# The diffuse supernova neutrino background as a probe of late-time neutrino
mass generation

André de Gouvêa<EMAIL_ADDRESS>Northwestern University, Department
of Physics & Astronomy, 2145 Sheridan Road, Evanston, IL 60208, USA  Ivan
Martinez-Soler<EMAIL_ADDRESS>Department of Physics &
Laboratory for Particle Physics and Cosmology, Harvard University, Cambridge,
MA 02138, USA  Yuber F. Perez-Gonzalez<EMAIL_ADDRESS>Institute for Particle Physics Phenomenology, Durham University, South Road
DH13EL, Durham, United Kingdom  Manibrata Sen<EMAIL_ADDRESS>Max-
Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany

###### Abstract

The relic neutrinos from old supernova explosions are among the most ancient
neutrino fluxes within experimental reach. Thus, the diffuse supernova
neutrino background (DSNB) could teach us if neutrino masses were different in
the past (redshifts $z\lesssim 5$). Oscillations inside the supernova depend
strongly on the neutrino mass-squared differences and the values of the mixing
angles, rendering the DSNB energy spectrum sensitive to variations of these
parameters. Considering a purely phenomenological parameterization of the
neutrino masses as a function of redshift, we compute the expected local DSNB
spectrum here on Earth. Given the current knowledge of neutrino oscillation
parameters, specially the fact that $|U_{e3}|^{2}$ is small, we find that the
$\nu_{e}$ spectrum could be significantly different from standard expectations
if neutrinos were effectively massless at $z\gtrsim 1$ as long as the neutrino
mass ordering is normal. On the other hand, the $\overline{\nu}_{e}$ flux is
not expected to be significantly impacted. Hence, a measurement of both the
neutrino and antineutrino components of the DSNB should allow one to test the
possibility of recent neutrino mass generation.

††preprint: IPPP/22/29

##  I Introduction

The discovery of neutrino oscillations in the last century established without
a doubt that neutrinos are massive. Neutrino oscillations provide precise
information on the neutrino mass-squared differences but are independent from
the absolute masses of the neutrinos. Data from neutrino oscillation
experiments can be used to constrain the sum of the neutrino masses to $\sum
m_{i}\gtrsim 0.058\,{\rm eV}$, $i=1,2,3$ in case of the Normal Mass Ordering
(NO) $(m_{3}>m_{2}>m_{1})$ or $\sum m_{\nu}\gtrsim 0.1\,{\rm eV}$ in the case
of the Inverted Mass Ordering (IO) $(m_{2}>m_{1}>m_{3})$ Tanabashi et al.
(2018). A kinematic upper bound to the neutrino masses, mostly model
independent, comes from the KATRIN experiment Aker et al. (2019, 2021), which
measures the beta-decay spectrum of tritium atoms. KATRIN is sensitive to a
linear combination of the neutrino masses; their most recent analysis yields
an upper limit of $\sum_{i}\sqrt{|U_{ei}|^{2}m_{i}^{2}}<0.9\,{\rm eV}$ ($90\%$
confidence level) Aker et al. (2021). The $U_{ei}$, $i=1,2,3$ elements of the
mixing matrix are measured with good precision by neutrino oscillation
experiments $|U_{e1}|^{2}\sim 0.7$, $|U_{e2}|^{2}\sim 0.3$, $|U_{e3}|^{2}\sim
0.02$ Esteban et al. (2020).

The most stringent bounds on neutrino masses come from indirect measurements
that rely on their effect on cosmological observables. Massless neutrinos are
hot dark matter candidates and mediate a “washing out” of small-scale
perturbations in the early Universe. Observations of the Cosmic Microwave
Background (CMB) by the Planck Satellite, combined with gravitational lensing
data, Baryon Acoustic Oscillations (BAO) and large-scale structure limit the
sum of neutrino masses $\sum_{i}m_{i}<0.13\,{\rm eV}$ Abbott et al. (2022).
Excluding BAO, this limit relaxes to $\sum m_{i}<0.24\,{\rm eV}$ Aghanim et
al. (2020). Adding Lyman alpha data and CMB temperature and polarization data
to the lensing and the BAO data further improves the bound to $0.09\,{\rm eV}$
Palanque-Delabrouille et al. (2020).

On the theoretical front, extending the SM to incorporate neutrino masses has
been a topic of intense research. The idea, if neutrinos are Majorana
fermions, is to augment the SM in a way so as to generate the effective
Weinberg operator $(LH)(LH)$ Weinberg (1979), where $L$ is the lepton doublet
containing the neutrino, and $H$ is the Higgs doublet. Popular mechanisms like
the seesaw models, radiative mass models, and several others, rely on the
introduction of new degrees of freedom at relatively high energy scales (see,
for example, de Gouvêa (2016) and references therein). These new massive
particles typically decouple from the cosmic plasma in the very early
Universe, and hence do not alter its evolution.

All current evidence of non-zero neutrino masses arises from experiments at
redshift $z=0$. In some sense, the “oldest” measurements of the non-zero
nature of neutrino masses comes from solar neutrinos. Data from cosmology do
not preclude a zero value of neutrino mass but only provide upper limits; the
vanilla $\Lambda$CDM cosmology is perfectly consistent with zero neutrino
masses Aghanim et al. (2020). As a result, scenarios where neutrinos are
massless in the early Universe and gain mass only after recombination are not
ruled out. Models predicting a late-time neutrino mass generation rely on
time-varying neutrino masses arising out of the neutrino coupling to some
time-varying scalar field Fardon et al. (2004); Krnjaic et al. (2018); Dev et
al. (2021), a late-time cosmic phase transition Lorenz et al. (2019), or the
gravitational anomaly Dvali and Funcke (2016). Using a combination of CMB
temperature and polarization power spectra, plus lensing data, the authors of
Lorenz et al. (2019) explored models where the neutrino masses are redshift
dependent. They report a slight preference for models of late-time neutrino
mass generated by a cosmic phase transition. In this scenario, due to the non-
trivial dynamics of the phase transition, the bound on the current sum of
neutrino masses is significantly weaker, $\sum m_{\nu}(z=0)<4.8\,$eV at
$95\%$CL. In a follow up work Lorenz et al. (2021), the authors extracted the
best-fit values of the neutrino masses as a function of redshift in a model-
independent manner, using CMB and BAO data and data from Type-IA SNe, and
found a significantly weaker bound, $\sum_{i}m_{\nu_{i}}(z=0)<1.46\,{\rm eV}$
(95$\%$ CL). These looser bounds indicate, for example, that the hypothetical
discovery of nonzero neutrino masses in future laboratory experiments
Formaggio et al. (2021) would be consistent with bounds from cosmic surveys if
we allow for late-time neutrino masses.

The CMB probes high redshifts $(z\sim 1000)$, and one may wonder if there are
other probes capable of testing the hypothesis that neutrino mass-generation
occurs at much smaller redshifts. The answer to this question may lie in the
diffuse supernova neutrino background (DSNB), a sea of MeV-neutrinos emerging
from all supernova (SN) explosions in the Universe since the moment of the
first stars $(z\lesssim 5)$ Lunardini (2006); Beacom (2010). This isotropic,
time-independent flux of neutrinos can be computed with precise knowledge of
the underlying cosmology and the rate at which SNe happen in the Universe. The
DSNB can be used as an excellent astrophysical laboratory to probe fundamental
particle physics de Gouvêa et al. (2020); Tabrizi and Horiuchi (2021); Das et
al. (2022).

The DSNB flux depends on whether the neutrinos are massive because of neutrino
oscillations. For massless neutrinos, flavor eigenstates trivially coincide
with mass eigenstates and will not undergo oscillations. However, the picture
changes if the neutrinos acquire mass at a certain redshift. This leads to a
scenario where the neutrino flavor and mass eigenstates are identical before a
certain redshift (hence the mixing matrix is diagonal) and, as soon as they
develop a non-zero mass, these two bases no longer coincide. This impacts the
DSNB flux that arrives at the Earth in a nontrivial way. Neutrinos that were
massless at the time of production would not suffer the usual effects that
arise from neutrino oscillations inside the SN. As a result, we expect the net
DSNB flux to be altered compared to what is predicted in the standard
scenario.

The detection of an altered DSNB flux can be used to probe such scenarios of
late neutrino mass generation. The Super-Kamiokande (SK) experiment Zhang et
al. (2015), enriched with Gadolinium, is ready to search for the DSNB, and is
expected to establish its existence within a decade Abe et al. (2021). Several
upcoming experiments like 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) will also be
instrumental in detecting the DSNB in the future. Moreover, the possibility of
observing the total –all flavors– DNSB flux via Coherent Elastic Neutrino-
Nucleus Scattering (CEvNS) has been recently demonstrated in Pattavina et al.
(2020); Suliga et al. (2022); Baum et al. (2022). As a result, the detection
of the DSNB in the next few decades will serve as a unique probe of the epoch
of neutrino mass-generation.

This work is organised as follows. We discuss our modelling of the DSNB flux
in Sec. II. We introduce our phenomenological approach to describing mass-
varying neutrinos in Sec. III. In Sec. IV, we determine the impact of late
neutrino-mass generation on the DSNB fluxes to be measured on the Earth. We
compute the expected event spectra in a DUNE-like detector in Sec. V. We
present our conclusions in Sec. VI. We use natural units where $\hbar=c=k_{\rm
B}=1$ throughout this manuscript.

##  II Modelling the DSNB flux

A prediction of the DSNB flux requires a good understanding of the evolution
of the Universe, including the rate of core-collapse supernova (CCSN) $R_{\rm
CCSN}$, as well as a handle on the flavor-dependent neutrino spectra from a
SN. The CCSN rate, in turn, depends on the history of the star-formation rate
(SFR), and has been measured by a number of independent astronomical surveys
Hopkins and Beacom (2006). The SFR data can be approximated by the following
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 the overall normalization of the rate and
$\alpha,\beta,\gamma$ indicate the relevant slopes at different values of $z$.
The parameters $B$ and $C$ 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)

We quote the parameters for the SFR used in our work in Tab. 1. A more
detailed discussion of these different parameters can be found in de Gouvêa et
al. (2020), and references therein. Using this, $R_{\rm CCSN}(z)$ can be
calculated as

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

where $\psi(M)\propto M^{-2.35}$ – the initial mass function (IMF) of stars –
gives the density of stars in a given mass range Salpeter (1955). The lower
limit on the IMF indicates tentatively the lowest mass at which a CCSN can
form (we neglect lower mass electron capture SNe), while the upper limit is
more ad hoc, including a reasonable fraction of failed SNe.

Table 1: Star formation rate parameters and their uncertainties used in this work. Parameter | Value
---|---
$\dot{\rho}_{0}$ | $0.0178^{+0.0035}_{-0.0036}~{}\mathrm{M_{\odot}~{}y^{-1}~{}Mpc^{-3}}$
$\alpha$ | $3.4\pm 0.2$
$\beta$ | $-0.3\pm 0.2$
$\gamma$ | $-3.5\pm 1.0$
$z_{1}$ | $1$
$z_{2}$ | $4$

Finally, neutrino emission from a SN can be parameterized by the well-known
alpha-fit spectra Tamborra et al. (2012)

$\displaystyle
F_{\nu_{\beta}}(E_{\nu})=\frac{1}{E_{0\beta}}\frac{(1+\alpha)^{1+\alpha}}{\Gamma(1+\alpha)}\left(\frac{E_{\nu}}{E_{0\beta}}\right)^{\alpha}e^{-(1+\alpha)\frac{E_{\nu}}{E_{0\beta}}},$
(II.5)

where $E_{0\beta}$ is the average energy for a flavor $\nu_{\beta}$ or
$\overline{\nu}_{\beta}$, $\beta=e,\mu,\tau$ and $\alpha$ is a parameter that
determines the width of the distribution. The DSNB spectra are dominated by
neutrino emission from the cooling phase, where the spectra are approximately
thermal. $\alpha=2.3$ approximates Eq. (II.5) as a Fermi-Dirac spectrum
Lunardini (2006).

With this information, in the absence of neutrino oscillations, the diffuse
neutrino flux from all past SNe, is Horiuchi et al. (2009); Beacom (2010);
Møller et al. (2018)

$\Phi_{\nu_{\beta}}^{0}(E)=\int_{0}^{z_{\rm max}}\frac{dz}{H(z)}R_{\rm
CCSN}(z)\phi_{\nu_{\beta}}^{0}(E(1+z))\,,$ (II.6)

where $H(z)=H_{0}\,\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}}$ is the Hubble
function with $H_{0}=67.36\,{\rm km\,s}^{-1}\,{\rm Mpc}^{-1}$ and
$\Omega_{m}$, $\Omega_{\Lambda}$ represent the matter and vacuum contribution
to the energy density, respectively Aghanim et al. (2018). The integral over
$z$ ranges up to the maximum redshift of star-formation $(z_{\rm max}\sim 5)$.
$\phi_{\nu_{\beta}}^{0}(E)$ in Eq. (II.6), contains contributions from CCSNe
and black-hole-forming (BHF) failed SNe,

$\displaystyle\phi_{\nu_{\beta}}^{0}(E)=f_{\rm CC}\,F_{\nu_{\beta}}^{\rm
CC}(E_{\nu})+f_{\rm BH}\,F_{\nu_{\beta}}^{\rm BH}(E_{\nu}),$ (II.7)

where $f_{\rm CC,BH}$ are the fraction of CC and BH-forming explosions, and
$F_{\nu_{\beta}}^{\rm CC,BH}(E_{\nu})$ are the time-integrated energy spectra
for CCSNe and BHF-SNe. In the following, we take $f_{\rm BH}=21\%,f_{\rm
CC}=1-f_{\rm BH}$. For the $F_{\nu_{\beta}}^{\rm CC,BH}(E_{\nu})$, we have
performed a fit of the time-integrated neutrino fluences obtained by the
Garching group results from (????) in the form of Eq. (II.5), taking as
benchmark the data for $12M_{\odot}$ for CCSNe and $40M_{\odot}$ as for BHF-
SNe. We present in Fig. 1 the unoscillated fluxes at the Earth, obtained from
Eq. (II.6) for $\nu_{e}$ (orange), $\overline{\nu}_{e}$ (green dashed), and
$\nu_{x}=\nu_{\mu},\nu_{\tau},\overline{\nu}_{\mu},\overline{\nu}_{\tau}$ and
the corresponding antineutrinos (purple dotted). The $\nu_{e}$ flux is about
twice the $\nu_{x}$ flux at $E\sim 3.5$ MeV. This difference arises mainly due
to the interactions with neutrons that render the average energy of the
$\nu_{e}$ flux smaller. Meanwhile, the $\overline{\nu}_{e}$ and $\nu_{x}$ have
closer average energies, making the fluxes much more similar. Such difference
will be crucial in our scenario of mass-varying neutrinos.

Figure 1: Unoscillated DSNB flux $\Phi_{\nu_{\beta}}^{0}$ for each neutrino
species, $\nu_{e}$ (orange), $\overline{\nu}_{e}$ (green dashed), and
$\nu_{x}$ (purple dotted), as function of the neutrino energy $E$. The bands
are associated to uncertainties in the star-formation rate.

The neutrino flux gets processed through oscillation effects inside the SN and
on the way to Earth. In this study, we neglect the effects of collective
neutrino oscillations arising out of neutrino self-interactions deep inside
the SN Duan et al. (2006); Hannestad et al. (2006). The quantitative impact of
collective oscillations is inconclusive to date, and we expect it to be
relatively smaller for neutrinos predominantly produced in the cooling phase.
The neutrino flux gets affected by adiabatic Mikheyev-Smirnov-Wolfenstein
(MSW) resonant flavor conversion Wolfenstein (1978); Mikheev and Smirnov
(1985). Assuming NO, this implies that the $\nu_{e}$ are primarily emitted as
$\nu_{3}$, while the non-electron neutrinos $\nu_{\mu,\tau}$ are emitted as
combinations of $\nu_{1}$ and $\nu_{2}$. In this case, the final $\nu_{e}$
flux at the Earth $\Phi_{\nu_{e}}(E)$ is given by

$\Phi_{\nu_{e}}(E)=|U_{e3}|^{2}\,\Phi_{\nu_{e}}^{0}+(1-|U_{e3}|^{2})\,\Phi_{\nu_{x}}^{0},\,$
(II.8)

where $\Phi_{\nu_{x}}^{0}=\Phi_{\nu_{\mu}}^{0}=\Phi_{\nu_{\tau}}^{0}$, and
$U_{\alpha i}$ is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix.
Clearly, the DSNB flux depends on the underlying neutrino oscillation
scenario. For example, if the neutrinos were massless at the time of the SN,
the flavor evolution of the $\nu_{e}$ and $\nu_{x}$ (and the antineutrinos)
would be trivial inside the explosion. On their way here, these would start
oscillating, fast, once the neutrino masses turn on. In this case, the
probability that a $\nu_{\alpha}$ is detected as a $\nu_{e}$ at the Earth is

$P_{\alpha e}=\sum_{i}|U_{\alpha i}|^{2}|U_{ei}|^{2}.$ (II.9)

Since SN neutrino energies are smaller than the muon mass, it is convenient to
define $P_{xe}=P_{\mu e}+P_{\tau e}=1-P_{ee}$ so

$\Phi_{\nu_{e}}(E)=P_{ee}\Phi_{\nu_{e}}^{0}+(1-P_{ee})\,\Phi_{\nu_{x}}^{0}\,.$
(II.10)

In the next sections, we discuss in detail how the DSNB is modified if a
fraction of it comes from neutrinos that were “born” with smaller masses or
different mixing parameters.

##  III Mass Varying neutrinos

Following Koksbang and Hannestad (2017); Dvali and Funcke (2016); Lorenz et
al. (2019), we assume that the neutrinos remain practically massless down to a
certain redshift $z_{s}$ and gain a non-zero mass for $z<z_{s}$. We further
assume the neutrino mass reaches its current value over a finite transition
period. This could happen due to neutrinos coupling to the
gravitational-$\theta$ term, causing a late phase transition in the Universe
Dvali and Funcke (2016), or to neutrinos coupled to a scalar background, which
evolves as a function of time Fardon et al. (2004); Berlin (2016); Krnjaic et
al. (2018); Dev et al. (2021). Here, we remain agnostic regarding the details
of mass-generation.

Assuming momentarily there is only one neutrino mass, we propose that it
varies as a function of redshift according to

$m_{\nu}(z)=\frac{m_{\nu}}{1+(z/z_{s})^{{\rm B}_{s}}}\,,$ (III.1)

where $m_{\nu}$ is the current mass of the neutrino, ${\rm B}_{s}$ is a
parameter which controls the width of the transition from a massless neutrino
to a massive neutrino, and $z_{s}$ is the redshift below which the neutrino
mass turns on. The specific form of the function is irrelevant and is chosen
just to present a smooth transition to a non-zero mass. The values of $z_{s}$
and ${\rm B}_{s}$ determine when and at what rate the neutrino mass turns on.

Since there are three neutrino masses, it is possible that they would “turn
on” at different $z_{s}$ and that the transition would be associated to a
different value of Bs. Here we assume a universal value for these two
phenomenological paramaters. It is also possible to imagine that, as the
neutrino mass turns on at $z_{s}$, so do all the PMNS mixing angles
$\\{\theta_{12},\theta_{13},\theta_{23}\\}$, and that these turn on in a way
that is also captured by Eq. (III.1). We will discuss this possibility later
but remain agnostic about the origin of such variations.

In the next section, we will detail the impact of redshift-dependent neutrino
masses and mixing angles on the flavor evolution of neutrinos within the SN,
as well as from the SN to Earth.

##  IV Impact of mass varying neutrinos on the DSNB

###  IV.1 Only masses

We first consider the case where the neutrino masses vary as a function of
red-shift while the elements of the mixing matrix are time independent.

####  IV.1.1 Calculation of the survival probability

In the standard three-massive-neutrinos paradigm, neutrinos produced via
charged-current weak interactions are described as superpositions of the three
neutrinos with well defined masses, $\nu_{\alpha}=U_{\alpha i}\nu_{i}$,
$\alpha=e,\mu,\tau$, $i=1,2,3$. During propagation, the fact that the neutrino
masses are different leads the neutrino flavor to oscillate; the associated
oscillation lengths are inversely proportional to the differences of the
squares of the neutrino masses. Global analysis of the present data indicate
that the two independent mass-squared differences are $\Delta m^{2}_{21}\sim
7.5\times 10^{-5}\text{eV}^{2}$ and $\Delta m^{2}_{31}\sim\pm 2.5\times
10^{-3}\text{eV}^{2}$(plus for NO, minus for IO) Esteban et al. (2020). In
matter, the neutrino flavor evolution is modified by the forward elastic
neutrino–electron interaction amplitude along the neutrino path.***As
mentioned earlier, we will ignore collective effects throughout. This
interaction is captured by a matter potential and modifies the effective
Hamiltonian that describes neutrino flavor evolution Wolfenstein (1978);
Mikheyev and Smirnov (1985). The matter potential depends on the electron
number density ($n_{e}$) along the neutrino path. For position-dependent
matter potentials, flavor evolution is rather involved. For certain matter
profiles, however, the phenomenon is well understood Parke (1986); Petcov
(1988); Krastev and Petcov (1988); Petcov (1988); Friedland (2001). The case
where neutrinos are produced in a region of space where $n_{e}$ is large and
propagate towards in the direction where $n_{e}$ falls roughly exponentially
is well known and applies to both solar neutrinos and neutrinos produced in
the core of SN explosions.

In the limit where $G_{F}n_{e}$, where $G_{F}$ is the Fermi constant, is much
larger than $|\Delta m^{2}/E|$, where $E$ is the neutrino energy and $\Delta
m^{2}$ are the neutrino mass-squared differences, electron neutrinos coincide
with one of the propagation-Hamiltonian eigenstates (the one with the largest
eigenvalue) in the production region. If neutrino flavor evolution is
adiabatic inside the medium, the electron neutrino exits the matter
distribution as a mass-eigenstate (eigenstate of the flavor-evolution
Hamiltonian in vaccum). This “mapping” between the electron neutrino and mass-
eigenstates depends on the mass ordering and whether we are considering
electron neutrinos or antineutrinos, keeping in mind that the matter potential
is positive for neutrinos, negative for antineutrinos.

Given what we know about the mass-squared differences, electron neutrinos, if
the flavor evolution inside the SN is adiabatic, exit the SN as $\nu_{3}$ for
NO and $\nu_{2}$ for IO. Electron antineutrinos, instead, exit the SN as
$\overline{\nu}_{1}$ for NO and $\overline{\nu}_{3}$ for IO. In the adiabatic
regime, it is easy to generalize this picture to the case where $G_{F}n_{e}$
is not much larger than one or both $\Delta m^{2}/E$: the flavor evolution
along the matter potential is just described by the effective mixing
parameters at neutrino production.

Figure 2: Constant crossing probability contours in the
$\sin^{2}2\theta\times\Delta m^{2}$–plane. These define three regions: (I)
$P_{c}<0.1$, (II) $0.1<P_{c}<0.9$, and (III) $P_{c}>0.9$. The color scale
indicates the values of the two independent mass-squared differences as a
function of the redshift of neutrino production. For the mass variation, we
make use of Eq. (III.1) with $z_{s}=0.32$ and $B_{s}=5$.

In the case of two neutrino flavors, if the electron number density decreases
roughly exponentially, adiabaticity is controlled by the “crossing
probability” $P_{c}$. When $P_{c}$ vanishes, the flavor evolution is perfectly
adiabatic. $P_{c}$ given by Petcov (1988); Krastev and Petcov (1988); Petcov
(1988)

$P_{c}=\frac{\exp^{-(\pi\gamma F/2)}-\exp^{-(\pi\gamma
F/2\sin^{2}\theta)}}{1-\exp^{-(\pi\gamma F/2\sin^{2}\theta)}},$ (IV.1)

where F depends on the matter distribution inside the supernova and the mixing
angle Kuo and Pantaleone (1989). The dependence of $P_{c}$ on the oscillation
parameters $\Delta m^{2}$ and $\theta$ is controlled by $\gamma$, which takes
the following expression around the resonant region, defined by $n_{e}$ values
that satisfy $2\sqrt{2}G_{F}En_{e}=\Delta m^{2}\cos 2\theta$.

$\gamma=\frac{\Delta m^{2}}{2E}\frac{\sin^{2}\theta}{\cos
2\theta}\left(\frac{1}{n_{e}}\frac{dn_{e}}{dr}\right)^{-1}.$ (IV.2)

If the variation of the effective mixing angles with the electron number
density is slower than the oscillation wavelength in matter, $\gamma F>>1$ and
the neutrino evolution is adiabatic.

It is easy to generalize the discussion to three flavors, taking advantage of
the fact that the magnitudes of the two known mass-squared differences differ
by two orders of magnitude. In this case, one can define two resonance regions
and two crossing probabilities: $P_{c}^{H}$ ($H$ for high), associated to
$\Delta m^{2}=\Delta m^{2}_{31}$ and $\theta=\theta_{13}$ and $P_{c}^{L}$ ($L$
for low), associated to $\Delta m^{2}=\Delta m^{2}_{21}$ and
$\theta=\theta_{12}$. In our computations, in the standard case and NO, we
use, for $P_{c}^{H}$, $\Delta m^{2}_{31}=2.57\times 10^{-3}\text{eV}^{2}$ and
$\theta_{13}=8.57^{\circ}$ and, for $P_{c}^{L}$, $\Delta m^{2}_{21}=7.42\times
10^{-5}\text{eV}^{2}$ and $\theta_{12}=33.44^{\circ}$ Esteban et al. (2020).

The flavor-at-production and the neutrino spectrum emitted by the supernova
depends on the evolution of the collapse of the star. After the shock-wave,
free electrons are captured by free protons generated by the dissociation of
nuclei yielding a $\nu_{e}$-rich flux – the neutronization burst. Thereafter,
a large fraction of the neutrinos are emitted during the cooling phase when
the supernova loses the remaining gravitation binding energy via the thermal
emission of neutrinos of all flavors. In this phase, the temperature of
$\nu_{e}$ is expected to be smaller than that of $\overline{\nu}_{e}$ and that
of $\nu_{x}$, since $\nu_{e}$ interacts more strongly with the production
medium. Most of the neutrinos are created deep inside the explosion where the
density is quite large. On their way out, neutrinos cross both the atmospheric
($\rho\sim 3\times 10^{3}~{}{\rm g/cm^{3}}$), and the solar resonances
($\rho\sim 40~{}{\rm g/cm^{3}}$) at lower densities. Both resonances happen
well outside of the neutrinospheres. In Fig. 2, we depict contours of constant
$P_{c}$ in the $\Delta m^{2}\times\sin^{2}2\theta$-plane. We identify three
qualitatively distinct regions: (I) $P_{c}<0.1$, where flavor-evolution
“through” the resonance is adiabatic, (II) $0.1<Pc<0.9$ (region II), and (III)
$P_{c}>0.9$, where neutrino flavor-evolution is highly non-adiabatic. Given
the current values of the mass-squared differences ($z=0$ in the figure),
flavor-evolution is very adiabatic through both the atmospheric and solar
resonances Dighe and Smirnov (2000). The large value of the density in the
region where the neutrinos are produced leads to, as discussed earlier,
$\nu_{e}$ being mapped to the most massive state (e.g. $\nu_{3}$ for NO) while
the $\nu_{x}$ is mapped into the lighter states (e.g., for NO, some
combination of $\nu_{1}$ and $\nu_{2}$). In this case, for NO, the flux of
electron neutrinos at the Earth is given by the projection of the three
massive states weighted by the initial flux – Eq. (II.8). The small value of
$|U_{e3}|^{2}\sim 0.02$ implies that most of the $\nu_{e}$ at the Earth
started out as a $\nu_{x}$ deep inside the explosion.

If the neutrino masses were smaller at a given time in the history of the
Universe, the flavor evolution inside the supernova might no longer be
adiabatic. This is depicted in Fig. 2, where we indicate the different values
of the mass-squared differences for different redshifts, following Eq. (III.1)
with $z_{s}=0.32$ and $B_{s}=5$. This allows the possibility that one massive
state “flip” into another one as the neutrinos propagate through the
supernova. In the case of a non-adiabatic evolution and NO, the initial
$\nu_{e}$ component of the flux will also be partially mapped to $\nu_{1}$ and
$\nu_{2}$ fluxes outside the SN. Following Dighe and Smirnov (2000), the
$\nu_{e}$ flux at the Earth in the case of a non-adiabatic evolution is given
by Eq. (II.10) where the $\nu_{e}$ survival probability $P_{ee}$ is given by

$P_{ee}=|U_{e1}|^{2}P^{H}_{c}P^{L}_{c}+|U_{e2}|^{2}(P^{H}_{c}-P^{H}_{c}P^{L}_{c})+|U_{e3}|^{2}(1-P^{H}_{c}).$
(IV.3)

In the adiabatic limit ($P^{L}_{c}=P^{H}_{c}=0$), we recover the standard
expression for the $\nu_{e}$ flux at the Earth, Eq. (II.8). As the neutrino
masses decrease, the atmospheric and solar resonances shift to lower
densities. Note that, if the neutrino mass is low enough, the neutrinos might
not “cross” one of the resonances on their way out of the SN. That will also
impact the final $\nu_{e}$ flux.

In the case of the normal mass ordering, the non-adiabatic evolution leads to
an enhancement of the $\nu_{e}$ flux because the initial $\nu_{e}$ flux is
larger than that of the other flavors. Fig. 3 depicts the electron-neutrino
survival probability on the Earth as a function of the redshift $z$ of the SN,
for $E_{\nu}=10$ MeV, $z_{s}=0.32$ and $B_{s}=5$. For this choice of mass-
varying parameters, the transition between massless and massive neutrinos
happens around $z\sim 1$. If the neutrino energy increases, the transition
shifts to lower redshifts. Around $z\sim 1$, we observe a small oscillatory
pattern in $P_{ee}$, highlighted in the inset. For those values of $z$,
$\Delta m^{2}_{21}\sim 10^{-8}$ eV2 and the associated oscillation length is
of order the size of the SN.

Figure 3: Electron-neutrino survival probability as function of the redshift
for $E_{\nu}=10$ MeV. We consider that the mass changes as a function of the
redshift according to Eq. (III.1) for $z_{s}=0.32$ and $B_{s}=5$.

####  IV.1.2 The DSNB $\nu_{e}$ flux on Earth

In order to include the possibility that neutrino masses are redshift-
dependent, Eq. (II.8) needs to be altered:

$\displaystyle\Phi_{\nu_{e}}(E)$ $\displaystyle=\int_{0}^{z_{\rm
max}}\frac{dz}{H(z)}R_{\rm
CCSN}(z)\left\\{P_{ee}(z)\phi_{\nu_{e}}^{0}+(1-P_{ee}(z))\phi_{\nu_{x}}^{0}\right\\}\,,$
(IV.4a) $\displaystyle\Phi_{\bar{\nu}_{e}}(E)$ $\displaystyle=\int_{0}^{z_{\rm
max}}\frac{dz}{H(z)}R_{\rm
CCSN}(z)\left\\{\overline{P_{ee}}(z)\phi_{\bar{\nu}_{e}}^{0}+(1-\overline{P_{ee}}(z))\phi_{\nu_{x}}^{0}\right\\}\,,$
(IV.4b) $\displaystyle\Phi_{\nu_{x}}(E)$ $\displaystyle=\int_{0}^{z_{\rm
max}}\frac{dz}{H(z)}R_{\rm
CCSN}(z)\frac{1}{4}\left\\{(1-P_{ee}(z))\phi_{\nu_{e}}^{0}+(1-\overline{P_{ee}}(z))\phi_{\bar{\nu}_{e}}^{0}+(2+P_{ee}(z)+\overline{P_{ee}}(z))\phi_{\nu_{x}}^{0}\right\\}\,,$
(IV.4c)

where, for clarity, we omitted the dependence of $\phi^{0}_{\nu_{\beta}}$ on
$(E,z)$. $P_{ee}(z)$ ($\overline{P_{ee}}(z)$) indicate the oscillation
probabilities for neutrinos (antineutrinos) from a SN explosion at a redshift
$z$ as described in the last subsection. It depends on the ${\rm B}_{s},z_{s}$
parameters, so the final DSNB flux will contain information regarding them.
The DSNB is an integrated flux, so, in principle, there is no an explicit way
to distinguish neutrinos that were emitted at higher redshifts from those
produced more recently. However, the energies of the neutrinos produced
earlier are more redshifted and hence we expect that time-dependent neutrino
masses will distort the DSNB energy spectrum.

Figure 4: (Left) The DSNB $\nu_{e}$ flux as a function of the neutrino energy
for different values of $z_{s}\in[10^{-2},1]$ (rainbow colored) and for the
standard case (orange) including the star-formation-rate uncertainty (orange
band). The dashed orange line indicates the DSNB flux assuming massless
neutrinos, i.e., $P_{ee}(z)=1$ for all $z$. (Right) Neutrino mass as a
function of redshift $z$, normalized to the current value of the mass,
$m_{\nu}(z)/m_{\nu}$, together with $R_{\rm CCSN}(z)/H(z)$. See text for
details. In both panels, ${\rm B}_{s}=5$. We assume the normal ordering for
the neutrino masses.

We first consider the case where only the neutrino masses change over time,
assuming the NO. Fig. 4, left, depicts the electron neutrino flux at the Earth
for different values of $z_{s}\in[10^{-2},1]$, for ${\rm B}_{s}=5$. The
standard flux, for constant neutrino masses, including uncertainties
associated to the SFR, is depicted as the orange band while the dashed orange
line corresponds to the unoscillated DSNB flux, $\Phi^{0}_{\nu_{e}}$ (i.e.,
expectations in the scenario where all neutrino masses are exactly zero). We
observe that the hypothesis that neutrino masses depend on the redshift can
significantly impact the DSNB electron-neutrino flux for $z_{s}\lesssim 0.5$.
In fact, for $E=3~{}{\rm\ MeV}$, the DSNB flux can be larger than standard
expectations by a factor of order $1.4$ for $z_{s}\sim 10^{-2}$. Moreover,
from a simple flux conservation argument, this also implies that the flux at
larger energies is reduced with respect to the standard case, see the inset
plot. Such neutrinos would have acquired their masses rather recently, when
the Universe was $13.652~{}{\rm Gyr}$ old (compare with the age of the
Universe $t_{0}=13.795~{}{\rm Gyr}$), therefore the DSNB was mostly produced
when neutrinos were virtually massless. The increment on the $\nu_{e}$ flux at
low energies is directly related to the difference between the unoscillated
$\nu_{e}$ and $\nu_{x}$ fluxes. In the standard scenario $P_{ee}\ll 1$, so the
$\nu_{e}$ flux at the Earth is basically the $\nu_{x}$ flux produced at the
neutrinosphere, which is much broader in energy. However, if $P_{ee}$
significantly differs from the standard case, the contribution from the
$\nu_{e}$ flux that exited the neutrinosphere becomes significant, thus
modifying the $\nu_{e}$ flux at the Earth.

For values of $0.1\lesssim z_{s}\lesssim 1$, the $\nu_{e}$ flux is still
larger than the SM flux at low energies. Meanwhile, for $z_{s}\gtrsim 1$, the
DSNB flux is basically indistinguishable from the standard case. To understand
the dependence on the values of $z_{s}$, we show in the right panel of Fig. 4
the redshift evolution of neutrino masses along with the factor $R_{\rm
CCSN}(z)/H(z)$, cf. Eq. (II.6). This object describes the SN neutrino
production as a function of redshift, including effects associated to the
expansion of the Universe. It reveals that most of the DSNB flux is produced
at $0.1\lesssim z\lesssim 5$. Thus, if $z_{s}\gtrsim 1$, the SNe matter
effects are basically the same as in the standard case, so we do not expect
any impact from the mass-varying hypothesis. On the other hand, if
$0.1\lesssim z_{s}\lesssim 1$, a significant fraction of the DNSB comes from
SN explosions that happened when the neutrino masses were significantly
smaller. The largest effects occur when neutrinos were effectively massless
during most of the history of the Universe, $z_{s}\lesssim 0.1$, as noted
above.

Fig. 5 captures the dependence of the DSNB flux on the parameter ${\rm B}_{s}$
for a fixed $z_{s}=0.32$. ${\rm B}_{s}$ controls how fast neutrino masses
increase, larger values associated to more abrupt transitions. For
$B_{s}\gtrsim 10$, the transition is almost instantaneous. Whenever the growth
of neutrino masses is rapid (${\rm B}_{s}\gtrsim 50$) the flux is relatively
larger (by the same factor, discussed earlier,
$\Phi_{\nu_{e}}/\Phi_{\nu_{e}}|_{\rm SM}\sim 1.4$). This dependence on ${\rm
B}_{s}$ again is understood by comparing the redshift dependence of both
$m_{\nu}(z)$ and $R_{\rm CCSN}(z)/H(z)$ (right panel). If the masses become
non-zero instantaneously, neutrinos emitted before the transition ($z>z_{s}$)
would have been effectively massless, and their contribution to the $\nu_{e}$
flux will be associated to the electron neutrino survival probability
$P_{ee}=\sum_{k}|U_{ek}|^{4}\approx 0.57$, characteristic of electron
neutrinos propagating very long distances in vacuum. After the masses “turn
on,” neutrinos will be subject to matter effects inside the SN and
$P_{ee}=|U_{e3}|^{2}$ in the NO, as discussed earlier. The final DSNB flux
will be an amalgam of neutrinos from two different epochs whose contributions
are weighted by the SFR divided by the expansion rate. If the transition is
not instantaneous (small Bs), the DSNB flux is reduced because matter effects
would impact the propagation inside the SN for a longer period of time and the
neutrino masses would be of order the current masses for an extended range of
redshifts (when the neutrino masses are $\gtrsim 10\%$ of the masses today,
the matter effects are very similar to the standard case.). Thence, the DSNB
flux in such cases is closer to the standard case, as can be observed in Fig.
5.

Figure 5: Same as Figure 4, for different values of ${\rm B}_{s}\in[1,100]$
and fixed $z_{s}=0.32$.

So far, we have focused on the impact of redshift-dependent neutrino masses on
the $\nu_{e}$ flux assuming NO. Instead the impact on the $\overline{\nu}_{e}$
spectrum is minimal. This is depicted in Fig. 6, left, for NO. This
indifference is not strongly dependent on the mass ordering and is mostly a
consequence of the fact that the original $\overline{\nu}_{e}$ and $\nu_{x}$
fluxes (keeping in mind that $\nu_{x}$ includes the antineutrino flavors) are
very similar, see Fig. 1. In this case, oscillation effects are invisible.
Nonetheless, it is worth discussing the oscillation of $\overline{\nu}_{e}$ in
a little more detail. The standard prediction assuming adiabatic propagation
indicates that $\overline{\nu}_{e}$ emerges from the SN as
$\overline{\nu}_{1}$ so $\overline{P_{ee}}=|U_{e1}|^{2}\approx 0.67$ at low
energies. The $\overline{\nu}_{e}$ flux at the Earth is, therefore, roughly an
equal admixture of $\Phi_{\overline{\nu}_{e}}^{0}$ and $\Phi_{\nu_{x}}^{0}$,
fluxes that are close to each other. Furthermore, if neutrino masses arise
later in the evolution of the Universe ($z_{s}\lesssim 0.1$),
$\overline{P_{ee}}\sim 0.57$ as in the $\nu_{e}$ case. The difference between
these fluxes is safely within the star-formation-rate uncertainty, making the
effect unobservable, even in the most optimistic cases. Similarly, $\nu_{x}$,
measurable only via NC interactions at these low energies, is also modified in
a negligible way, as can be observed in the right panel of Fig. 6, for NO.
Since it contains the contributions of both neutrinos and antineutrinos, the
modification of this flux is at most $10\%$, within the star-formation-rate
uncertainty.

Figure 6: The DSNB $\overline{\nu}_{e}$ flux (left) and $\nu_{x}$ flux (right)
as a function of the neutrino energy for different values of
$z_{s}\in[10^{-2},1]$ (rainbow colored) and fixed ${\rm B}_{s}=5$. The fluxes
in the standard case, including the star-formation-rate uncertainty, define
the green (left) and purple (right) bands. The dashed lines indicate the DSNB
flux assuming massless neutrinos. We assume the normal ordering for the
neutrino masses.

For the IO, the situation changes significantly for $\nu_{e}$. In the standard
scenario, the MSW effect predicts that the a $\nu_{e}$ created at the
neutrinosphere leaves the SNe as a $\nu_{2}$ mass eigenstate, so
$P_{ee}=|U_{e2}|^{2}\sim 0.3$. Meanwhile, if neutrinos only acquired their
masses recently ($z_{s}\lesssim 1$), we would have the same probability as in
the NO, $P_{ee}=\sum_{k}|U_{ek}|^{4}\approx 0.57$. Thus, we find that any
possible modification on the DSNB energy spectrum in the IO will lie within
the current star-formation uncertainty band. On the other hand, for
antineutrinos and the IO, we have $\overline{P_{ee}}=|U_{e3}|^{2}$, so, if
matter effects inside the SNe were significantly different at some point of
the evolution of the Universe, $\overline{P_{ee}}$ would be considerably
different from the standard value. Nevertheless, since $\Phi^{0}_{\nu_{x}}$
and $\Phi^{0}_{\overline{\nu}_{e}}$ are very similar, cf. Fig. 1, any imprint
of the mass-varying hypothesis would be very difficult to measure.

###  IV.2 Masses and mixing

If one allows for the possibility that the neutrino masses are redshfit-
dependent, it is reasonable to ask whether the the neutrino mixing parameters
also depend on the redshift. We address this possibility in this subsection.

####  IV.2.1 Calculation of the survival probability

In the case where both the neutrino masses and the mixing parameters depend on
the redshift, the adiabaticity of the neutrino flavor-evolution inside the SN
is modified relative to the case where only the masses depend on the redshift.
Similar to Fig. 2, Fig. 7 depicts contours of constant $P_{c}$ in the $\Delta
m^{2}\times\sin^{2}2\theta$-plane along with the $z$-dependent values of the
oscillation parameters. Here, however, both the masses and mixing angles go to
zero as $z$ grows. Explicitly, we postulate that the redshit-dependent mixing
angles $\theta_{ij}$, $ij=12,13,23$ are

$\theta_{ij}(z)=\frac{\theta_{ij}}{1+(z/z_{s})^{{\rm B}_{s}}}\,.$ (IV.5)

Similar to the mass varying scenario, the non-adiabatic evolution of the
neutrinos for smaller masses and mixing angles will lead to an enhancement of
the $\nu_{e}$ flux in the Earth. The lower values of the mixing parameters
imply that the flavor resonances happen at lower electron number densities.
The minimum densities considered here are around $2{\rm\ g}/{\rm cm^{3}}$. If
the MSW resonance happens at lower densities, neutrinos will not “cross” them
as they exit the supernova. In this case, flavor-evolution resembles the
vacuum case Friedland (2000). For $B_{s}=5$ and $z_{s}=0.32$, this happens for
$z\sim 1.3$ for the “atmospheric” resonance and $z\sim 0.6$ for the solar one.

Figure 7: Constant crossing probability contours in the
$\sin^{2}2\theta\times\Delta m^{2}$–plane. These define three regions: (I)
$P_{c}<0.1$, (II) $0.1<P_{c}<0.9$, and (III) $P_{c}>0.9$. The color scale
indicates the values of the two independent sets of oscillation parameters –
$\Delta m^{2}_{31}$ and $\sin^{2}2\theta_{13}$ (Atm) and $\Delta m^{2}_{21}$
and $\sin^{2}2\theta_{12}$ (Sol) – as a function of the redshift of neutrino
production. For the mass variation, we make use of Eq. (III.1) with
$z_{s}=0.32$ and $B_{s}=5$.

####  IV.2.2 The DSNB $\nu_{e}$ flux at Earth

We compute the DNSB flux as discussed around Eq. (IV.4a), this time including
in the $z$-dependency of $\theta_{ij}(z)$ ($ij=\\{12,13,23\\}$). As in the
previous subsection, we concentrate on the NO and on electron neutrinos, where
we anticipate the strongest effects. Similar to the results presented in the
last subsection, Fig. 8 depicts the DSNB $\nu_{e}$ flux as function of the
neutrino energy for different values of $z_{s},{\rm B}_{s}$. In the left
panel, we fix ${\rm B}_{s}=5$, and vary $z_{s}\in[10^{-2},10]$. In the right
panel, we fix $z_{s}=0.3$, and vary ${\rm B}_{s}\in[1,100]$. We observe that
the enhancement of $\Phi_{\nu_{e}}/\Phi_{\nu_{e}}|_{\rm SM}\sim 1.5$ at
$E=3{\rm\ MeV}$, larger than what we found in the mass-only varying case. At
higher energies, instead, the flux is relatively suppressed, by a factor
ranging from, roughly, $0.6$ for $E=20{\rm\ MeV}$ to $0.4$ at $E=50{\rm\
MeV}$. Hence, the fact that the mixing angles also decrease with increasing
redshift leads to more pronounced effects. Taking as example $z_{s}=0.05,{\rm
B}_{s}=5$, we find that a $\nu_{e}$ emitted at redshifts $z\gtrsim z_{s}$ will
exit mostly as a $\nu_{1}$ (assuming the normal mass ordering) since the
mixing angles are small enough that the PMNS matrix is effectively diagonal.
After exiting the supernova, neutrino masses turn on in such a way that they
remain a $\nu_{1}$ throughout. Thus, at the Earth, the electron survival
probability is simply $P_{ee}=|U_{e1}|^{2}\approx 0.67$, which enhances the
observable $\nu_{e}$ flux at lower energies. For the same reason, the flux is
suppressed at higher energies $E\gtrsim 10$ MeV, such that, for $E\gtrsim
20{\rm\ MeV}$, the flux lies below the smallest value allowed by the
uncertainty on the SFR, see the inset in the left panel.

Figure 8: (Left) The DSNB $\nu_{e}$ flux as a function of the neutrino energy
for different values of $z_{s}\in[10^{-2},1]$ (rainbow colored) and fixed
${\rm B}_{s}=5$ and for the standard case (orange) including the star-
formation-rate uncertainty (orange band). The dashed orange line indicates the
DSNB flux assuming massless neutrinos, i.e., $P_{ee}(z)=1$ for all $z$.
(Right) The DSNB $\nu_{e}$ flux as a function of the neutrino energy for
different values of ${\rm B}_{s}\in[1,100]$ (rainbow colored) and fixed
$z_{s}=0.32$ and for the standard case (orange) including the star-formation-
rate uncertainty (orange band). The dashed orange line indicates the DSNB flux
assuming massless neutrinos, i.e., $P_{ee}(z)=1$ for all $z$. We assume the
normal ordering for the neutrino masses.

The dependence on $z_{s}$ and ${\rm B}_{s}$ of the final flux is similar to
the masses-only varying case. If $z_{s}\lesssim 0.1$, the DSNB is mostly
composed of neutrinos that were emitted when their masses and mixing angles
where small. On the other hand, if $z_{s}\gtrsim 2$, the largest contribution
to the DSNB comes from neutrinos produced with masses and mixing angles
similar to the ones observed today. In the latter case, the DSNB will be
consistent with standard values. On the other hand, depending on how fast the
transition between almost massless neutrinos and the observed mixing pattern
occurs, parametrized by ${\rm B}_{s}$, the flux is enhanced at low energies.
If the transition is rather sharp ($B_{s}\gtrsim 10$), the DSNB is simply the
superposition of a nearly massless component coming from SN explosions with
$z>z_{s}$, and a standard part emitted when $z<z_{s}$. For smaller values of
${\rm B}_{s}$, the dependence on redshift is smoother, leading to a small
variation of the masses as a function of redshift. For instance, for ${\rm
B}_{s}=1$, $\Delta m_{ij}^{2}(z=10)/\Delta m_{ij}^{2}\sim 0.02$ and the
propagation in the SN is still adiabatic. In this case, there are no
significant changes to the DSNB spectrum.

The $\overline{\nu}_{e}$ spectra in this case is virtually unaltered. The MSW
adiabatic flavor conversion predicts that $\overline{P_{ee}}=|U_{e1}|^{2}$,
value equal to the probability obtained when the mixing angles are small.
Since $\overline{\nu}_{e}$ would be mostly composed by $\overline{\nu}_{1}$,
because the PMNS matrix would be close to diagonal, the predicted antineutrino
flux at the Earth would be identical to the standard case. As before, a
measurement of both neutrinos and antineutrinos from the DNSB would be crucial
to test this scenario.

##  V Event Spectra in a DUNE-like detector

The detection of the DSNB is one of the main goals of current and future
experiments, including SK and HK Abe et al. (2021, 2018), JUNO An et al.
(2016), and DUNE Abi et al. (2020). Our results from the last section can be
summarized as follows. If the neutrino mass varies as a function of redshift
around $z\sim 1$ and if the neutrino mass ordering is normal, we expect the
DNSB $\nu_{e}$ flux to be very different from standard expectations. The DSNB
$\overline{\nu}_{e}$ flux, on the other hand, is quite indifferent to the
potential $z$-dependency of neutrino masses. These two facts point towards a
simple strategy for testing the hypothesis that neutrino masses “turn on” as a
function of time. The detection of the DSNB $\overline{\nu}_{e}$ flux in
experiments like SK and HK,†††These experiments, along with scintillator
experiments, predominantly detect the DSNB via inverse beta-decay. can be used
to normalize the total flux, thus reducing systematic uncertainties, including
those related to uncertainties in the SFR. Meanwhile, data from an experiment
like DUNE, which can detect electron neutrinos instead of antineutrinos, can
be used to provide information on whether the $\nu_{e}$ spectrum is consistent
with standard expectations.

Of course, a measurement of the DSNB in either Cherenkov or Liquid Argon
detectors is not an easy task. There are many sources of uncertainty and
backgrounds that will impact the search for the DSNB. We do not address these
in any detail here but instead, provide a simple example. We compute the
number of events at a DUNE-like detector fixing ${\rm B}_{s}=5,z_{s}=0.05$,
assuming an exposure of 400 kton-years, and considering as detection channel
the process $\nu_{e}+\ ^{40}{\rm Ar}\rightarrow\ ^{40}{\rm K^{*}}+e^{-}$. As
far as other characteristics of the DUNE-like detector, we repeat the
assumptions we made in Ref. de Gouvêa et al. (2020). Fig. 9 depicts event
spectra as a function of the electron kinetic energy. The standard case is
depicted in orange along with the uncertainties associated to our imperfect
understanding of the SFR (orange region). We consider the case where only the
neutrino masses vary (Tyrian purple) and the one where both the masses and
mixing angles vary (green). The blue curve corresponds to the expected
$\nu_{e}$ flux under the assumption that the neutrinos are massless. The gray
bands correspond to regions where background events are expected to be
dominant. If the neutrino masses “turn on” at a finite redshift, the DSNB
$\nu_{e}$ flux is significantly smaller relative to standard expectations, as
observed in the previous section. For $E_{e^{-}}\gtrsim 30$ MeV, in the case
where both masses and mixing angles vary (green), the flux is expected to lie
slightly below the orange-shaded standard region. Fig. 9 reveals that a better
understanding of systematic uncertainties is crucial to test the hypothesis
that the neutrino oscillation parameters is $z$-dependent. As previously
mentioned, a high-statistics measurement of the DSNB antineutrino flux should
play a decisive role in reducing uncertainties.

Figure 9: $\nu_{e}$ DSNB event spectra in a DUNE-like detector, assuming 400
kton-years of exposure, as function of the recoil-electron energy. The
spectrum in the standard case is in orange, together with the uncertainty
associated to the star-formation rate (orange region). Other spectra
correspond to the case of mass-varying neutrinos (Tyrian purple), the case
where both masses and mixing angles vary (green), for $z_{s}=0.05$ and ${\rm
B}_{s}=5$, and the case where all neutrino masses are zero (cyan). We assume
the normal ordering for the neutrino masses. The gray regions correspond to
those where the expected number of background events is dominant.

##  VI Conclusions

After more than two decades, the mystery surrounding the origin of the
neutrino mass persists. A popular direction to pursue is to introduce new
physics at high or very high energy scales but it is clear that very light new
physics can also do the job. There is the possibility that the neutrinos are
effectively massless at high redshifts and gain masses only recently, at very
low redshifts, due to some exotic new physics operating at these scales. Such
low-scale physics only affects the evolution of the Universe after photon
decoupling and hence is completely compatible with observations of the CMB and
other cosmic surveys.

In this work, we propose that imprints of such low redshift neutrino-mass
generation can be found on the diffuse supernova neutrino background (DSNB).
The DSNB consists of neutrinos from all past supernovae (SNe), since the birth
of star formation (redshifts around $5$). If neutrino masses are generated at
relatively low redshifts, neutrino flavor-evolution through the SN is
different from standard expectations. These effects can lead to significant
changes to the flavor content of the neutrinos arriving at the Earth. Using a
phenomenological parametrization for the redshift evolution of neutrino masses
and mixing angles, we computed the DSNB spectra at the Earth. We found that
the DSNB $\nu_{e}$ spectral shape is sensitive to the epoch of neutrino mass
generation: the peak can be, roughly, larger by up to a factor $1.5$, while
the tail can be suppressed, leading to a more pinched spectrum. We also
identified scenarios where, earlier in the history of the Universe, neutrino
flavor propagation is completely non-adiabatic inside the SN. Finally, we
simulated DSNB event spectra in a DUNE-like detector for different hypotheses
concerning the time-dependency of the neutrino oscillation parameters and
demonstrated that redshift-varying neutrino masses and mixing angles can lead
to the suppression of the $\nu_{e}$ event spectrum. We find that there are
circumstances under which effects due to time-dependent oscillation parameters
are significant even if one includes uncertainties associated with our current
understanding of the SFR, especially in the higher energy bins. The concurrent
measurement, with enough statistics, of both electron neutrinos and
antineutrinos from the DSNB, should allow one to make relatively robust claims
about the constancy of neutrino masses.

Measurements of the DSNB are possibly the only way to test scenarios where the
mass-generation of neutrinos occurred only recently in the history of the
Universe. The Super-Kamiokande experiment, doped with gadolinium, is expected
to make a compelling discovery of the DSNB within this decade Abe et al.
(2021). Future experiments like Hyper-Kamiokande (also doped with gadolinium)
and DUNE are expected to collect a significant sample of DSNB events. On the
astrophysical front, we expect the uncertainties on the SFR to go down in the
coming decades. As a result, it is exciting to wonder whether a measurement of
the DSNB can shed some light on the origin of the neutrino mass.

## Acknowledgements

We would like to thank Pedro Machado for illuminating discussions on mass-
varying neutrinos. IMS, YFPG, and MS would like thank Northwestern University
where part of the work was done. IMS and YFPG would like to thank the Fermilab
theory group where this work started and for the lovely years well spent
there. This work was supported in part by the US Department of Energy (DOE)
grant #de-sc0010143 and in part by the National Science Foundation under Grant
Nos. PHY-1630782 and PHY-1748958.

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