# Neutrino Flavor Conversions in High-Density Astrophysical and Cosmological
Environments

Francesco Capozzi  Instituto de Física Corpuscular, Universidad de Valencia &
CSIC, Edificio Institutos de Investigación, Calle Catedrático José Beltrán 2,
46980 Paterna, Spain  Ninetta Saviano  Scuola Superiore Meridionale,
Università degli studi di Napoli “Federico II”, Largo San Marcellino 10, 80138
Napoli, Italy  INFN - Sezione di Napoli, Complesso Univ. Monte S. Angelo,
I-80126 Napoli, Italy

(February 2, 2022)

###### Abstract

Despite being a well understood phenomenon in the context of current
terrestrial experiments, neutrino flavor conversions in dense astrophysical
environments probably represent one of the most challenging open problems in
neutrino physics. Apart from being theoretically interesting, such a problem
has several phenomenological implications in cosmology and in astrophysics,
including the primordial nucleosynthesis of light elements abundance and other
cosmological observables, nucleosynthesis of heavy nuclei and the explosion of
massive stars. In this review, we briefly summarize the state of the art on
this topic, focusing on three environments: early universe, core-collapse
supernovae and compact binary mergers.

##  I Introduction

Neutrino flavor conversions, or oscillations, are a genuine quantum mechanical
phenomenon, for which a flavor eigenstate $\nu_{\alpha}$ ($\alpha=e,\mu,\tau$)
is converted to $\nu_{\beta}$ ($\beta\neq\alpha$) during propagation, due to
it being an admixture of different mass (or propagation) eigenstates $\nu_{i}$
($i=1,2,3$). This has been firmly established experimentally with atmospheric
RevModPhys.88.030501  and solar neutrinos RevModPhys.88.030502 , proving that
neutrinos are massive particles and leading to the award of the Nobel Prize in
Physics 2015 to the experiments involved in this discovery. Among the
parameters describing such a phenomenon, the three mixing angles
$(\theta_{12},\theta_{13},\theta_{23})$ and the two mass differences $(\Delta
m^{2}_{21},\Delta m^{2}_{31})$ have been measured with a precision between 1
and 10% Capozzi:2021fjo ; Gonzalez-Garcia:2021dve ; deSalas:2017kay . The mass
ordering (normal for $\Delta m^{2}_{31}>0$ or inverted for $\Delta
m^{2}_{31}<0$) is still unknown, but there is a $3\sigma$ hint in favor of the
normal one. The CP phase $\delta$, which leads to the violation of the CP
symmetry if $\delta\neq 0,\pi$, is also largely unknown, but the CP conserving
values seem to be disfavoured at 90% confidence level and the best fit seems
to lie in the range [$\pi,2\pi$]. Despite the presence of some partially
unknown parameter and of few anomalies LSND:2001aii ; MiniBooNE:2018esg ;
Abdurashitov:2005tb ; Kaether:2010ag ; Mention:2011rk , which still need to be
confirmed and require further investigations, flavor conversions relevant in
current experiments are theoretically under control.

Our understanding becomes less solid the higher the density of neutrinos being
considered. In particular, when the interactions among neutrinos are no longer
negligible, their flavor evolution becomes deeply non linear and cannot be
treated in the standard way used in the context of terrestrial experiments.
The environments where such conditions occur are the early universe, core-
collapse supernovae and merger events between two compact astrophysical
objects. Apart from representing an extremely interesting theoretical problem
to solve, flavor conversions in dense environments have a much deeper
relevance. In particular, neutrino flavor conversions in the early universe
are a fascinating problem involving collisional damping, refractive effects
from charged leptons, and neutrino self-interactions Dolgov:2002ab . In this
context, there is a strong interest for active-sterile neutrino oscillation in
the early universe in a broad range of mass for the sterile neutrino.
Concerning supernovae, how neutrinos are distributed among all flavors affects
the amount of energy deposited by these particles just below the shock wave in
a supernova, potentially changing a successful explosion into a failed one, or
viceversa. Furthermore, the large amount of information carried by supernova
neutrinos can be fully exploited by their detection only if we can precisely
predict both the original fluxes for each flavor and how they are modified
during propagation.

In this review our goal is to provide a short summary of the state of the art
of the available literature on flavor conversions in dense environments,
focusing on both the theoretical understanding of the equations of motion and
applications to phenomenological aspects of the environments under
consideration. In Section II we present a common theoretical framework based
on a Boltzmann equation of the neutrino density matrix. In Section III, IV and
V we discuss old and recent results concerning flavor conversions in the early
universe, core-collapse supernovae and compact binary mergers, respectively.
Finally we give our conclusions and future perspectives.

##  II Equations of motions

The simple formalism of the Schrödinger equation for the treatment of neutrino
mixing is not very practical whenever we are concerned with the evolution of a
statistical ensemble of neutrinos simultaneously mixing and scattering in a
medium. In such cases the evolution cannot be easily understood in terms of
the propagation of a beam. Indeed, while this treatment allows to obtain the
transition probabilities between single-particle states and to study the
oscillations between field amplitudes,it cannot be directly applied to many-
particle states. Specifically, we need to handle the interactions between the
medium and the coherent superposition of states that are involved. In presence
of a propagation medium it is important to follow the evolution of the whole
ensemble, including those particles scattered out of the beam. Two types of
effects have to be considered. One concerns the refractive effect namely the
potential due to coherent forward scattering on charged leptons present in
ordinary matter Wolfenstein:1977ue ; Mikheyev:1985zog . The second effect is
due to collisions which destroy the coherence of the evolution and can
influence the behavior of the mixing process Stodolsky:1986dx .

In addition to the interactions with the external medium, one has also to
consider the interactions of neutrinos among themselves (_self-interaction_).
Indeed, as pointed out by Pantaleone in the 1992 Pantaleone:1994ns  , in the
deepest region of the Supernova (SN) and in the early universe, the neutrinos
gas is so dense that the neutrinos themselves form a background medium leading
to intriguing non-linear effects in the neutrino flavor conversions.

A proper treatment of all these effects requires to exploit the density matrix
formalism, in which mixed quantum states for neutrinos and possible loss of
coherence due to real collisions are described together. Strictly following
the derivation of Sigl:1993ctk  and also Raffelt:1991ck ; Raffelt:1992uj ;
Dolgov:1980cq ; Barbieri:1990vx , an ensemble of neutrinos and antineutrinos
can be characterized by the $n\times n$ density matrices
$\varrho_{\mathbf{p,x}}(t)$ (and an analogue for antineutrinos
$\overline{\varrho}_{\mathbf{p,x}}(t)$) defined as

$\displaystyle\small{\varrho_{\mathbf{p,x}}=\left(\begin{array}[]{ccc}\varrho_{ee}&\varrho_{e\mu}&\varrho_{e\tau}\\\
\varrho_{\mu e}&\varrho_{\mu\mu}&\varrho_{\mu\tau}\\\ \varrho_{\tau
e}&\varrho_{\tau\mu}&\varrho_{\tau\tau}\end{array}\right)}~{},$ (4)

where the diagonal elements of $\varrho_{\mathbf{p,x}}$ are the usual particle
distribution functions (occupation numbers) for the corresponding neutrino
species, while the off-diagonal ones encode phase information and vanish for
zero mixing. In general $\varrho_{\mathbf{p,x}}$ depends on the momentum ${\bf
p}$ and on the coordinate ${\bf x}$).

The entries of $\varrho_{\mathbf{p},\mathbf{x}}$ in Eq. 4 are intended as the
expectation values of the neutrino field bilinears. Under this approximation,
labelled as the mean field approach, one neglects possible entanglement
effects. Despite the wide use of this approach, it has been debated in a
series of papers Friedland:2003eh ; Friedland:2003dv ; Balantekin:2006tg ;
Pehlivan:2011hp ; Birol:2018qhx ; Patwardhan:2019zta ; Cervia:2019res ;
Rrapaj:2019pxz ; Roggero:2021asb ; Roggero:2021fyo .

In this review we will focus on the results obtained in the mean field
approximation. In this case the flavor evolution for the density matrices
$\varrho_{\bf p,x}$ and $\bar{\varrho}_{\bf p,x}$ in dense environments is
governed by the Boltzmann collision equations

$\hat{L}[\varrho_{\bf p,x}]=-i[{\Omega}_{\bf p,x},\varrho_{\bf
p,x}]+\hat{\mathcal{C}}[\varrho_{\bf p,x}]$ (5)

where at r.h.s. ${\Omega}_{\bf p,x}={\Omega}^{\rm vac}_{\bf p}+{\Omega}^{\rm
ref}_{\bf p,x}$ and ${\Omega}^{\rm ref}_{\bf p,x}={\Omega}^{\rm mat}_{\bf
p,x}+{\Omega}^{\rm\nu\nu}_{\bf p,x}$. ${\Omega}^{\rm mat}_{\bf p,x}$ describes
the interactions with all other particles of the medium except for neutrinos
and ${\Omega}^{\rm\nu\nu}_{\bf p,x}$ represents the “self-interactions” with
the other neutrinos in the medium and, depending on the neutrino ensemble
$\varrho_{\mathbf{p,x}}$ itself, it makes the problem non-linear. Finally, the
last term at the right-hand side (r.h.s) of the Eq. 5 is the second order in
the perturbative expansion ($\propto G^{2}_{F}$), known as collisional term,
responsible for the breaking of the coherence of the neutrino ensemble. The
l.h.s. of Eq. 5 contains the Liouville operator

$\hat{L}[\varrho_{\bf p,\bf x}]=\partial_{t}\varrho_{\bf p,x}+{\bf v}_{\bf
p}\cdot{\nabla}_{\bf x}\,\varrho_{\bf p,x}+\dot{\bf p}\cdot{\nabla}_{\bf
p}\,\varrho_{\bf p,x},$ (6)

which includes temporal evolution and spatial transport phenomena. In
particular, the first term represents an explicit time dependence, the second
a drift caused by the particles free-streaming, and the third the effects of
external macroscopic forces, for example gravitational deflection. The
equation for $\bar{\varrho}_{\bf p,x}$ is similar to the equation 5, where
inside the commutator the relative sign of ${\Omega}^{\rm vac}$ and
${\Omega}^{\rm ref}$ changes.

##  III Flavor Conversions in the early universe

The evolution in time in the form of Boltzmann-like equations applied to the
early universe, safely considered isotropic and homogeneous at large scale,
reduces to

$\partial_{t}\varrho_{\bf p}=-i[{\Omega}_{\bf p},\varrho_{\bf
p}]+\hat{\mathcal{C}}[\varrho_{\bf p}]~{},$ (7)

where

$\partial_{t}\rightarrow\partial_{t}-Hp~{}\partial_{\mathbf{p}}$ (8)

with $H$ the Hubble parameter which encodes the information about the universe
expansion. The quantity $\Omega_{p}$ includes the _vacuum term_ , the
_Refractive matter term_ and the _neutrinos self-interactions_.

For the Refractive matter term, since the electron-positron density is
expected to be of the same order of the baryons-antibaryons ones which is
subdominant. Therefore, it is necessary to consider the higher order which
depends on the sum of the electron and positron energy densities. Concerning
the self-interaction term, the contributions to neutrino-neutrino forward
scatterings come at leading order from a term
$\Omega_{\mathbf{p,x}}^{\nu\nu(0)}\propto G_{F}(\varrho-\overline{\varrho})$,
and at higher order from a term
$\Omega_{\mathbf{p,x}}^{\nu\nu(1)}\propto\frac{G_{F}}{m^{2}_{Z}}(\varrho+\overline{\varrho})$
111In order to include the correction due to the non-local nature of the Z
boson propagator which mediate forward scattering on neutrinos of the same
species.. In absence of a neutrino-antineutrino asymmetry, as expected in the
standard case, the only contribution is given by
$\Omega_{\mathbf{p,x}}^{\nu\nu(1)}$ which is subleading for the neutrino
evolution, since its numerical value is small. Conversely, in the case of
large neutrino asymmetries, $\Omega_{\mathbf{p,x}}^{\nu\nu(0)}$ becomes
important and the evolution is dominated by the effect of _synchronized_
oscillations, i.e. the self-potential forces all neutrino modes to follow the
same oscillation pattern Dolgov:2002ab . Finally, the _collisional term_
$\hat{\mathcal{C}}[\varrho_{\bf p}]~{}$ at high temperature of the primordial
plasma, is very important since it breaks the coherence of the neutrino
ensemble. It damps the off diagonal terms of the density matrix
$\varrho_{\mathbf{p}}$ and it pushes the diagonal terms towards their
equilibrium distributions.

In the case of instantaneous neutrino decoupling (at temperature around 1 MeV)
from the primordial plasma of particles, the relic neutrinos of each flavor
have the same momentum distributions making the effect of the oscillations
irrelevant except for an exchange of equal neutrino spectra. However, the
neutrino oscillations become important in some situations in which unequal
neutrino distributions arise, such as small flavour-dependent distortions due
to not-instantaneous decoupling, non zero neutrino-antineutrino asymmetry, or
sterile species mixing with the active ones Lesgourgues:2013sjj ;
Dolgov:2002wy . In such situations, neutrino oscillation could modify the non-
electromagnetic contribution of the neutrino heating to the total relativistic
energy density, defined in term of the effective number of neutrinos species

$\epsilon_{R}=\epsilon_{\gamma}\left(1+\frac{7}{8}\left(\frac{4}{11}\right)^{4/3}~{}N_{\textrm{eff}}\right)$
(9)

where $\epsilon_{R}$ and $\epsilon_{\gamma}$ are the total energy density of
radiation and the energy density of photon, respectively. The factor 4/11
comes form the heating of the photons due to the $e^{+}-e^{-}$ annihilation.
In the case of small non-thermal distortions plus oscillations, the value of
this parameter is estimated to be $\rm N_{eff}=3.046$ Mangano:2005cc . The
slightly excess with respect to the value $3$ is to due to the non
instantaneous neutrino decoupling thanks to which neutrinos share a small part
of the entropy release after the $e^{+}-e^{-}$ annihilation. More recently new
calculations, where finite-temperature effects in the quantum electrodynamics
plasma and a full evaluation of the neutrino–neutrino collision integral are
taken into account, provide a number very close to the original one
Bennett:2020zkv ; Froustey:2020mcq .

An important implication of active neutrino oscillations at temperature of MeV
of the Universe is the evolution of a possible neutrino-antineutrino
asymmetry, denoted by

$L_{\nu_{\alpha}}=\frac{n_{\nu_{\alpha}}-n_{\bar{\nu_{\alpha}}}}{n_{\gamma}}.$
(10)

Considering the very small value of the baryonic asymmetry,
$\eta_{\beta}=(n_{B}-n_{\bar{B}})/n_{\gamma}\simeq 6\times 10^{-10}$, it is
reasonable to expect, for the charge neutrality, the same order of magnitude
for the charged lepton asymmetry. In the neutrino case, being neutral
particles, the constrains on $L_{\nu_{\alpha}}$ are quite loose, allowing
values for $L_{\nu_{\alpha}}$ order of magnitude larger then $\eta_{\beta}$.
The presence of $L_{\nu_{\alpha}}$ implies a degeneracy parameter in the
neutrinos spectra, $\xi_{\alpha}=\mu_{\nu_{\alpha}}/T_{\nu}$. A significant
value of a cosmological neutrino-antineutrino asymmetry implies an extra
contribution for $\rm N_{eff}$ without to introduce additional relativistic
degrees of freedom. An important issue is to compute the evolution of
$L_{\nu_{\alpha}}$ in the epoch of the Universe before Big Bang
Nucleosynthesis (BBN). A combined analysis of active neutrino flavor
oscillations and BBN has led to an almost standard value for the effective
number also in the presence of neutrino asymmetries, with $N_{\rm eff}<3.2$ at
$95\%$ C.L. Mangano:2011ip .

###  III.1 Active-sterile neutrino oscillations

The existence of sterile neutrinos is investigated in a very broad range of
mass, from the GUT (Grand Unification Theory) scale to the eV scale. Some of
them are theoretically very well motivated, other are more suggested by
possible experimental hints. In this context, a special and interesting case
is represented by neutrino oscillations among active and sterile neutrinos at
different mass scales.

eV sterile neutrinos-Light sterile neutrinos with a mass around $\sim 1$ eV,
which are mixing with the active ones, have been suggested to solve different
anomalous results observed in $\nu_{\mu}\to\nu_{e}$ (LSND LSND:2001aii  and
MiniBooNE MiniBooNE:2018esg ) and $\nu_{e}\to\nu_{e}$ (SAGE
Abdurashitov:2005tb  and GALLEX Abdurashitov:2005tb ) short-baseline
oscillations, as well as in $\bar{\nu}_{e}\to\bar{\nu}_{e}$ reactor neutrino
experiments Mention:2011rk . Many analyses have been performed to explain the
anomalies and scenarios with one (dubbed “3+1”) or two (“3+2”) sub-eV sterile
neutrinos Giunti:2011gz ; Kopp:2013vaa ; Dentler:2018sju ; Gariazzo:2017fdh ;
Diaz:2019fwt ; Boser:2019rta  have been proposed to fit the different data.
The search for sterile neutrinos in laboratory experiments is still
undergoing. The $\nu_{\mu}\to\nu_{e}$ has been recently tested in MicroBooNE
MicroBooNE:2021nxr ; MicroBooNE:2021rmx ; MicroBooNE:2021sne ;
MicroBooNE:2021jwr , whose current data cannot exclude the full parameter
space hinted at by MiniBooNE Arguelles:2021meu  and if combined with MiniBooNE
leads to a preference for a 3+1 scenario with a best fit at $\Delta m^{2}\sim
0.2$ eV2 and very large mixing MiniBooNE:2022emn . The $\nu_{e}\to\nu_{e}$ has
now been updated by the data collected by BEST Barinov:2021asz , which
strongly favors rather large mixing angle and masses larger than 1 eV2.
Concerning the reactor anomalies, a recent reevaluation of the neutrino fluxes
Kopeikin:2021ugh  has led to new global analyses Giunti:2021kab ;
Berryman:2021yan  showing that combined reactor data are consistent with no
sterile-neutrino oscillations. Despite the recent progress in all oscillation
channels, new experimental data is still needed to fully assess the sterile
neutrino option.

In this context, cosmological observations represent a valid complementary
tool to probe these scenarios, being sensitive to the number of neutrinos and
to their mass at eV scale Hagstotz:2020ukm . In fact, the sterile states can
be produced in the early universe via oscillations with active neutrinos and
can modify cosmological observables. In particular, adding extra contribution
to the radiation content in the Universe, expressed in terms of the effective
number of neutrino species, $N_{\rm eff}$, has indeed a strong impact on both
BBN primordial yields and the Cosmic Microwave Background (CMB) anisotropy.
Moreover, the mass of sterile species can impact the structure formation (LSS)
spectrum Mangano:2011ar ; Giusarma:2011ex ; Lesgourgues:2012uu ; Riemer-
Sorensen:2013iql ; Hagstotz:2020ukm . From recent astrophysical data of
Deuterium and Helium, as well as last constraints on radiation and mass, even
a single extra thermalized sterile neutrino with mass $m\sim 1$ eV appears to
be inconsistent BBN, CMB and LSS data Cooke:2017cwo ; Grohs:2019cae ;
Hsyu:2020uqb ; Fields:2019pfx ; Planck:2018vyg ; Mirizzi:2013kva ;
Gariazzo:2019gyi . A possible escape-route to reconcile the eV sterile
neutrino interpretation with cosmology would be to suppress the sterile
neutrino thermalization and therefore non standard scenarios have to be
invoked to alleviate the tension. One of the proposed mechanism is to consider
a primordial asymmetry $L_{\nu}$ between active neutrinos and antineutrinos
which implies an additional “matter term potential” in the equations of
motion. If sufficiently large, it can block the active-sterile flavor
conversions via the in-medium suppression of the mixing angle. However, this
term can also generate MSW-like resonant flavor conversions among active and
sterile neutrinos, enhancing the sterile neutrino production. A detailed study
of the kinetic equations for active-sterile neutrino oscillations is therefore
mandatory to assess which of the two effects dominates. Recent studies have
shown that asymmetry $L_{\nu}\sim 10^{-2}$ is required in order to achieve a
sufficient suppression of the sterile neutrino abundance, Mirizzi:2012we .
However, such a large value delays the oscillations at temperature close to
neutrino decoupling one inducing distortions on the active neutrino spectra
with non-trivial effects on BBN. Consequently, the tensions with cosmology re-
emerges Saviano:2013ktj . Recently, different groups have proposed and
investigated an alternative method to suppress the sterile neutrino production
based on the introduction of new “secret” self- interactions among sterile
neutrinos, mediated by new gauges boson Chu:2015ipa ; Hannestad:2013ana . Also
in this case , the self-interactions would generate a matter potential in the
flavor evolution equations potentially suppressing the sterile neutrino
abundance. Depending on the selected theoretical model, we have more or less
severe constraints on light sterile neutrinos induced by the cosmological
data. BBN, CMB and neutrino mass bounds strongly constrain the model with
vector boson, disfavoring all the values of the mass of the new gauge boson.
Indeed, while the effective temperature-dependent potential produced by secret
interactions can efficiently suppress active–sterile neutrino mixing in the
early universe down to lower temperatures, the momentum spectra of active
neutrinos will result distorted due to delayed oscillations with impact on the
production of BBN yields Saviano:2014esa . Moreover, efficient collisional
production of $\nu_{s}$ can still occur (depending on the neutrino temperature
and on the mass of the mediator) violating the cosmological constraints on CMB
and neutrino mass bound of cosmological structures Mirizzi:2014ama ;
Forastieri:2017oma  . In particular, scattering processes mediated by the new
particles can be strongly enhanced by the s-channel resonance and by collinear
enhancement in the forward direction. For the mass and mixing parameter
preferred by laboratory anomalies, all values of the vector boson and the
corresponding coupling constants are disfavored Chu:2018gxk , as shown in Fig.
1.

Figure 1: The parameter space of the secret interactions model mediated by a
vector boson of mass M and the corresponding fine structure constant
$\alpha_{s}$ ($\alpha_{s}=g^{2}_{s}/(4\pi)$). A sterile neutrino mass $m_{s}$
= 1 eV and a vacuum mixing angle $\theta_{s}$= 0.1 are assumed. The colored
regions are ruled out by different cosmological observations. This figure is
adapted from Chu:2018gxk .

The situation can be in part different in the case of a very light (or even
massless) pseudoscalar particle mediating the secret interactions, due to its
late time phenomenology, in particular since the collisional term becomes
relevant at late times Archidiacono:2014nda ; Archidiacono:2016kkh ;
Archidiacono:2020yey . Performing a series of extensive analyses fitting
various combinations of CMB data and combining with other cosmological data,
it is still debated if eV sterile neutrinos could be accommodated within the
pseudoscalar interaction model Corona:2021qxl . Another chance to possibly
alleviate the eV-sterile neutrino controversy could be to consider minimal
dark energy models which modify only late time physicsDiValentino:2021rjj .

keV sterile neutrinos\- KeV sterile neutrinos could represent a valid
candidate of Dark Matter since they are neutral, cosmologically stable,
sufficiently non-interacting, and massive Drewes:2016upu ; Abazajian:2017tcc ;
Boyarsky:2018tvu . A renewed interest has been recently sparked by new
appealing and controversial indications for an unidentified X-ray line at 3.5
keV in the spectra of a large number of galaxy clusters and in Andromeda,
possibly consistent with a radiative decay of 7 keV sterile neutrinos.
Bulbul:2014sua ; Boyarsky:2014jta . Sterile neutrinos in this mass range are
usually classified as warm candidates of dark matter because, when they become
non-relativistic, their free-streaming suppresses the matter density
perturbations at the dwarf galaxies scales, more in agreement with the
observations and cosmological simulations. However, the free-streaming length
not only depends on the particle mass, but also on the momentum, and then on
the particle phase-space distribution Murgia:2017lwo . Depending on the
production mechanism for sterile neutrinos, the distribution function can be
more or less warm, with a different impact on the structure formation. The
most natural production mechanism to produce keV sterile neutrinos is by
mixing with active species. The mixing angle which allows the decay of sterile
neutrinos also enables an efficiently production of sterile species in the
early universe. The non-resonant production, linked to DM by Dodelson and
Widrow Dodelson:1993je , is a collision-dominated production via neutrino
oscillations, which could gradually produce enough sterile neutrinos to
explain the observed amount of DM, in according to the production rate

$\Gamma_{\rm prod}=\frac{1}{2}\sin^{2}2\theta_{m}\Gamma$ (11)

driven by the active neutrino interaction rate

$\Gamma\sim G_{F}^{2}T^{5}.$ (12)

However, in the absence of new physics, this mechanism results to be similar
to the hot DM case and in tension with various current astrophysical
observations Tremaine:1979we ; Boyarsky:2008ju ; Gorbunov:2008ka ;
Horiuchi:2013noa ; Ng:2019gch . However, this situation could change in the
case of a resonant enhancement in the active-sterile transitions due to a
lepton number asymmetry $L_{\nu}$ possibly present in the early universe.
Indeed, as pointed out by Shi-Fuller Shi:1998km , a suitable value of such an
asymmetry produces sterile neutrinos at a specific combination of momentum and
temperature resulting in colder DM distribution and so improving the
compatibility with astrophysical and cosmological data. Anyway, also this
scenario is strongly constrained Cherry:2017dwu ; Wang:2017hof ;
Schneider:2017qdf ; Schneider:2016uqi ; Drewes:2016upu  and the 7 keV sterile
neutrino interpretation of the observed 3.5 keV line is not favoured
Dessert:2018qih ; Enzi:2020ieg . As recently speculated by different groups,
the allowed mass and mixing parameter space could be extended in presence of
new interactions among only active neutrinos DeGouvea:2019wpf ; Kelly:2020pcy
or among only sterile species Johns:2019cwc  which imply extra terms in the
interaction rate $\Gamma$ (eq. (12)).

MeV sterile neutrinos\- They emerge rather naturally in theories beyond the
Standard Model, like low scale seesaw models in the Neutrino Minimal Standard
Model being related to fundamental open problems of particle physics such as
the origin of neutrino mass, the baryon asymmetry in the early universe and
the nature of dark matterAsaka:2005an ; Asaka:2005pn . Depending on the mixing
with active neutrinos, the parameter space of sterile neutrino in MeV mass
range is strong constrained by: collider and beam-dump experiments
Alekhin:2015byh , searches of decays of D mesons and $\tau$ leptons
Chun:2019nwi ; Orloff:2002de  and core-collapse supernova limits due to energy
loss which would short the observed neutrino burst Dolgov:2000jw ;
Fuller:2008erj ; Mastrototaro:2019vug . Further and complementary constraints
can also come from cosmological observations. Indeed, sterile neutrinos
produced in the early universe via collisional processes involving active
neutrinos, can decay into lighter species injected into the primordial plasma
altering both the effective number of neutrino species $N_{\rm eff}$ and the
abundance of the primordial yield. Solving the exact Boltzmann equation for
sterile and active neutrino evolution is possible to set constraints on the
mixing angles or lifetimes Mastrototaro:2021wzl .

##  IV Flavor Conversions in Core-Collapse Supernovae

Core-collapse supernovae are the final explosion of a star with a mass greater
than 8 $M_{\odot}$. After extinguishing all the nuclear fuel, the core of the
star begins to collapse under the effect of gravity until it reaches the
Chandrasekhar limit. Then the density rapidly increases until the core reaches
the nuclear density ($3\times 10^{14}$ g/cm3), when matter becomes
incompressible and the collapse stops, producing a rebound Colgate:1960zz .
The bounce generates a shock wave travelling outwards and ejecting outer
layers. This shock may stall without driving off the stellar mantle and
envelope. According to the “delayed explosion scenario” Colgate:1966ax ;
Bethe:1985sox , since 99% of the liberated gravitational energy ($10^{53}$
ergs) is in the form of neutrinos Gamow:1940eny ; Gamow:1941gis  such
particles might deposit enough energy below the shock wave in order to
revitalize the explosion.

Neutrino emission can be divided in three main phases (see e.g.,
Mirizzi:2015eza ; Janka:2017vlw ; Burrows:2020qrp  for detailed overviews).
The neutronization burst lasts $\sim 10$ ms and consists in the rapid electron
capture on dissociated nuclei. This leads to a sudden rise in the luminosity
of $\nu_{e}$ up to $10^{53}$ erg/s, while for the other flavors it is
negligible. The accretion phase lasts a few hundreds of ms and during this
time material continues to fall onto the core and accretes on it, leading to a
flux of neutrinos and antineutrinos of all species. The high density traps
neutrinos Mazurek1 ; Freedman:1973yd ; Mazurek2 ; Mazurek3 , which escape from
the last scattering surface, which is usually called the neutrinosphere. In
the cooling phase the remaining proto-neutron star cools by neutrino emission
of all flavors. During this phase the luminosity is approximately equal for
all flavors.

Apart from inside the neutrinosphere, in general the propagation of supernova
neutrinos is not isotropic. Consequently, the self-interaction potential has
the following form

$\Omega_{\mathbf{p,x}}^{\nu\nu}=\sqrt{2}G_{F}\int\,\frac{\mathrm{d}^{3}\mathbf{q}}{(2\pi)^{3}}\,(\varrho_{\mathbf{q,x}}-\bar{\varrho}_{\mathbf{q,x}})(1-\mathbf{q}\cdot\mathbf{p})\,,$
(13)

whereas for the isotropic case of the early universe the
$(1-\mathbf{p}\cdot\mathbf{q})$ term is absent. The phenomenology of flavor
conversions is strongly dependent on the relative size of
$\Omega_{\mathbf{p,x}}^{\nu\nu}$ and $\Omega_{\mathbf{p,x}}^{\rm mat}$, which
changes as a function of both time after bounce and radial distance from the
centre. In general, a few thousands kilometers from the centre of the
supernova the neutrino density is low enough to make
$\Omega_{\mathbf{p,x}}^{\nu\nu}$ negligible. In this case
$\Omega_{\mathbf{p,x}}^{\rm mat}$ can give rise to the MSW resonance
Wolfenstein:1977ue ; Mikheyev:1985zog . Deeper inside the supernova,
$\Omega_{\mathbf{p,x}}^{\nu\nu}$ can be of the same order or even larger than
the matter potential, making flavor conversions a deeply non linear phenomenon
and giving rise to the so called collective effects. Such a name stems from
the fact that neutrinos can change their flavor in a coherent fashion,
regardless of their original energy.

A remark is in order. The role of flavor conversions in a SN is not important
just for a correct determination of the expected neutrino signal, but also in
the context of astrophysical processes, such as the nucleosynthesis of heavy
elements Burbidge:1957vc ; Qian1993 ; Chakraborty:2009ej ; Duan:2010af ;
Xiong:2020ntn  and the explosion itself Fuller1992 , since neutrino energy
deposition is flavor dependent. Below we briefly review all types of flavor
conversions happening in a supernova, focusing on both theoretical and
phenomenological aspects.

###  IV.1 MSW resonances

An MSW resonance occurs when the vacuum oscillation frequency is equal to the
spatially dependent matter potential Wolfenstein:1977ue ; Mikheyev:1985zog .
Such a condition is equivalent to $\omega_{E}=\pm V_{\rm mat}(\mathbf{x})$,
where $\omega_{E}=\Delta m^{2}/(2E_{\nu})$ is the vacuum oscillation frequency
of neutrinos, and $V_{\rm mat}(\mathbf{x})=\sqrt{2}G_{F}N_{e}(\mathbf{x})$ is
the potential from coherent forward scattering on electrons in a medium, with
$N_{e}(\mathbf{x})$ being the spatially dependent electron number density. The
$+$ ($-$) sign refers to neutrinos (antineutrinos). If $\Delta m^{2}=\Delta
m^{2}_{31}$, then the resonance occurs for neutrinos in normal mass ordering
and for antineutrinos in inverted mass ordering. If $\Delta m^{2}=\Delta
m^{2}_{21}>0$ then the resonance can only happen for neutrinos. The final
survival probability of electron neutrinos is given by Dighe:1999bi

$P_{ee}(E_{\nu})=\left\\{\begin{array}[]{llll}\sin^{2}\theta_{12}P_{H}\quad(\nu,\text{
NO})\\\ \sin^{2}\theta_{12}\quad(\nu,\text{ IO})\\\
\cos^{2}\theta_{12}\quad(\bar{\nu},\text{ NO})\\\
\cos^{2}\theta_{12}P_{H}\quad(\bar{\nu},\text{ IO})\\\ \end{array}\right.$
(14)

where $P_{H}$ represents the probability that a given mass eigenstate is
transformed to a different one when crossing the resonance point. This is also
known as the crossing probability and it is equal to 0 when the propagation is
adiabatic, whereas it must be determined (numerically or analytically) in the
opposite case. In particular, when the resonance crosses the shock wave, which
introduces a sharp variation in the electron density, the propagation becomes
non adiabatic. This is a time dependent feature, whose potential observation
has been studied in Schirato:2002tg ; Fogli:2003dw ; Fogli:2004ff ;
Tomas:2004gr ; Dasgupta:2005wn ; Choubey:2006aq ; Kneller:2007kg ;
Friedland:2020ecy .

The presence of turbulence in the matter density can introduce stochastic
fluctuations and thus induce random temporal variations in the occurrence of
the MSW resonance Fogli:2006xy ; Friedland:2006ta ; Kneller:2010sc ;
Lund:2013uta ; Loreti:1995ae ; Choubey:2007ga ; Benatti:2004hn ;
Kneller:2013ska . If the fluctuations are of sufficiently large amplitude they
might lead to $P_{H}\to 1/2$ Fogli:2006xy .

###  IV.2 Collective effects

In order to study the phenomenology of supernova neutrino flavor conversions
in the presence of a large self-interaction potential one needs to solve Eq.
5, with $\Omega_{\mathbf{p,x}}^{\nu\nu}$ given by Eq. 13. Being a
7-dimensional system of coupled differential equations, it has been solved
only imposing some simplifying symmetries.

Even without solving the system of partial differential equations, some
information can be obtained through a linear stability analysis Sawyer:2008zs
; Banerjee:2011fj . In this case one expands the density matrix as

$\varrho_{\mathbf{p,x}}(t)=\frac{f_{\nu_{e}}+f_{\nu_{x}}}{2}+\frac{f_{\nu_{e}}-f_{\nu_{x}}}{2}\,\begin{pmatrix}s_{\mathbf{p}}(t,\mathbf{x})&S_{\mathbf{p}}(t,\mathbf{x})\\\
S^{*}_{\mathbf{p}}(t,\mathbf{x})&-s_{\mathbf{p}}(t,\mathbf{x})\end{pmatrix}\,,$
(15)

where $f_{\nu_{\alpha}}$ represents the number of neutrinos with flavor
$\alpha$. Since neutrinos are produced in flavor eigenstates, one expects that
initially $s_{\mathbf{p}}\ll S_{\mathbf{p}}$, which justifies an expansion of
Eq. 5 at linear order in $S_{\mathbf{p}}$. Then, one looks for plane wave
solutions of the form
$S_{\mathbf{p}}(t,\mathbf{x})=Q_{\mathbf{p,k}}\,e^{-i(k_{0}t-\mathbf{k}\cdot\mathbf{x})}$,
where $k_{0}$ is the temporal frequency of the flavor wave and $\mathbf{k}$ is
its spatial wave vector. Plugging the plane wave expansions in Eq. 5 one
arrives at a dispersion relation $\rm{det}[\Pi^{\mu\nu}]=0$, i.e. the
determinant of a $4\times 4$ matrix $\Pi^{\mu\nu}$ is equal to 0, where

$\Pi^{\mu\nu}=\eta^{\mu\nu}+\int_{0}^{+\infty}\frac{E^{2}dE}{2\pi^{2}}\int\frac{d\mathbf{v}}{4\pi}\,g_{E,\mathbf{v}}\,\frac{v^{\mu}v^{\nu}}{v_{\alpha}\,k^{\alpha}-\omega_{E}}\,.$
(16)

Here, $v^{\mu}=(1,\mathbf{v})$, $k^{\mu}=(k^{0},\mathbf{k})$,
$\mathbf{v}=\mathbf{p}/|\mathbf{p}|$ and $g_{E,\mathbf{v}}$ is defined as

$g_{E,\mathbf{v}}=\sqrt{2}G_{F}[f_{\nu_{e},E,\mathbf{v}}-f_{\nu_{x},E,\mathbf{v}}-(f_{\bar{\nu}_{e},E,\mathbf{v}}-f_{\bar{\nu}_{x},E,\mathbf{v}},)]\,,$
(17)

where $f_{\nu_{\alpha},E,\mathbf{v}}$ is the energy and angular distribution
of $\nu_{\alpha}$. If for some real $k_{0}$ the dispersion relation is solved
with an imaginary $\mathbf{k}$, the system has a spatial instability.
Conversely, if for some real $\mathbf{k}$ the dispersion relation is solved
with an imaginary $k_{0}$, the system has a temporal instability. This method
allows one to know whether a system is unstable or not, possibly leading to
significant flavor conversions. However, the final amount of flavor
conversions to be expected is not accessible through this method.

A widely used classification of instabilities is done according to the growth
rate of unstable modes, i.e. the imaginary part of $k$, ${\rm Im}(k)$. We
define slow instabilities those having ${\rm
Im}(k)\propto\sqrt{\omega_{E}\mu}$, while we define fast instabilities those
having ${\rm Im}(k)\propto\mu$. The latter can reach ${\rm Im}(k)\sim O(1)$
cm-1, if they occur deep inside the supernova core, where $\mu$ can be
$O(10^{5})$ cm-1. This is why fast instabilities are usually considered to be
the most likely to affect astrophysical processes occurring in a supernova,
provided that they are triggered in the first place.

It has been proven in Morinaga:2021vmc  that a necessary and sufficient
condition for having fast instabilities is an angular crossing in
$g_{E,\mathbf{v}}$ defined in Eq. 17. Assuming to have azimuthal symmetry
around a specific direction, angular crossing means that there is a $v_{0}$
for which $g_{E,v>v_{0}}\times g_{E,v<v_{0}}<0$. In Dasgupta:2021gfs  such a
proof has been made more general: flavor instabilities, either fast or slow,
can arise only if there is a spectral crossing, either in angle or energy.

In the following we review the state of the art for both slow and fast
instabilities. Other reviews focused on slow modes are Mirizzi:2015eza ;
Chakraborty:2016yeg , whereas the only one already available on fast modes is
Tamborra:2020cul .

####  IV.2.1 Slow Flavor Conversions

The seminal papers investigating collective effects assumed the so called bulb
model, where neutrinos are emitted uniformly and half-isotropically from the
surface of a spherical neutrinosphere. Moreover, neutrino emission is taken to
be azimuthally symmetric and that the conditions in the star depend only on
the radial distance. In this framework, Eq. 5 becomes a system of first order
ordinary differential equations that has been solved numerically by different
groups Duan:2005cp ; Duan:2006an ; Duan:2006jv ; Hannestad:2006nj ;
Fogli:2007bk ; Duan:2007mv . Some understanding can already be obtained
analytically through an analogy with a gyroscopic pendulum in flavor space
Hannestad:2006nj ; Fogli:2007bk ; Duan:2007mv . In normal mass ordering the
pendulum starts in a downward (stable) position where the small value of the
mixing angle induces only negligible flavor changes. Conversely, in inverted
mass ordering the pendulum starts in upward (unstable) position in and it
induces maximal flavor conversions
$\nu_{e}\bar{\nu}_{e}\to\nu_{x}\bar{\nu}_{x}$, conserving lepton number. The
main feature in this context is a spectral split: $\bar{\nu}_{e}$ are
completely converted to $\bar{\nu}_{x}$, whereas for $\nu_{e}$ this happens
for $E>E_{c}$. This happens because the the initial energy spectra of
$\nu_{e}$ are lager than the ones of $\bar{\nu}_{e}$ and $\bar{\nu}_{x}$ and
the total lepton number must be conserved.

For a while the presence of spectral splits Duan:2007bt ; Raffelt:2007cb ;
Fogli:2008pt ; Raffelt:2007xt ; Galais:2011gh  in inverted ordering and the
absence of significant flavor conversions in normal ordering has been
considered to be the paradigm for slow conversions. It was later realized that
the outcome strongly depends on the many assumptions entering the
calculations. Here we briefly provide a list of them.

  * •

Flavor asymmetries Esteban-Pretel:2007jwl ; Fogli:2009rd ; Chakraborty:2014lsa
; Dasgupta:2009mg . The absence of conversions in normal ordering seems to
hold only for a large hierarchy between neutrino fluxes ($f_{\nu_{e}}\gg
f_{\bar{\nu}_{e}}\gg f_{\nu_{x}}$, typical of the accretion phase), whereas
for more similar fluxes ($f_{\nu_{e}}\gtrsim f_{\bar{\nu}_{e}}\gtrsim
f_{\nu_{x}}$) one can find multiple splits and also in normal ordering.

  * •

Large matter potential Chakraborty:2011nf ; Chakraborty:2011gd ;
Dasgupta:2011jf ; Sarikas:2011am ; Saviano:2012yh ; Chakraborty:2014nma . A
matter potential larger than the neutrino-neutrino interaction potential leads
to a suppression of flavor conversions. This is typically happening closer to
the supernova core and during the accretion phase.

  * •

Multi angle effects Mirizzi:2010uz ; Cherry:2010yc ; Mirizzi:2011tu ;
Mirizzi:2012wp . When the flavor asymmetries are mild the phase dispersion
induced by different propagation lengths of neutrinos can smear or completely
remove the effects of the spectral splits, eventually leading to complete
decoherence, i.e. all flavors are equilibrated up to lepton number
conservation.

  * •

Three flavors effects Dasgupta:2007ws ; Duan:2008za ; Fogli:2008fj ;
Dasgupta:2010ae ; Friedland:2010sc ; Dasgupta:2010cd . As for the multi angle
effects, small flavor asymmetries can induce complete spectral swaps or even
flavor equilibration among all three flavors.

  * •

Sponteneous breaking of azimuthal Raffelt:2013rqa ; Raffelt:2013isa ;
Duan:2013kba ; Mirizzi:2013rla ; Chakraborty:2013wsu , spatial homogeneity
Mangano:2014zda ; Duan:2014gfa ; Mirizzi:2015fva ; Mirizzi:2015hwa ;
Duan:2015cqa ; Abbar:2015fwa ; Chakraborty:2015tfa  and stationarity
Dasgupta:2015iia ; Capozzi:2016oyk . It has been realized that the symmetries
used as initial conditions of neutrino emission (azimuthal symmetry, spatial
homogeineity, stationarity) are spontaneously broken during neutrino
propagation. This leads to new instabilities, that can develop in both mass
orderings, but that are in general suppressed when the matter potential is
dominating. Nevertheless, it has been shown Dasgupta:2015iia ; Capozzi:2016oyk
that self-interacting neutrinos can generate a pulsating solution with a
frequency that effectively compensates the phase dispersion associated with
the large matter term, lifting the suppression and making collective
oscillations possible deep inside the supernova. However, because the matter
potential changes during neutrino propagation, it is not clear whether a
flavor wave with a specific pulsation can have enough time to grow and lead to
significant flavor conversions. The presence of turbulent variations of the
matter potential may introduce a coupling among flavor waves with different
$k$, so making it more likely to have an instability even when neutrinos are
propagating away in a supernova Abbar:2020ror .

  * •

Neutrino halo effect Cherry:2012zw ; Cherry:2013mv ; Sarikas:2012vb ;
Zaizen:2019ufj ; Cherry:2019vkv ; Zaizen:2020mun . Neutrinos are not
completely free-streaming after the neutrinosphere and even a small fraction
of scattering neutrinos can produce a small “neutrino halo”. Such inward going
neutrinos can modify the outcome of conversions and the shape of spectral
splits, if present, but, according to the latest simulations, the effects have
been found to be relatively small.

####  IV.2.2 Fast Flavor Conversions

For slow conversions it is usually assumed that neutrinos are emitted half
isotropically, making it intrinsically impossible to create an angular
crossing in $g_{\mathbf{p}}$. The importance of a non-trivial angular
distributions of neutrinos was first recognized in Sawyer:2005jk ;
Sawyer:2008zs ; Sawyer:2015dsa . It was realized that angular crossings can
lead to new instabilities, developing on extremely short time scales, and that
can occur even in the absence of neutrino mixing. In this case, flavor
conversions is entirely due to pairwise interactions of the type
$\nu_{e}(\mathbf{p})+\nu_{x}(\mathbf{q})\to\nu_{e}(\mathbf{q})+\nu_{x}(\mathbf{p})$
and
$\nu_{e}(\mathbf{p})+\bar{\nu}_{e}(\mathbf{q})\to\nu_{x}(\mathbf{q})+\bar{\nu}_{x}(\mathbf{p})$.
The potential impact that a change of flavors happening on such short time
scales can have on the explosion and on the nucleosynthesis of heavy nuclei
has lead to an extensive research activity by the entire community, which is
still underway.

Some work has provided a detailed characterization of fast conversions using
the linear stability analysis Izaguirre:2016gsx ; Capozzi:2017gqd ; Yi:2019hrp
; Airen:2018nvp ; Capozzi:2019lso ; Chakraborty:2019wxe . Useful insights has
also been obtained analytically, usually working under the assumption of
spatial homogeneity and azimuthal symmetry Dasgupta:2017oko ; Johns:2019izj ;
Padilla-Gay:2021haz . In particular in Padilla-Gay:2021haz  it has been
pointed out that the analogy with a pendulum in flavor space works also in the
case of fast conversions. However, differently from what happens for slow
conversions, the real part of the pulsation of the flavor wave resulting from
the linear analysis acts as the pendulum spin, and plays role in determining
the final amount of conversions.

With a significant development of the stability analysis, it has become
possible to apply this tool directly to data provided by supernova
hydrodynamical simulations, with the purpose of assessing whether the
conditions for fast flavor instabilities are actually satisfied. This is the
closest one can get to knowing whether fast instabilities are possible in real
supernovae. However, simulations usually do not provide the angular
distributions of the neutrino fluxes, but only their angular moments defined
as:

$\mathbf{I}_{n}=\int_{-\infty}^{+\infty}\frac{E^{2}dE}{2\pi^{2}}\int\frac{d\mathbf{v}}{4\pi}\,g_{E,\mathbf{v}}\mathbf{v}^{n}\,.$
(18)

A few methods Dasgupta:2018ulw ; Abbar:2020fcl ; Nagakura:2021txn ;
Nagakura:2021suv ; Nagakura:2021hyb , have been proposed to perform a
stability analysis using only the angular moments $\mathbf{I}_{n}$ and they
have been recently applied to a plethora of supernova simulations. The first
investigations in Tamborra:2017ubu ; DelfanAzari:2019epo  found no evidence of
instabilities, but they were either limited to 1D simulations or to very
specific time and space locations of the supernova. More recent studies
reported that crossings are possible in the following areas of a supernova:

  * •

Proto-neutron star DelfanAzari:2019tez ; Abbar:2019zoq ; Glas:2019ijo . The
physical origin of the crossings is a strong convective activity happening in
the proto-neutron star, which can generate large amplitude modulations in the
spatial distributions of $\nu_{e}$ and $\bar{\nu}_{e}$ number densities. The
physical implications are not very clear due to the nearly equal distributions
of neutrinos and antineutrinos of all flavors.

  * •

Neutrino decoupling region Abbar:2018shq ; Nagakura:2019sig ; Abbar:2019zoq .
Their existence can be explained by the neutron richness of matter, which
induces a later decoupling of $\nu_{e}$ with respect to $\bar{\nu}_{e}$ and,
thus, a more forward peaked angular distributions of $\bar{\nu}_{e}$. Another
possibility is the presence of LESA Tamborra:2014aua ; OConnor:2018tuw ;
Glas:2018vcs ; Nagakura:2019evv , i.e. an asymmetric emission of lepton
number, or other phenomena Nagakura:2019evv .

  * •

Free streaming regime Morinaga:2019wsv ; Capozzi:2020syn ; Abbar:2020qpi .
Crossings can be generated by neutrino backward scatterings off heavy nuclei
and their size seems to get larger for smaller radial distances
Morinaga:2019wsv ; Capozzi:2020syn ; Abbar:2020qpi . Such crossings are
ubiquitous in the pre-shock region, but they can also occur in the post-shock
region. In the former case, there is hardly an impact on astrophysical
processes and on the detection at Earth Zaizen:2021wwl ; Abbar:2021lmm , both
because of the slower growth rates of the instabilities and the very small
size of the amount of conversions expected. On the other hand, the
instabilities developing in the post-shock region might produce an observable
effect.

Even if only under some simplifying assumptions, fast flavor instabilities
have been also simulated numerically in order to assess what is the final
amount of conversions expected and how it depends on the initial conditions.
It was originally thought that such instabilities lead to equilibration among
all flavors Sawyer:2005jk ; Sawyer:2008zs ; Sawyer:2015dsa ; Dasgupta:2016dbv
, up to conservation of the total lepton number. This conclusion was
considered to hold for all neutrino energies and mass orderings, but it has
been recently called into questions. Here we report a brief summary of the
findings obtained through numerical simulations:

  * •

Neutrino propagation Shalgar:2019qwg . Even if a system of neutrinos has an
initial crossing and thus develops a fast flavor instability, the propagation
in space and time may cancel the crossing Shalgar:2019qwg .

  * •

Propagation of the power of the instability to small angular scales
Abbar:2018beu ; Martin:2019gxb ; Johns:2020qsk ; Bhattacharyya:2020dhu ;
Bhattacharyya:2020jpj ; Wu:2021uvt . During its evolution, the power of a fast
instability is moved from large scales to small ones in momentum space,
accelerating the decoherence of the system (and the equilibration of flavors)
Abbar:2018beu ; Martin:2019gxb ; Johns:2020qsk ; Bhattacharyya:2020dhu ;
Bhattacharyya:2020jpj ; Wu:2021uvt . Moreover, when considering a coarse-
graining over space, fast conversions seem to reach eventually a steady state
Bhattacharyya:2020dhu ; Bhattacharyya:2020jpj ; Wu:2021uvt . In particular in
Bhattacharyya:2020jpj  an approximate analytical formula has been derived for
calculating the amount of decoherence reached in fast conversions and, thus,
the final amount of flavor conversions. This formula depends on the
propagation angle of neutrinos and on the initial asymmetry between the total
number of neutrinos and antineutrinos. If the asymmetry is extremely small, as
expected for the crossings generated by backward scattering of neutrinos in
the free streaming regime, then the amount of flavor conversions is expected
to be negligible, as confirmed in Zaizen:2021wwl ; Abbar:2021lmm . In
Bhattacharyya:2020jpj  the analytical formula is independent from the type of
perturbation used to seed the flavor instabilities. However, in Wu:2021uvt  a
distinction is made between localized and random seeds. In the first case a
coherent behaviour in the space and momentum evolution of the flavor wave is
retained for a longer time. The difference between Bhattacharyya:2020jpj  and
Wu:2021uvt  can be associated to heterogeneous numerical methods employed for
calculating spatial derivatives.

  * •

Spontaneous symmetry breaking Shalgar:2021wlj . As it happens to slow
conversions, also in the context of fast ones there is a spontaneous breaking
of the symmetries imposed in the initial conditions. This has been shown to
happen for the azimuthal symmetry in Shalgar:2021wlj  and in Capozzi:2020kge ,
though in the second reference it was not explicitly stated in the
conclusions.

  * •

Dependence on neutrino energy Shalgar:2020xns . It has been proposed that the
outcome of fast conversions depend on the size of the mass differences between
mass eigenstates and on their ordering Shalgar:2020xns . This claim has been
criticized in Martin:2021xyl  where only a modest dependence has been
observed. However, the first simulation has been performed considering an
homogeneous system, whereas the second has taken into account also the spatial
evolution.

  * •

Impact of inelastic collisions Capozzi:2018clo ; Shalgar:2020wcx ;
Sasaki:2021zld ; Martin:2021xyl ; Sigl:2021tmj . Since the conditions for fast
conversions have been found even in locations where neutrinos are still
partially or completely coupled to the plasma, there have been a few studies
implementing collisions in numerical simulations. In this context
Shalgar:2020wcx ; Sasaki:2021zld  reported the possibility of enhancement of
flavor conversions, assuming only evolution in time. On the other hand in
Martin:2021xyl ; Sigl:2021tmj  both time and space evolution has been taken
into account and it was observed that collisions might cause flavor
depolarization, thus suppressing conversions, but that a much larger mean free
path than expected is required in order for this happen. This is in agreement
to what found in Capozzi:2018clo , where the role of collisions in the
generation of crossing has been also pointed out.

  * •

Dependence on the number of crossings Bhattacharyya:2021klg . Multiple
crossings can inhibit flavor conversions Bhattacharyya:2021klg . This is
especially relevant when considering fast conversions in the early universe,
where the neutrino angular distributions are expected to be more similar
compared to the supernova case.

  * •

Dependence on the number of neutrino species Capozzi:2020kge ; Shalgar:2021wlj
. Considering six neutrino species the crossings and, thus, fast instabilities
can occur in one (or more) of the three sectors $e\mu,e\tau,\mu\tau$ and then
propagate to other ones Capozzi:2020kge . Moreover, even considering the
distributions of $\nu_{\mu},\nu_{\tau},\bar{\nu}_{\mu},\bar{\nu}_{\tau}$ to be
the same, the outcome of flavor conversions is different Shalgar:2021wlj  with
respect to what obtained by using only the equation of motions for three
species, as usually done in literature.

  * •

Impact of new physics Dighe:2017sur . Fast conversions can be affected by the
presence of new physics, such as non-standard interactions Dighe:2017sur .

##  V Fast Flavor Conversions in Compact Binary Mergers

Neutrinos are copiously produced also in merger events between two neutron
stars or a neutron star and a black hole, reaching a total luminosity of $\sim
10^{53}$ erg Ruffert:1996by ; Foucart:2015vpa . Despite their low chance of
detectability, these neutrinos play a fundamental role in cooling the
accretion torus that is created around the central remnant (a massive neutron
star or a black hole), and the nucleosynthesis of heavy nuclei through
$r$-processes. Ruffert:1996by ; Foucart:2015vpa ; Kulkarni:2005jw ;
2010MNRAS.406.2650M ; Perego:2014fma . Therefore, as it happens for core-
collapse supernovae, a full understanding of neutrino flavor conversions is
mandatory.

A peculiarity of such environments is the excess of $\bar{\nu}_{e}$ over
$\nu_{e}$. A first consequence is the possible occurrence of the “matter-
neutrino resonance” Malkus:2014iqa ; Malkus:2015mda ; Wu:2015fga , i.e. the
almost complete cancellation of the neutrino self interaction potential
$\Omega_{\mathbf{p}}^{\nu\nu}$ with the potential due to interactions with
electrons $\Omega_{\mathbf{p}}^{\rm mat}$. A second consequence is the
ubiquity of crossings and thus of fast flavor instabilities in the accretion
disk, first recognized in Wu:2017qpc . Indeed, as it happens in supernovae,
$\nu_{e}$ are coupled to the plasma in a larger volume, thus, despite the
overall excess of $\bar{\nu}_{e}$, for specific backward going propagation
directions there may be a local excess of $\nu_{e}$. However, the large region
of instability reduces its size after $O(10)$ ms from the formation of the
accretion disk. This time interval can be larger in case of a massive neutron
star remnant George:2020veu . Following this finding, the impact of fast
instabilities on nucleosynthesis has been assessed in Wu:2017drk  and
George:2020veu . Both references assumed complete flavor equilibration. The
first found that the fraction of lanthanides produced in the neutrino-driven
wind may be increased by a factor of $O(10^{3})$. On the other hand, the
second reference found an enhancement of the iron peak abundances and a
reduction of the first peak abundances, whereas lanthanides are marginally
affected. Flavor equilibration was assumed also in Li:2021vqj , but,
differently from the previous works, a magneto-hydrodynamic simulation of a
typical post-merger accretion disk was used. Their results show that fast
conversions can increase the production of the lanthanides and of elements in
the third peak to a level comparable to solar abundances. This is an
interesting hint for massive post-merger disks being a major production site
for heavy elements.

All the results on the impact of fast conversions on nucleosynthesis in post-
merger accretion disks are based on the assumption of flavor equilibration.
However, as discussed in the context of supernova neutrinos, such assumption
might not always be justified. For instance, in Padilla-Gay:2020uxa  a
simplified toy model was studied numerically, finding a conversion of less
than 1%. despite the large growth rate of fast instabilities.

##  VI Conclusions and future perspectives

Neutrino flavor conversions are relevant in cosmological and astrophysical
dense environments, such as the early universe, core-collapse supernovae and
the merger of compact objects. Indeed, a full understanding of these
phenomenona is mandatory for a correct interpretation of both the
corresponding neutrino signals and (or) the astrophysical processes developing
in these environments. The flavor content of a system of neutrinos is usually
represented by a density matrix $\varrho_{\mathbf{p,x}}$, whose time and space
evolution are described by Eq. 5, where $\Omega_{\mathbf{p,x}}^{\nu\nu}$
represents the potential stemming from neutrino interactions among themselves.
Such a term makes the evolution equation a non linear system of coupled
partial differential equations, which has never been solved in its entire
form.

The early universe represents a peculiar environment for testing nonlinear
neutrino oscillations in high-density conditions. Indeed, the primordial
nucleosynthesis (BBN), the leptogenesis, the cosmic background radiation
(CMB), the formation of cosmological structures (LLS) can be profoundly
influenced by the presence of neutrinos, by their properties such
oscillations, interactions and non-zero mass. A special and interesting case
is represented by the oscillations of an between active neutrinos and a
sterile one, where the latter has a mass at the eV scale. The presence of such
a light new degree of freedom is suggested by intriguing, but controversial,
indications from short-baseline neutrino oscillation experiments. These
sterile neutrinos are produced through mixing and contribute to the radiation
content beyond photons and ordinary neutrinos. They also leave an imprint on
different cosmological observables. In order to determine their relic
abundance an accurate solution of Eq. 5 is mandatory. For the mass and mixing
parameters suggested by experimental analysis, the eV sterile neutrinos should
be produced with the same number density of the active states. This result is
however in disagreement with recent cosmological data on the radiation and on
neutrino mass constraints. Extra ingredient should be added in the model of
the flavor evolution in order to try to alleviate this tension. A full multi-
flavor and multi momentum treatment of the active-sterile flavor evolution is
mandatory in order to confirm or reject the eV sterile neutrino hypothesis. At
the moment, the eV scenario is strongly disfavored by cosmological
observations.

Sterile neutrinos with mass of the order of keV can represent natural dark
matter candidates. A complete characterization of the production mechanisms
for keV sterile neutrinos is extremely complex requiring the inclusion of
hadronic contributions still not perfectly known, detailed solutions of the
kinetic equations as well as some model building. The results obtained should
be then compared with astrophysical and cosmological observables. This
comparison requires the use of signatures from different physical probes
(X-ray, counting galaxy, Lyman-alpha) and different distance scales. New
cosmological and astrophysical data will help shed light on the keV sterile
neutrino paradigm.

Differently from the early universe, in the context of core-collapse
supernovae and mergers of compact objects, flavor conversions are not fully
understood even in the context of the Standard Model. In these cases different
types of flavor instabilities can develop according to the shape of
$g_{E,\mathbf{v}}$ (Eq. 17), i.e. the difference of the energy and angular
distributions of neutrinos and antineutrinos, which are usually not equal to
each other. Two approaches are usually employed to study these instabilities:
linearized analyses and numerical solutions of a simplified version of Eq. 5.
The first tool is computationally more manageable, but it only allows to know
whether a flavor instability can develop given a set of initial conditions for
$\varrho_{\mathbf{p,x}}$. Its application has shown that instabilities
requires that $g_{E,\mathbf{v}}$ has a crossing either in energy $E$ or in the
propagation direction $\mathbf{v}$. Instabilities can be classified according
to their growth rate: slow conversions grow with a rate
$\propto\sqrt{n_{\nu}\omega_{E}}$, whereas fast conversions with $\propto
n_{\nu}$. The latter can develop on time scales as small as few nanoseconds,
if they occur close to the neutrinosphere, with a potential major impact on
the supernova explosion. The application of the linear stability analysis on
real data provided by hydrodynamical simulations has shown that crossings are
indeed a relatively common feature in supernovae. These searches have been
performed by several groups on different data and a general consensus has been
achieved. Nevertheless, a systematic study of how crossing generation depends
on the characteristics of supernova progenitors is still lacking, especially
for three-dimensional models which are in principle the most realistic
representation of the real explosion event, but at the same time are more time
consuming in terms of simulations.

A lot of work has been dedicated to numerically solving Eq. 5 and it is now
clear that the results obtained can deviate from the real non linear dynamics
of the system when the equation is simplified by imposing some symmetries.
This has been observed for slow conversions, for which the spectral splits
originally found within the bulb model seem to be washed out when allowing
space or time instabilities to develop. Similarly, it was thought that fast
conversions lead to flavor equilibration, but more recently the dependence of
this result on the presence of collisions, the symmetries of the system and
the neutrino energy has been highlighted. Nevertheless, results obtained by
different calculations are not always in perfect agreement. Future studies
should continue investigating the phenomenology of flavor conversion when
symmetry assumptions are relaxed, as well as the possible interplay between
fast and slow conversion (and non standard physics), which is still largely
unknown. Ultimately, an implementation of flavor conversions into
hydrodynamical simulations is necessary in order to assess what is the impact
on the astrophysical processes happening in the exploding star. This is of
course computationally extremely challenging, if not even unfeasible at the
moment. However, a first step in this direction can be similar to the one
adopted for mergers of compact objects, where complete flavor conversions has
been assumed at those locations in space and time where conditions for flavor
instabilities are satisfied. A more realistic approach can be either moving
away from flavor equilibration, or writing Eq. 5 in terms of the angular and
energy moments of the density matrix $\varrho_{\mathbf{p,x}}$. The latter
possibility is numerically more manageable and it has been recently adopted
Myers:2021hnp . Finally, we underline the importance of cross checks between
different numerical codes.

## Acknowledgements

We thank Alessandro Mirizzi for valuable suggestions. The work of FC is
supported by GVA Grant No.CDEIGENT/2020/003. This work was partially supported
by the research grant number 2017W4HA7S “NAT-NET: Neutrino and As- troparticle
Theory Network” under the program PRIN 2017 funded by the Italian Ministero
dell’Universit‘a e della Ricerca (MUR). The authors also acknowledge the
support by the research project TAsP (Theoretical Astroparticle Physics)
funded by the Istituto Nazionale di Fisica Nucleare (INFN).

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