Review Shin’ichiro AndolabelA,labelBNick EkangerlabelC Shunsaku
HoriuchilabelC,labelB Yusuke KoshiolabelD,labelB [labelA]GRAPPA Institute,
University of Amsterdam, 1098 XH Amsterdam, The Netherlands [labelB]Kavli
Institute for the Physics and Mathematics of the Universe (WPI), University of
Tokyo, Chiba 277-8583, Japan [labelC]Center for Neutrino Physics, Department
of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA [labelD]Department
of Physics, Okayama University, Okayama, Okayama 700-8530, Japan Shin’ichiro
Ando<EMAIL_ADDRESS>
# Diffuse neutrino background from past core-collapse supernovae

###### Abstract

Core-collapse supernovae are among the most powerful explosions in the
universe, emitting thermal neutrinos that carry away the majority of the
gravitational binding energy released. These neutrinos create a diffuse
supernova neutrino background (DSNB), one of the largest energy budgets among
all radiation backgrounds. Detecting the DSNB is a crucial goal of modern
high-energy astrophysics and particle physics, providing valuable insights in
both core-collapse modeling, neutrino physics, and cosmic supernova rate
history. In this review, we discuss the key ingredients of DSNB calculation
and what we can learn from future detections, including black-hole formation
and non-standard neutrino interactions. Additionally, we provide an overview
of the latest updates in neutrino experiments, which could lead to the
detection of the DSNB in the next decade. With the promise of this
breakthrough discovery on the horizon, the study of DSNB holds enormous
potential for advancing our understanding of the Universe.

###### keywords:

Supernova; neutrino

##  1 Introduction

It has been 35 years since a supernova exploded in the neighbourhood of our
Milky-Way galaxy. This supernova, SN 1987A, which happened in the Large
Magellanic Cloud 50 kpc away from the Earth, gave the first detected neutrino
signal from beyond the solar system. In total, 11, 8, and 5 neutrino events
were detected by Kamiokande-II[1], IMB[2], and Baksan[3] detectors,
respectively, enabling intensive discussions on the core-collapse mechanisms
and neutrino physics. The next Galactic core-collapse supernova will be a
spectacular event for not just neutrino telescopes and various traditional
astronomical observatories but also gravitational wave detectors; it will
quite possibly be the first multi-messenger transient where all three
messengers are detected. If this happens at the Galactic center, this will
yield thousands of neutrino events at the current generation of water
Cherenkov detectors such as Super-Kamiokande. Other detectors will also detect
hundreds of neutrino events, covering a wide energy range and different flavor
species altogether. For example IceCube, a Gton-scale neutrino telescope at
the south pole, and its planned successor, IceCube-Gen2, are sensitive to the
time evolution of neutrino luminosity very precisely, even though they do not
have sensitivity to measure the energy of each neutrino. However, since the
supernova rate is estimated to be a few per century[4], the probability of
having such an event over the coming decade is unfortunately small, around
10–30%.

Even in an unlucky situation where we do not have a Galactic supernova in the
near future, detector sensitivities have improved substantially such that we
are in a position to detect neutrinos from supernovae at cosmological
distances. Within the Hubble volume, the rate of core-collapse supernovae is
on the order of one per second, which compensates the reduction of flux from
each supernova. Furthermore, the detection of this Diffuse Supernova Neutrino
Background (DSNB) is just around the corner, especially after the gadolinium
upgrade of the Super-Kamiokande detector. By measuring the spectrum of the
DSNB, one can study various important physics of both the core-collapse
supernovae and neutrinos themselves. For example, the DSNB spectrum will
contain information on the cosmic history of supernova rate evolution,
neutrino generation mechanism, and the fraction of black holes that form as a
result of the core collapse (in which case there may not be a supernova
associated to it). It also provides an ideal setup for the longest baseline
low-energy neutrino experiments to test various new interactions in the
neutrino sector with unprecedented sensitivity.

All these physics ingredients are combined to predict the flux of the DSNB
spectrum. Since there is no time or directional information available for
measurement111Supernovae are stochastic events. However, since their frequency
of $\sim$1 Hz is way higher than frequency of the DSNB detection, the DSNB
events will look isotropic and diffuse., only the energy distribution (i.e.,
spectrum) is the observable for the DSNB. Therefore, there is a high degree of
degeneracy for these model ingredients. To enable detailed interpretations,
one must have sufficient statistics and also the most accurate theoretical
models. The former can be achieved with large-volume neutrino telescopes. A
prime example is the Super-Kamiokande experiment based in Japan, which
recently restarted taking neutrino data with Gadolinium to substantially
reduce the backgrounds. The latter relies on progress in supernova physics
mainly driven by numerical simulations of the core-collapse process and
neutrino physics that progressed substantially over the previous decades,
especially for the oscillation parameters, as well as astronomical observables
of supernova and associated quantities.

In this article, we review the studies of the DSNB by going over the latest
progress in the field (see also Refs [5, 6, 7] for earlier reviews). We will
also cover the state-of-the-art of the Super-Kamiokande detector and its
sensitivity to study supernova and neutrino physics.

##  2 Formalism

###  2.1 Brightness of the extragalactic neutrino sky

Before deriving the rigorous formula with which one can compute the DSNB flux,
we begin with a simple order-of-magnitude estimate. Each core-collapse
supernova releases the 99% of its gravitational binding energy on the order of
$E_{b}=3GM_{\rm NS}^{2}/(5R_{\rm NS})=3\times 10^{53}$ erg in the form of
thermal neutrinos. Here $M_{\rm NS}=1.4M_{\odot}$ and $R_{\rm NS}=10$ km are
the mass and radius of a newly born neutron star, respectively. The supernova
rate in the Milky Way is estimated to be $\sim 0.02$ yr-1. Combined with the
local number density of galaxies that is estimated to be $0.01$ Mpc-3, the
local supernova rate density is $\rho_{\rm SN}=2\times 10^{-4}$ yr-1 Mpc-3.
The Universe has kept injecting supernova neutrinos since the beginning of
star formation, and the time scale can be estimated as the Hubble time
$t_{H}\equiv H_{0}^{-1}$, where $H_{0}$ is the Hubble constant. By multiplying
the three we obtain the energy density of the DSNB as $\epsilon_{\rm
DSNB}=E_{b}\rho_{\rm SN}t_{H}\approx 3\times 10^{-14}$ erg cm-3. This is
likely a conservative estimate because the rate of supernovae is known to be
larger in the past, by up to an order of magnitude at the redshifts $z=1$–2.

The brightest radiation component in the Universe is the cosmic microwave
background (CMB) — the radiation from the Big Bang — whose energy density is
$\epsilon_{\rm CMB}=4\times 10^{-13}$ erg cm-3. The second brightest component
after the CMB is the extragalactic background light (EBL; i.e., emissions from
stars and those reprocessed via dust absorption and re-emission) covering the
infrared, optical, and ultraviolet wavebands and its energy density is
$\epsilon_{EBL}=(2$–$3)\times 10^{-14}$ erg cm-3, whereas other components at
different frequencies (X rays, gamma rays, etc.) are much more subdominant by
orders of magnitude. Therefore, the supernova neutrinos, even though we have
not seen them yet, make up of one of the brightest radiation components in the
Universe.

We can reach the same conclusion by another simple consideration. For each
galaxy, its time-averaged luminosity in supernova neutrinos is $L_{\rm
DSNB,gal}=E_{b}R_{\rm SN,gal}=2\times 10^{44}$ erg s-1, where $R_{\rm
SN,gal}=0.02$ yr-1 is the supernova rate per galaxy. On the other hand, the
typical luminosity of galaxies (in star light) is $L_{\star,{\rm
gal}}=10^{10}L_{\odot}=4\times 10^{43}$ erg s-1. Therefore, if we average over
a long time scale (such as Hubble time), then the Milky-Way Galaxy shines
brighter with supernova neutrinos than its optical photons emitted by stars by
a factor of a few. We must, therefore, find this component observationally.

###  2.2 Kinetic equation

The differential intensity of the neutrino beam $I(E,t)$ — the number of
neutrinos received per unit area, unit time, unit energy range, and unit solid
angle — is the most relevant quantity for the DSNB. It is related to the
number density per unit energy range, $n(E)$, through $I(E)=n(E)/(4\pi)$,
where we adopt the natural unit of $c=\hbar=1$. Because of the homogeneity and
isotropy of the Universe, the differential number density of the neutrinos,
$n(E)$, is independent of spatial coordinates but is a function of the
neutrino energy $E$. The current and near future detectors will have no
directional sensitivity to enable study of the DSNB anisotropies, and thus the
differential flux $F(E)=4\pi I(E)=n(E)$ has been used in the literature.

We want to understand the time evolution of the differential number density of
the DSNB, $n(E,t)$, which we define as a number of the neutrinos per comoving
volume per unit energy range. It is related to the phase space density
$f(E,t)$ through,

$n(E,t)=\frac{E^{2}}{2\pi^{2}}f(E,t)a^{3}(t),$ (1)

where $a(t)$ is the scale factor of the Universe, which coverts the physical
density to the comoving density. The phase space density $f(E,t)$ depends on
$(E,t)$ as the distribution is assumed homogeneous and isotropic, and
neutrinos are relativistic, $E=|\vec{p}|$.

The evolution of $f(E,t)$ is described by the following kinetic equation:

$\hat{L}[f(E,t)]=\mathcal{C}[f(E,t)],$ (2)

where $\hat{L}$ is the Liouville operator and $\mathcal{C}$ is the collisional
term.

The Liouvillle operator $\hat{L}$ in generic spacetime is

$\hat{L}=p^{\alpha}\frac{\partial}{\partial
x^{\alpha}}-\Gamma^{\alpha}_{\beta\gamma}p^{\beta}p^{\gamma}\frac{\partial}{\partial
p^{\alpha}},$ (3)

where $\Gamma^{\alpha}_{\beta\gamma}$ is the Christoffel symbol.[8] By
applying this to the Friedmann-Robertson-Walker metric, we have

$\hat{L}[f(E,t)]=E\frac{\partial f}{\partial t}-H(t)E^{2}\frac{\partial
f}{\partial E},$ (4)

where $H(t)=\dot{a}(t)/a(t)$ is the Hubble function. Rewriting this in terms
of $n(E,t)$, then, the Liouville operator acting on $n(E,t)$ is

$\hat{L}[n(E,t)]=\left[\frac{\partial}{\partial
t}-H(t)E\frac{\partial}{\partial E}-H(t)\right]n(E,t).$ (5)

The “collisional” term on the right-hand side is then given by

$\mathcal{C}[n(E,t)]=R_{\rm SN}(t)\frac{dN(E)}{dE},$ (6)

where $R_{\rm SN}(t)$ is the supernova rate per comoving volume, and $dN/dE$
is the neutrino spectrum per each supernova. To summarize, the neutrino
kinetic equation is

$\left[\frac{\partial}{\partial t}-H(t)E\frac{\partial}{\partial
E}-H(t)\right]n(E,t)=R_{\rm SN}(t)\frac{dN(E)}{dE},$ (7)

###  2.3 Solution

We largely follow Ref. [9] for solving the kinetic equation. First, instead of
$(E,t)$ variables, we adopt $(\varepsilon,z)$, where $z$ is the cosmological
redshift and $\varepsilon=E/(1+z)$. Then, the derivative operators
$\partial_{t}$ and $\partial_{E}$ can be written as
$\partial_{t}=-H(z)[(1+z)\partial_{z}-\varepsilon\partial_{\varepsilon}]$ and
$\partial_{E}=(1+z)^{-1}\partial_{\varepsilon}$. With these new variables, Eq.
(7) can be rewritten as

$-H(z)\frac{\partial}{\partial z}\left[(1+z)n(\varepsilon,z)\right]=R_{\rm
SN}(z)\left.\frac{dN}{dE}\right|_{E=(1+z)\varepsilon},$ (8)

which can be integrated to obtain

$n(\varepsilon,z)=\frac{1}{1+z}\int_{z}^{\infty}\frac{dz^{\prime}}{H(z^{\prime})}R_{\rm
SN}(z^{\prime})\left.\frac{dN}{dE}\right|_{E=(1+z^{\prime})\varepsilon}.$ (9)

We are interested in the differential flux of the neutrinos at $z=0$, and
since $F(E)=n(E,z=0)$, we finally obtain

$\displaystyle F(E)$ $\displaystyle=$
$\displaystyle\frac{c}{H_{0}}\int_{0}^{\infty}\frac{dz}{\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}}}$
(10) $\displaystyle{}\times R_{\rm
SN}(z)\left.\frac{dN}{dE^{\prime}}\right|_{E^{\prime}=(1+z)E}.$

Here we note that we recovered $c$ explicitly and used the Friedmann equation
in the flat expanding Universe,
$H(z)=H_{0}\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}}$, which is dominated by
matter and the cosmological constant $\Lambda$ with their density parameters,
$\Omega_{m}$ and $\Omega_{\Lambda}$, respectively. Throughout this review, we
adopt $H_{0}=67.4~{}\mathrm{km~{}s^{-1}~{}Mpc^{-1}}$, $\Omega_{m}=0.315$, and
$\Omega_{\Lambda}=0.685$, which are compatible with the latest Planck
measurements of the cosmological parameters.[10]

##  3 Theoretical model ingredients

###  3.1 Cosmic supernova rate density

####  3.1.1 Direct measurements

While supernovae have been directly observed for centuries, systematic
observations which allow their volumetric rates to be inferred has been driven
more recently by the rise of large surveys. Since supernovae are transient
phenomena, their search requires multiple scans of the same target(s). There
are broadly two survey strategies to search for supernovae: targeted and non-
targeted.

In the targeted strategy, a list of targets (e.g., galaxies) is pre-compiled
and surveyed repeatedly over time. Early surveys[11, 12, 13] often adopted
this strategy because the number of discovered supernovae can be maximized by
targeting large star-forming galaxies. Such surveys therefore often report
supernova rates per unit galaxy luminosity in a particular band, e.g., SNuB =
1 SN $(100\,{\rm yr})^{-1}(10^{10}L_{\odot}^{\rm B})^{-1}$ where
$L_{\odot}^{\rm B}$ is the solar B-band luminosity. To convert this to the
_volumetric_ rate needed for the DSNB, one must account for the luminosity
density of the Universe at the redshift in question which need to be
independently measured, e.g., $j_{B}(z)=(1.03+1.76z)\times
10^{8}L_{\odot}^{B}\,{\rm Mpc}^{-3}$[13]. One concern is that targeted surveys
would miss bright supernovae in smaller galaxies not included in the target
list, even if they are within the survey’s flux sensitivity and volume range.
This must be corrected for. In the more recent survey by the Lick Observatory
Supernova Search (LOSS)[14, 15, 16], over 10,000 galaxies were monitored over
the course of 11 years, leading to the discovery of the “rate-size relation”
where the supernova frequency is not linearly proportional to the size of the
host galaxy, but some power (of 0.4–0.6), the exact number depending on the
supernova type and galaxy Hubble type. Thus, correcting for missing supernovae
is far from a simple scaling exercise.

In the non-targeted strategy, patches of the sky are pre-selected and surveyed
repeatedly over time. This locates all supernovae only limited by flux, i.e.,
regardless of the supernova host galaxy. This means the volumetric rate is
more readily estimated. On the other hand, a large field of view must be
surveyed to acquire a large sample of supernovae, especially for nearby
distances. Nevertheless, most recent surveys have adopted this method. The
Rubin Observatory Legacy Survey of Space and Time (LSST), for example, is
expected to survey $\sim 20,000\;\mathrm{deg}^{2}$, slightly less than half of
the sky, with a planned revisit time of $\sim 3$ days on average per 10,000
deg2 with 2 visits per night.[17] This will result in orders of magnitude
increase in the number of supernovae discovered [18, 17].

An important systematic consideration affecting both survey strategies is that
of dust extinction. Due to flux sensitivity limits, supernova surveys will
miss the faintest supernovae[19]. Over a population, the supernova luminosity
has some intrinsic distribution, so it is natural some will fall below the
survey sensitivity; but intrinsically luminous supernovae can also appear
significantly fainter due to dust extinction. Typically, the supernova
luminosity function is constructed using a volume-limited sample (for example,
either obtained independently or using a sub-set of nearby supernovae within
the survey) and used to infer the missing faint end in larger flux-limited
samples. However, dust extinction is redshift-dependent and its modeling for
supernovae correction has been updated over the last decades.[20, 19] As
explained later in Section 3.1.3, it has been pointed out that the direct
supernova measurements are systematically lower than the birth rates of
massive stars[21]. This discrepancy between the birth rates and supernova
rates can be remedied by such improved dust modeling.

Furthermore, the DSNB is sensitive to the total core collapse rate, regardless
of whether the collapse yields a luminous supernova or not. In this context,
direct measurements of supernovae is likely to fall short, since the collapse
to black holes are expected to yield systematically dimmer, longer, and redder
transients[22, 23] which are not efficiently detected by supernova surveys[24,
25, 26].

####  3.1.2 Measurements of birth rates

An alternative method to measure the cosmic supernova rate is indirectly from
the cosmic birth rate of stars: the mass of the total stars formed per unit
time per unit comoving volume at a given redshift $z$. This strategy works
since the lifetimes of massive stars which undergo supernova explosions
(heavier than approximately $\sim 8M_{\odot}$ in mass) is less than $O(100)$
Myr which is short on cosmological timescales. In other words, all massive
stars born in a particular redshift range will expend their nuclear fuel and
collapse within the same redshift bin, without a cosmologically significant
delay (the same cannot be said about thermonuclear supernovae[27], which can
be delayed by billions of years since the birth of their progenitor stars).

The core-collapse rate density is then inferred as follows assuming the
stellar initial mass function (IMF) $\phi(M)$:

$R_{\rm
CC}(z)=\frac{\int_{8M_{\odot}}^{125M_{\odot}}dM\phi(M)}{\int_{0.1M_{\odot}}^{125M_{\odot}}dMM\phi(M)}R_{\rm
SF}(z),$ (11)

where Salpeter mass function ($\phi(M)\propto M^{-2.35}$) has been adopted
often in the literature[28]. Here, the stellar mass distribution spans
$0.1M_{\odot}$ to $125M_{\odot}$, with the supernova progenitor ranging from
$8M_{\odot}$ to $125M_{\odot}$. In reality, the Salpeter IMF is known to be
inaccurate especially at the low-mass end, but it is still useful as a common
way to compare. Various modifications to the Salpeter IMF exist, among the
more frequently used including those by Chabrier[29] and Kroupa.[30]

The cosmic birth rate of stars is inferred from the number of massive stars in
a galaxy at a given time. From this, the mean star formation rate of stars
during the time window equivalent to the lifetimes of massive stars can be
estimated. It is, however, not feasible to observe individual massive stars in
distant galaxies. So instead, aggregate galactic observables are used to
estimate the number of massive stars: so-called “star-formation
indicators.”[31] For example, massive stars have high surface temperature and
thus typically dominate a galaxy’s ultraviolet (UV) emission (barring emission
from non-stellar origins, e.g., active galactic nuclei, AGN). Thus,
observations of a galaxy’s UV emission can be combined with knowledge of
massive stars’ UV luminosities to estimate the recent star formation rate. The
conversion to star formation rate is known as the “calibration factor” and its
modeling typically relies on stellar population modeling. In practice, when
the star-formation rate estimated using, e.g., the UV indicator, the observed
UV luminosity is appropriately dust-corrected and AGN corrected (when
necessary), then multiplied by the UV calibration factor. A recent collection
of star formation rate density measurements is shown in Fig. 1 (including UV
and other measurements), alongside two commonly used functional fits to the
data from Yuksel et al. 2008[32] and Madau and Dickinson 2014[33]. In the
redshift range of a few relevant for the DSNB, the spread in measurements is a
factor of 2–3.

Figure 1: Star formation rate density (SFRD) as a function of redshift. SFRD
points collected from the right panel of Figure 6 of Audcent-Ross et al.
2018[34], entitled “Near-identical star formation rate densities from
H$\alpha$ and FUV at redshift zero”, published in MNRAS volume 480 no. 1. Fits
to other data sets performed in Madau and Dickinson 2014[33] and Yuksel et al.
2008[32] with Horiuchi et al. 2008[35] parameters are plotted for comparison.
All assume the Salpeter IMF.

Calibration factors have been computed for the various star-formation
indicators but requires a great deal of complex physics, including challenging
problems such as the evolution of massive stars and its dependence on stellar
parameters, such as mass, metallicity, and rotation; modeling of stellar
atmospheres; the binary fraction and the resulting interactions; the shape of
the stellar IMF; whether star formation is continuous or variable; and so on.
The status was summarized by Kennicutt in 1998,[31] which continues to be used
for ease of comparison with other estimates, but have also been updated for
various specific indicators and circumstances.

One of the significant uncertainties arises from the IMF. Physically, this is
because an extrapolation in mass is needed, which in turn depends on the IMF.
More specifically, although the measurement of star formation is based on the
population of massive stars, the total star-formation rate is dominated by
low-mass stars. Uncertainties in the shape of the IMF introduces a factor
$\sim 2$ or more effect on the inferred SFR. Fortunately, this uncertainty
does not propagate directly to the DSNB, because the DSNB is not powered by
low-mass stars. Instead, the DSNB is powered by the same massive stars used to
measure the star formation, thus the effect of the IMF uncertainty on the
supernova rate is at the level of $<10$ percent.[35] This is also corroborated
by a study of the implications of a non-Universal IMF, where even extreme IMF
variations led to small variations in the supernova rate, at least in the low
redshifts of relevance for the DSNB.[36]

####  3.1.3 Core collapse to black holes

As mentioned earlier, the core collapse of a massive star need not necessarily
cause a supernova explosion. In fact, the observation of stellar-mass black
holes—from microquasars to gravitational waves—motivate that a non-negligible
fraction of massive stars should collapse to black holes, creating potentially
weak or negligible supernova explosions. However, they are still powerful
neutrino sources.

So far, the in-situ formation of a black hole has not been directly observed.
Theoretical investigations suggest the direct collapse of the core of a large
red supergiant to a black hole could yield a weak, red, and long-lived
transient due to the unbinding of the hydrogen envelope[22, 23]. However,
transient surveys are not tuned to capture such long and red explosions.
Alternatively, if an explosion is hard to observe, the disappearance of a
massive stars might be easier. Such a survey has in fact been in operation for
about a decade. This is the “survey about nothing,” searching for the
disappearance of massive stars without a supernova explosion[37]. These stars
are then strong candidates for stars which undergo core collapse into black
holes. The general strategy is to monitor a sufficient number of massive stars
such that on average one of them will core collapse in any given year. In 11
years running, the survey has 9 massive stars coincident in time and location
with supernovae, and 2 massive stars disappearing without such coincident
supernovae[24, 38, 39, 40, 25]. Naively, this implies a
$23.6^{+23.3}_{-15.7}$% fraction of massive stars undergoing collapse to a
black hole; note that if only one of the disappearing candidate is included,
the fraction is $16.2^{+23.2}_{-12.5}$%[25]. Even for the more conservative
latter estimate, it is still a substantial enough fraction to be of interest
for the DSNB. While the uncertainty is large, this is a non-negligible
fraction for the DSNB. One of the benefits of using the star formation rate is
that it does not bias against collapse to black holes.

In principle, precisely knowing the cosmic star-formation rate and the cosmic
supernova rate independently can yield the fraction of core collapse to black
holes. Early measurements suggested a systematic discrepancy between them, at
the factor of $\sim 2$ level,[21] that naively could be interpreted as some
half of massive stars are collapsing to black holes without luminous
supernovae. However, supernovae could be faint for other reason, chief among
them dust extinction (see section 3.1.1). Indeed, the authors[21] found that
at nearby distances $<20$ Mpc, where even dust extinguished supernovae could
be more easily detected, the fraction of faint supernovae were much higher
than at larger distances, suggesting dust to be an important effect. Since
then, updated dust models have been applied to supernova rate measurements,
which have bridged the gap between the supernova rate and star-formation
rates[41]. Given the existing uncertainties, fractions of $\sim 10$–$40$% seem
to be allowed, consistent with searches of disappearing stars.

Theoretically predicting the fraction of stars which collapse to black holes
is very uncertain at present, due to the unknown nature of the explosion
mechanism and the need to survey a vast landscape of massive star properties.
Attempts have been made assuming the delayed-neutrino heating explosion
mechanism and using simpler core-collapse simulation treatments to survey
large numbers of progenitors. For example, a recent analysis by the Garching
group yields fractions ranging from 17.8% up to 41.7%[42]. As a result, most
studies therefore parameterize the fraction of core collapse into black holes.

###  3.2 Neutrino spectrum from core-collapse supernova explosions

Broadly speaking, in predicting the DSNB we need to account for: (i) the
emission of neutrinos from the vast range of properties of massive stars in
nature, (ii) the time-integrated neutrino emission, often going beyond the
time ranges studied by numerical simulations, (iii) the emission of neutrinos
from both core collapse to neutron stars and black holes, and (iv) a
consideration for neutrino oscillations in the progenitor.

####  3.2.1 Progenitors

It is generally accepted that stars heavier than $\sim 8M_{\odot}$ end their
life triggered by a core collapse, leaving a neutron star or a black hole
behind. How the core collapse depends on properties of progenitors such as
their mass and rotation has been investigated extensively; however it is still
far from complete. For example, the conditions for black-hole formation after
the core collapse was once considered to depend mostly on the mass and
metallicity of its progenitor star, with massive and low metals being the
important criteria[43]. This implied for galaxies dominated by solar metal
stars, the fraction of massive stars collapsing to black holes was only a few
percent.[44] It turned out, however, to be not so simple. Shock revival is
highly sensitive to the energy transport by neutrinos, which in turn depends
on both the microphysics governing the emission from the hot dense core and
the three-dimensional turbulence structure determining the neutrino capture
(and hence heating) rate. In other words, whether an explosion occurs is
sensitive to the initial conditions which appear to be not simply monotonic
with e.g., the stellar mass.[45, 46, 47, 48, 42] As a result, black-hole
formation and hence a failed supernova can happen even for low-mass
progenitors as light as $\sim 15M_{\odot}$, while very heavy stars can still
cause successful explosions.

On the other hand, the neutrino emission can be more accurately characterized.
For example, the stellar compactness parameter,[44]

$\xi_{M}=\frac{M/M_{\odot}}{R(M_{\rm bary}=M)/\mathrm{1000\,km}},$ (12)

has been discussed in the literature. Here, $R(M_{\rm bary}=M)$ is a radius
enclosing a baryonic mass $M$. The compactness at $M=2.5M_{\odot}$,
$\xi_{2.5}$ is a suitable criteria for the black-hole formation since it is
close to the maximal mass of a neutron star[44], although other masses can
provide clearer distinguishing power[47]. Reference [49] compiled core-
collapse simulations of 8–$100M_{\odot}$ progenitor masses, and found
extensive dependence of neutrino emission spectrum on $\xi_{2.5}$.
Approximately, a higher compactness star has a large massive core, hence a
longer lasting intense mass accretion onto the collapsed core. This accretion
powers strong neutrino emissions, and thus both the total energy liberated in
neutrinos and the neutrino mean energy grows with compactness. However, at
some stage, the mass accretion would become too intense and the shock revival
cannot be achieved by the neutrinos. This may suggest a critical compactness
necessary for collapse to black holes. In reality, this simplistic picture is
complicated by many factors, which smears the predictability noticeably. For
example, the progenitor density shift at the silicon shell appears to play an
important role in shock revival[50], which is not sufficiently captured in a
simple compactness picture.

Another important consideration for supernova progenitors is binary effects.
Observations of stars in the Milky Way and nearby stars reveal that the
majority (>50%) of massive stars have experienced binary effects.[51] Binary
interactions, especially mass transfer and stellar mergers, can significantly
change the masses of core-collapse progenitors and strongly influence the
DSNB. For example, stars which are born below the core collapse threshold
($\sim 8M_{\odot}$) may eventually exceed the threshold by, e.g., merging with
another star. By adopting population synthesis of binary-star systems, binary
effects were found to enhance the DSNB flux by 15–20% in favor of future
detection.[52]

####  3.2.2 Neutrino emission

Neutrinos are emitted from the surface of the proto-neutron star. Therefore,
neutrinos—along with gravitational waves—is one of the only ways to probe the
condition of the collapsing core. To date, the only observational dataset is
the approximately two dozen neutrinos from SN1987A[1, 2]; thus, for predicting
the DSNB we must in general rely on results of numerical simulations of
stellar core collapse. They are, however, one of the most challenging problems
in computational astrophysics[53, 54, 55, 56], as they involve all the four
fundamental forces of nature (electromagnetic, gravitational, strong, and weak
forces) playing key roles and one must address microphysics of neutrino
propagation, with gravito-magnetohydrodynamics part of the code. Especially,
for obtaining the most realistic outcomes, we should perform the simulations
in the three-dimensional setup, including both the coordinate space and
neutrino momentum space, which makes it computationally expensive.
Nevertheless, researchers from multiple groups have tackled this problem and
have reached reliable results. For example, they now successfully lead to
supernova explosions especially for low-mass progenitor models. Recent three-
dimensional simulations of the $\sim 20M_{\odot}$ progenitor with elaborate
neutrino transfer scheme successfully obtained the explosion energy of $\sim
10^{51}$ erg that matches supernova observations well through neutrino-heating
mechanism.[57]

Despite this incredible progress in the core-collapse simulations, most
simulations are still limited to within the initial one second or so. This is
mainly due to the fact that successful shock revival must occur during the
initial second while the neutrino emission is most intense. However, the time-
integrated spectrum is relevant for the DSNB flux estimate, and indeed, a
substantial fraction of the total neutrino emission attributes to the later
phase of the proto-neutron-star cooling after one second (typically $>50\%$).
We discuss this deficit in detail in Section 3.2.4.

Multiple independent groups compared one-dimensional results from each of
their simulations, and confirmed that, even with different approximations,
etc., the results are in good agreement. For the neutrino emission, the
agreement level is within a few tens of percent.[58] This is confirmed as well
in Fig. 2, where we compare the time-integrated spectra from five simulation
models of a $20\,M_{\odot}$ progenitor (with the exception of Bollig et
al.[54], which is a $27\,M_{\odot}$ progenitor) up to the first $300\,{\rm
ms}$ post core collapse.222All DSNB flux spectra figures are created using our
publicly available PyDSNB code, found at
https://github.com/shinichiroando/PyDSNB/tree/main. This code makes use of and
is consistent with the most recent version of SNEWPY (version 1.3).[59, 60]

Figure 2: DSNB spectra of different models for the hydrodynamic phase of
protoneutron-star evolution; each model shown here has been integrated up to
the first $300\,{\rm ms}$ of each simulation for $20\,M_{\odot}$ progenitors
(except for Bollig et al. 2016[54] which is a $27\,M_{\odot}$ progenitor and
the Nakazato et al. 2013[61] model has metallicity $Z=0.02$). We compare these
against the Fornax 2021[56, 62], Kuroda et al. 2020[63], and Tamborra et al.
2014[64] models.

Since the neutrino spectrum from each core-collapse event depends on
characteristics of progenitor stars such as their masses, for the estimate of
the DSNB flux, one must take the average of $dN/dE$ over mass with the weight
of the stellar IMF:

$\frac{dN}{dE}=\frac{\sum_{i}(dN/dE)_{i}\int_{M_{i}^{l}}^{M_{i}^{u}}dM\phi(M)}{\int
dM\phi(M)}.$ (13)

Here one can use a model spectrum $(dN/dE)_{i}$ in a mass bin $i$ between the
lower and upper mass bounds, $M_{i}^{l}$ and $M_{i}^{u}$, respectively.

####  3.2.3 Neutrino oscillations

It is strongly established that the neutrinos have masses and different
flavors mix during propagation. These phenomena called neutrinos oscillations
are well understood in vacuum and matter-dominated environments, particularly
because all the mixing angles and mass squared differences have been measured
precisely. There is still uncertainty related to the mass hierarchy that can
be either normal (NH) or inverted (IH). The MSW mechanism of the matter-
induced neutrino oscillations mixes the electron flavors ($\nu_{e}$ and
$\bar{\nu}_{e}$) with heavy-lepton flavors ($\nu_{x}$ collectively
representing $\mu$ and $\tau$ flavor neutrinos and anti-neutrinos) as follows:

$F_{\nu_{e}}=\left\\{\begin{array}[]{lll}F_{\nu_{x}}^{0}&\mbox{(NH)},\\\
\sin^{2}\theta_{12}F_{\nu_{e}}^{0}+\cos^{2}\theta_{12}F_{\nu_{x}}^{0}&\mbox{(IH)},\end{array}\right.$
(14)

for the neutrino sector, and

$F_{\bar{\nu}_{e}}=\left\\{\begin{array}[]{lll}\cos^{2}\theta_{12}F_{\bar{\nu}_{e}}^{0}+\sin^{2}\theta_{12}F_{\nu_{x}}^{0}&\mbox{(NH)},\\\
F_{\nu_{x}}^{0}&\mbox{(IH)},\end{array}\right.$ (15)

for the anti-neutrino sector. Here the superscript 0 represent the flux at
production (i.e., before propagation through the stellar envelope and
interstellar space).

Around the neutrinosphere where the density of the neutrinos is large,
collective oscillations triggered by the neutrino-neutrino interactions can be
important. This is likely to be important in a time-dependent manner for
example in the next Milky Way core-collapse supernova, but could be smeared
out in the DSNB spectrum where the signature is integrated over time.[65, 66]
However, the jury is still out given the dearth of predictable collective
oscillation patterns. We thus neglect the effect of collective neutrino
oscillation on the DSNB flux, and focus only on the MSW effect. In Fig. 3, we
show the DSNB flux spectra for each flavor with a Fermi-Dirac distribution at
temperatures of 3.4, 4.1, and $4.1\,{\rm MeV}$ for $\nu_{e}$, $\bar{\nu}_{e}$,
and $\nu_{x}$, respectively, based on the analysis done in Ekanger et al.[67].
Because the $\bar{\nu}_{e}$ and $\nu_{x}$ spectra are so similar, the spectrum
after MSW oscillation looks very similar for normal and inverted hierarchies.

Figure 3: DSNB spectra for the Fermi-Dirac model with parameters derived from
the model of Ekanger et al.[67] which have $E_{\nu,e}=5.8\times 10^{52}\,{\rm
erg}$, $E_{\bar{\nu},e}=6\times 10^{52}\,{\rm erg}$, and $E_{\nu,x}=5\times
10^{52}\,{\rm erg}$, and temperatures $T_{\nu,e}=3.4\,{\rm MeV}$ and
$T_{\bar{\nu}_{e}}=T_{\nu,x}=4.1\,{\rm MeV}$. Because $\bar{\nu}_{e}$ and
$\nu_{x}$ have very similar temperatures, the spectrum after oscillation due
to the MSW effect looks very similar for normal and inverted hierarchies.

####  3.2.4 Late-time evolution

Approximately half of the total neutrinos emitted by a stellar core collapse
is during the so-called cooling phase, when the protoneutron star cools and
shrinks by the emission of mostly neutrinos. Simulations dedicated to the
cooling phase have been performed, e.g., exploring the impacts of the
protoneutron star’s interior turbulence, the equation of state of hot-dense
matter, and other nuclear physics. However, due to the considerably different
scales from the core-collapse phase, it is not easy to simulate the cooling
phase in the same code as the core-collapse phase. Instead, groups have tied
together such simulations in various ways. For example, Nakazato et al.[61]
performed an analytic “match” using a user-defined explosion time. In Fig. 4,
we show the DSNB flux spectra for the $20\,M_{\odot}$, $Z=0.02$, $300\,{\rm
ms}$ revival time model from Nakazato et al.[61] broken into the early
hydrodynamical phase, the late cooling phase, and the total spectrum. The
later cooling phase is characterized by lower energy neutrinos and contributes
significantly to the overall spectrum. Recently, Ekanger et al.[67]
implemented different schemes to model the cooling phase neutrino emissions on
the DSNB, and found a factor of $\sim 2$ difference in DSNB rates between
them. Fig. 5 shows the differences in flux spectra when combining the early
hydrodynamical phase from Nakazato et al.[61] and estimating the late phase
with four different schemes. While the estimated neutrino luminosities were
different between the schemes implemented, the differences in neutrino
energies had a larger impact on the predicted DSNB rates.

Figure 4: Comparison of the DNSB spectrum from the early hydrodynamic, late
cooling, and total phases of the Nakazato et al.[61] (or N13) model. This
assumes a revival time of $300\,{\rm ms}$, metallicity of $Z=0.02$, and
progenitor mass of $20\,M_{\odot}$. Figure 5: Comparison of the DSNB spectrum
of the sum of early hydrodynamic phase plus four different late phase
strategies. The data from the hydrodynamic phase is from N13 in this figure,
using the same revival time, metallicity, and progenitor mass as Fig. 4.

####  3.2.5 Core collapse to black holes

The neutrino emission from a core collapse to a black hole is expected to be
systematically different to that from a core collapse to a neutron star. To
understand why, consider that the ultimate energy source of neutrino is
gravitational binding energy released as the stellar core collapses to a more
compact object. A collapse to a black hole, which has larger mass/radius than
a neutron star, therefore releases more energy. More energy density equates to
higher temperatures and luminosities. In reality, the situation is complicated
by the time evolution of the core and the fact that once a black hole is
formed, the neutrinos from within the event horizon cannot escape and
therefore do not contribute to the total neutrino emission. As a result,
simulations generally find that compared to collapse to neutron stars,
collapse to black holes show (i) higher neutrino energies, especially the
heavy-lepton flavor neutrinos whose neutrinospheres are smallest and subject
to interior temperatures the most, and (ii) the time-integrated energy
released as neutrinos decreases, due to neutrinos not being able to escape. A
quicker collapse to a black hole exacerbates these features [49].

How long a certain core will take to collapse to a black hole depends on both
the progenitor and on the equation of state of hot-dense matter. For a given
equation of state, it is well described by the progenitor’s core compactness
[44, 49]. While the progenitor compactness in large part sets the mass
accretion rate and thus the evolution of the mass of the protoneutron star,
the equation of state sets the maximum mass of the collapsing central compact
object that can be supported. This in turn depends on the stiffness of the
equation of state, the amount of trapped leptons, as well as the temperature
in the accreting protoneutron star. While various equations of state models
have been explored over the years, the addition of more neutron star
mass/radii measurements, as well as the discovery of a neutron-star merger,
has helped to narrow the range of possibilities [68].

Figure 6: Neutrino number spectrum of successful and failed models of Nakazato
et al. (N13)[61] and of SN 1987A.[69] Here, successful N13 model is of
$20\,M_{\odot}$ progenitor, revival time is $300\,{\rm ms}$, metallicity is
0.02, and the equation of state is Shen[70]. Failed model is $30\,M_{\odot}$
progenitor, with 0.004 metallicity, and the equation of state is Shen. Also
plotted are Fermi-Dirac distributions for 4 and $8\,{\rm MeV}$ ($3.2\times
10^{53}\,{\rm erg}$ total energy liberated) to represent common estimations of
successful and failed supernovae, respectively.

Figure 6 shows neutrino spectrum of $\bar{\nu}_{e}$ flavor for various models
of both successful and failed supernovae. We also compare the models with
those reconstructed from the SN 1987A neutrino data as well as conventional
Fermi-Dirac distribution (with zero chemical potential) with the temperature
of 4 MeV and 8 MeV. If the core collapse leaves a black hole, then the
temperature is higher, yielding the neutrino spectrum close to the Fermi-Dirac
distribution with 8 MeV temperature. The core collapse into a neutron star is
well approximated as a Fermi-Dirac distribution with 4 MeV temperature.

##  4 DSNB spectrum

Because of the redshift of neutrino energies, the DSNB flux is dominated by
the contribution from supernovae with $z<1$ (Fig. 7). At energies above 10
MeV, which is the main target of major detectors such as Super-K, the DSNB
spectrum falls exponentially. Therefore, the DSNB spectrum is often plotted
with a linear scale for the horizontal axis representing the neutrino energy
and with a linear or log scale for the vertical axis representing the flux.

With the current detection technology, one can adopt neither time nor spatial
information, and thus must rely solely on the energy information to probe
physics and astrophysics relevant to the DSNB. Yet, with the high statistics
expected with the future generation of neutrino detectors, one can extract
various important information on the supernova rate, black-hole formation, and
neutrino physics. We elaborate on each of them in the following.

Figure 7: Contributions to the DSNB spectrum from different redshift ranges.
The DSNB flux is dominated mostly by supernovae with $z<1$. The Fermi-Dirac
distribution with $E_{\rm tot}=3.2\times 10^{53}\,{\rm erg}$ and
$T_{\bar{\nu}_{e}}=T_{\nu_{x}}=4.1\,{\rm MeV}$ has been adopted.[67]

###  4.1 Neutrino spectrum of various flavors

Even though the dominant detection channel for the DSNB is the inverse beta
decay for the $\bar{\nu}_{e}$ flavor, it is desirable to detect all the
flavors ($\nu_{e}$, $\bar{\nu}_{e}$, and $\nu_{x}$) to reach a comprehensive
understanding of the supernova explosion. We can study $\bar{\nu}_{e}$ fluxes
with Super-K and Hyper-K, while a good option for the $\nu_{e}$ flux would be
DUNE, which can reach DSNB sensitivity if backgrounds can be controlled.[71]
The non-electron flavors $\nu_{x}$ is the hardest to detect, as one must rely
on neutral current interactions. These are not efficient at MeV energies
because either the cross section is suppressed (for neutrino-electron
scatterings) or the target is too heavy (for neutrino-nucleon scatterings).
This challenge can be achieved with existing and future dark matter direct-
detection detectors such as XENON, LZ, and DARWIN. Especially with the
ultimate dark matter detector, DARWIN, one can reach the sensitivity about an
order of magnitude larger than theoretical predictions.[72] It is also
possible to adopt the neutral-current interaction off the protons in
detectors: $\nu p$ scattering. It has been studied that if the backgrounds
against detecting these scattering events could be well understood and
suppressed, one could constrain the $\nu_{x}$ flavor content using JUNO-like
detectors.[71] Another suggested opportunity is to use ancient minerals as
track detectors, i.e., paleo detectors,[73] which would record supernova
neutrinos over geological timescales. Since the relevant interaction for
supernova neutrinos is coherent scattering, all flavor information will be
obtained, which can be disentangled by a combined analysis with $\nu_{e}$ and
$\bar{\nu}_{e}$ results.[74] However, it should be noted that paleo detectors
will record more the historical record of Milky Way supernova neutrinos rather
than the DSNB.[75] This means that while still an average over many
supernovae, it is not the true DSNB; the travelled baseline and progenitor
samples will be different.

###  4.2 Tests of neutrino physics

Because the DSNB can be regarded as the longest-baseline (at a cosmological
distance scale) low-energy (MeV) neutrino experiment, one can probe unique new
physics beyond the standard model in the neutrino sector. Non-radiative
neutrino decays induced by the interactions with scalar or pseudo-scalar
fields such as Majoron is one such example, where the DSNB can improve upon
the sensitivity to neutrino lifetimes by orders of magnitude.[76, 9, 77]
Similarly, one can test the pseudo-Dirac nature of neutrinos as it would yield
oscillation between the active and sterile species during propagation, and a
unique window in the relevant mass-scale parameter $m_{k}=10^{-25}$–$10^{-24}$
eV can be tested.[77] If the sterile neutrinos at eV mass scales exist and
they interact with a new unknown gauge vector boson $\phi$, then such a
scenario can be tested using the DSNB. If $M_{\phi}=5$–10 keV and the coupling
to the sterile neutrinos $g_{s}=10^{-4}$–$10^{-2}$, it would show a
characteristic dip in the DSNB energy spectrum that should be captured by the
detectors.[78]

###  4.3 Cosmology and astrophysics

The DSNB spectrum could also be used to test cosmological models. By using the
DSNB data, one can test the standard $\Lambda$CDM model including the
measurement of the Hubble constant[77] or other exotic models such as a
logotropic universe and a bulk viscous matter-dominated universe.[79]

###  4.4 Core collapse to black holes

As shown in Sec. 3.2.5, the neutrino spectrum (both the normalization and
spectral shape) depends on the final state of collapsed object — a neutron
star or a black hole. One can use the DSNB spectrum to test the black-hole
formation after the core collapse, which is challenging using the optical
observations alone. The latest upper limits from the Super-K on the DSNB
flux[80] already excludes models of a black-hole formation with a few
equation-of-state models.[81] This means that one can already constrain
fraction of formation of black holes or high-mass neutron stars. Future
observations with SK-Gd and Hyper-K can constrain these parameters to be
smaller than tens of percent levels.[66]

##  5 Detection prospects

###  5.1 Detection principle and the history of the observation

There are mainly two types of detectors for DSNB observation. One is the water
Cherenkov detector and the other is the liquid scintillator detector. In a
water Cherenkov detector, photo detectors, such as photomultiplier tubes,
detect the Cherenkov light produced by charged particles scattered by
neutrinos in water. Cherenkov light is emitted when a charged particle
traverses a medium at a velocity greater than the speed of light reduced by
the refractive index. The Cherenkov light is emitted at an angle to the
direction of the charged particle due to the refractive index and its
velocity, thus, it is detected as a ring image. If the charged particle is
traveling at nearly the speed of light in water, this angle is 42 degrees. In
a liquid scintillator detector, the scintillation light generated by the
charged particles as they pass through the material is detected by photo
detectors. Since the light yield is higher than the water Cherenkov detector,
it has a lower energy threshold and higher neutron tagging efficiency after
neutrino interactions.

The dominant detection reaction with the largest interaction cross section in
the DSNB energy region is the inverse beta decay (IBD):
$\bar{\nu_{e}}+p\rightarrow e^{+}+n$. The total cross section is roughly
calculated as $\sigma_{\rm tot}\sim 9.52\times 10^{-44}({p_{e}E_{e}}/{\rm
MeV}^{2})$ cm2, and the detailed calculations have been published by several
authors[82, 83]. In this interaction, positron as a prompt signal is detected,
in several detectors, followed by gamma rays from neutron capture on proton or
nucleus as a delayed signal. Here, the gamma-ray energy depends on what the
neutron is captured by, for example, it is 2.2 MeV in the case of proton
capture. This detection method is called delayed coincidence (DC) and is
effective in the reduction of background events compared to measuring only a
prompt signal. In this interaction, the neutrino energy is easily
reconstructed from the prompt positron energy using the following equation:
$E_{\nu}\sim E_{e^{+}}+m_{n}-m_{p}$, where $m_{n}$ and $m_{p}$ are the neutron
and proton mass, respectively.

Since the 1980s, DSNB searches have been made with various detectors and, in
this section, we briefly introduce each detector and the results.
Unfortunately, all experiments have not observed a significant signal of DSNB
and only set a 90% C.L. upper limit on it. There are two methods for
calculating the limit. One is the so-called ‘model-independent search’, which
is estimated from the number of observed events and the expected background
rate without assuming any DSNB or other physics models, which means a flux
upper limit of electron antineutrinos. The other is the so-called ‘model-
dependent search’, which is derived by fitting the observed energy spectrum to
the spectral shape of the signal from a DSNB model and the background. Here,
we will focus on the results of the model-independent search, i.e., the
electron antineutrino flux upper limit. In addition, we will comment on the
results of the model-dependent search for some detectors.

The first observation of the DSNB was reported by Kamiokande-II in 1988 [84].
It was a 2,140 ton water Cherenkov detector in the Kamioka mine, Japan. It is
famous for the first detecton of neutrinos from a supernova explosion [1]. As
for the DSNB observation, the limit of the electron antineutrinos in the
energy region between 19 and 35 MeV was set to 226 cm-2 s-1 by the exposure of
0.58 kton $\cdot$ yr.

The next observation was reported by LSD in 1992 [85]. It was a liquid
scintillation detector, which can tag neutron signals generated via IBD using
the DC technique by proton capture. The upper limit of electron antineutrinos
with the exposure of 0.0939 kton $\cdot$ yr was $9.0\times 10^{4}$ and
$8.2\times 10^{3}$ cm-2 s-1 for $9<E_{\nu}/{\rm MeV}<50$ and $20<E_{\nu}/{\rm
MeV}<50$, respectively. They also reported the results of neutrino
interactions with carbon nuclei which are sensitive to other flavors of
neutrinos although the cross sections are about one order of magnitude lower
and the energy thresholds are higher than those of IBD.

The situation, that the upper limits are two orders of magnitude higher than
the theoretical expectations, has drastically changed since Super-Kamiokande
(SK) began in 1996. It is a 50 kton water Cherenkov detector in the Kamioka
mine, Japan. It is famous for the discovery of neutrino oscillation in
atmospheric neutrinos [86]. The first result of the DSNB observation in SK was
reported in 2003 [87]. In this paper, the first phase of SK was used, and the
upper limit of electron antineutrinos above 19.3 MeV was 1.2 cm-2 s-1 by the
exposure of 92.2 kton $\cdot$ yr. The second report appeared in 2012 [88],
where the combined results of phase I through III in SK with the exposure of
176 kton $\cdot$ yr using only prompt signals were reported. Figure 8 shows
the flux upper limit of electron antineutrinos as a function of neutrino
energy. Since the fourth phase of SK (SK-IV) started in 2008, new electronics
were installed [89]. It enabled the background events reduced by detecting 2.2
MeV gamma-ray from proton capture of a neutron. SK demonstrated successful
detection of the neutron capture signal, although the efficiency was low
($\sim$20%) [90]. The results using the full data set of SK-IV were reported
in 2021 [91]. A new analysis method for efficient background reduction and an
improved neutron tagging algorithm allowed to lower the energy threshold to
9.3 MeV. The flux upper limit is shown in the figure by the exposure of 183
kton $\cdot$ yr. Furthermore, the combined analysis of all phases of SK was
conducted, and the flux limit for electron antineutrinos above 17.3 MeV was
set to 2.7 cm-2 s-1 with the exposure of 359 kton $\cdot$ yr. In this paper, a
model-dependent search with 21 modern DSNB predictions was also conducted. A
1.5$\sigma$ level of excess was observed over the background prediction across
all the models, and the DSNB flux limits were between 2.5 and 2.8 cm-2 s-1.

Another water Cherenkov detector, Sudbury Neutrino Observatory (SNO), also
reported some results of DNSB observations. SNO was located in the Inco, Ltd.
Creighton mine near Sudbury, Ontario, Canada, which used ultrapure heavy
water. By using heavy water, SNO enabled very unique neutrino observations,
which could measure the charged current ($\nu_{e}+d\rightarrow p+p+e^{-}$) and
neutral current ($\nu+d\rightarrow\nu+p+n$) interactions independently. This
led to conclusive evidence that neutrino oscillation also occurs in solar
neutrinos [92]. As for the DSNB measurements, SNO showed the unique results
that were the flux limit of electron neutrinos using the charged current
interaction, instead of the electron antineutrinos which was rather high [93,
94]. The result in the energy region between 22.9 and 36.9 MeV was 70 cm-2 s-1
with the exposure of 0.65 kton $\cdot$ yr.

Recently, the results of DSNB observations by two liquid scintillator
detectors were shown; one was KamLAND and the other was Borexino. The KamLAND
experiment uses 1 kton of ultrapure liquid scintillator located in Kamioka
mine, Japan. It started in 2002, including KamLAND-Zen, which set the balloon
filled xenon-loaded liquid scintillator at the center in 2011 for the
discovery of the neutrinoless double beta decay. For the DSNB observations,
the first results were published in 2012 [95], followed by the results using
more data and improved analysis methods in 2022 [96]. In the latest result,
the flux upper limit in the energy region between 8.3 and 30.8 MeV with the
exposure of 6.72 kton $\cdot$ yr was shown in the Figure 8. It gave the most
stringent flux limit below 13 MeV, though it was still an order of magnitude
larger than DSNB theoretical expectations. The Borexino experiment used 278
tons of ultrapure liquid scintillator located in the underground hall C of
Gran Sasso, Italy. Thanks to the exceptionally low level of radiopurity
Borexino realized a low energy threshold. It led to the first discovery of
low-energy solar neutrinos such as 7Be, pep, pp, and CNO cycle. For the DSNB
search, neutrinos in a wide energy range from 1.8 to 16.8 MeV were searched
[97]. The flux upper limit with the exposure of 1.494 kton $\cdot$ yr is shown
in the Figure 8. The flux limit below 8.3 MeV was set only by Borexino.

Figure 8: The model-independent upper limits of electron antineutrino flux at
90% C.L. from various recent experiments, SK-I/II/III [88], SK-IV [91], SK-VI
[98], KamLAND [96], and Borexino in the case including atmospheric neutrino
background [97], overlayed with various DSNB theoretical predictions. In
particular, three models computed as part of this review are shown in color: a
Fermi-Dirac distribution with total energy $3.2\times 10^{53}$ erg and
temperature 4.1 MeV (FD+BH), Nakazato et al. [61] IMF-weighed spectrum
(N13+BH), and Fornax (2021)[56, 62] IMF-weighed spectrum (F21+BH). Here, we do
not assume a late-phase treatment. A Salpeter IMF is used, the black hole
fraction is fixed to be 23.6%, and the black hole model is from Nakazato et
al. [61] assuming a Shen EOS.

###  5.2 Backgrounds

As described in the previous section, the DSNB has not yet been discovered. It
is limited by background events. The DC method is effective for DSNB analysis
to search for IBD events, and it has been recently become available for the
water Cherenkov detectors as well as liquid scintillator detectors.
Nevertheless, there are still background events that can mimic IBD events. At
first, there are two major background sources: one is from spallation products
induced by cosmic muons, and the other is from atmospheric neutrinos. They are
background events common to the two types of detectors, but there are also
differences in the events observed in each type of detector. In addition,
reactor neutrinos can be background events in lower energy regions, and
accidental coincidence backgrounds, which mimic DC events, should be
considered in both types of detectors. For liquid scintillator detectors, fast
neutrons are also one of the backgrounds. In this section, these backgrounds
in the DSNB analysis are described.

####  5.2.1 Spallation products

Cosmic ray muons can penetrate detectors located deep underground, although
the rate is drastically reduced relative to sea level. These muons interact
with the nuclei in the detector and produce various radioactive isotopes
called ‘spallation products’. The energies of the spallation products, $\beta$
and/or $\gamma$, are similar to those of positrons from DSNBs. Most of these
events can be removed using temporal and spatial correlations with muons.
Nevertheless, long-lived products are not easy to remove since it is quite
difficult to identify their parent muon, and can eventually become background
events in DSNB analysis. In particular, 9Li, which undergoes beta decay (35%
of which emit one neutron) and has a higher production yield, is one of the
remaining background events.

####  5.2.2 Atmospheric neutrinos

When primary cosmic rays, which are mainly composed of protons, collide with
nuclei in the Earth’s atmosphere, several hadrons, such as pions and kaons,
are generated. Atmospheric neutrinos are generated when these hadrons decay.
The energy of atmospheric neutrinos are on the order of 100 MeV to PeV,
peaking at several hundred MeV. Particles generated by atmospheric neutrino
interactions in the detector have a wide range of energies. If they have a
reconstructed energy in the DSNB analysis region and neutrons are emitted
together, will constitute a background event. Both charged current (CC) and
neutral current (NC) interactions cause such events. Here, targets of neutrino
interaction in the detector are nuclei (oxygen for water, carbon for liqud
scintillator) and free protons.

As for the water Cherenkov detector, the most serious background is the
neutral current quasi-elastic (NCQE) interaction, which knocks out a nucleon
in oxygen nuclei. If the knocked-out nucleon is a neutron and the excited
nucleus generates gamma rays on deexcitation, it is indistinguishable from
DSNB signal because the gamma-ray energy is similar to that of positron in IBD
[99]. Accurate estimation of this background is not easy because of several
uncertainties in the nuclear models. In addition, other interactions, such as
an NC interaction with one pion ($\nu+N\rightarrow\nu+N\textquoteright+\pi$)
or a charged current quasi-elastic interaction (CCQE, $\nu+N\rightarrow
l+N\textquoteright$, where $N$ and $N\textquoteright$ are nucleons and $l$ is
a lepton), could be backgrounds. However, this is the case if neutrons are
emitted together.

For liquid scintillator detectors, estimating the background NC interaction
with carbon is also challenging due to its uncertainties. The most dominant
process is $\nu(\bar{\nu})+^{12}C\rightarrow\nu(\bar{\nu})+n+^{11}C+\gamma$,
but all possible interaction modes are considered in Ref. [95]. Compared to
NC, the contribution of CC is not much. In the case of CC, interaction with
free proton is dominant over that with carbon nuclei because of their cross
section [100]. Here, the muon antineutrino interaction is more likely to be
the background in the DSNB analysis energy region than electron antineutrinos
because a large fraction of the initial neutrino energy is spent to generate
muon and the observed energy is shifted to low.

####  5.2.3 Reactor neutrinos

Reactor neutrinos are produced by the beta decay of neutron-rich fission
fragments mainly from the following four isotopes: 235U, 238U, 239Pu, and
241Pu. The flux is calculated from reactor operating data such as thermal
power generation, fuel burn, shutdown, and fuel reload schedules. The energy
spectrum is calculated based on experimental results [101]. Most reactor
neutrinos have energies below 10 MeV, where the DSNB flux is still analyzed.

####  5.2.4 Accidental coincidence

In actual data acquisition, the prompt signal can be paired with a delayed
signal that is misidentified due to radioactive event or noise, which is
called ‘accidental coincidence’. This background event is estimated for each
experiment.

####  5.2.5 Fast neutrons

In liquid scintillator detector, fast neutrons produced by cosmic muons in the
surrounding rock and water and introduced into the detector can also be
background events. Prompt events can be mimicked by the scattering of neutron
on protons or carbon nuclei, followed by a delayed event when the neutron is
thermalized and captured on a proton or carbon nucleus.

###  5.3 Recent upgrade of Super-Kamiokande

In the summer of 2020, Super-Kamiokande moved to a new phase with the start of
SK-Gd, which drastically improves the detection sensitivity of DSNB by adding
gadolinium. As described above, neutron tagging in IBD is crucial for
background reduction in water Cherenkov detector. The original idea for this
purpose in Super-Kamiokande was proposed in 2004 [102]. Gadolinium has the
largest neutron capture cross section of all elements, giving, for example, a
neutron tagging efficiency of 90% at a mass concentration of only 0.1%. In
addition, the delayed gamma-ray energy is about 8 MeV in total, which is high
enough to be detected by Super-Kamiokande. After years of research and
development [103] and a tank refurbishment work with the main purpose of
repairing a water leak in 2018, 13 tons of gadolinium sulfate equivalent to
0.011% mass concentration were loaded in 2020 [104] as a first step, and this
phase is called SK-VI.

The first result using about 1.5 years of data from SK-VI appeared in [98].
This analysis did not assume any spectral shape of the astrophysical electron
antineutrino sources as well as the DSNB. Thanks to a significant increase in
neutron tagging efficiency $(35.6\pm 2.5\%)$ mainly with gadolinium capture
and low misidentification probability ($\sim$10-4), the IBD signal efficiency
was twice that of the previous pure water phase. The events observed in the
searched neutrino energy range of 9.3 MeV to 31.3 MeV were consistent with the
expected background, mainly due to atmospheric neutrinos and spallation
products like 9Li. Though no significant signal was discovered from this
result, its detection sensitivity was comparable to that of the previous pure
water phase, SK-IV, by only one-fifth of statistics. The flux upper limit with
34.0 kton $\cdot$ yr in SK-VI is shown in Figure 8.

In the summer of 2022, an additional 26 tons of gadolinium sulfate were loaded
into the detector as a second step. The loading work was successfully
completed in about a month. As a result, gadolinium equivalent to a mass
concentration of approximately 0.03% was introduced into SK, and this new
experimental phase is called SK-VII. The neutron tagging efficiency was
confirmed to be 1.5 times higher than in SK-VI, which was consistent with
expectation. In addition to the improving neutron tagging efficiency, this
phase will allow analysis with large statistics data assuming the spectral
shape of each DSNB model.

###  5.4 Experiments in the near future

####  5.4.1 JUNO

JUNO (Jiangmen Underground Neutrino Observatory) is the next generation
neutrino detector, which is located at Jiagmen in Guangdong province, China.
It is the largest liquid scintillator detector ever, with the primary goal of
accurately measuring neutrino oscillations from reactor antineutrinos. In
addition, it will be a pioneering experiment to observe the DSNB signal for
the first time together with SK-Gd in the next decade. After many years of
construction, it will be operational in 2023. The detector is 20 kton of
liquid scintilator with 17,612 20-inch and 25,600 3-inch photomultiplier tubes
which are installed in the gaps between the 20-inch photomultiplier tubes to
improve the energy resolution. The signal detection method is the same, i.e,
the DC technique with proton capture via IBD.

Backgrounds for the DSNB search to be considered are described above. First,
the $\bar{\nu}_{e}$ from reactor and atmospheric neutrinos are inevitable. The
energy region of reactor neutrinos is less than 10 MeV, while the energy
region of atmospheric neutrinos increases with the neutrino energy. Thus, the
search for is DSNB is in the energy region in between. Other backgrounds could
be long-lived isotopes due to muon spallation, especially 8He and 9Li. This is
because the $\beta$-n decay of these isotopes is very similar to the IBD
signal. Nevertheless, the energy of the prompt signal due to the beta decay is
relatively low, allowing this background to be ignored if an appropriate
energy threshold is set. External fast neutrons could also be a background in
liquid scintillator detectors. They are, however, expected to be removed by
appropriate fiducial volume cuts. As described above, estimation and reduction
of background due to atmospheric neutrino NC interaction with carbon is
challenging. The experimental group has devised various methods to reduce this
background and has demonstrated their effectiveness through simulation studies
[105].

####  5.4.2 Hyper-Kamiokande

Hyper-Kamiokande is the next generation water Cherenkov detector located 8 km
from the SK site. It is now under construction and will start the operation in
2027. It is based on a well-established technology with a fiducial volume of
187 ktons, which is 8.3 times larger than that of SK [106]. This huge volume
of the detector allows DSNB observations with large statistics as well as
other physics targets, thereby promising, for example, the observation of the
DSNB spectrum. The first phase uses pure water, thus, the energy threshold is
as high as about 16 MeV due to the effect of spallation background.
Nevertheless, if a DSNB signal with sufficient statistics is observed in the
higher energy region, for example, it is sensitive to the black-hole formation
rate and can provide information on the history of star formation and its
metallicity. In the future, if the gadolinium loading is realized, the energy
threshold for a DSNB search can be lowered to about 10 MeV. This would make it
possible to explore the history of supernova explosions back to an age with a
redshift of about 1. Thus, Hyper-Kamiokande has the potential to perform
neutrino astronomy and cosmology with DSNB observations.

##  6 Outlook and conclusions

This is an exciting era for neutrino astrophysics. The detection of
extraterrestrial neutrinos from an astrophysical origin associated with
stellar explosions is just around the corner. The diffuse supernova neutrino
background (DSNB) provides valuable information about past core collapse,
including the fraction of black-hole forming collapse and neutrino physics
through flavor oscillations or other exotic interactions over cosmological
baselines. While detecting the DSNB and studying supernova physics and
neutrino properties is a challenging goal, recent progress in neutrino
experiments makes this an achievable goal in the next decade. For instance,
the recent upgrade of the Super-Kamiokande detector has significantly reduced
background noise, bringing the first detection of DSNB within reach. This
review has gathered essential information on DSNB physics and the latest
updates from DSNB search experiments. In parallel, it is likely that we will
see in the next decades the second detection of neutrinos from a core-collapse
supernovae after SN 1987A that happened more than a quarter century ago.
Besides the mere excitement of detection, it will bring plenty of physics and
astrophysics to be studied using the rich multi-messenger observational data.
The coming decades therefore hold great promise for detecting and studying
supernova neutrinos, with potential exciting breakthroughs.

## Acknowledgments

This work was supported by MEXT KAKENHI Grant Numbers, JP20H05850, JP20H05861
(SA), NSF Grant No. AST-1908960 (NE, SH), No. PHY-2209420 (NE, SH), NSF Grant
No. PHY-1914409 (SH), the U.S. Department of Energy Office of Science under
award number DE-SC0020262 (SH), JSPS KAKENHI Grant Number JP22K03630 (SH),
20H00162 (YK) and World Premier International Research Center Initiative
(WPI), MEXT, Japan (SA, SH, and YK).

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