (2018) Driver S. P., et al., 2018, Mon. Not. Roy. Astron. Soc., 475, 2891
  * Duan et al. (2010) Duan H., Fuller G. M., Qian Y.-Z., 2010, Ann. Rev. Nucl. Part. Sci., 60, 569
  * El Hedri et al. (2021) El Hedri S., Ashida Y., Giampaolo A., 2021, in 37th International Cosmic Ray Conference. Berlin, Germany, p. 1139, doi:10.22323/1.395.1139, https://hal.archives-ouvertes.fr/hal-03373391
  * Ertl et al. (2016) Ertl T., Janka H. T., Woosley S., Sukhbold T., Ugliano M., 2016, Astrophys. J., 818, 124
  * Fardal et al. (2007) Fardal M. A., Katz N., Weinberg D. H., Davé R., 2007, Mon. Not. Roy. Astron. Soc., 379, 985
  * Farzan & Palomares-Ruiz (2014) Farzan Y., Palomares-Ruiz S., 2014, JCAP, 06, 014
  * Ferreras et al. (2012) Ferreras I., Barbera F. L., Rosa I. G. d. l., Vazdekis A., Carvalho R. R. d., Falcón-Barroso J., Ricciardelli E., 2012, Monthly Notices of the Royal Astronomical Society: Letters, 429, L15–L19
  * Ferré-Mateu et al. (2013) Ferré-Mateu A., Vazdekis A., de la Rosa I. G., 2013, Monthly Notices of the Royal Astronomical Society, 431, 440–454
  * Fioc & Rocca-Volmerange (2019) Fioc M., Rocca-Volmerange B., 2019, Astronomy & Astrophysics, 623, A143
  * Fishbach et al. (2018) Fishbach M., Holz D. E., Farr W. M., 2018, Astrophys. J. Lett., 863, L41
  * Fogli et al. (2004) Fogli G. L., Lisi E., Mirizzi A., Montanino D., 2004, Phys. Rev. D, 70, 013001
  * Fontanot et al. (2017) Fontanot F., De Lucia G., Hirschmann M., Bruzual G., Charlot S., Zibetti S., 2017, Mon. Not. Roy. Astron. Soc., 464, 3812
  * Fontanot et al. (2018) Fontanot F., Lucia G. D., Xie L., Hirschmann M., Bruzual G., Charlot S., 2018, Monthly Notices of the Royal Astronomical Society
  * Freese et al. (2016) Freese K., Rindler-Daller T., Spolyar D., Valluri M., 2016, Reports on Progress in Physics, 79
  * Fuller (1982) Fuller G. M., 1982, Astrophys. J., 252, 741
  * Garching Core-Collapse Supernova Archive (2022) Garching Core-Collapse Supernova Archive 2022, , https://wwwmpa.mpa-garching.mpg.de/ccsnarchive/
  * Geha (2013) Geha M., 2013, A Non-Universal Initial Mass Function in the Ultra-Faint Galaxy Coma Berenices, HST Proposal ID 13449. Cycle 21
  * Gerke et al. (2015) Gerke J., Kochanek C., Stanek K., 2015, Mon. Not. Roy. Astron. Soc., 450, 3289
  * Goldberg et al. (2006) Goldberg H., Perez G., Sarcevic I., 2006, JHEP, 11, 023
  * Gruppioni et al. (2013) Gruppioni C., et al., 2013, Mon. Not. Roy. Astron. Soc., 432, 23
  * Gunawardhana et al. (2011) Gunawardhana M. L. P., et al., 2011, Monthly Notices of the Royal Astronomical Society, 415, 1647
  * Harayama et al. (2008) Harayama Y., Eisenhauer F., Martins F., 2008, The Astrophysical Journal, 675, 1319–1342
  * Heger et al. (2003) Heger A., Fryer C. L., Woosley S. E., Langer N., Hartmann D. H., 2003, Astrophys. J., 591, 288
  * Hopkins (2018) Hopkins A. M., 2018, Publications of the Astronomical Society of Australia, 35
  * Hopkins & Beacom (2006) Hopkins A. M., Beacom J. F., 2006, Astrophysical Journal, 651, 142
  * Horiuchi et al. (2009) Horiuchi S., Beacom J. F., Dwek E., 2009, Phys. Rev. D, 79, 083013
  * Horiuchi et al. (2011) Horiuchi S., Beacom J. F., Kochanek C. S., Prieto J. L., Stanek K. Z., Thompson T. A., 2011, Astrophys. J., 738, 154
  * Horiuchi et al. (2013) Horiuchi S., Beacom J. F., Bothwell M. S., Thompson T. A., 2013, Astrophys. J., 769, 113
  * Horiuchi et al. (2014) Horiuchi S., Nakamura K., Takiwaki T., Kotake K., Tanaka M., 2014, Mon. Not. Roy. Astron. Soc., 445, L99
  * Horiuchi et al. (2018) Horiuchi S., Sumiyoshi K., Nakamura K., Fischer T., Summa A., Takiwaki T., Janka H.-T., Kotake K., 2018, Mon. Not. Roy. Astron. Soc., 475, 1363
  * Horiuchi et al. (2021) Horiuchi S., Kinugawa T., Takiwaki T., Takahashi K., Kotake K., 2021, Phys. Rev. D, 103, 043003
  * Ivezić et al. (2019) Ivezić Ž., et al., 2019, Astrophysical Journal, 873, 111
  * Janka (2017) Janka H.-T., 2017, Neutrino-Driven Explosions. Springer International Publishing, Cham, pp 1095–1150, doi:10.1007/978-3-319-21846-5_109, https://doi.org/10.1007/978-3-319-21846-5_109
  * Jeong et al. (2018) Jeong Y. S., Palomares-Ruiz S., Reno M. H., Sarcevic I., 2018, JCAP, 06, 019
  * Keehn & Lunardini (2012) Keehn J. G., Lunardini C., 2012, Phys. Rev. D, 85, 043011
  * Keil (2003) Keil M. T., 2003, Other thesis (arXiv:astro-ph/0308228)
  * Keil et al. (2003) Keil M. T., Raffelt G. G., Janka H.-T., 2003, Astrophys. J., 590, 971
  * Kennedy et al. (2013) Kennedy G. M., Wyatt M. C., Kalas P., Duchene G., Sibthorpe B., Lestrade J.-F., Matthews B. C., Greaves J., 2013, Monthly Notices of the Royal Astronomical Society: Letters, 438, L96
  * Kennicutt (1998) Kennicutt Robert C. J., 1998, Annual Review of Astronomy and Astrophysics, 36, 189
  * Kennicutt & Evans (2012) Kennicutt R. C., Evans N. J., 2012, Annual Review of Astronomy and Astrophysics, 50, 531–608
  * Kobayashi (2010) Kobayashi C., 2010, in Debattista V. P., Popescu C. C., eds, American Institute of Physics Conference Series Vol. 1240, Hunting for the Dark: the Hidden Side of Galaxy Formation. pp 123–126 (arXiv:1002.4475), doi:10.1063/1.3458465
  * Kochanek et al. (2008) Kochanek C., Beacom J., Kistler M., Prieto J., Stanek K., Thompson T., Yuksel H., 2008, Astrophys. J., 684, 1336
  * Koekemoer et al. (2019) Koekemoer A. M., et al., 2019, An Ultra Deep Field survey with WFIRST (arXiv:1903.06154)
  * Krauss et al. (1984) Krauss L. M., Glashow S. L., Schramm D. N., 1984, Nature, 310, 191
  * Kresse et al. (2021) Kresse D., Ertl T., Janka H.-T., 2021, Astrophys. J., 909, 169
  * Kroupa (2001) Kroupa P., 2001, Mon. Not. Roy. Astron. Soc., 322, 231
  * La Barbera et al. (2019) La Barbera F., et al., 2019, Monthly Notices of the Royal Astronomical Society, 489, 4090–4110
  * Larson (1998) Larson R. B., 1998, Mon. Not. Roy. Astron. Soc., 301, 569
  * Lattimer & Swesty (1991) Lattimer J. M., Swesty F. D., 1991, Nucl. Phys., A535, 331
  * Laureijs et al. (2011) Laureijs R., et al., 2011, Euclid Definition Study Report (arXiv:1110.3193)
  * Leitherer et al. (1999) Leitherer C., et al., 1999, Astrophys. J. Suppl., 123, 3
  * Li & Draine (2001) Li A., Draine B. T., 2001, The Astrophysical Journal, 554, 778
  * Li et al. (2022) Li Y.-F., Vagins M., Wurm M., 2022, Universe, 8, 181
  * Libanov & Sharofeev (2022) Libanov A., Sharofeev A., 2022, ArXiv e-print
  * Lien et al. (2010) Lien A., Fields B. D., Beacom J. F., 2010, Phys. Rev., D81, 083001
  * Lunardini (2009) Lunardini C., 2009, Phys. Rev. Lett., 102, 231101
  * Lunardini (2016) Lunardini C., 2016, Astropart. Phys., 79, 49
  * Lunardini & Tamborra (2012) Lunardini C., Tamborra I., 2012, JCAP, 07, 012
  * Madau & Dickinson (2014) Madau P., Dickinson M., 2014, Ann. Rev. Astron. Astrophys., 52, 415
  * Magnelli et al. (2011) Magnelli B., Elbaz D., Chary R. R., Dickinson M., Le Borgne D., Frayer D. T., Willmer C. N. A., 2011, Astronomy & Astrophysics, 528, A35
  * Magnelli et al. (2013) Magnelli B., et al., 2013, Astronomy & Astrophysics, 553, A132
  * Malek et al. (2003) Malek M., et al., 2003, Physical Review Letters, 90
  * Maoz et al. (2014) Maoz D., Mannucci F., Nelemans G., 2014, Annual Review of Astronomy and Astrophysics, 52, 107–170
  * Mathews et al. (2014) Mathews G. J., Hidaka J., Kajino T., Suzuki J., 2014, Astrophys. J., 790, 115
  * Mattila et al. (2012) Mattila S., et al., 2012, The Astrophysical Journal, 756, 111
  * Mazzali et al. (2007) Mazzali P. A., R opke F. K., Benetti S., Hillebrandt W., 2007, Science, 315, 825–828
  * Mirizzi et al. (2016) Mirizzi A., Tamborra I., Janka H.-T., Saviano N., Scholberg K., Bollig R., Hudepohl L., Chakraborty S., 2016, Riv. Nuovo Cim., 39, 1
  * Møller et al. (2018) Møller K., Suliga A. M., Tamborra I., Denton P. B., 2018, JCAP, 1805, 066
  * Nakazato (2013) Nakazato K., 2013, Phys. Rev. D, 88, 083012
  * Nakazato et al. (2015) Nakazato K., Mochida E., Niino Y., Suzuki H., 2015, Astrophys. J., 804, 75
  * Neustadt et al. (2021) Neustadt J. M. M., Kochanek C. S., Stanek K. Z., Basinger C. M., Jayasinghe T., Garling C. T., Adams S. M., Gerke J., 2021, Mon. Not. Roy. Astron. Soc., 508, 516
  * Offner et al. (2014) Offner S. S. R., Clark P. C., Hennebelle P., Bastian N., Bate M. R., Hopkins P. F., Moreaux E., Whitworth A. P., 2014, Protostars and Planets VI
  * Padoan et al. (1997) Padoan P., Nordlund A., Jones B. J. T., 1997, Mon. Not. Roy. Astron. Soc., 288, 145
  * Perrett et al. (2012) Perrett K., et al., 2012, The Astronomical Journal, 144, 59
  * Petrushevska et al. (2016) Petrushevska T., et al., 2016, Astron. Astrophys., 594, A54
  * Priya & Lunardini (2017) Priya A., Lunardini C., 2017, JCAP, 11, 031
  * Razzaque et al. (2009) Razzaque S., Dermer C. D., Finke J. D., 2009, The Astrophysical Journal, 697, 483–492
  * Regős & Vinkó (2019) Regős E., Vinkó J., 2019, The Astrophysical Journal, 874, 158
  * Rose et al. (2021) Rose B. M., et al., 2021, A Reference Survey for Supernova Cosmology with the Nancy Grace Roman Space Telescope, doi:10.48550/ARXIV.2111.03081, https://arxiv.org/abs/2111.03081
  * Salpeter (1955) Salpeter E. E., 1955, Astrophysical Journal, 121, 161
  * Sawatzki et al. (2021) Sawatzki J., Wurm M., Kresse D., 2021, Phys. Rev. D, 103, 023021
  * Sharda & Krumholz (2021) Sharda P., Krumholz M. R., 2021, Monthly Notices of the Royal Astronomical Society
  * Singh & Rentala (2021) Singh R., Rentala V., 2021, JCAP, 08, 019
  * Smith (2014) Smith N., 2014, Ann. Rev. Astron. Astrophys., 52, 487
  * Strolger et al. (2015) Strolger L.-G., et al., 2015, Astrophys. J., 813, 93
  * Strolger et al. (2020) Strolger L.-G., Rodney S. A., Pacifici C., Narayan G., Graur O., 2020, The Astrophysical Journal, 890, 140
  * Sukhbold et al. (2016) Sukhbold T., Ertl T., Woosley S., Brown J. M., Janka H. T., 2016, Astrophys. J., 821, 38
  * Suliga et al. (2022) Suliga A. M., Beacom J. F., Tamborra I., 2022, Phys. Rev. D, 105, 043008
  * Tabrizi & Horiuchi (2021) Tabrizi Z., Horiuchi S., 2021, JCAP, 05, 011
  * Tamborra & Shalgar (2021) Tamborra I., Shalgar S., 2021, Ann. Rev. Nucl. Part. Sci., 71, 165
  * Tamborra et al. (2012) Tamborra I., Muller B., Hudepohl L., Janka H.-T., Raffelt G., 2012, Phys. Rev. D, 86, 125031
  * Turatto (2003) Turatto M., 2003, Lecture Notes in Physics, p. 21–36
  * Vitale et al. (2019) Vitale S., Farr W. M., Ng K. K. Y., Rodriguez C. L., 2019, The Astrophysical Journal, 886, L1
  * Weingartner & Draine (2001) Weingartner J. C., Draine B. T., 2001, The Astrophysical Journal, 548, 296
  * Wilkins et al. (2008a) Wilkins S. M., Trentham N., Hopkins A. M., 2008a, MNRAS, 385, 687
  * Wilkins et al. (2008b) Wilkins S. M., Hopkins A. M., Trentham N., Tojeiro R., 2008b, MNRAS, 391, 363
  * Wilkins et al. (2019) Wilkins S. M., Lovell C. C., Stanway E. R., 2019, Monthly Notices of the Royal Astronomical Society, 490, 5359–5365
  * Wilson et al. (1986) Wilson J. R., Mayle R., Woosley S. E., Weaver T., 1986, Annals N. Y. Acad. Sci., 470, 267
  * Zhang et al. (2015) Zhang H., et al., 2015, Astropart. Phys., 60, 41
  * Zubko et al. (2004) Zubko V., Dwek E., Arendt R. G., 2004, The Astrophysical Journal Supplement Series, 152, 211
  * de Gouvêa et al. (2022) de Gouvêa A., Martinez-Soler I., Perez-Gonzalez Y. F., Sen M., 2022, ArXiv e-print
  * de Gouvêa et al. (2020) de Gouvêa A., Martinez-Soler I., Perez-Gonzalez Y. F., Sen M., 2020, Physical Review D, 102
  * van Dokkum & Conroy (2010) van Dokkum P. G., Conroy C., 2010, Nature, 468, 940
  * van Son et al. (2021) van Son L. A. C., et al., 2021, ArXiv e-print

##  Appendix A Supernova Neutrino Spectra

In order to estimate the DSNB flux, we model the CCSN population by relying on
the outputs of one-dimensional, hydrodynamic supernova simulations with
Boltzmann neutrino transport from the Garching group (Garching Core-Collapse
Supernova Archive, 2022). Following Møller et al. (2018), three reference CCSN
models are used to account for the variations in neutrino emission depending
on the mass and fate of the progenitor star. For CCSNe leading to the
formation of a neutron star as the compact object remnant, we use models with
initial masses of $9.6$ and $27\,{\rm M}_{\odot}$, whereas for stellar
collapses leading to the formation of black holes, we use the 40 ${\rm
M}_{\odot}$ “low” mass accretion rate model (Mirizzi et al., 2016). In all
three models, the nuclear equation of state is assumed to be that of Lattimer
and Swesty, with a nuclear incompressibility modulus $K=220$ MeV (LS220 EoS)
(Lattimer & Swesty, 1991).

The related neutrino energy distributions are well described by a pinched
Fermi-Dirac distribution (Keil et al., 2003; Keil, 2003; Tamborra et al.,
2012):

$\left(\frac{\mathrm{d}n}{\mathrm{d}E}\right)_{\bar{\nu}_{e}}=E_{\nu}^{\rm
tot}\frac{\left(1+a\right)^{1+a}}{\Gamma(1+a)}\frac{E^{a}}{\langle
E_{\nu}\rangle^{2+a}}e^{\left[-\left(1+a\right)\frac{E}{\langle
E_{\nu}\rangle}\right]}\,.$ (12)

Here, the parameters $E_{\nu}^{\rm tot}$, $a$, and $\langle E_{\nu}\rangle$
represent the total energy emitted in (anti-electron) neutrinos; a parameter
that describes the spectral shape, related to the pinching parameter; and the
average energy of the emitted neutrinos, respectively.

While the mean energy, luminosity (energy emitted per time), and pinching
parameter are time-dependent quantities, we are interested in the time-
integrated neutrino energy distributions. Therefore, we report the time-
integrated characteristic quantities in Table 2 for all three models adopted
to model the DSNB and show these energy spectra in Fig. 11. Note that these
simulations do not include the effects of neutrino flavor mixing, which we do
not take into account throughout this work.

| $E_{\nu}^{\rm tot}\,[\mathrm{MeV}]$ | $a$ | $\langle E_{\nu}\rangle\,[\mathrm{MeV}]$
---|---|---|---
Neutron Star ($10\mathrm{\,M}_{\odot}$) | $1.890\times 10^{58}$ | 2.355 | 12.620
Neutron Star ($27\mathrm{\,M}_{\odot}$) | $3.435\times 10^{58}$ | 2.307 | 13.856
Black Hole ($40\mathrm{\,M}_{\odot}$) | $4.426\times 10^{58}$ | 2.083 | 17.943
Table 2: Best fit parameters for the pinched Fermi-Dirac distribution used to
describe the numerically generated time-integrated neutrino spectra. Figure
11: Electron antineutrino energy spectra at the source: Here we show the time-
integrated anti-electron neutrino emission spectra of the three reference CCSN
models that we use in this work. The blue solid and orange dashed curves
correspond to supernovae that result in neutron stars, whose stellar
progenitors have $9.7$ and $27\,{\rm M}_{\odot}$ masses, respectively. The
green dotted curve represents a supernova that results in the formation of a
black hole and whose stellar progenitor has a mass of $40\,{\rm M}_{\odot}$.
Note that these are the spectra at the source; when observed, these spectra
will be redshifted by a factor $1+z$, so that supernovae at high redshift only
have non-negligible contributions to the observed spectra at low energies.