# On the occurrence of fast neutrino flavor conversions
in multidimensional supernova models

Sajad Abbar  Astro-Particule et Cosmologie (APC), CNRS UMR 7164, Université
Denis Diderot, 75205 Paris Cedex 13, France  Department of Physics &
Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA
Huaiyu Duan  Department of Physics & Astronomy, University of New Mexico,
Albuquerque, New Mexico 87131, USA  Kohsuke Sumiyoshi  Numazu College of
Technology, Ooka 3600, Numazu, Shizuoka 410-8501, Japan  Tomoya Takiwaki
National Astronomical Observatory of Japan, Osawa, Mitaka, Tokyo 181-8588,
Japan  Maria Cristina Volpe  Astro-Particule et Cosmologie (APC), CNRS UMR
7164, Université Denis Diderot, 75205 Paris Cedex 13, France

###### Abstract

The dense neutrino medium in a core-collapse supernova or a neutron-star
merger event can experience fast flavor conversions on time/distance scales
that are much smaller than those of vacuum oscillations. It is believed that
fast neutrino flavor transformation occurs in the region where the angular
distributions of $\nu_{e}$ and $\bar{\nu}_{e}$ cross each other. We present
the first study of this crossing phenomenon and the fast neutrino flavor
conversions in multidimensional (multi-D) supernova models. We examine the
neutrino distributions obtained by solving the Boltzmann transport equation
for several fixed profiles which are representative snapshots taken from
separate 2D and 3D supernova simulations with an $11.2M_{\odot}$ progenitor
model. Our research shows that the spherically asymmetric patterns of the
$\nu_{e}$ and $\bar{\nu}_{e}$ fluxes in multi-D models can assist the
appearance of the crossing between the $\nu_{e}$ and $\bar{\nu}_{e}$ angular
distributions. In the models that we have studied, there exist unstable
neutrino oscillation modes in and beyond the neutrino decoupling region which
have amplitude growth rates as large as an $e$-fold per nanosecond. This
finding can have important consequences for the explosion mechanism,
nucleosynthesis, and neutrino signals of core-collapse supernovae.

##  I Introduction

At the exhaustion of its nuclear fuel, the stellar core of a massive star
collapses under its own gravity into a neutron star or black hole, and the
rest of the star either explodes as a supernova or falls back to become a part
of the black hole. In a successful explosion, the hot proto-neutron star (PNS)
formed at the center of the (core-collapse) supernova quickly cools down by
emitting $\sim 10^{58}$ neutrinos in all flavors in just tens of seconds.
Although the exact mechanism of the supernova explosion remains elusive, the
neutrino reactions

$\displaystyle\bar{\nu}_{e}+p\rightleftharpoons
n+e^{+}\quad\text{and}\quad\nu_{e}+n\rightleftharpoons p+e^{-}$ (1)

are known to be important in the heating and cooling of the matter deep inside
the supernova and can play an important role in exploding the star Bethe and
Wilson (1985); Janka (2012); Burrows (2013). Supernova ejecta are also one of
the few places in the universe where the heavy elements can be abundantly
produced. An important factor that regulates the nucleosynthesis in the
supernova ejecta is its electron fraction $Y_{e}$ (defined as the ratio of the
net number density of the electrons to that of the baryons or
$n_{e}/n_{\mathrm{b}}$) Qian and Woosley (1996) which in turn is determined by
the neutrino processes in Eq. (1). Because these processes involve only
$\nu_{e}$ and $\bar{\nu}_{e}$, and because the neutrinos in different flavors
are emitted in different intensities and energies in a supernova, the
transformation or oscillations between different neutrino flavors
($\nu_{e}\rightleftharpoons\nu_{\mu/\tau}$ and
$\bar{\nu}_{e}\rightleftharpoons\bar{\nu}_{\mu/\tau}$) near or even inside the
surface of the PNS can have a significant impact on the nucleosynthesis in the
supernova ejecta and the supernova dynamics. Understanding the flavor
transformation of the neutrinos inside the supernova is also crucial to the
prediction of the neutrino signals from future galactic supernovae and the
neutrino diffuse background Gava  _et al._ (2009); Horiuchi  _et al._ (2009);
Beacom (2010); Mirizzi  _et al._ (2016); Horiuchi  _et al._ (2018).

Because of the coherent forward scattering by the dense neutrino medium
surrounding the PNS, supernova neutrinos can experience collective flavor
transformation Pastor and Raffelt (2002); Duan  _et al._ (2006a, b, 2010);
Chakraborty  _et al._ (2016a) which occurs much deeper than does the flavor
conversion induced by matter alone through the Mikheyev-Smirnov-Wolfenstein
(MSW) mechanism Wolfenstein (1978, 1979); Mikheyev and Smirnov (1985).
However, based on the stationary neutrino bulb model Duan  _et al._ (2006a) in
which all neutrinos are decoupled from matter at a single sharp neutrino
sphere, it was found that collective flavor transformation is suppressed
during the accretion phase of the supernova due to the large density of the
ambient matter Esteban-Pretel  _et al._ (2008); Sarikas  _et al._ (2012);
Chakraborty  _et al._ (2011); Zaizen  _et al._ (2018) and/or even the
neutrinos themselves Duan and Friedland (2011). Although, in the more
realistic models that evolve with time and does not have the spherical
symmetry, this suppression can be lifted for the collective modes that
oscillate rapidly in space and time Duan and Shalgar (2015); Chakraborty  _et
al._ (2016b); Abbar  _et al._ (2015); Abbar and Duan (2015); Dasgupta and
Mirizzi (2015), the physical conditions can change significantly before these
modes have a sufficient growth in amplitudes to engender significant flavor
conversions.

In a real supernova, the neutrinos of different flavors (and energies)
decouple from matter at different depths of the PNS and have different angular
distributions outside the PNS. As a result, fast neutrino flavor conversions
can be driven by the neutrino density $n_{\nu}$ which occur on a
characteristic length scale $l_{\text{fast}}\sim(\hbar
c)^{-2}G_{\mathrm{F}}^{-1}n_{\nu}^{-1}$ with $G_{\mathrm{F}}$ being the Fermi
coupling constant of the weak interaction Sawyer (2005, 2016); Chakraborty
_et al._ (2016c); Izaguirre  _et al._ (2017); Wu and Tamborra (2017); Capozzi
_et al._ (2017); Dasgupta  _et al._ (2017); Abbar and Duan (2018); Abbar and
Volpe (2019). Such flavor transformation is rightly called fast because
$l_{\text{fast}}$ is much shorter than the typical wavelength of vacuum
oscillations ($\sim\mathcal{O}(1)$ km for a 10 MeV neutrino with the
atmospheric neutrino mass splitting) and even the mean free path of the
neutrino inside the PNS Capozzi  _et al._ (2019). It is believed that fast
neutrino flavor conversions occur only where the angular distributions of
$\nu_{e}$ and $\bar{\nu}_{e}$ cross each other Dasgupta  _et al._ (2017);
Izaguirre  _et al._ (2017). However, an earlier study of the one-dimensional
(1D) supernova models of the Garching group with the Boltzmann neutrino
transport did not reveal any scenario with such crossed neutrino angular
distributions or fast neutrino flavor conversions Tamborra  _et al._ (2017).
It has been speculated Chakraborty  _et al._ (2016c); Dasgupta  _et al._
(2017) that the multi-D models exhibiting the lepton-emission self-sustained
asymmetry (LESA) Tamborra  _et al._ (2014) may have regions that foster fast
flavor conversions. But the neutrino transports in most of the existing
multi-D supernova simulations are implemented with various approximations and
do not provide detailed angular distributions of the neutrinos.

Recently, the full information of the neutrino angular distributions became
available to multi-D supernova models by directly solving the Boltzmann
equation in three momentum dimensions Sumiyoshi and Yamada (2012). This
approach has been used to study the neutrino transport in supernova cores
Sumiyoshi  _et al._ (2015) and its impact on the explosion dynamics Nagakura
_et al._ (2018).

In this paper, we present the first survey of the neutrino angular
distributions in multi-D supernova models to ascertain the physical conditions
under which fast neutrino flavor conversions may occur.

##  II Neutrino transport and fast flavor conversions

At each space-time point $(t,\mathbf{r})$, the flavor content of the neutrino
medium in the momentum mode $\mathbf{p}$ can be specified by its flavor
density matrix $\varrho_{\mathbf{p}}(t,\mathbf{r})$ Sigl and Raffelt (1993)
which, for two flavors ($e$ and $x$) and in the weak-interaction basis, is
Banerjee  _et al._ (2011)

$\displaystyle\varrho=\frac{f_{\nu_{e}}+f_{\nu_{x}}}{2}+\frac{f_{\nu_{e}}-f_{\nu_{x}}}{2}\begin{bmatrix}s&S\\\
S^{*}&-s\end{bmatrix},$ (2)

where $f_{\nu_{e}/\nu_{x}}$ are the initial occupation numbers in the
corresponding flavors, and the complex and real scalar fields $S$ and $s$
describe the flavor coherence and the flavor conversion of the neutrino,
respectively. A similar expression exists for the flavor density matrix
$\bar{\varrho}$ of the antineutrino. The flavor evolution of the neutrino
medium is governed by the Liouville-von Neumann equation Sigl and Raffelt
(1993); Strack and Burrows (2005); Cardall (2008); Volpe  _et al._ (2013);
Vlasenko  _et al._ (2014)

$i(\partial_{t}+\mathbf{v}\cdot\bm{\nabla})\varrho_{\mathbf{p}}=\left[\frac{\mathsf{M}^{2}}{2E_{\nu}}+\frac{\lambda}{2}\sigma_{3}+\mathsf{H}_{\nu\nu,\mathbf{p}},\varrho_{\mathbf{p}}\right]+\mathcal{C}[\varrho_{\mathbf{p}}],$
(3)

where we have adopted the natural units $c=\hbar=1$. In the above equation,
$E_{\nu}=|\mathbf{p}|$, $\mathbf{v}=\mathbf{p}/E_{\nu}$, and $\mathsf{M}^{2}$
are the energy, velocity, and mass-square matrix of the neutrino,
respectively, $\lambda=\sqrt{2}G_{\mathrm{F}}n_{e}$ is the matter potential
Wolfenstein (1978); Mikheyev and Smirnov (1985), $\sigma_{3}$ is the third
Pauli matrix,

$\mathsf{H}_{\nu\nu,\mathbf{p}}=\sqrt{2}G_{\mathrm{F}}\int\\!\frac{\mathrm{d}^{3}p^{\prime}}{(2\pi)^{3}}(1-\mathbf{v}\cdot\mathbf{v}^{\prime})(\varrho_{\mathbf{p}^{\prime}}-\bar{\varrho}_{\mathbf{p}^{\prime}}).$
(4)

is the neutrino potential stemming from the neutrino-neutrino forward
scattering Fuller  _et al._ (1987); Nötzold and Raffelt (1988); Pantaleone
(1992), and $\mathcal{C}[\varrho_{\mathbf{p}}]$ denotes the collision,
emission and absorption of the neutrinos.

To study fast neutrino flavor conversions, we ignore the vacuum oscillation
Hamiltonian $\mathsf{M}^{2}/2E_{\nu}$ so that the flavor transformation of the
neutrinos becomes energy independent. We also ignore the collision term
$\mathcal{C}[\varrho_{\mathbf{p}}]$ and assume that the physical conditions
are homogeneous and stationary for the distance and time scales of interest
except for the small but rapidly varying flavor mixing amplitude
$S_{\mathbf{v}}(t,\mathbf{r})$. This approximation is valid before significant
flavor conversion has occurred so that $|S_{\mathbf{v}}|\ll 1$ and $s\approx
1$. In this scenario, it is useful to define the angular distribution of the
electron lepton number (ELN) of the neutrino flux as Izaguirre  _et al._
(2017)

$G_{\mathbf{v}}=\sqrt{2}G_{\mathrm{F}}\int_{0}^{\infty}\frac{E_{\nu}^{2}\mathrm{d}E_{\nu}}{(2\pi)^{3}}[f_{\nu_{e}}(\mathbf{p})-f_{\bar{\nu}_{e}}(\mathbf{p})],$
(5)

where we have assumed $f_{\nu_{x}}(\mathbf{p})=f_{\bar{\nu}_{x}}(\mathbf{p})$.
Keeping only the terms in Eq. (3) of magnitude $\mathcal{O}(|S_{\mathbf{v}}|)$
or larger, one obtains Banerjee  _et al._ (2011); Väänänen and Volpe (2013);
Izaguirre  _et al._ (2017)

$i(\partial_{t}+\mathbf{v}\cdot\bm{\nabla})S_{\mathbf{v}}=(\epsilon_{0}+\mathbf{v}\cdot\bm{\epsilon})S_{\mathbf{v}}-\int\\!\mathrm{d}\Gamma_{\mathbf{v}^{\prime}}(1-\mathbf{v}\cdot\mathbf{v}^{\prime})G_{\mathbf{v}^{\prime}}S_{\mathbf{v}^{\prime}},$
(6)

where $\mathrm{d}\Gamma_{\mathbf{v}^{\prime}}$ is the differential solid angle
in the direction of $\mathbf{v}^{\prime}$,
$\epsilon_{0}=\lambda+\int\\!\mathrm{d}\Gamma_{\mathbf{v}^{\prime}}G_{\mathbf{v}^{\prime}}$,
and
$\bm{\epsilon}=\int\\!\mathrm{d}\Gamma_{\mathbf{v}^{\prime}}G_{\mathbf{v}^{\prime}}\mathbf{v}^{\prime}$.

Collective flavor transformation is induced by the normal modes in the
neutrino medium which, in the linear regime (where $|S_{\mathbf{v}}|\ll 1$),
are of the form $S_{\mathbf{v}}(t,\mathbf{r})=Q_{\mathbf{v}}\,e^{-i\Omega
t+i\mathbf{K}\cdot\mathbf{r}},$ where $Q_{\mathbf{v}}$, $\Omega$ and
$\mathbf{K}$ are constant with the latter two being also independent of
$\mathbf{v}$. The flavor mixing amplitude $S_{\mathbf{v}}$ remains small
unless for a real wave vector $\mathbf{K}$ the corresponding frequency has a
positive imaginary component, i.e.
$\Omega_{\mathrm{i}}=\mathrm{Im}(\Omega)>0$. In this case the wave amplitude
can grow exponentially on the time scale of $l_{\text{fast}}/c$. Such unstable
normal modes with fast growing amplitudes can exist when the ELN distribution
$G_{\mathbf{v}}$ crosses 0 at some angle(s) Dasgupta  _et al._ (2017);
Izaguirre  _et al._ (2017).

Figure 1: The neutrino number densities $n_{\nu_{e}}$ and $n_{\bar{\nu}_{e}}$
(left panels) and their ratio $\alpha=n_{\bar{\nu}_{e}}/n_{\nu_{e}}$ (right
panels) in the $t_{\text{pb}}=200$ ms snapshot of the 2D supernova model. The
lengths and orientations of the arrows in the left panels indicate the
magnitudes and directions of the average neutrino flux densities
$\mathbf{j}_{\nu_{e}}$ and $\mathbf{j}_{\bar{\nu}_{e}}$ in the corresponding
spatial zones. The white crosses in the right panels mark the zones where the
ELN crossing occurs. There is almost no difference between angle averaged
properties of the neutrinos solved in the low and high resolutions (with
$N_{\theta_{\nu}}=6$ and 36, respectively) except for the spatial extent over
which the ELN crossing occurs.

##  III ELN crossings in multi-D supernova models

We study the angular distributions of the neutrinos which are obtained by
solving the Boltzmann transport equation (without any flavor transformation)
for several fixed supernova profiles Sumiyoshi  _et al._ (2015). These
profiles were taken from the representative snapshots at $t_{\text{pb}}=100$,
150 and 200 ms post the core bounce, respectively, of a 2D and a 3D supernova
simulations by using the Lattimer & Swesty equation of state Lattimer and
Swesty (1991) with an approximate neutrino transport and with an
$11.2M_{\odot}$ progenitor model Takiwaki  _et al._ (2012, 2014). The spatial
resolutions of the Boltzmann calculations are $(256,64,1)$ and $(256,64,32)$
for the 2D and 3D models, respectively, for radius up to $2613$ km from the
original simulations, where $(N_{r},N_{\Theta},N_{\Phi})$ are the numbers of
spatial zones in the spherical coordinates $(r,\Theta,\Phi)$. The momentum
resolution $(N_{E_{\nu}},N_{\theta_{\nu}},N_{\phi_{\nu}})$ of the neutrino
flux in each spatial zone is $(14,6,12)$, where $\theta_{\nu}$ and
$\phi_{\nu}$ are the zenith and azimuthal angles of the neutrino velocity
$\mathbf{v}$ with respect to the radial direction, respectively.

Out of the three snapshots of the 2D model we find regions with ELN crossings
within and above the decoupling region in the one at $t_{\text{pb}}=200$ ms.
At this time, the deformed shock has reached over 500 km and is poised to
explode the star with the help of a bipolar growth of the hydrodynamic
instabilities. Depending on their flavors and energies, neutrinos decouple
from matter at radius $\sim 50-70$ km which can be viewed as the “surface” or
neutrino sphere of the PNS. In the first two panels of Fig. 1 we show the
number densities
$n_{\nu}=\int\\!\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}f_{\nu}(\mathbf{p})$ and
average flux densities
$\mathbf{j}_{\nu}=\int\\!\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}f_{\nu}(\mathbf{p})\mathbf{v}$
in this snapshot for $\nu=\nu_{e}$ and $\bar{\nu}_{e}$, respectively. Both
$n_{\nu}$ and $\mathbf{j}_{\nu}$ are mostly spherically symmetric in this
snapshot with $\mathbf{j}_{\nu}$ generally pointing in the radial direction
with some cases of non-radial fluxes. Naively, one may think that ELN
crossings may occur below the neutrino sphere where $\mathbf{j}_{\nu_{e}}$ and
$\mathbf{j}_{\bar{\nu}_{e}}$ can point in very different directions. This is
not the case, however, because $f_{\nu}(\mathbf{p})$ is highly isotropic in
this region, and we find no ELN crossing here.

In our study, ELN crossings usually begin to appear in the region where the
neutrinos begin to decouple from matter as shown in the right panels of Fig.
1. Furthermore, we find that, at the radii where ELN crossings do occur, they
usually appear in the angular zones with the $\bar{\nu}_{e}$-to-$\nu_{e}$
ratio $\alpha=n_{\bar{\nu}_{e}}/n_{\nu_{e}}$ close to 1. The correlation
between $\alpha$ and ELN crossings is not really a surprise. In the neutrino
decoupling region, $f_{\nu}(\mathbf{p})$ becomes more and more peaked in the
forward direction. Because the PNS is rich in neutrons, $\bar{\nu}_{e}$’s
decouple from matter at smaller radii than $\nu_{e}$’s do and thus obtain a
more forwardly peaked distribution. However, this difference in
$f_{\nu_{e}}(\mathbf{p})$ and $f_{\bar{\nu}_{e}}(\mathbf{p})$ is usually not
large enough to result in an ELN crossing unless $\bar{\nu}_{e}$ has a flux
density very close to that of $\nu_{e}$. This is likely the reason why no ELN
crossing was found in a previous study of 1D supernova models Tamborra  _et
al._ (2017). In the 2D model presented in Fig. 1, however, $\alpha$ can vary
across angular zones at the same radius which leads to ELN crossings in some
regions.

Figure 2: The neutrino angular distributions $f_{\nu_{e}}(\theta_{\nu})$ and
$f_{\bar{\nu}_{e}}(\theta_{\nu})$ (left) and the ELN distribution
$G(\theta_{\nu})$ (middle) solved in two angular resolutions,
$N_{\theta_{\nu}}=6$ and $36$, respectively, and the corresponding exponential
growth rates $\Omega_{\mathrm{i}}$ as functions of the real wave number $K$ of
the fast neutrino oscillation modes propagating in the radial direction. The
results are for the spatial zone centered at $r=65.6$ km and $\cos\Theta=0.96$
in the 2D model shown in Fig. 1. Figure 3: The Mollweide projection of the
$\bar{\nu}_{e}$-to-$\nu_{e}$ density ratio $\alpha$ at $r=65.6$ km (left) and
the electron fractions $Y_{e}$ at $r=65.6$ km and $30.5$ km (middle and
right), respectively, in the $t_{\text{pb}}=200$ ms snapshot of the 3D
supernova model. The white crosses in the left panel mark the spatial zones
where ELN crossings occur.

To check the sensitivity of our results on the angular resolution of the
neutrino distributions, we also solve the neutrino transport for the
$t_{\text{pb}}=200$ ms snapshot of the 2D model with $N_{\theta_{\nu}}=36$. We
find that, although the angle averaged properties of the neutrinos such as
$n_{\nu}$, $\mathbf{j}_{\nu}$ and $\alpha$ are almost identical for the two
calculations, the region with ELN crossings is much wider in the high-
resolution calculation than in the one with a lower resolution (right panels
of Fig. 1). To find out the reason why some spatial zones show ELN crossings
in the high-resolution calculation only, we compare the ELN distributions
$G_{\mathbf{v}}$ of these zones in the two calculations. Because
$f(\mathbf{p})$ are approximately axially symmetric about the radial direction
in our models, we integrated them over $\phi_{\nu}$ and calculated
$f_{\nu}(\theta_{\nu})=\iint\frac{E_{\nu}^{2}\mathrm{d}E_{\nu}\mathrm{d}\phi_{\nu}}{(2\pi)^{3}}f_{\nu}(\mathbf{p})$
and $G(\theta_{\nu})=\int_{0}^{2\pi}\mathrm{d}\phi_{\nu}G_{\mathbf{v}}$. We
plot $f_{\nu_{e}}(\theta_{\nu})$, $f_{\bar{\nu}_{e}}(\theta_{\nu})$ and
$G(\theta_{\nu})$ for the spatial zone centered at $r=65.6$ km and
$\cos\Theta=0.96$ in the first two panels of Fig. 2. From this figure one sees
that the coarse angular resolution is sufficient to capture the overall shape
of $f_{\nu}(\theta_{\nu})$, and it is more accurate in the backward directions
than in the forward directions. However, because the ELN distribution
$G(\theta_{\nu})$ is sensitive to the small difference between
$f_{\nu_{e}}(\theta_{\nu})$ and $f_{\bar{\nu}_{e}}(\theta_{\nu})$, and because
it is likely to cross 0 near the radial direction, a high angular resolution
in the forward directions is needed to accurately describe the crossing in the
neutrino distributions. This issue becomes more severe at large radii where
$f_{\nu_{e}}(\theta_{\nu})$ and $f_{\bar{\nu}_{e}}(\theta_{\nu})$ are both
highly peaked in the forward direction, and they may cross each other at an
angle beyond the last $\theta_{\nu}$ bin.

We calculate the exponential growth rates $\Omega_{\mathrm{i}}$ of the
unstable fast oscillation modes as functions of the real wave number $K$ for
the (interpolated) ELN distributions plotted in Fig. 2, and the results are
shown in the right panel in the same figure. We assume the axial symmetry
about the radial direction in calculating $\Omega_{\mathrm{i}}$. We also
assume that the wave vector of the collective flavor oscillation wave is along
the radial direction. In this particular example, $S_{\mathbf{v}}$ can grow by
an $e$-fold within $\sim 1$ nanosecond which is indeed much faster than the
typical changing rates of the physical conditions inside the supernova.

We also find ELN crossings in the 3D model which seem to be more common than
in the 2D model. This observation can be understood by noting that the spatial
asymmetries in the neutrino emission might evolve faster in the case of the 3D
simulations Vartanyan  _et al._ (2019). They appear in all the three snapshots
that we have studied. In the $t_{\text{pb}}=200$ ms snapshot, ELN crossings
begin to occur at $r=46$ km in the neutrino decoupling region. We show
$\alpha$ in this snapshot at $r=65.6$ km in the left panel of Fig. 3 and mark
with crosses the spatial zones where ELN crossings occur. As in the 2D model,
ELN crossings are more likely to appear in the regions with $\alpha$ close to
1. The asymmetry pattern in the emissions of $\nu_{e}$ and $\bar{\nu}_{e}$ is
associated with a similar pattern in $Y_{e}$ which is present even well below
the neutrino sphere (middle and right panels of Fig. 3). The region with lower
$Y_{e}$ and more neutron-rich matter emits more $\bar{\nu}_{e}$’s and absorbs
more $\nu_{e}$’s through the reactions in Eq. (1). This pattern of $Y_{e}$ and
$\alpha$ is likely caused by the active matter convection inside the PNS as in
the LESA phenomenon Tamborra  _et al._ (2014).

Figure 4: The optical depths $\tau$ for the neutrinos of different flavors and
with energy $E_{\nu}=20.9$ MeV (the first three panels) and the matter density
(the very right panel) in the $t_{\text{pb}}=200$ ms snapshot of the 2D
supernova model shown in Fig. 1. The solid, dashed and dot-dashed circles in
the right panel indicate the location of the neutrinospheres of these
neutrinos.

Our study highlights the need of more multi-D supernova simulations with the
neutrino Boltzmann transport like those performed in Ref. Nagakura  _et al._
(2018). Our results show that the neutrino distributions with even a very
coarse angular resolution can be used to identify the ELN crossings in the
neutrino decoupling region, although a better resolution in the radial
direction is needed to accurately describe the crossings especially at large
radii. Although we have found fast growing neutrino oscillations in the
multi-D models that we have studied, it remains to be seen if the conditions
for the ELN crossing will be met in the more self-consistent multi-D supernova
simulations with accurate neutrino transport, and if the amplitudes of the
fast neutrino oscillation modes can continue to grow into the nonlinear regime
and cause significant flavor conversions deep inside the supernova. If fast
neutrino flavor conversions are indeed found to occur near the surface of the
PNS, they will have profound implications for the dynamics, nucleosynthesis,
and neutrino signals of the supernova.

## Acknowledgments

We thank J. Martin and C. Yi for valuable discussions. This work is supported
by the US DOE EPSCoR grant DE-SC0008142 and the NP grant DE-SC0017803 at UNM
for S. A. and H. D., “Gravitation et physique fondamentale” (GPHYS) of the
Observatoire de Paris for S. A. and M. C. V., and JSPS and MEXT KAKENHI Grant
Numbers (JP15K05093, JP17H01130, JP17K14306, 18H01212) and (JP26104006,
17H05206, JP17H06357, JP17H06364, JP17H06365) for K. S. and T. T.,
resepctively. K. S. and T. T. acknowledge the support on the computing
resources at NAOJ, KEK, JLDG, YITP, RCNP, UT as well as Post-K and K-computer
of the RIKEN AICS.

##  Appendix A The optical depths

In this appendix, we provide the information of the optical depths in the
$t_{\text{pb}}=200$ ms snapshot of the 2D supernova model shown in Fig. 1. The
optical depth of a neutrino is defined as

$\tau(s)=\int_{s}^{\infty}\chi(s^{\prime})\mathrm{d}s^{\prime},$ (7)

where $\chi(s)$ is the neutrino opacity, and the integration is performed
along the radial coordinate for the neutrinos in the forward angle bin. In the
first three panels of Fig. 4, we show the optical depths of the neutrinos of
different flavors and all with energy $E_{\nu}=20.9$ MeV. The very right panel
of the figure shows the matter density in this snapshot and the location of
the neutrinospheres (where $\tau=2/3$) for these neutrinos. The average
energies of the neutrinos are in the range of $10-25$ MeV in this snapshot.

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