NUHEP-TH/20-08, FERMILAB-PUB-20-353-T

# Fundamental physics with the diffuse supernova background neutrinos

André de Gouvêa<EMAIL_ADDRESS>Northwestern University, Department
of Physics & Astronomy, 2145 Sheridan Road, Evanston, IL 60208, USA  Ivan
Martinez-Soler<EMAIL_ADDRESS>Northwestern University,
Department of Physics & Astronomy, 2145 Sheridan Road, Evanston, IL 60208, USA
Theory Department, Fermi National Accelerator Laboratory, P.O. Box 500,
Batavia, IL 60510, USA  Colegio de Física Fundamental e Interdisciplinaria de
las Américas (COFI), 254 Norzagaray street, San Juan, Puerto Rico 00901.
Yuber F. Perez-Gonzalez<EMAIL_ADDRESS>Northwestern University,
Department of Physics & Astronomy, 2145 Sheridan Road, Evanston, IL 60208, USA
Theory Department, Fermi National Accelerator Laboratory, P.O. Box 500,
Batavia, IL 60510, USA  Colegio de Física Fundamental e Interdisciplinaria de
las Américas (COFI), 254 Norzagaray street, San Juan, Puerto Rico 00901.
Manibrata Sen<EMAIL_ADDRESS>Northwestern University, Department of
Physics & Astronomy, 2145 Sheridan Road, Evanston, IL 60208, USA  Department
of Physics, University of California Berkeley, Berkeley, California 94720, USA

###### Abstract

The Universe is awash with tens-of-MeV neutrinos of all species coming from
all past core-collapse supernovae. These have never been observed, but this
state of affairs will change in the near future. In the less than ten years,
the Super-Kamiokande experiment, loaded with gadolinium, is expected to
collect dozens of events induced by the scattering of neutrinos from the
diffuse supernova neutrino background (DSNB). Next-generation projects,
including Hyper-Kamiokande and Theia, are expected to collect data samples
with hundreds of DSNB events after a decade of running. Here, we study
quantitatively how well the DSNB, including its energy spectrum, will be
measured by different current or upcoming large neutrino detectors. We analyze
the simulated data in order to estimate how well measurements of the DSNB can
be used to inform research topics in cosmology – including measurements of the
Hubble parameter – astrophysics – including the star-formation-rate – and
particle physics – including the neutrino lifetime and the possibility that
neutrinos are pseudo-Dirac fermions.

##  I Introduction

A core-collapse supernova (CCSN) emits almost all its binding energy in the
form of neutrinos. Observations of these neutrinos can provide crucial
information on both the dynamics of CCSNe Janka et al. (2007); Horiuchi et al.
(2018) and the properties of neutrinos, including the neutrino lifetimes Ando
(2004), magnetic moments Ando and Sato (2003); Balantekin et al. (2007); de
Gouvêa and Shalgar (2012), and the number of neutrino species Raffelt and Zhou
(2011). To date, we have only observed neutrinos from one CCSN: SN1987A Hirata
et al. (1987); Bionta et al. (1987), in the Large Magellanic Cloud. In spite
of the limited statistics, these provided priceless information and solidified
the current understanding of the dynamics of CCSNe.

Galactic CCSNe, however, are very rare; their rate is, on average, around one
to three per century Diehl et al. (2006). There is, however, an additional,
strongly related but continuous source of astrophysical neutrinos: the diffuse
supernova background (DSNB). The DSNB consists of neutrinos and antineutrinos
emitted cumulatively from all past CCSNe in the observable Universe (see
Lunardini (2016); Beacom (2010) for a detailed review). The DSNB is expected
to be, to a very good approximation, isotropic and time-independent. It can be
determined from the rate of CCSNe in our Universe Horiuchi et al. (2011) and
the flavor-dependent time-integrated neutrino spectra from CCSNe. We discuss
the different ingredients that go into simulating the DSNB in Section II and
provide a brief summary in the next paragraphs.

The CCSN rate can be determined from observations of the cosmic star-formation
rate (SFR), which can be deduced from measurements of the distributions of
stars as a function of their luminosities and masses Hopkins and Beacom
(2006); Horiuchi et al. (2009); Lunardini and Tamborra (2012). With these
observations, an analytical fit to the SFR can be obtained Yuksel et al.
(2008); Horiuchi et al. (2009). This method depends on numerous astrophysical
inputs, and is fraught with uncertainties Nakazato et al. (2015). The neutrino
flux from CCSNe is also not very well known, keeping in mind the very limited
statistics from SN1987A Hirata et al. (1987); Bionta et al. (1987). One has to
resort to large scale hydrodynamic simulations in order to estimate the energy
and flavor spectra of neutrinos from CCSNe Horiuchi et al. (2018). Different
groups agree, with varying degrees of precision, that the neutrino spectra are
quasi-thermal and can be described as pinched Fermi-Dirac spectra Keil et al.
(2003). Finally, the neutrino fluxes are processed, deep inside the CCSN, by
collective oscillations Duan et al. (2006); Hannestad et al. (2006); Fogli et
al. (2007); Dasgupta et al. (2009); Esteban-Pretel et al. (2007); Dasgupta et
al. (2008, 2012); Chakraborty et al. (2016); Dasgupta et al. (2017); Izaguirre
et al. (2017); Capozzi et al. (2019); Bhattacharyya and Dasgupta (2020);
Capozzi et al. (2020); Johns et al. (2020) and, a little further out, by
oscillations in the presence of ordinary matter with a decreasing number
density. The latter are described well by the Mikheyev-Smirnov-Wolfenstein
(MSW) resonant flavor conversion mechanism Wolfenstein (1978); Mikheev and
Smirnov (1985). Uncertainties in these different flavor conversion mechanisms
also translate into uncertainties in the DSNB flux.

Previous studies have tried to address the uncertainties that plague the DSNB.
A detailed study of the SFR Horiuchi et al. (2009), incorporating multiple
observations, confirmed the analytic fit and demonstrated that uncertainties
in the SFR translate into an uncertainty of $40\%$ in the DSNB fluxes. On the
neutrino-flavor-conversion side, a comprehensive study of the non-trivial
effects of collective oscillations and MSW transitions on the DSNB neutrino
flux have been performed in Chakraborty et al. (2008, 2011); Møller et al.
(2018). Overall, non-trivial qualitative progress will only be made with a
direct measurement of the DSNB.

Several publications have focussed on the prospects for the detection of the
DSNB in current and future generation experiments including Super-Kamiokande
(SK) Zhang et al. (2015), Hyper-Kamiokande (HK) Abe et al. (2018), the
Jiangmen Underground Neutrino Observatory (JUNO) An et al. (2016), and the
Deep Underground Neutrino Experiment (DUNE) Abi et al. (2020). The sensitivity
of the DSNB spectra to the local supernova-neutrino rate using a combination
of CCSNe and failed supernovae have been studied in Møller et al. (2018). The
strongest upper-limit on the DSNB flux, imposed by SK, hovers about a factor
of a few above current predictions Lunardini and Peres (2008); Zhang et al.
(2015). Enrichment with gadolinium (Gd) will improve SK’s sensitivity
drastically, and is predicted to lead to the detection of the DSNB with three-
sigma significance Beacom and Vagins (2004); Simpson (2019) in about five
years. Future experiments like HK Abe et al. (2018) and Theia Askins et al.
(2020), a very large, water-based liquid-scintillator detector, have very good
prospects for detecting the DSNB and, assuming a discovery in SK, measuring
some of its properties.

Detecting the DSNB would open up new avenues in multi-messenger astronomy. In
particular, it offers the unique opportunity to probe properties of the cosmos
using neutrinos (as opposed to photons). Given current and near-future
experimental capabilities, the sensitivity of neutrino observables is inferior
to that of photons but the probes are qualitatively distinct and, in some
circumstances, complementary. Measurements of the DSNB, for example, allow one
to “look back” at the Universe with neutrinos and constrain the properties of
its expansion. Instead, with precise knowledge of the expansion rate of the
Universe, one can use the DSNB observation to inform the SFR. And finally,
given external knowledge of the underlying cosmology and the SFR, one can
probe new neutrino properties with the DSNB. In sum, the detection of the DSNB
offers a unique way to pursue complementary avenues in particle astrophysics
and cosmology using only neutrinos as probes (for example, see Barranco et al.
(2018); Møller et al. (2018); Jeong et al. (2018); Yang et al. (2019);
Dasgupta et al. (2019); Bell et al. (2020)).

In this work, we explore some these avenues. We simulate the DSNB event-rate
in future neutrino detectors, particularly HK and Theia, and demonstrate how
the DSNB can be utilized to inform the parameters of the Standard Model of
Cosmology ($\Lambda$CDM) Aghanim et al. (2018), the SFR, or new neutrino
properties. Details of our simulations are provided in Sec. III. In Sec. IV,
we assume that there is no new physics beyond the Standard Model other than
nonzero neutrino masses and study how well one can probe $\Lambda$CDM given
our current knowledge of the SFR, and vice-versa. In Sec. V, we ask the
opposite question: given our current understanding of $\Lambda$CDM and the
SFR, how well can the DSNB probe new neutrino properties? We consider two
different scenarios: (i) neutrino decay, and (ii) pseudo-Dirac neutrinos. We
find that the DSNB can provide very nontrivial information in both cases. We
summarize our results in Sec. VI.

##  II Modelling the DSNB flux

The DSNB flux is a function of the rate of CCSNe $(R_{\rm CCSN})$. This, in
turn, depends on the history of star formation. An analytical fit of the
redshift evolution of the co-moving SFR yields Yuksel et al. (2008); Horiuchi
et al. (2009)

$\dot{\rho}_{*}(z)=\dot{\rho}_{0}\left[(1+z)^{-10\,\alpha}+\left(\frac{1+z}{B}\right)^{-10\,\beta}+\left(\frac{1+z}{C}\right)^{-10\,\gamma}\right]^{-1/10}\,,$
(II.1)

where $\dot{\rho}_{0}$ is an overall normalization factor and
$\alpha,\beta,\gamma$ are dimensionless exponents. The analytical fit to data
is essentially a broken power-law, with $B$ and $C$ labeling the functional
breaks at characteristic redshifts $z_{1}$ and $z_{2}$. They are defined as

$\displaystyle B$ $\displaystyle=$
$\displaystyle(1+z_{1})^{1-\alpha/\beta}\,,$ (II.2) $\displaystyle C$
$\displaystyle=$
$\displaystyle(1+z_{1})^{(\beta-\alpha)/\gamma}(1+z_{2})^{1-\beta/\gamma}\,.$
(II.3)

The redshift breaks are defined to occur at $z_{1}=1$ and $z_{2}=4$. A fit to
data from different astronomical surveys reveals
$\dot{\rho}_{0}=0.0178^{+0.0035}_{-0.0036}\,{\rm
M_{\odot}\,yr^{-1}\,Mpc^{-3}},\,\alpha=3.4\pm 0.2,\,\beta=-0.3\pm 0.2$ and
$\gamma=-3.5\pm 1$. The variability in the comoving SFR considered in Ref.
Horiuchi et al. (2009) takes into account the local rates estimated by
different astronomical surveys. The negative-ten factors in Eq. (II.1) are
chosen to smoothen the fit.

$R_{\rm CCSN}$ also depends on the initial mass function (IMF) of stars
$\psi(M)$, which gives the density of stars in a given mass range. The IMF is
assumed to have a power-law behavior; according to Salpeter, $\psi\propto
M^{-2.35}$ Salpeter (1955). With this information,

$R_{\rm
CCSN}(z)=\dot{\rho}_{*}(z)\frac{\int_{8}^{50}\psi(M)\,dM}{\int_{0.1}^{100}M\psi(M)\,dM}\,.$
(II.4)

The IMF in the numerator of Eq. (II.4) is integrated over the mass range of
$8\,M_{\odot}-50\,M_{\odot}$ to account for the stars that can undergo a core-
collapse, while the denominator accounts for all stars up to $100\,M_{\odot}$.
The fraction of stars undergoing a collapse to a SN corresponds to
$0.007/M_{\odot}$. Stars with masses greater than $50\,M_{\odot}$ lead to the
formation of a black hole Møller et al. (2018) and are neglected in this
analysis.

The diffuse differential neutrino flux, associated to an energy $E$ at the
Earth, is

$\Phi_{\nu}(E)=\int_{0}^{z_{\rm max}}\frac{dz}{H(z)}R_{\rm
CCSN}(z)F_{\nu}(E^{\prime})\,,$ (II.5)

where

$\displaystyle
H(z)=H_{0}\,\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}(1+z)^{3(1+w)}+(1-\Omega_{m}-\Omega_{\Lambda})(1+z)^{2}}$
(II.6)

is the Hubble function determined from the Friedmann equation. The Hubble
parameter is taken to be, unless otherwise noted, $H_{0}=67.36\,{\rm
km\,s}^{-1}\,{\rm Mpc}^{-1}$, in agreement with the best fit to the latest
Planck data Aghanim et al. (2018). $z_{\rm max}\equiv 5$ is the maximum
redshift up to which there is a reasonable amount of star formation. Here,
$\Omega_{m}$ and $\Omega_{\Lambda}$ refer to the matter and vacuum
contributions to the energy density, in units of the total energy density, and
$w$ is the equation-of-state parameter for the dark energy. $E^{\prime}$ is
the neutrino energy at the source, relative to which $E$ is redshifted,
$E=E^{\prime}/(1+z)$.

$F_{\nu}(E)$ is the supernova neutrino spectrum. Most of the neutrino flux is
emitted during the cooling phase of the supernova, which lasts for, very
roughly, ten seconds post-bounce. This phase is characterized by approximately
equal luminosities for all neutrino flavors so each neutrino species has, to a
good approximation, the same total energy $E_{\nu}^{\rm tot}/6$. Neutrino
energy spectra from these CCSNe are approximately thermal and can be described
by a non-degenerate time-integrated Fermi-Dirac distribution Beacom (2010),

$F_{\nu}(E)=\frac{E_{\nu}^{\rm
tot}}{6}\,\frac{120}{7\pi^{4}}\,\frac{E_{\nu}^{2}}{T_{\nu}^{4}}\,\frac{1}{e^{E_{\nu}/T_{\nu}}+1}\,.$
(II.7)

$T_{\nu}$ is a species-dependent quantity as it is set by the neutrino
interactions within the medium. Since temperatures inside the supernova allow,
to a good-enough approximation, only the electron-flavored neutrinos to
undergo charged-current interactions, the temperature hierarchy
$T_{\nu_{e}}<T_{\bar{\nu}_{e}}<T_{\nu_{x}}$ is generically expected Horiuchi
et al. (2018). Here $\nu_{x}$ refers to either $\nu_{\mu}$ or $\nu_{\tau}$ or
their antiparticles. The neutron richness of the supernova environment causes
the $\nu_{e}$ to interact more than the $\bar{\nu}_{e}$, accounting for the
lower value of $T_{\nu_{e}}$ relative to $T_{\bar{\nu}_{e}}$. These are the
spectra at the so-called neutrinospheres (one for each species
$\nu_{e},\bar{\nu}_{e},\nu_{x}$). Neutrino oscillations both inside the CCSN
and between the CCSN and the Earth lead to “mixed” spectra at the time of
detection. We will discuss our treatment of oscillation effects momentarily.

Figure 1: The DSNB flux, excluding oscillation effects, for each neutrino
species, as a function of the neutrino energy $E$ at the Earth.
$T_{\nu_{e}}=6.6\,{\rm MeV},\,T_{\bar{\nu}_{e}}=7\,{\rm MeV}$ and
$T_{\nu_{x}}=10\,{\rm MeV}$. The hatched region indicates current bounds from
Super-Kamiokande Zhang et al. (2015). The shading around the bands indicates
the corresponding uncertainty arising from the SFR. The regions of energy-
space where significant backgrounds are expected at the future facilities of
interest are indicated with a gray background.

Fig. 1 depicts the DSNB spectra as a function of $E$ for all three species of
neutrinos, discounting oscillation effects. The shaded regions around the
spectra indicate the corresponding uncertainty arising from the SFR. The
hotter spectra extend out to higher energies while the cooler spectra have a
more pronounced peak at lower energies. the hatched region indicates current
bounds from Super-Kamiokande Zhang et al. (2015). The regions of energy-space
where significant backgrounds are expected at the future facilities of
interest are indicated with a gray background. These will be discussed in
detail in the next section.

Neutrino-oscillation effects inside the CCSN are significant and diverse; they
depend, for example, on whether the neutrinos were produced earlier or later
in the explosion. Neutrino interactions with the medium and neutrino
interactions with other neutrinos lead to a rich oscillation pattern that is
still the subject of intense theoretical and phenomenological investigation
Duan et al. (2006); Hannestad et al. (2006); Fogli et al. (2007); Dasgupta et
al. (2009); Esteban-Pretel et al. (2007); Dasgupta et al. (2008, 2012);
Chakraborty et al. (2016); Dasgupta et al. (2017); Izaguirre et al. (2017);
Capozzi et al. (2019); Bhattacharyya and Dasgupta (2020); Capozzi et al.
(2020); Johns et al. (2020). Here, we ignore collective oscillations and
assume that a strong, adiabatic MSW effect dominates neutrino flavor evolution
for all energies of interest. We further assume that the neutrino mass
ordering is, unless otherwise noted, known to be normal so all neutrinos that
exit the neutrinosphere as $\nu_{e}$ will exit the CCSN as $\nu_{3}$ while all
antineutrinos that exit the neutrinosphere as $\bar{\nu}_{e}$ will exit the
CCSN as $\bar{\nu}_{1}$.***For the inverted neutrino mass ordering, still
allowed by current data Kelly et al. (2020), all neutrinos that exited the
neutrinosphere as $\nu_{e}$ would exit the CCSN as $\nu_{2}$ while all
antineutrinos that exited the neutrinosphere as $\bar{\nu}_{e}$ would exit the
CCSN as $\bar{\nu}_{3}$. The other states exit the CCSN as incoherent mixtures
of $\nu_{1}$, $\nu_{2}$, $\bar{\nu}_{2}$, and $\bar{\nu}_{3}$. In summary, we
assume an incoherent mixture†††In Sec. V.2 we explore the hypothesis that
neutrinos are pseudo-Dirac fermions and need to revisit this hypothesis more
carefully. of neutrino and antineutrino mass eigenstates arrives at the
surface of the Earth. Their energy distributions are given by Eq. (II.7) with
$T_{\nu_{3}}=T_{\nu_{e}}$, $T_{\nu_{2}}=T_{\nu_{1}}=T_{\nu_{x}}$,
$T_{\bar{\nu}_{1}}=T_{\bar{\nu}_{e}}$, and
$T_{\bar{\nu}_{2}}=T_{\bar{\nu}_{3}}=T_{\nu_{x}}$.

As we will discuss in the next section, we will concentrate on detectors that
are sensitive to the $\nu_{e}$ and $\bar{\nu}_{e}$ flavors. A mass eigenstate
$\nu_{i}$ or $\bar{\nu}_{i}$ ($i=1,2,3$) arriving at the Earth’s surface will
be detected as a $\nu_{e}$ or a $\bar{\nu}_{e}$, respectively, with
probability $P_{ei}$ and $\bar{P}_{ei}$. If the neutrinos do not travel a fair
amount of the Earth in order to get to the detector,
$P_{ei}=\bar{P}_{ei}=|U_{ei}|^{2}$ ($i=1,2,3$), where $U_{ei}$ are elements of
the leptonic mixing matrix. Instead, if the neutrinos have to travel a fair
amount of Earth in order to reach the detector, Earth matter effects need to
be taken into account. Assuming the DSNB flux is isotropic, these effects can
be included via the average probabilities (as a function of the neutrino
energy) $\langle P_{ei}\rangle$ and $\langle\bar{P}_{ei}\rangle$. We checked
that the impact of Earth matter-effects is safely negligible for our purposes
and have not included them here.

It turns out that neutrino oscillation effects have a small impact in the
results we discuss here. In the energy range of interest (see Fig. 1), the
different $F_{\nu}(E)$ for the different neutrino species are rather similar
given the large SFR uncertainties. Had they been exactly the same, oscillation
effects would have been completely absent so the impact of the oscillations,
very generically, are proportional to the differences among the initial
spectra. Since that these are small given the large uncertainties, oscillation
effects are also small.

##  III Detecting the DSNB

The DSNB flux is very small, concentrated around neutrino energies of order
tens of MeV. Large neutrino detectors sensitive to these neutrino energies
have to contend with several different sources of background Lunardini (2016)
that render the unambiguous detection of the DSNB a difficult task Zhang et
al. (2015). It is, however, widely expected that the next generation of
experiments will be both large and well-equipped enough to identify signal
from background to detect the DSNB in the next five to ten years Beacom and
Vagins (2004); Simpson (2019). Indeed, several upcoming or planed experiments
should be able to not only detect but also measure some of the properties of
the DSNB. These include, in the near future, Super-Kamiokande Simpson (2019)
and JUNO An et al. (2016) and, further down the road, the Hyper-Kamiokande Abe
et al. (2018) and Theia Askins et al. (2020) detectors.

In order to estimate the sensitivity to the DSNB, we compute the expected
event rate for each experiment as a function of the reconstructed energy.
Those events are distributed in an array of bins of reconstructed energy, the
size of the bin depending on the energy resolution of each proposed
experiment. Inside a given bin i, the expected total number of events is

$N_{i}=N_{\rm tar}T\int
dE^{r}dE^{t}\Phi_{\alpha}\sigma_{\alpha}\epsilon(E^{t},E^{r})+\text{Bkg}_{i}\,,$
(III.8)

where $\sigma_{\alpha}$ is the cross-section of the main detection channel,
$\alpha=e,\mu,\tau$ is a flavor index, $\Phi_{\alpha}$ is the flux of each
neutrino species at the detector, and $\epsilon(E^{t},E^{r})$ correlates the
true neutrino energy $E^{t}$ with the experimentally reconstructed energy
$E^{r}$. The integrals are over the values of $E^{r}$ that define the bin and
over all $E^{t}$. The normalization constants $N_{\rm tar}$ and $T$ correspond
to the number of targets in the detector and the running time, respectively.
Here, we assume 10 years of data taking unless otherwise noted. The
correlation matrix $\epsilon$ for each experiment was obtained by Montecarlo
integration. $\text{Bkg}_{i}$ is the number of expected background events in
the $i$th energy bin after all background-reduction information has been
applied. These also depend, of course, on the type of experiment.

The dominant background to detect the DSNB depends on the energy. For energies
above $\sim 40$ MeV, atmospheric neutrinos make up the majority of the
neutrino flux Battistoni et al. (2005). Atmospheric muons also contribute to
the background via muon-spallation Li and Beacom (2014); Abe et al. (2018). At
lower energies, neutrinos from the Sun or nearby nuclear reactors become a
significant component of the neutrino flux. For energies below $10$ MeV, these
are expected to overwhelm the DSNB.

We will concentrate on the detection of $\bar{\nu}_{e}$ via inverse beta-decay
(IBD) in water and liquid scintillator experiments and on the detection of
$\nu_{e}$ in liquid Argon experiments, discussed later. For neutrino energies
typical of the DSNB, the main detection channel for electron antineutrinos is
the IBD interaction with free protons, $\bar{\nu}_{e}+p\rightarrow n+e^{+}$
Strumia and Vissani (2003); unless otherwise noted, we ignore all
$\bar{\nu}_{e}$ subdominant detection channels in the experiments we will be
considering here. The identification of the positrons created in the IBD
interactions of the DSNB is challenging due to the large number of background
sources of electrons or positrons in the same energy range, around tens of
MeV. For energies below 10 MeV, the dominant irreducible background consists
of $\bar{\nu}_{e}$ from nuclear reactors. We assume those overwhelm the DSNB
for all experimental setups of interest. For higher energies, the reactor flux
is significantly smaller than the expected DSNB flux and the main background
sources are muon-spallation products, atmospheric $\bar{\nu}_{e}$ – this
background is irreducible – the decay of invisible muons produced in the
charged current interactions of atmospheric muon neutrinos and antineutrinos,
and neutral-current scattering of atmospheric neutrinos of all species. Some
of these backgrounds are, however, reducible if one is able to identify the
neutrons produced in the IBD process: neutron tagging. These neutrons are
associated with the IBD positron both in time and space. Efficient neutron
tagging is fundamental for the detection of $\bar{\nu}_{e}$ from the DSNB.

In what follows, we describe the main techniques proposed for the measurement
of the DSNB, focusing in the different detector technologies that will be used
by future experiments and projects.

_Water doped with gadolinium – SK and HK_ — Starting this calendar year,
Super-Kamiokande (SK), a 50 kton water Cherenkov detector running in Japan,
will be doped with gadolinium. The main purpose of the gadolinium doping is to
increase SK’s sensitivity to the DSNB Beacom and Vagins (2004). Other
immediate applications include precision measurements of the flux of
antineutrinos produced in nuclear reactors around SK Beacom and Vagins (2004);
de Gouvêa and Martinez-Soler (2020). In the future, Hyper-Kamiokande (HK),
another water Cherenkov detector proposed in Japan, with a fiducial volume of
187 kton Abe et al. (2018), is also expected to be doped with gadolinium.

In water, neutrons are absorbed by hydrogen, emitting a photon with $2.2$ MeV.
Using neutron tagging with hydrogen, SK has searched for the DSNB flux Zhang
et al. (2015). Remaining backgrounds were such that SK was restricted to look
for the DSNB in the energy window depicted in Fig. 1. Unlike hydrogen,
gadolinium has a large cross-section for capturing neutrons and these are
associated with the emission of $8$ MeV of photons. Both of these facts
increase the neutron-tagging capabilities of large water Cherenkov detectors.
The impact of the neutron tagging with Gd is expected to reduce the background
signal below that of the expected DSNB flux for energies between $10$ MeV and
$~{}40$ MeV. The reduction is particularly significant for the muon-spallation
background, and only spallation nuclei created together with a neutron, like
9Li, may still be misidentified as an IBD signal. It is expected that with the
Gd-doping, SK will be sensitive to the DSNB for antineutrino energies
immediately above the reactor background ($E\gtrsim 10$ MeV). For antineutrino
energies above, roughly, 40 MeV, the invisible antimuon-decay background
overwhelms the DSNB signal.

In order to estimate the sensitivity of SK and HK to the DSNB flux, we compute
the number of events for neutrinos in the energy range from $10$ MeV to $50$
MeV. We distributed those events in energy bins of $8$ MeV; the size of the
bins is two times the uncertainty in the energy reconstruction at the maximum
energy considered. We assume the energy resolution to be
$\sigma_{E}=0.6\sqrt{E/{\rm MeV}}$, corresponding to an uncertainty of $20\%$
at 10 MeV Abe et al. (2018) for both SK and HK. Fig 2 (left) depicts the
number of events expected for HK (purple) together with the contribution of
the different backgrounds. The shading indicates the uncertainly associated
with the SFR.

Figure 2: Number of events expected for HK (left) and Theia (right) in 10
years, from different sources. The DSNB is in purple, this shading indicating
the corresponding modeling uncertainty, while different background sources
correspond to differently colored histograms. For HK, background sources are
reactor antineutrinos (red), atmospheric $\bar{\nu}_{e}$ (cyan) and the decay
of invisible atmospheric-muons (green). For Theia, the background sources are
atmospheric $\bar{\nu}_{e}$ (cyan), neutral-current atmospheric neutrino
scattering on carbon (orange) and reactor antineutrinos (red). Some of these
backgrounds are reducible. See text for details.

_Liquid Scintillator (LS) – JUNO_ — Organic liquid scintillators are another
popular medium for the detection of electron antineutrinos with tens of MeV of
energy via IBD. JUNO is a $20$ kton LS detector under construction in China
and expected to be the largest LS detector for the foreseeable future. Its
main mission is to precisely measure the flux of electron antineutrinos from
nearby nuclear reactors. The nuclear power plants at Taishan and Yanjiang,
located $53$ km from JUNO, limit the detection of the DSNB flux to energies
above $10$ MeV. At higher energies, the main backgrounds come from atmospheric
muons and neutrinos. Backgrounds from muon-spallation products in LS detectors
dominate below $10$ MeV and are less significant in the energy range of
interest. In addition to the charged-current (CC) interactions of atmospheric
$\bar{\nu}_{e}$ (irreducible background), the neutral-current (NC)
interactions of high-energy atmospheric neutrinos with the carbon nuclei can
lead to the emission of a neutron and to an excited nucleus that will emit
$\gamma$-rays as it de-excites Gando et al. (2012). That signal will mimic the
IBD events of interest. The NC background can be reduced using timing
information of the de-excitation of the carbon nucleus and the shape of the
signal Möllenberg et al. (2015). LS is more efficient in the identification of
muons than water Cherenkov experiments, thanks to the detection of the Michel
electron associated to the muon decay at rest and to more information on the
shape of the event. This allows one to effectively eliminate all backgrounds
associated to muons interacting or decaying inside the detector. Muons that
interact outside the detector, however, can produce fast neutrons that can
penetrate the detector volume and interact with protons. The recoil of the
proton and the absorption of the neutron will lead to a signal that can mimic
the IBD. A shape-analysis and reduction in the fiducial volume – the short
penetration length of the neutrons implies that those events are concentrated
towards the boundary of the detector – will reduce this background.

When simulating the JUNO detector, we consider a fiducial volume of 17 kton.
For energy resolution, we consider an uncertainty $\sigma_{E}=0.03\sqrt{E/{\rm
MeV}}$. We consider the DSNB flux from $10$ MeV to $50$ MeV, distributing the
events in bins of $1$ MeV.

_Water-based Liquid Scintillator (WbLS) – Theia_ — Some liquid scintillating
materials can be dissolved in water. The detection material, therefore,
consist of a solution of $1\%-10\%$ liquid scintillator in water. Theia is a
proposal for one of the four modules in the DUNE far detector complex and
plans to use WbLS as its detector material. There are two proposals for the
size-configuration, 25 kton and 100 kton. The results presented here are based
on the larger realization. The advantage of using this kind of target material
is the possibility to combine the reconstruction techniques developed for
water Cherenkov and liquid scintillator detectors. In a nutshell, water
Cherenkov detectors are good at reconstructing event topologies whereas the
liquid scintillator provides precise information on the energy of the event.
In addition to the irreducible backgrounds – neutrinos from reactors at low
energies (below $\sim 10$ MeV) and atmospheric antineutrinos at high energies
(above $\sim 40$ MeV) – similar backgrounds to those at LS detectors will
affect WbLS detectors. In the energy region of interest, the dominant
backgrounds include backgrounds of atmospheric origin including muon-
spallation 9Li, the NC interactions of atmospheric neutrinos, or the fast
neutrons created outside the detector that penetrate and interact elastically
with free protons. Studying the topology and the time distribution of each
event, together with the ratio between the Cherenkov and scintillation light,
WbLS detector appear to be capable of eliminating most of these sources of
background, reducing remaining sources by more than one order of magnitude.
The sources that cannot be eliminated are the atmospheric CC $\bar{\nu}_{e}$
and NC neutrino interactions Orebi Gann (2015); Askins et al. (2020).

When simulating events at Theia, we consider a fiducial volume of $80$ kton
and $6.2\times 10^{33}$ targets. We compute the number of events in the energy
range from $10$ MeV to $50$ MeV using bins of $1$ MeV. For energy resolution,
we assume $\sigma_{E}=0.03\sqrt{E/\text{MeV}}$. The expected event
distribution (purple) as well as the main background expectations are depicted
in Fig. 2 (right). The energy resolution for Theia will depend on the fraction
of scintillator dissolved in the water. The value we use here corresponds to
the most optimistic expected energy resolution.

_Liquid Argon (LAr) – DUNE_ — Liquid-argon detectors are currently the subject
of intense experimentation. The DUNE far detector Abi et al. (2020), which
will be ready before the end of decade, is expected to be the largest such
detector for the foreseeable future with 40 ktons of LAr. LAr is mainly
sensitive to the $\nu_{e}$ component of the DSNB flux. These are measured via
neutrino absorption in 40Ar, yielding an electron and an excited state of
potassium, $\nu_{e}+^{40}{\rm Ar}\rightarrow^{40}{\rm K}^{*}+e^{-}$. In the
de-excitation process, the potassium nucleus emits a cascade of photons. Both
the electron and the photons are measured as MeV blips in the LAr. The
dominant backgrounds here – mostly irreducible – include the flux of $\nu_{e}$
from the atmosphere, for energies above $40$ MeV, and the solar neutrinos, for
energies below $16$ MeV.

When simulating events in the DUNE far detector, we consider a fiducial volume
of $40$ ktons. To simulate the interaction of the DSNB flux with LAr, we used
MARLEY Gardiner (2018), a Montecarlo event generator that provides the final
states and their energies after the neutrino interaction for a given neutrino
energy. We assume that electrons and photons are observed if their energies
are above $2$ MeV. Regarding the energy resolution, we assume the same
precision demonstrated in previous LAr experiments Amoruso et al. (2004),
$\sigma_{E}=0.11\sqrt{E/\text{MeV}}+0.2(E/\text{MeV})$. The expected events
are distributed in energy bins of 5 MeV.

For the analysis of all simulated data, we include systematic uncertainties
associated to the DSNB flux in the form of a 40% uncertainty in the overall
normalization and include uncertainties associated to the current knowledge of
the leptonic mixing matrix from the results of the global fit in Ref. Esteban
et al. (2019). The latter have a negligible impact in our results, presented
in the next two sections.

##  IV Cosmology and astrophysics potential of the DSNB

Eq. (II.5) reveals that the DSNB fluxes (different flavors and energies)
broadly depend on three input quantities:

  1. 1.

the rate of core-collapse supernovae $R_{\rm CCSN}$, which in turn depends on
the SFR. This depends on various astrophysical inputs, as discussed earlier;

  2. 2.

the cosmological history of the Universe, through the parameters $\Omega_{m}$,
$\Omega_{\Lambda}$, and $H_{0}$, assuming the $\Lambda$CDM model;

  3. 3.

the spectra of the emitted neutrinos $F_{\nu}(E)$.

These three ingredients are entwined in the observables related to the DSNB we
can hope to measure in the future, i.e., energy spectra inside a limited
energy window for different neutrino species. Assuming external knowledge on
subsets of these parameters, however, measurements of the DSNB can be used to
inform the complementary subset in a nontrivial way.

In this section, we look at the first two directions. We assume that there is
no exotic physics in the neutrino sector, i.e., the only new physics we assume
is that neutrinos have nonzero masses and oscillate, so the input that goes
into $F_{\nu}(E)$ is as described in Eq. (II.7) and Sec. II. In Sec. IV.1, we
assume that the SFR is indeed given by Eq. (II.1) and constrained as discussed
in Sec. II, and try to constrain some of the different parameters of
$\Lambda$CDM cosmology using the simulated events. In Sec. IV.2, we ask the
reverse question: if cosmology is indeed governed by $\Lambda$CDM with the
different parameters well-determined by Planck Aghanim et al. (2018) and other
cosmic surveys, how well can we constrain the SFR parameters?

###  IV.1 Probing $\Lambda$CDM Cosmology with the DSNB

The DSNB, composed of neutrinos coming from all possible CCSNe at redshifts
$z\lesssim 5$, is sensitive to the low redshift expansion of the Universe.
Therefore, measurements of the DSNB serve as unique tools to constrain the
underlying six-parameter $\Lambda$CDM cosmology using neutrinos. While, for
example, measurements of different properties of the cosmic microwave
background (CMB) already provide excellent measurements of these parameters
Aghanim et al. (2018), these measurements are essentially properties of the
high-redshift $(z\simeq 1100)$ Universe. The DSNB offers a very distinct probe
of the same parameters at low redshifts. In particular, with the tension
between the extracted value of the Hubble parameter $H_{0}$ inferred from CMB
data Aghanim et al. (2018) and those from low redshift measurements of Type Ia
supernovae Riess et al. (2016); Bernal et al. (2016); Birrer et al. (2019);
Riess et al. (2019), an independent, low-redshift measurement of $H_{0}$ with
neutrinos from CCSNe would play a role in resolving the tension. While the
Planck measurements are expected to be much more precise owing to the high
statistics of CMB photons, future measurements of the DSNB offer an
intriguing, albeit, a lot less precise, way of looking at the early Universe
using neutrinos.

Figure 3: $\Delta\chi^{2}$ (relative to the minimum value) as a function of
the Hubble parameter $H_{0}$ (left) and the equation of state parameter for
the dark energy $w$ (right), for ten years of simulated data at Theia (blue)
and HK enriched with Gd (green). The vertical bands in the left-panel indicate
the current bound from CMB observations by Planck (gray) Aghanim et al. (2018)
and observations at low redshifts (orange) Riess et al. (2016); Bernal et al.
(2016); Birrer et al. (2019); Riess et al. (2019). The vertical band in the
right-panel indicates the current bound from Ref. Aghanim et al. (2018).

The left panel of Fig. 3 depicts the sensitivity of HK and Theia to the Hubble
parameter, $H_{0}$, measured using the DSNB. Since $H_{0}$ controls only the
absolute normalization to the DSNB flux, it is more sensitive to the number of
events than their energy distribution. In the fit, we assume the parameters
associated with the SFR and neutrino mixing are known perfectly and agree with
their current best-fit values. As discussed earlier, the 40% systematic
uncertainty assumed for the overall neutrino flux partially takes care of
uncertainties associated to the SFR and neutrino mixing. Both HK and Theia can
rule out, at more than the two sigma level, $H_{0}<10\,{\rm km\,s}^{-1}\,{\rm
Mpc}^{-1}$ and can hence confirm that the universe is expanding. On the other
hand, very large values of $H_{0}$ can be ruled out at more than the three-
sigma level. As advertised, DSNB measurements are not competitive with those
using photons at either high Aghanim et al. (2018) or low redshifts Riess et
al. (2016); Bernal et al. (2016); Birrer et al. (2019); Riess et al. (2019).
Nonetheless, it is remarkable that one can obtain a nontrivial, independent
measurement of $H_{0}$ using the DSNB flux.‡‡‡Strictly speaking, we are
measuring the impact of the expansion rate of the universe on the propagation
of neutrinos between the source and the detector. In our analysis, we assume
that these effects are divorced from the impact of the expansion rate when it
comes to determining the SFR, for example. The figure reveals that after ten
years, both HK and Theia would be able to measure $H_{0}$ at the 40% level.
Given the 40% systematic uncertainty on the expected flux assumed here, this
means the ten-year result is already systematics dominated. A significantly
more precise measurement would require qualitatively better understanding of
the theoretical expectations for the DSNB flux.

The DSNB signal is not very sensitive to the energy density parameters
$\Omega_{m,\Lambda}$; it can, however, be used to constrain the Dark Energy
equation-of-state parameter $w$. The right panel Fig. 3 depicts the
sensitivity of HK and Theia to $w$. Assuming the true value of $w=-1$
(cosmological constant), both HK and Theia can exclude $w\gtrsim 0.5$ at
around the three-sigma level. Due to higher statistics, HK does a marginally
better job than Theia.

###  IV.2 Probing the star-formation rate with the DSNB

One of the major inputs that go into calculating the DSNB spectra is the rate
of CCSNe in the observable Universe. $R_{\rm CCSN}$ is governed by the star-
formation ratio, given by Eq. (II.1). Data from different astronomical surveys
provide key insights into the cosmic SFR up to redshifts $z\simeq 6$ Hopkins
and Beacom (2006); Horiuchi et al. (2009); Lunardini and Tamborra (2012). The
uncertainties arising from these surveys translate into uncertainties in the
fit parameters $(\dot{\rho}_{0},\alpha,\beta,\gamma)$, as outlined in Sec. II
(also see Table I of Ref. Horiuchi et al. (2009)).

It is interesting to consider the prospects of constraining the SFR parameters
through the detection of the DSNB neutrinos in HK Mathews et al. (2014) and
Theia Askins et al. (2020). A recent discussion of the capabilities of HK can
be found in Ref. Riya and Rentala (2020). We simulate events in these
detectors, assuming the SFR is described by the current best-fit values of
$(\dot{\rho}_{0},\alpha,\beta,\gamma)$, and allow these parameters to vary one
at a time when analyzing the simulated data. In the fit, we assume the
parameters associated with the expansion rate of the universe and neutrino
mixing are known perfectly and agree with their current best fit value. The
same is true of the SFR parameters not under investigation. As discussed
earlier, the 40% systematic uncertainty assumed for the overall neutrino flux
partially includes the uncertainties associated to the SFR, cosmological
parameters and neutrino mixing. We find that the DSNB measurements are most
sensitive to $\dot{\rho}_{0}$ and $\alpha$, and these parameters can be non-
trivially constrained. Fig. 4 depicts the $\Delta\chi^{2}$ as a function of
$\dot{\rho}_{0}$ (left panel), and $\alpha$ (right panel). $\dot{\rho}_{0}$
functions as an overall normalization, and both HK and Theia have similar
sensitivity to it, as we observed in the $H_{0}$ discussion in the previous
subsection. On the other hand, when it comes to constraining $\alpha$, HK has
slightly better sensitivity than Theia. The bounds arising from fits to the
astronomical survey data, presented in Sec. II, are depicted in gray. At the
two-sigma level, the results obtained from the measurement of the DSNB are
almost competitive with those obtained from astronomical surveys.

Figure 4: $\Delta\chi^{2}$ (relative to the minimum value) as a function of
$\dot{\rho}_{0}$ (left) and $\alpha$ (right) for HK with Gd (green), and Theia
(blue). The vertical shaded regions indicate the current allowed range for the
relevant parameters, obtained from fits to astronomical observations Hopkins
and Beacom (2006); Horiuchi et al. (2009). See text for details.

We also studied the sensitivity of the DSNB signal to $\dot{\rho}_{0}$ and
$\alpha$ simultaneously. Fig. 5 depicts the allowed contours in the
$\dot{\rho}_{0}-\alpha$ plane for HK (left panel) and Theia (right panel).
Both HK and Theia exhibit a clear correlation between these two SFR
parameters. With access to more data, one can expect the contours to shrink,
especially in the $\alpha$ direction. As before, in the fit, all other
parameters are held fixed.

Figure 5: Allowed regions of the $\dot{\rho}_{0}$ vs $\alpha$ parameter space,
from ten years of HK with Gd (left) and Theia (right), at the $1,2,3-\sigma$
confidence levels. See text for details.

##  V Constraining exotic neutrino physics with the DSNB

In the previous section, we demonstrated how the DSNB can be used to put
constraints on cosmological as well as astrophysical parameters, assuming one
has a clear understanding of the underlying neutrino physics. However, one
might turn the argument around and use the DSNB as a probe of exotic physics
in the neutrino sector. This, of course, requires knowledge of the underlying
cosmological model and the star-formation rate. In this section, we assume
that the underlying cosmology is governed by $\Lambda$CDM and that the vacuum
and matter energy density fractions are well-constrained by Planck Aghanim et
al. (2018) and other cosmic surveys. Furthermore, we assume the SFR is
governed by Eq. (II.1) and that the phenomenological parameters are within the
range quoted in Sec. II.

One can use the naturally long baseline offered by the DSNB to constrain
“slow” neutrino dynamical processes. These include testing the hypothesis –
from Standard-Model-only interactions and massive neutrinos – that the
neutrino lifetime is much longer than the age of the Universe, and looking for
new neutrino oscillation-lengths associated to tiny new mass-squared
differences. The latter searches allow one to constrain the hypothesis that
massive neutrinos are pseudo-Dirac fermions.

###  V.1 Probing neutrino decays with the DSNB

Massive neutrinos can decay into lighter ones. Within the SM, these decays
take place via weak electromagnetic-dipole-diagrams and weak box-diagrams, and
the expected lifetimes are much longer than the age of the Universe. However,
if the neutrino has interactions beyond those in the SM then it can decay
faster. Such non-radiative decays of the neutrino have been studied
extensively, in the context of terrestrial and astrophysical neutrinos
Berezhiani et al. (1992); Fogli et al. (1999); Choubey et al. (2000); Lindner
et al. (2001); Beacom and Bell (2002); Joshipura et al. (2002); Bandyopadhyay
et al. (2003); Ando (2004); Beacom et al. (2004); Berryman et al. (2015);
Picoreti et al. (2016); Frieman et al. (1988); Mirizzi et al. (2007);
Gonzalez-Garcia and Maltoni (2008); Maltoni and Winter (2008); Baerwald et al.
(2012); Broggini et al. (2012); Dorame et al. (2013); Gomes et al. (2015);
Abrahão et al. (2015); Coloma and Peres (2017); Choubey et al. (2018); de
Salas et al. (2019); Aharmim et al. (2019); Funcke et al. (2019). Mostly
model-independent bounds on the decays of $\nu_{2}$, using the ${}^{8}{\rm B}$
solar neutrino data, constrain $\tau_{2}/m_{2}>10^{-3}\,{\rm s/eV}$ Berryman
et al. (2015); Aharmim et al. (2019). Similarly, one can use the ${}^{7}{\rm
Be}$ and low energy $pp$ neutrino data Bellini et al. (2014) to constrain
$\tau_{1}/m_{1}>10^{-4}\,{\rm s/eV}$ Berryman et al. (2015). The tiny
magnitude of $\theta_{13}$ makes it very difficult to use solar neutrinos to
constrain the $\nu_{3}$ lifetime. Bounds on $\tau_{3}$ from atmospheric
neutrino data are much weaker, $\tau_{3}/m_{3}>10^{-10}\,{\rm s/eV}$ Gonzalez-
Garcia and Maltoni (2008); Gomes et al. (2015). Astrophysical bounds arising
from the observation of neutrinos from SN1987A Kachelriess et al. (2000) are
quite uncertain, and cannot be treated at the same level of accuracy.

Model-dependent bounds are somewhat stronger. Recently, for example, the
authors of Funcke et al. (2019) proposed model-dependent bounds on
$\tau_{3}/m_{3}>2.2\times 10^{-5}\,{\rm s/eV}$ using KamLAND solar neutrino
data. It is also expected that next-generation neutrino oscillation
experiments like JUNO An et al. (2016) and DUNE Acciarri et al. (2016) can do
a better job constraining invisible $\nu_{3}$ decays Abrahão et al. (2015).
Astrophysical neutrinos have longer baselines, and hence can be used to
constrain longer lifetimes. Constraints from the observation of ultrahigh-
energy neutrinos by ICECUBE are potentially stronger, but they suffer from
uncertainties in the neutrino flavor composition Beacom et al. (2003);
Bustamante et al. (2017); Denton and Tamborra (2018); Abdullahi and Denton
(2020). Measurements of the cosmic microwave background can be used to put
very strong bounds on the neutrino lifetimes, of order $10^{8}$ s Escudero and
Fairbairn (2019); Chacko et al. (2019), but these are indirect probes of
neutrino decay in the sense that one is using information on the expansion
rate of the universe to infer whether the neutrino component of the universe
in changing.

Typically, neutrino flavor evolution in a supernova is very involved and is
marred by the details of collective flavor oscillations. One can circumvent
some of these complications by concentrating on the neutrinos emitted in the
shock-breakout epoch Ando (2003, 2004) that are usually free from such
effects. The impact of neutrino decays in current and forthcoming experiments
was recently explored by some of the authors in de Gouvêa et al. (2020).
Depending on the mass-ordering, neutrino decay can either enhance or deplete
the neutrino flux in an observable manner. It was inferred that one can
constrain neutrino lifetimes as long as $\tau/m\sim 10^{5}\,{\rm s/eV}$ de
Gouvêa et al. (2020) from the neutronization-burst neutrino spectra. These
bounds are model-dependent, and depend on whether the neutrinos are Dirac or
Majorana fermions.

One can extend the same line of argument for the DSNB. Here the typical
baseline is even longer than what is expected of an observable individual
CCSN. Effects of neutrino decay on the DSNB signal were studied in Ando
(2003); Fogli et al. (2004) where it was concluded that such decays could be
observable for neutrino lifetimes as long as $\tau/m\sim 10^{10}\,{\rm s/eV}$.
The bounds quoted in these works were relatively simple estimates that did not
take into account realistic detector-related effects, including systematics
and the effect of finite detector energy resolution. In this section, we
revisit the sensitivity of measurements of the DSNB to the neutrino lifetime
and calculate the signals that can be observed in current and near-future
generation detectors. The specifics are, of course, model-dependent, and also
depend on whether neutrinos are Dirac or Majorana fermions de Gouvêa et al.
(2020). For concreteness, we consider a scenario where the neutrinos are
Majorana fermions and interact with an almost-massless scalar $\varphi$. The
interaction is described by

$\mathcal{L}\supset\frac{f_{ij}}{2}(\nu_{L})_{i}(\nu_{L})_{j}\varphi+{\rm
H.c.}\,.$ (V.9)

At the tree level, this operator allows a heavy neutrino $\nu_{j}$ to decay
into a lighter neutrino $\nu_{i}$, and $\varphi$. For concreteness, we
consider the scenario where the heaviest neutrino decays into the lightest
one, i.e., if the neutrino mass ordering is normal (inverted), a
$\nu_{3}\,(\nu_{2})$ decays into a $\nu_{1}\,(\nu_{3})$. Although the neutrino
mass ordering is still not known, global fits of neutrino oscillation data
favor the normal mass ordering Esteban et al. (2019) (see, however, Kelly et
al. (2020)), which will be assumed throughout. Results for the inverted mass
ordering are qualitatively similar.

Assuming the $\nu_{1}$ is massless, the rate of decay is given by

$\Gamma(E_{3})=2\times\frac{f^{2}m_{3}^{2}}{64\pi E_{3}}\,,$ (V.10)

where the factor of 2 arises because both the left helicity (L) and right
helicity (R) final states contribute. Depending on whether the daughter
neutrinos have the same helicity as the parent (helicity-conserving–h.c.) or
different helicity (helicity-flipping–h.f.), the energy distribution of the
daughter particles is,

$\displaystyle\psi_{\rm
h.c.}(E_{3},E_{1})\equiv\frac{1}{\Gamma}\,\frac{d\,\Gamma}{d\,E_{1}}$
$\displaystyle\propto$
$\displaystyle\frac{2E_{1}}{E_{3}^{2}}\qquad\qquad\qquad\,\,{\rm
for\,\,}\nu_{3_{L}}\rightarrow\nu_{1_{L}}+\varphi_{0}\,\,,$
$\displaystyle\psi_{\rm
h.f.}(E_{3},E_{1})\equiv\frac{1}{\Gamma}\,\frac{d\,\Gamma}{d\,E_{1}}$
$\displaystyle\propto$
$\displaystyle\frac{2}{E_{3}}\left(1-\frac{E_{1}}{E_{3}}\right)\qquad{\rm
for\,\,}\nu_{3_{L}}\rightarrow\nu_{1_{R}}+\varphi_{0}\,\,.$ (V.11)

Since we are assuming the neutrinos are Majorana neutrinos, the “wrong”
helicity neutrinos act as what we typically refer to as “antineutrinos” (and
vice-versa), and will interact with the detector even in the limit where the
daughter neutrino is massless. On the other hand, for Dirac neutrinos, the
“wrong”-helicity neutrinos (right-handed neutrino states and left-handed
antineutrino states) are essentially inert.

To calculate the daughter $\nu_{1}$ flux from the decay of the $\nu_{3}$, one
needs to compute a Boltzmann transfer equation for $\nu_{1}$, involving terms
which can create or destroy $\nu_{1}$. The flux of the mass eigenstates
arriving at the Earth is

$\displaystyle\Phi_{\nu_{3}}(E)$ $\displaystyle=$
$\displaystyle\int_{0}^{z_{\rm max}}\
\frac{dz^{\prime}}{H(z^{\prime})}\,R_{\rm
CCSN}(z^{\prime})\,F_{\nu_{3}}\left(E(1+z^{\prime})\right)e^{-\Gamma(E)\zeta(z^{\prime})}$
$\displaystyle\Phi_{\nu_{2}}(E)$ $\displaystyle=$
$\displaystyle\int_{0}^{z_{\rm max}}\
\frac{dz^{\prime}}{H(z^{\prime})}\,R_{\rm
CCSN}(z^{\prime})\,F_{\nu_{2}}\left(E(1+z^{\prime})\right)$
$\displaystyle\Phi_{\nu_{1}}(E)$ $\displaystyle=$
$\displaystyle\int_{0}^{z_{\rm max}}\ \frac{dz^{\prime}}{H(z^{\prime})}R_{\rm
CCSN}(z^{\prime})\,F_{\nu_{1}}\left(E(1+z^{\prime})\right)+$
$\displaystyle\int_{0}^{z_{\rm max}}\
\frac{dz^{\prime}}{H(z^{\prime})}\int_{E(1+z^{\prime})}^{\infty}\,dE^{\prime}\,\bigl{[}\Phi_{\nu_{3}}(E^{\prime})\,\Gamma(E^{\prime})\,\psi_{\rm
h.c.}(E^{\prime},E(1+z^{\prime}))+\Phi_{\bar{\nu}_{3}}(E^{\prime})\,\Gamma(E^{\prime})\,\psi_{\rm
h.f.}(E^{\prime},E(1+z^{\prime}))\bigr{]}\,,$

where
$\zeta(z)=\int_{0}^{z}dz^{\prime}\,H^{-1}(z^{\prime})(1+z^{\prime})^{-2}$, and
$z_{\rm max}=5$. Similar expressions hold for the antineutrinos. For a
detailed derivation of these transfer equations, see Fogli et al. (2004).

Using this formalism, we explore the consequences of neutrino decay on the
DSNB flux. Fig. 6 depicts the expected neutrino spectra in HK (left) and Theia
(right) for three different lifetimes for the heaviest mass eigenstate,
$\nu_{3}$. As expected, the number of events in each bin increases if the
neutrino decays faster. While HK profits from larger statistics, Theia has
better energy resolution and is more sensitive to distortions in the spectral
shape.

Figure 6:  Expected event rates as a function of the reconstructed positron
energy after ten years of data-taking in HK (left) and Theia (right) for
different values of the heaviest neutrino lifetime. See text for details.

We consider the data to be consistent with the hypothesis that the neutrinos
do not decay, and perform a simple $\chi^{2}$ analysis to estimate the bounds
forthcoming experiments can set on the neutrino lifetime. Fig. 7 depicts
$\Delta\chi^{2}$ as a function of the lifetime over mass $\tau_{3}/m_{3}$ for
some of the current and forthcoming experiments sensitive to the DSNB. Near-
future experiments like SK and JUNO (and DUNE), are expected to observe only a
few events and hence are not very sensitive to a finite $\nu_{3}$ lifetime. HK
(doped with Gd) and Theia, on the other hand, are sensitive, at more than the
two-sigma level, to lifetimes as long as $10^{9}\,{\rm s/eV}$. In the fit, we
assume the parameters associated with the SFR, the time evolution of the
universe, and neutrino mixing are known perfectly and agree with their current
best fit value. As discussed earlier, the 40% systematic uncertainty assumed
for the overall neutrino flux partially takes care of uncertainties associated
to the SFR, cosmology, and neutrino mixing. This sensitivity, while somewhat
limited, rivals some of the best bounds on neutrino lifetimes, and would
complement bounds inferred indirectly from cosmology Escudero and Fairbairn
(2019); Chacko et al. (2019). The lack of higher statistics prevents us from
getting a higher sensitivity to neutrino lifetimes even in these experiments.
Note that these sensitivity estimates are around one order of magnitude
smaller than those quoted in Fogli et al. (2004) that are less detailed when
it comes to more specific experimental issues, including backgrounds and the
energy resolution at different future experiments. More statistics leads to
better sensitivity. After 20 years of data taking, for example, we estimate
that HK can reach three-sigma sensitivity to $\tau_{3}/m_{3}\lesssim 5\times
10^{9}$ s/eV.

Figure 7: $\Delta\chi^{2}$ (relative to the minimum value) as a function of
the $\nu_{3}$ lifetime-over-mass $\tau_{3}/m_{3}$, for ten years of simulated
data at different current and future experiments.

###  V.2 Probing pseudo-Dirac neutrinos with the DSNB

Neutrinos are the only Standard Model fermions whose fundamental nature is
still unknown. They can be either Dirac or Majorana particles, depending on
whether total lepton number is exactly conserved in nature. An intriguing
possibility is that neutrinos are Majorana fermions but the lepton-number
breaking structure of the theory is such that neutrinos behave, for the most
part, as Dirac fermions Wolfenstein (1981); Petcov (1982); Bilenky and
Pontecorvo (1983); Kobayashi and Lim (2001); de Gouvea et al. (2009). Under
these circumstances, neutrinos are referred to as pseudo-Dirac fermions.

If neutrinos are pseudo-Dirac fermions, right-handed neutrino fields $N_{i}$
($i=1,2,\ldots$, at least two of them) exist and couple of the Standard Model
lepton doublets $L_{\alpha}$ ($\alpha=e,\mu,\tau$) and the Higgs double $\Phi$
via Yukawa interactions (proportional to Yukawa couplings $Y_{i\alpha}$). In
the absence of more sources of mass, after electroweak symmetry breaking,
neutrinos are Dirac fermions with mass $M_{D}=Yv/\sqrt{2}$, where $v/\sqrt{2}$
is the vacuum expectation value of the neutral component of the Higgs doublet.
If, along with $M_{D}$, there is another source for the neutrino masses – in
the form of either Majorana masses $M_{R}$ for the right-handed neutrinos or
Majorana masses for the left-handed neutrinos – these would arise, after
integrating out new, heavy degrees of freedom, via the Weinberg operator,
$(L\Phi)^{2}$ – and the new source of mass is subdominant to $M_{D}$,
neutrinos are pseudo-Dirac fermions.

For concreteness, we concentrate on the so-called Type-I seesaw scenario,
where, after electroweak symmetry breaking, the neutrino Majorana mass Matrix
$M_{ij}$ ($i,j=1,2,3,\ldots,3+n_{R}$, where $n_{R}$ is the number of right-
handed neutrino fields), is

$\displaystyle M=\begin{pmatrix}\mathbb{0}_{3}&M_{D}\\\
M_{D}&M_{R}\end{pmatrix}\,,$ (V.13)

where $\mathbb{0}_{3}$ is a $3\times 3$ matrix of zeros. In the pseudo-Dirac
regime, $M_{R}\ll M_{D}$ and we will assume $n_{R}$ is equal to three
henceforth.

It is straight forward to diagonalize the $6\times 6$ mass matrix in Eq.
(V.13) via $V^{\dagger}(M^{\dagger}M)V=M_{\rm diag}^{2}$ perturbatively in
this regime. The unitary $6\times 6$ matrix $V$ can be written as Kobayashi
and Lim (2001)

$\displaystyle V=\frac{1}{\sqrt{2}}\begin{pmatrix}U&0\\\
0&U_{R}\end{pmatrix}\cdot\begin{pmatrix}\mathbb{1}_{3}&i\mathbb{1}_{3}\\\
\varphi&-i\varphi\end{pmatrix}\,,$ (V.14)

where $\mathbb{1}_{3}$ is the $3\times 3$ identity matrix, $U$ is a $3\times
3$ unitary matrix that is equivalent of the leptonic mixing matrix in the
limit $M_{R}\rightarrow 0$, $U_{R}$ is a (unphysical) $3\times 3$ matrix which
diagonalizes the “sterile” sector mass matrix, and $\varphi$ depends on
unknown phases $\varphi={\rm
diag}(e^{-i\phi_{1}},e^{-i\phi_{2}},e^{-i\phi_{3}})$. The active neutrinos can
be expressed as linear superpositions of the six mass eigenstates,
$\\{\nu_{js},\nu_{ja}\\},j=1,2,3$:

$\displaystyle\nu_{\alpha L}$ $\displaystyle=\frac{1}{\sqrt{2}}U_{\alpha
j}(\nu_{js}+i\,\nu_{ja})\,.$ (V.15)

These, in turn, have masses-squared

$\displaystyle m_{ks}^{2}=m_{k}^{2}+\frac{1}{2}\delta m_{k}^{2},\quad
m_{ka}^{2}=m_{k}^{2}-\frac{1}{2}\delta m_{k}^{2}\,,$ (V.16)

$k=1,2,3$. $m_{k}$ and $\delta m_{k}^{2}$ are functions of the elements of
$M_{D}$ and $M_{R}$. In the limit $M_{R}\rightarrow 0$, $\delta
m_{k}^{2}\rightarrow 0$ and $m_{k}^{2}$ are the eigenvalues of
$M_{D}M_{D}^{\dagger}$. In this limit, it is easy to see that the neutrinos
are Dirac fermions.

The pseudo-Dirac limit, $M_{R}\ll M_{D}$, can be understood as follows.
Neutrino mass eigenstates are organized into three pairs of quasi-degenerate
states, separated by small mass-squared differences $\delta m^{2}_{k}$, which
are much smaller than the known “solar” $\Delta m^{2}_{21}$ and “atmospheric”
$|\Delta m^{2}_{31}|$. These degenerate pairs are fifty-fifty linear
combinations – maximal mixing – of active and sterile neutrinos. These small
mass-squared differences will lead to oscillations at very large distances.
Limits on this scenario are obtained by considering solar neutrino
oscillations, sensitive to mass-squared differences down to $\delta
m_{k}^{2}\sim 10^{-12}{\rm\ eV}^{2}$ de Gouvea et al. (2009); Donini et al.
(2011), and from atmospheric neutrinos, affected if $\delta m_{k}^{2}\gtrsim
10^{-4}{\rm\ eV}^{2}$ Beacom et al. (2004). High-energy astrophysical
neutrinos constrain $10^{-18}{\rm\ eV}^{2}\lesssim\delta m_{k}^{2}\lesssim
10^{-12}{\rm\ eV}^{2}$, but these bounds depend strongly on knowledge of the
Gamma Ray Burst mechanism and the neutrino energy Beacom et al. (2004);
Esmaili and Farzan (2012).

The DSNB was emitted, on average, at large redshifts and hence allows one to
probe baselines of order a few Gpc. Therefore, we expect sensitivity to very
tiny, new mass-squared differences. Oscillation probabilities can be obtained
in the standard way Kobayashi and Lim (2001); de Gouvea et al. (2009), with a
few extra ingredients that are worthy of note. The initial states are,
according to our oscillation discussion in Section II, the “active” mass
eigenstate $\nu_{ia}$, $i=1,2,3$, which will be detected as a $\nu_{\alpha}$
on Earth with probability $|U_{\alpha i}|^{2}$ (ignoring Earth matter-
effects). These are the states that oscillate into sterile states (with
maximal mixing). The oscillation phase has to be corrected in order to include
the energy redshift since neutrinos are propagating in an expanding Universe
Esmaili and Farzan (2012); Beacom et al. (2004). Furthermore, given the
cosmological distances, wave-packet-separation decoherence effects may also
play an interesting role and are included.

The oscillation probability for $\nu_{i}\rightarrow\nu_{\beta}$, as function
of the neutrino energy today and the redshift, is

$\displaystyle P_{i\beta}(z,E)=\frac{1}{2}|U_{\beta
k}|^{2}\left(1+e^{-\mathscr{L}_{k}^{2}(z)}\cos\left(\frac{\delta
m_{k}^{2}}{2E}L_{2}(z)\right)\right)$ (V.17)

where the decoherence factor $\mathscr{L}_{k}(z)$ is Giunti and Kim (1998)

$\displaystyle\mathscr{L}_{k}(z)$ $\displaystyle=\frac{\delta
m_{k}^{2}}{4\sqrt{2}E^{2}}\frac{L_{3}(z)}{\sigma_{x}}$ (V.18)

and $\sigma_{x}$ the initial size of the wave packet. The distances $L_{n}(z)$
are Esmaili and Farzan (2012); Beacom et al. (2004)

$\displaystyle L_{n}(z)$
$\displaystyle=\frac{1}{H_{0}}\int_{0}^{z}\frac{dz^{\prime}}{(1+z^{\prime})^{n}}\frac{1}{\sqrt{\Omega_{m}(1+z^{\prime})^{3}+\Omega_{\Lambda}}}$
$\displaystyle=\frac{1}{H_{0}}\left\\{{}_{2}F_{1}\left[1,(5-2n)/6;(4-n)/3;-\Omega_{m}/\Omega_{\Lambda}\right]-(1+z)^{1-n}\,_{2}F_{1}\left[1,(5-2n)/6;(4-n)/3;-(1+z)^{3}\Omega_{m}/\Omega_{\Lambda}\right]\right\\},$
(V.19)

where ${}_{2}F_{1}[a,b;c;z]$ is the hypergeometric function.

Oscillation and decoherence effects are important for mass-squared differences
and wave-packet sizes of order

$\displaystyle L_{\rm osc}$ $\displaystyle=\frac{4\pi E}{\delta
m_{k}^{2}}\approx 8.03{\rm\ Gpc}\left(\frac{E}{10{\rm\
MeV}}\right)\left(\frac{10^{-25}{\rm\ eV}^{2}}{\delta m_{k}^{2}}\right),$
(V.20a) $\displaystyle L_{\rm coh}$
$\displaystyle=\frac{4\sqrt{2}E^{2}}{|\delta m_{k}^{2}|}\sigma_{x}\approx
180{\rm\ Gpc}\left(\frac{E}{10{\rm\ MeV}}\right)^{2}\left(\frac{10^{-25}{\rm\
eV}^{2}}{\delta m_{k}^{2}}\right)\left(\frac{\sigma_{x}}{10^{-12}{\rm\
m}}\right).$ (V.20b)

Hence, measurements of the DSNB are sensitive to mass-squared differences as
small as $\mathscr{O}(10^{-25}{\rm\ eV}^{2})$. For mass-squared differences
$\delta m_{k}^{2}\lesssim 10^{-28}{\rm\ eV}^{2}$, the naive oscillation length
is larger than the size of the observable Universe and, for neutrino energies
$E\sim\mathscr{O}(10{\rm\ MeV})$, such mass-squared differences are simply not
accessible.

Decoherence effects may be significant if $\sigma_{x}$ is small enough. For an
initial wave packet size of $\sigma_{x}\sim 1{\rm\ fm}$ and $\delta
m^{2}_{k}\sim 10^{-25}~{}{\rm{\rm\ eV}}^{2}$, for example, the DSNB neutrinos
arrive at the Earth as incoherent superpositions of active and sterile states.
In this case, the expect flux is 50% that of not-pseudo-Dirac neutrinos. In
fact, if the $\sigma_{x}$ values were of order one fermi, decoherence effects
would always supersede oscillation effects for typical DSNB energies,

$\displaystyle\frac{L_{\rm coh}}{L_{\rm
osc}}=\frac{\sqrt{2}}{\pi}E\sigma_{x}\approx 0.0023\left(\frac{E}{10{\rm\
MeV}}\right)\left(\frac{\sigma_{x}}{10^{-15}{\rm\ m}}\right)\,.$

In this regime, it is unfair to refer to the phenomenon in question as quantum
mechanical oscillations. Incoherent mixtures of active and sterile states
would be a more appropriate description. For the rest of this discussion, we
will be agnostic concerning the value of $\sigma_{x}$. We assume that it
cannot be smaller than one fermi – the size of a proton – given how CCSN
neutrinos are produced and the conditions under which they are produced. In
fact, we expect $\sigma_{x}\gg 1$ fm; thoughtful studies in the literature
have argued for $\sigma_{x}\sim 10^{-13}$ m Kersten and Smirnov (2016).

Fig. 8 depicts the modification of the DSNB active $\bar{\nu}_{1}$ flux due to
active-sterile oscillations in the pseudo-Dirac scenario. In the left panel,
$\sigma_{x}=10^{-10}$ m is held fixed and the different curves correspond to
different values of $\delta m_{k}^{2}$. We observe an oscillatory behavior for
the active flux on Earth for $\delta m^{2}_{k}\lesssim 10^{-24}$ eV2. For
larger values, the oscillations average out and one is left with half of the
unoscillated flux. In the right panel, $\delta m_{k}^{2}=3\times 10^{-25}{\rm\
eV}^{2}$ is held fixed and the different curves correspond to different values
of $\sigma_{x}$.

Figure 8: The impact of active–sterile oscillations on the expected active
DSNB $\bar{\nu}_{1}$ flux, assuming the neutrinos are pseudo-Dirac fermions.
The left panel includes different values of $\delta m_{k}^{2}$ for a fixed
value of $\sigma_{x}=10^{-10}{\rm\ m}$. The right panel includes different
values of the wave-packet size $\sigma_{x}$ for a fixed value of $\delta
m_{k}^{2}=3\times 10^{-25}{\rm\ eV}^{2}$.The green shaded region indicates the
modeling uncertainty on the normalization of the DSNB.

Assuming data consistent with no active-sterile oscillations, we perform a
$\chi^{2}$ analysis asking how future facilities could constrain the new mass-
squared differences $\delta m_{k}^{2}$, for fixed values of the initial size
of the wave-packet. Here, we assume the three new mass-squared difference are
the same, $\delta m^{2}_{1}=\delta m^{2}_{2}=\delta m^{2}_{3}$. Fig. 9 depicts
the sensitivity to the pseudo-Dirac scenario for SK, (left), HK (center) and
Theia (right). When the decoherence effects are not important, i.e., for
$\sigma_{x}=10^{-10}{\rm\ m}$, SK is sensitive to a new mass-squared
difference at the $2\sigma$ C.L. if $\delta m_{k}^{2}\gtrsim 1.8\times
10^{-25}{\rm\ eV}^{2}$, while HK (Theia) would be sensitive to $\delta
m_{k}^{2}\gtrsim 10^{-25}{\rm\ eV}^{2}$ ($\delta m_{k}^{2}\gtrsim 1.3\times
10^{-25}{\rm\ eV}^{2}$). Moreover, since the new oscillations would induce a
large modification of the shape of the fluxes in the observable region, we
find a significant enhancement in the sensitivity for $10^{-25}{\rm\
eV}^{2}\lesssim\delta m_{k}^{2}\lesssim 10^{-24}{\rm\ eV}^{2}$, reaching
values of $\Delta\chi^{2}=70~{}(30)$ for HK (Theia). Ten years of SK data can
rule out $\delta m_{k}^{2}\in\\{2.5,7.9\\}\times 10^{-24}{\rm\ eV}^{2}$ at the
three sigma level. For values of $\delta m_{k}^{2}\lesssim 10^{-26}{\rm\
eV}^{2}$, the oscillation lengths are larger than the cosmological baselines
of the DSNB, so the sensitivity is effectively lost. On the other hand, if the
mass-squared difference is large, $\delta m_{k}^{2}\gtrsim 10^{-24}{\rm\
eV}^{2}$, the oscillations are averaged out, and neutrinos arrive basically as
incoherent superpositions of active and sterile states. The sensitivity in
that case does not depend on $\delta m^{2}_{k}$, plateauing at
$\Delta\chi^{2}=\\{4.3,5.9,5.5\\}$ for SK, HK and Theia, respectively.

Figure 9: $\Delta\chi^{2}$ (relative to the minimum value) as a function of
$\delta m_{k}^{2}$, for ten years of simulated data at SK (left), HK (middle),
and Theia (right), for three different values of the decoherence parameter
$\sigma_{x}=(10^{-15}\,{\rm m},10^{-14}\,{\rm m},10^{-12}\,{\rm m})$. Figure
10: Sensitivity on the ($\sigma_{x}$, $\delta m^{2}_{k}$) parameter space from
ten years of simulated data at SK (left), HK (middle), and Theia (right). The
regions that can be tested at less than or equal to $1\sigma,2\sigma,3\sigma$
level correspond to the blue, green and yellow areas, respectively. The white
regions have a sensitivity $>3\sigma$ C.L.

One interesting effect appears when the decoherence due to the initial wave
packet size is important. Since the oscillations are erased by the separation
of the wave packets, the modification of the DSNB spectra becomes less
pronounced, thus reducing the overall sensitivity for all three experiments
considered. However, we find that the sensitivity is shifted to lower values
of $\delta m_{k}^{2}$ because the decoherence effects are present even for
cases in which the oscillations would be absent. The effect is maximal for
$\delta m_{k}^{2}\sim 10^{-25}{\rm\ eV}^{2}$ and $\sigma_{x}=10^{-15}{\rm\ m}$
in all experiments considered since, for these values of the relevant
parameters, decoherence effects impact the shape of the DSNB. This enlarges
the accessible region of parameter space, and one is sensitive to $\delta
m_{k}^{2}\sim 5\times 10^{-25}{\rm\ eV}^{2}$ for HK and Theia at the $2\sigma$
C.L., while for SK the sensitivity remains below the $2\sigma$ level for
$\delta m_{k}^{2}\lesssim 10^{-24}{\rm\ eV}^{2}$. The sensitivity to pseudo-
Dirac neutrinos as a function of $\sigma_{x}$ and $\delta m^{2}_{k}$ is
depicted in Fig. 10 for SK, (left), HK (center) and Theia (right). We observe
that the sensitivity is independent of $\sigma_{x}$ for $\sigma_{x}\gtrsim
10^{-14}{\rm\ m}$ at the three experiments. For some range of mass-squared
differences, all experiments can rule out the pseudo-Dirac neutrino hypothesis
at more than the three-sigma level as long as $\sigma_{x}$ is large enough. In
the case of HK, even when decoherence effects are strong, one can rule out the
pseudo-Dirac neutrino hypothesis at more than the three-sigma level for some
range of mass-squared differences.

##  VI Conclusions

Since the dawn of the first stars, core-collapse supernovae have created a
plethora of tens-of-MeV neutrinos of all species. This steady source of CCSN
relic neutrinos, the diffuse supernova neutrino background (DSNB), serves as a
complementary probe of certain aspects of cosmology, astrophysics, and
particle physics. As Super-Kamiokande, doped with gadolinium, prepares to take
data, the detection of the DSNB appears to be just a matter of time. As a
result, a detailed study of the role played by the DSNB in this era of multi-
messenger astronomy is timely. This is what we pursued here.

Predicting the DSNB flux requires inputs from three different disciplines: (i)
the cosmological history of the Universe, (ii) the rate of CCSNe, which
depends on the star-formation rate and is constrained with astrophysical data
and modeling, and (iii) the neutrino energy spectra from CCSNe that depend on
simulations of CCSNe and the particle-physics properties of neutrinos. As a
result, successful detection of the DSNB in currently running and future
detectors can help to constrain different cosmological parameters, the star-
formation history, as well as different types of beyond-the-Standard-Model
particle physics. In this work, we have analyzed the potential of terrestrial
detectors to constrain the above, using the DSNB.

In ten years, JUNO and SK are expected to accumulate a few to several tens of
events. In ten years, HK and Theia should have hundreds of events in hand.
These samples dictate what information one can hope to extract from
measurements of the DSNB. Before HK and Theia are available, combined SK and
JUNO can already yield interesting results. In Fig. 11(left) we depict how
well combined JUNO and SK data, collected over a five-year period, can
constrain the hypothesis that neutrinos are pseudo-Dirac fermions. We find
that a small range of new mass-squared differences (around several times
$10^{-25}$ eV2) can be ruled out at the three-sigma level. We refer to Sec.
V.2 for more details.

Figure 11: Left: $\Delta\chi^{2}$ (relative to the minimum value) as a
function of $\delta m_{k}^{2}$, for five of years of combined simulated data
at SK and JUNO, for three different values of the decoherence parameter
$\sigma_{x}=(10^{-15}\,{\rm m},10^{-14}\,{\rm m},10^{-10}\,{\rm m})$. Right:
current measurements of the Hubble constant $H_{0}$ from Planck Aghanim et al.
(2018), DES+BAO+BBN Abbott et al. (2018), SH0ES Riess et al. (2016, 2019),
CCHP Freedman et al. (2019), MIRAS Huang et al. (2019), H0LICOW Wong et al.
(2019), MCP Reid et al. (2009), and SBF Potter et al. (2018), along with our
estimates for the precision with which $H_{0}$ can be extracted from future
DSNB data.

More statistics allow one to consider more ambitious measurements. We argued
that HK and Theia, after a decade of running, can provide a “neutrino-
measurement” of the expansion rate of the universe. Fig. 11(right) depicts the
current “electromagnetic” measurements of the Hubble constant at both low and
high redshift, along with our estimate for how precisely HK and Theia can
measure $H_{0}$ after ten years of DSNB data are collected. The latter are
expected to be systematics dominated; improved estimates of the star-
formation-rate allow for more precise measurements of $H_{0}$. While it is
clear that DSNB measurements of $H_{0}$ will not compete, precision-wise, with
current measurements of $H_{0}$ – they probably will not be able to seriously
inform the current tension between low- Riess et al. (2016, 2019); Freedman et
al. (2019); Huang et al. (2019); Wong et al. (2019); Reid et al. (2009) and
high-redshift data Aghanim et al. (2018); Abbott et al. (2018); Potter et al.
(2018) – it is also clear that a neutrino-measurement of the expansion of the
Universe would be invaluable. Given expectations that $H_{0}$ will be measured
using the detection of gravity-waves from the collisions of ultra-dense
objects Schutz (1986); Holz and Hughes (2005); Chen et al. (2018), it seems we
are only a decade away from doing multi-messenger cosmology with photons,
gravitons, and neutrinos!

## Acknowledgements

We thank Boris Kayser for illuminating discussions on decoherence. The work of
AdG is supported in part by the DOE Office of Science award #DE-SC0010143. MS
acknowledges support from the National Science Foundation, Grant PHY-1630782,
and to the Heising-Simons Foundation, Grant 2017-228. Fermilab is operated by
the Fermi Research Alliance, LLC under contract No. DE-AC02-07CH11359 with the
United States Department of Energy.

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