# On probing turbulence in core-collapse supernovae in upcoming neutrino
detectors

Mainak Mukhopadhyay<EMAIL_ADDRESS>Department of Physics; Department of
Astronomy & Astrophysics; Center for Multimessenger Astrophysics, Institute
for Gravitation and the Cosmos, The Pennsylvania State University, University
Park, PA 16802, USA.  Manibrata Sen<EMAIL_ADDRESS>Max-Planck-
Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany.

###### Abstract

Neutrino propagation through a turbulent medium can be highly non-adiabatic
leading to distinct signatures in the survival probabilities. A core-collapse
supernova can be host to a number of hydrodynamic instabilities which occur
behind the shockfront. Such instabilities between the forward shock and a
possible reverse shock can lead to cascades introducing turbulence in the
associated matter profile, which can imprint itself in the neutrino signal. In
this work, we consider realistic matter profiles and seed in the turbulence
using a randomization scheme to study its effects on neutrino propagation in
an effective two-flavor framework. In particular, we find that the double-dip
feature, originally predicted in the neutrino spectra associated with forward
and reverse shocks, can be completely washed away in the presence of
turbulence, leading to total flavor depolarization. We also study the
sensitivity of upcoming neutrino detectors - DUNE and Hyper-Kamiokande- to the
power spectrum of turbulence to check for deviations from the usual Kolmogorov
($5/3$) inverse power law. We find that while these experiments can
effectively constrain the parameter space for the amplitude of the turbulence
power spectra, they will only be mildly sensitive to the spectral index.

##  I Introduction

The detection of neutrinos from SN1987A can be regarded as one of the first
instances of multi-messenger astronomy, where neutrino events were observed
hours before visible light. Although limited in statistics - a total of
$\mathcal{O}(30)$ neutrino events in the Kamiokande detector [1, 2], the IMB
detector [3] and the Baksan detector [4] were observed - this data has been
used widely by the particle and astroparticle physics community to probe
important questions of fundamental physics.

It is already well-known that neutrinos from a galactic core-collapse
supernova (SN) will be our best bet to probe the supernova dynamics. These
neutrinos originate from the deepest regions of the SN and propagate through
the entire matter envelope before exiting the SN and arriving at the Earth.
The flavor composition of the neutrinos is extremely sensitive to the
background fermion density it encounters en route. Close to the
neutrinosphere, the neutrino density is large enough, so neutrino forward
scattering on a neutrino background is the dominant contribution to the
propagation Hamiltonian. This can lead to collective flavor conversion - both
“slow” [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
24] and ’‘fast” [25, 26, 27, 28, 29, 25, 26, 27, 30, 31, 32, 31, 33, 34, 35,
36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54,
55, 56, 57, 58, 59, 60, 61, 32, 62, 63, 64, 65] (see [66, 67, 68, 69] for
detailed reviews on the subject)- which is the major mechanism of flavor
transformation within radii of $\sim\mathcal{O}(100)\,$km from the
neutrinosphere.

At larger radii, neutrino density falls off and matter effects, due to
coherent forward scattering on background electrons, become important.
Multiple studies have demonstrated the sensitivity of the neutrino spectra to
the underlying SN density profile through the Mikheyev-Smirnov-Wolfenstein
(MSW) resonances [70, 71], including signatures of the shockwave propagation
at late times [72, 73, 74, 75, 76, 77, 78, 79]. In the absence of the
shockwave, neutrino propagation is shown to be adiabatic, governed only by the
vacuum mixing angle and the matter density. However, violations of
adiabaticity can arise due to an abrupt change in density at the shockfront.
This non-adiabaticity leaves its imprint on the flavor transition and can be
used to reconstruct a real-time propagation of the shockwave.

A distinctive signature of the non-adiabaticity introduced by the shockwave
propagation can manifest in the form of dips in the energy-dependent survival
probabilities of neutrinos. A few seconds post the SN core bounce, a possible
reverse shockwave can arise if a supersonic neutrino-driven wind collides with
the ejecta, as shown in Fig. 1. Note that this is very sensitive to whether
the neutrino-driven wind is supersonic or subsonic, for a detailed discussion,
see [80]. In [77], it was demonstrated that for neutrinos passing through the
two density discontinuities, characterised by the reverse and the forward
shocks, a “double-dip” feature would be present in the neutrino survival
probability as a function of energy. Such a signal can be present in electron
neutrinos for normal mass ordering (NMO) and electron antineutrinos for
inverted mass ordering (IMO) and can be used to trace the position of the two
shockfronts.

However, this is not the complete story. A SN progenitor is host to a variety
of hydrodynamic instabilities, which take place behind the shockfront. This
leads to turbulence in the ejecta, causing stochastic fluctuations in the
matter density. Turbulence is generally expected to be present in aspherical
simulations of SNe, and can play an important role in seeding the explosion
[81]. These stochastic fluctuations in the matter density have a non-trivial
impact on neutrino flavor evolution.

Neutrino flavor evolution in a turbulent medium has been studied extensively
to determine its impact on the neutrino spectra from the next Galactic
supernova [82, 83, 84, 85, 86, 87, 88, 89, 90, 91]. . Generically, the effect
of turbulence on flavor propagation can be broadly classified into two types:
(i) ‘stimulated’ or non-adiabatic transitions between different propagation
eigenstates if the stochastic density fluctuations are present near the MSW
resonance region [84], and (ii) phase effects due to the presence of semi-
adiabatic resonances [78]. The latter effect is more subtle and can be
suppressed if the amplitude of fluctuations is large.

It is numerically challenging to obtain realistic turbulent matter density
profiles from aspherical (3D) hydrodynamic simulations. Theoretically, the
effect of turbulence has been modelled by adding disturbances to a smooth
matter profile obtained from coarse, spherically symmetric simulations. This
can be done in a number of different ways - for example, by adding delta
correlated noise [82], or by modelling the turbulent density fluctuations
through a Kolmogorov-type power spectrum [83]. In particular, recent 2D
hydrodynamic simulations have shown the presence of a power-law dependence of
the power spectrum on the wavelength of fluctuations. The long-wavelength
fluctuation indeed shows a power-law dependence with the mean spectral index
given by $p=1.85^{+0.54}_{-0.77}$, thus indicating that 2D turbulence might be
closer to having a Kolmogorov-type power spectrum [85]. However, simulations
also indicate that the turbulence might not be fully developed, hence other
values of the spectral indices cannot be ruled out. Moreover, it is well known
that the cascading of turbulence in 3D is different from 2D [92, 93]. However,
the lack of sufficient resolution makes it difficult to recover the Kolmogorov
power law behaviour from 3D simulations.

A theoretical study of the impact of turbulence on the neutrino survival
probability was performed in [83], where power-law fluctuations in the density
profile were assumed. It was found that the electron neutrino survival
probability exhibits a power law when the amplitude of turbulence is increased
and a corresponding analytical estimate to explain the power law for the
simple turbulence model was derived. This was followed by a series of works
[88, 89, 90], where the authors used rotating wave approximation to predict
the final survival probability amongst other techniques both numerical and
analytical. The latter formalism was also used to predict the dependence of
the neutrino flavor transition on the amplitude and the spectral index of
turbulence. However, various unanswered questions remain. For example, does
the “double-dip” feature pointed out in [77] survive in the presence of
turbulence? Can turbulence inside a SN media follow a different power-law
index than the Kolmogorov prediction, and with what amplitude? Can future-
generation neutrino experiments designed to detect thousands of neutrinos from
the next galactic supernova actually distinguish between different parameters
characterising the turbulence in a supernova?

We aim to shed some light on these questions in this study through a detailed
effective two-flavor study of neutrino propagation through a typical SN media
profile, characteristic of the cooling phase. We add turbulence to the smooth
density profile following the prescription in [86]. We find that the addition
of turbulence with an amplitude beyond $10\%$ washes out the double-dip
feature completely. We simulate neutrino events in the upcoming DUNE and
Hyper-Kamiokande experiment and find that they can distinguish between a
turbulent and a non-turbulent SN matter profile with high significance. On the
other hand, they do not have enough sensitivity to probe the index of the
power spectrum.

Our paper is organized as follows. We discuss turbulence in supernovae and the
slab-approximation used as the numerical technique in Sec. II. The algorithm
to seed the turbulence along with its impact on neutrino propagation is
discussed in Sec. III. The effects of turbulence on the double-dip feature in
the neutrino spectra are studied in Sec. III.3. In Sec. IV, we discuss the
event rates in DUNE and Hyper-Kamiokande along with their potential to
constrain the physical parameters associated with turbulence for a galactic
scale supernova. We conclude and discuss the implications of our work in Sec.
V. The numerical algorithm developed for this work is summarized in Appendix.
A.

##  II Turbulence in supernovae

We work in the approximate framework of 2-neutrino oscillations, where we have
the electron neutrinos ($\nu_{e}$) and $\nu_{x}$ where, $x=\\{\mu,\tau\\}$,
along with each of their antineutrino components. The Equation of Motion (EoM)
for a neutrino flavor $|\nu_{\alpha}\rangle$ can be written as

$\mathrm{i}\frac{d}{dt}|\nu_{\alpha}\rangle=\left[UH_{\rm
vac}U^{\dagger}+V(r)\right]|\nu_{\alpha}\rangle\,,$ (1)

where $H_{\rm vac}={\rm diag}(\Delta m^{2}/(2E),-\Delta m^{2}/(2E))$ is the
vacuum Hamiltonian and $U$ is the $2\times 2$ unitary mixing matrix. The
matter potential $V(r)$ has a contribution from the forward scattering of
neutrinos off the background fermions and is given by
$V_{0}(r)=\sqrt{2}G_{F}n_{e}(r){\rm diag}(1,0)$, where $G_{F}$ is the Fermi
constant and $n_{e}(r)$ is the electron number density at a given distance
$r$. This is extracted from a snapshot of the matter density at a certain time
from a simulation. We further neglect the contributions from the non-electron
flavor neutrinos, which cancel at the tree-level and arise only at the loop-
level [94, 95, 96]. Neutrinos close to the neutrinosphere also experience a
forward scattering potential due to their scattering on other background
neutrinos [97]. This potential is given by
$V_{\nu\nu}\simeq\sqrt{2}G_{F}n_{\nu}$, where $n_{\nu}$ is the total
background neutrino density. This neutrino-neutrino potential dominates deep
inside the SN, and can lead to fascinating collective effects. However, in the
region of our interest, close to the shockfront, this neutrino potential is
subdominant, and collective oscillations can be safely ignored.

Figure 1:  The density profile snapshot at time $t=5$ s post collapse [77].
The various regions of interest in the shock wave profile are labelled.

The turbulence is included in our model by adding a random field $F(r)$ as

$V(r)=\big{(}1+F(r)\big{)}V_{0}(r)$ (2)

The random field $F(r)$ is defined in the following way

$F(r)=\begin{cases}C_{*}\tanh{\bigg{(}\frac{r-r_{\rm
rs}}{\lambda}\bigg{)}}\tanh{\bigg{(}\frac{r_{\rm
fs}-r}{\lambda}\bigg{)}}\times F_{\rm rand}(r),\hskip 5.69046ptr_{\rm rs}\leq
r\leq r_{\rm fs}\,,\\\ 0,\hskip 5.69046pt\text{elsewhere}\,,\end{cases}$ (3)

where $C_{*}$ is the amplitude of the fluctuations, $F_{\rm rand}(r)$ is a
real-valued homogenous Gaussian scalar random field. The reverse shock
position is given by $r_{\rm rs}$ while the forward shock position is $r_{\rm
fs}$. The $\tanh$ terms suppress the fluctuations close to the shocks and
prevent discontinuities at $r_{\rm rs}$ and $r_{\rm fs}$. The parameter
$\lambda$ is the length scale over which the fluctuations reach their extent.
We choose $\lambda=100$ km.

The standard way of calculating the evolution of the neutrino wavefunctions
involves numerically solving the Schr$\ddot{\rm o}$dinger equation directly.
Unfortunately for irregular potentials (matter densities), this method is not
efficient. In particular, the current scenario has turbulences making the
matter potential highly noisy and the standard scenario for solving such
problems is unsuitable. Hence, we use the _slab approximation_ [98] to
calculate the spatial evolution of the neutrino wavefunctions.

In this approximation, the density profile is approximated by a series of
slabs of constant density. We discretize the spatial dimension of physical
size $L$ into $n$ slabs, thus the width of each slab is, $\Delta x=L/n$. In
such a discretization scheme, the starting point is given by $x_{0}$ and the
boundaries of the slabs are at $x_{1},x_{2},\dots,x_{n}$. The mass eigenstates
of the neutrinos propagate as plane waves in constant density regions with
phases given by ${\rm exp}{(\pm i\Delta m_{M}^{2}\Delta x/4E)}$, where the
effective mass-squared difference in matter is given by

$\Delta m^{2}_{M}=\sqrt{\Big{(}\hbox{$\Delta m^{2}$
}\cos{(2\theta)}-A_{CC}\Big{)}^{2}+\Big{(}\hbox{$\Delta m^{2}$
}\sin{(2\theta)}\Big{)}^{2}}\,,$ (4)

with $\theta$ being the vacuum mixing angle. $A_{CC}$ is the effective
effective charge current potential,

$A_{CC}(x)=2\sqrt{2}EG_{F}n_{e}(x)\,,$ (5)

All the information about the matter density profile is provided by this term.

The wavefunction in the flavor basis is given by,

$\bm{\Psi}_{e}=\langle x|\nu_{e}\rangle=\begin{pmatrix}\psi_{ee}\\\
\psi_{ex}\end{pmatrix}$ (6)

The wavefunction at the boundaries is joined according to the following scheme

$\bm{\Psi}_{e}(x_{n})=\left[U_{M}\ \mathcal{U}_{M}(x_{n}-x_{n-1})\
U_{M}^{\dagger}\right]_{(n)}\left[U_{M}\ \mathcal{U}_{M}(x_{n-1}-x_{n-2})\
U_{M}^{\dagger}\right]_{(n-1)}\dots\left[U_{M}\ \mathcal{U}_{M}(x_{1}-x_{0})\
U_{M}^{\dagger}\right]_{(1)}\bm{\Psi}_{e}(x_{0})\,,$ (7)

where,

  * •

the unitary evolution operator in each slab diagonal basis
$\mathcal{U}_{M}(\Delta x)$ is given by,

$\mathcal{U}_{M}(\Delta x)=\rm diag\Bigg{(}exp{(+i\Delta m_{M}^{2}\Delta
x/4E)},exp{(-i\Delta m_{M}^{2}\Delta x/4E)}\Bigg{)}\,,$ (8)

  * •

$U_{M}$ is the unitary mixing matrix in matter given by

$U_{M}=\begin{pmatrix}\cos{\theta_{M}}&\sin{\theta_{M}}\\\
-\sin{\theta_{M}}&\cos{\theta_{M}}\end{pmatrix}\,,$ (9)

where the effective mixing angle in matter $\theta_{M}$ is

$\tan{(2\theta_{M})}=\frac{\tan{(2\theta)}}{1-\frac{A_{CC}}{\Delta
m^{2}\cos{(2\theta)}}}\,.$ (10)

  * •

The notation $\left[\dots\right]_{(i)}$ indicates that all the matter-
dependent quantities inside the brackets need to be evaluated with the matter
density profile of the $i$-th slab, where the slab goes from $x_{i-1}$ to
$x_{i}$.

The wavefunction in the matter basis can be easily obtained using (7) and (9)
as

$\bm{\Phi}=\begin{pmatrix}\phi_{1}\\\
\phi_{2}\end{pmatrix}=U_{M}^{-1}\bm{\Psi}_{e}\,.$ (11)

The probability that a neutrino of flavor $\alpha$ is found in some flavor
$\beta$ after passing through the matter profile is given by

$P_{\nu_{\alpha}\rightarrow\nu_{\beta}}=|\psi_{\alpha\beta}|^{2}\,.$ (12)

Similarly, the corresponding survival probability of the matter eigenstate is
given by $P_{\nu_{i}\rightarrow\nu_{j}}=|\bm{\Phi}_{i}|^{2}$. The adiabaticity
parameter, which is an important quantity to understand the changes in
oscillation and transition probabilities related to neutrino propagation, is
defined as [98]

$\gamma=\frac{\Delta m^{2}_{M}}{4E_{\nu}|d\theta_{M}/dx|}=\frac{(\Delta
m_{M}^{2})^{2}}{2E_{\nu}\ ({\rm sin}\ 2\theta_{M})\ |dA_{{\rm CC}}/dx|}\,.$
(13)

Armed with this, we can investigate the propagation of neutrinos in an
irregular potential.

##  III Turbulence: Numerical ansatz

In this section, we discuss the Randomization Method used to implement
stochastic fluctuations following a power-law on top of a smooth matter
profile.

###  III.1 The Randomization Method: Generating the turbulence

In previous works, the authors of [84] used the _Randomization Method_ [99] to
generate and sample the turbulence. In [86], a slightly advanced version of
the randomization method - _‘Variant C’_ was used. For this work, we will use
the Variant C of the randomization method to generate the turbulence. It is
important to note here that in this method the modes are distributed in a
logarithmic scale which makes the distribution scale-invariant by ensuring
that the realizations are uniform over the decades of length scales considered
in the work. We summarize the method below.

Let the wavenumber space from which we draw the wavenumbers have a size
$\Delta$, such that, $\Delta=[0,\infty)$. For numerical implementation, the
space $\Delta$ is divided into $n$ non-intersecting intervals $\Delta_{j}$,
such that, $\Delta=\cup_{j=1}^{n}\Delta_{j}$. We then select $n_{0}$
independent random wavenumbers from each of these intervals $\Delta_{j}$,
which can now be indexed as $k_{jl}$, where, $l=1,2,\dots,n_{0}$. The
selection of the $k_{jl}$’s are according to the probability distribution
function $p_{j}(k)$ defined as

$p_{j}(k)=\begin{cases}2E(k)/\sigma_{j}^{2},\hskip 14.22636pt\text{for
}k\in\Delta_{j}\,,\\\ 0,\hskip 14.22636pt\text{for
}k\notin\Delta_{j}\,,\end{cases}$ (14)

where $E(k)$ is the power spectrum, which we assume to be of the following
form,

$E(k)=(\alpha-1)\bigg{(}\frac{k_{*}}{k}\bigg{)}^{\alpha}\,.$ (15)

In the above definition of the power spectrum, $\alpha$ is the spectral index,
$k_{*}=2\pi/\lambda_{*}$ is the wavenumber cutoff, i.e., the cutoff that
decides the longest non-zero turbulence wavelength ($\lambda_{*}$) and hence
the smallest wavenumber. Thus, it can be referred to as the driving scale for
our turbulence model [87]. For this work, we treat the spectral index $\alpha$
as a free parameter that is varied and the corresponding sensitivity of the
next-generation neutrino detectors is tested. We choose, $\lambda_{*}=2(r_{\rm
fs}-r_{\rm rs})$, which is twice the distance between the positions of the
reverse and forward shocks. In Eq. (14),
$\sigma_{j}^{2}=2\int_{\Delta_{j}}E(k)dk$.

Figure 2:  The effective matter potential profile for fixed $\alpha=-5/3$ and
different values of the normalization $C_{*}$. The case where turbulence is
absent is represented by $C_{*}=0$ (solid red curve). The value of
$\omega=\Delta m^{2}/(2E_{\nu})$ is also shown as a dashed green line
corresponding $\Delta m^{2}_{\rm atm}=2.6\times 10^{-3}\ \rm eV^{2}$, $\Delta
m^{2}_{\rm sol}=6.9\times 10^{-5}\ \rm eV^{2}$ for $E_{\nu}=10\ \rm MeV$.

We can now define $F_{\rm rand}(r)$ (see Eqs. (2) and (3)) as

$F_{\rm
rand}(r)=\sum_{j=1}^{n}\frac{\sigma_{j}}{\sqrt{n_{0}}}\sum_{l=1}^{n_{0}}\left[\xi_{jl}\
\text{cos}\big{(}2\pi k_{jl}r\big{)}+\eta_{jl}\ \text{sin}\big{(}2\pi
k_{jl}r\big{)}\right]\,,$ (16)

where, $\xi_{jl}$ and $\eta_{jl}$ are standard Gaussian random variables,
which are independent of each other and of $k_{jl}$, have zero mean, and unit
variance. Note that the indices $j=1,2,\dots,n$ and $l=1,2,\dots,n_{0}$. In
this strategy of stratified sampling of the wavenumber bins, we choose the
sampling bins such that they are equally spaced in the log-space, that is with
respect to $\text{ln }k$. Thus we have, for a given index $j$,
$\hat{k}_{j+1}=q\hat{k}_{j}$, where $q$ is some ratio parameter of the
geometric distribution which defines the wavenumber intervals
$\Delta_{j}=(\hat{k}_{j},\hat{k}_{j+1}]$. Since we have a minimum wavelength
in our model, we have $\hat{k}_{1}=k_{*}$. The bounded domain for $k$ also
introduces the constraint, $k\leq k_{\max}$, where we choose $k_{\max}=100$,
such that $\hat{k}_{n+1}=k_{\max}$. For this work, we chose $n=4$ and
$n_{0}=40$. We have also verified that pushing $n$ and $n_{0}$ to higher
values of $5$ and $50$ respectively does not change the results. It is
important to note here that this strategy of stratified sampling is efficient
in way that allows us to work with a comparatively low value of total number
of modes, $\mathcal{O}(100)$, to give us accurate and physical scale
independent results, as compared to just sampling the modes trivially, thus,
gaining some computational advantage. The density profile extends from $r_{\rm
start}=0.105\times 10^{4}$ km to $r_{\rm end}=11.35\times 10^{4}$ km. This
region is sampled in spatial steps of $\delta x=10$ km. The turbulence is
introduced to this density profile, starting at the reverse shock position
which is at, $r_{\rm rs}=0.39\times 10^{4}$ km and it is terminated at the
position of the forward shock, $r_{\rm fs}=5.6\times 10^{4}$ km. The region of
turbulence has the same spatial discretization scale as the original density
profile ($=10$ km).

As stated earlier, the primary parameters that we vary and examine in this
work are the amplitude associated with the turbulence $C_{*}$ and the power
spectral index $\alpha$. In Fig. 2, we show the effective matter potential for
a fixed $\alpha=-5/3$ and various values of $C_{*}$. The red thick line is the
case when turbulence is absent, that is, $C_{*}=0$. This is essentially the
same curve as shown in Fig. 1 (solid blue curve) rescaled to convert the
matter density to effective potential using natural units $\hbar$ and $c$. As
the amplitude of turbulence increases, the fluctuations grow as expected. The
effects of this variation in the amplitude on the survival probabilities for
neutrinos are discussed in the following section. It is important to note here
that we want to focus on the effects associated solely with the non-
adiabaticity at the forward and reverse shocks, hence we artificially smoothed
the contact discontinuity so as to avoid any non-adiabaticity effects arising
from it.

###  III.2 Results

(a) (a)

(b) (b)

(c) (c)

(d) (d)

Figure 3:  Comparison between the scenarios when turbulence is present
($C_{*}=0.3,\alpha=-5/3$) (solid orange curves) and when turbulence is absent
($C_{*}=0$) (dashed purple curves). (a) The $\nu_{e}$ survival probability,
$P_{ee}(r)$. (b) The survival probability for the first instantaneous matter
eigenstate, $P_{11}(r)$. (c) The effective mixing angle in matter,
$\theta_{M}(r)$ from Eq. (10); (d) The adiabaticity parameter, $\gamma(r)$,
from Eq. (13). The reverse shock (RS) and forward shock (FS) are shown as dot-
dashed dark blue lines. We assume $\theta_{0}=0.09967$ rads, $\Delta
m_{vac}^{2}=2.6\times 10^{-3}\rm eV^{2}$, and $E_{\nu}=10$ MeV.

In this section, we discuss the results obtained by solving for neutrino
flavor propagation through the matter profile as shown in Fig. 2. The two
primary scenarios we are interested in are when turbulence is absent and when
turbulence has a high amplitude. Thus, we compare the results for $C_{*}=0$
(which we label as _without turbulence_) and $C_{*}=0.3$ (which we label as
_with turbulence_) respectively to implement the above scenarios. The choice
of $C_{*}=0.3$ is representative of an extreme case where complete flavor
depolarization occurs and the oscillation signatures are washed out.

The results are shown in Fig. 3 for the electron neutrino survival probability
($P_{ee}(r)$), the survival probability for the first instantaneous matter
eigenstate ($P_{11}(r)$), the effective mixing angle in matter ($\theta_{M}$)
and the adiabaticity parameter ($\gamma(r)$) as $\nu_{e}$ propagates through
the effective matter profile from $r_{\rm start}$ to $r_{\rm end}$. In Fig. 3a
we show the survival probability ($P_{ee}(r)$) for $\nu_{e}$ as a function of
distance for the cases of with (solid orange) and without (dotted purple)
turbulence. We note that the survival probability initially starts from unity,
corresponding to a pure $\nu_{e}$ state. The position of reverse shock
introduces a drop in the survival probability. Between the forward and the
reverse shocks, the behaviour of $P_{ee}(r)$ is clearly distinct for the cases
with and without turbulence. The propagation of the neutrinos in the presence
of turbulence is extremely non-adiabatic due to the highly fluctuating
background matter density. A better understanding can be achieved by studying
$P_{11}(r)$ (in Fig. 3b), which is expected to track the changes in the
density. The matter eigenstate survival probability shows a jump due to non-
adiabatic evolution corresponding to the density jumps. While the case without
turbulence has jumps only at the positions of the reverse and forward shocks,
the case with turbulence has numerous jumps.

A similar trend can be seen in the effective mixing angle in the matter for
the two cases, as shown in Fig. 3c. The case without turbulence exhibits a
maximal change in $\theta_{M}$ at the locations of the forward and reverse
shocks, whereas, in the presence of turbulence, $\theta_{M}$ has large jumps
throughout the density profile. To demonstrate the non-adiabaticity associated
with the neutrino flavor evolution, Fig. 3d shows the associated adiabaticity
parameter, $\gamma(r)$. A highly non-adiabatic evolution is characterized by
$\gamma(r)<<1$. We see that for the case with turbulence between the reverse
and the forward shocks, the adiabaticity parameter is much smaller than 1,
implying a highly non-adiabatic evolution. The turbulence is only seeded
between the forward and reverse shocks, so beyond the forward shock, the
evolution once again becomes adiabatic in the absence of any further shocks or
turbulence. This can be seen from Figs. 3b and 3d, where in the former case,
the transition probability does not show any jumps beyond the forward shock,
whereas in the latter case, $\gamma(r)>>1$ in the same region, implying
adiabatic evolution.

###  III.3 On the double-dip associated with shocks

The presence of shocks characterized by rapid changes in the density profile
can have imprints on the neutrino oscillation probabilities. The MSW resonance
associated with neutrino oscillations is given by, $\Delta m^{2}{\rm
cos}2\theta/(2E_{\nu})=\pm\sqrt{2}G_{F}Y_{e}\rho$, where $G_{F}$ is the Fermi
constant, $Y_{e}$ is the electron fraction, and $\rho$ is the matter density.
Depending on the mass-squared difference corresponding to solar or atmospheric
neutrinos, the resonances are classified as L-resonance (low density) and
H-resonance (high density) respectively. The propagation of the shockwave
through the densities corresponding to H-resonance regions can break
adiabaticity leading to distinctive features in the neutrino energy spectra.

As discussed in Sec. I, numerical simulations hint towards the presence of
both forward and reverse shocks in the matter density profile, resulting in a
_double-dip_ feature in the neutrino spectra [77].

Figure 4:  The average survival probability ($\langle P_{ee}\rangle$) with
neutrino energy ($E_{\nu}$) for different amplitudes of turbulence associated
with the density profile (see Fig. 2).

In Fig. 4 we show the average survival probability for $\nu_{e}$ as a function
of the neutrino energy ($E_{\nu}$). Note the presence of the double-dip
feature (solid red curve) in the neutrino spectra corresponding to the density
profile (in Fig. 2) in the absence of turbulence $(C_{*}=0)$. The first peak
corresponds to $E_{\nu}\sim 18$ MeV and is due to the forward shock. Following
the first peak, the adiabatic transitions corresponding to the forward and
reverse shocks partially cancel out to give the valley. Finally, a second peak
appears for $E_{\nu}\gtrsim 60$ MeV which is due to the neutrinos crossing one
of the forward or reverse shocks.

We now aim to study the effects of the presence of turbulence on the double-
dip feature in the neutrino spectra. The results are shown in Fig. 4 for a
range of turbulence amplitudes given a fixed value of $\alpha=-5/3$.
Interestingly, we find that the double-dip feature vanishes as the amplitude
of turbulence increases leading to highly non-adiabatic flavor evolution. For
$C_{*}>0.01$, the first peak disappears. As the value of $C_{*}$ is increased,
a gradual wash-out of the feature is seen. In fact, for the maximum value of
$C_{*}$ used in this work, i.e., $C_{*}=0.3$, the survival probability
averages to $1/2$, signifying complete flavor depolarization. Thus, the
presence of turbulence can completely wash away the double-dip feature in the
neutrino energy spectra associated with forward and reverse shocks. This can
be used to probe the presence of hydrodynamic turbulence in core-collapse
supernovae.

###  III.4 Revisiting previous analytical results for a simplified model of
turbulence

In [83], the effects of a simple model for turbulence on neutrino propagation
in supernovae were studied. In particular, analytical estimates corresponding
to neutrino flavor depolarization in the presence of a Kolmogorov-type
turbulence were derived. The authors argued that evidence from numerical
simulations hints towards the development of Kolmogorov cascades due to
various instabilities. In such a turbulence scenario, the Fourier transform of
the velocity correlator has the following form,

$C(k)\equiv\int dx\langle\delta n(0)\delta n(x)\rangle
e^{-ikx}=C_{0}k^{\alpha}\,,$ (17)

where $\alpha=-5/3$. The matter potential, $A(x)$, was assumed to be linear
and turbulent random noise was introduced as

$A(x)=x+F\sum_{k=1}^{600}k^{\beta}\cos{\Big{(}kx+\phi(k)\Big{)}}\,,$ (18)

where $F$ is the amplitude or normalization factor for the noise, $k$ is the
mode number, $\beta=\alpha/2$, and $\phi(k)$ is a random phase which is a
random value between $0$ and $2\pi$ for every mode $k$. The above equation can
be compared to Eq. (2) used in this work.

We reproduce the numerical results from [83] using the same parameters to
check our numerical code. We show the result in Fig. 5a. The spatial
coordinate goes from $x=-100$ to $x=100$, $\Delta=50$, and $\theta=0.1$. We
choose $\Delta x=0.1$. The agreement between our results and the same obtained
by the previous work is good. We find that for low turbulence amplitudes, the
effect on survival probability is less pronounced, which is expected and
matches with our own analysis discussed above. When the turbulence amplitude
is increased from $F\sim 0.2$ to higher values, a power law develops. The
power law rise seen in the figure can be explained using an analytical
estimate derived in [83],

$P^{\rm rise}\approx 0.84\ \frac{G_{F}C_{0}(2\Delta{\rm sin}\
2\theta_{13})^{-2/3}}{\sqrt{2}|n_{0}^{\prime}|}\,,$ (19)

where $n_{0}^{\prime}$ is related to the spatial derivative of the matter
density. We note that for values of $F\gtrsim 0.05$, a complete _flavor
depolarization_ happens, implying the flavor constitution becomes equal for
$\nu_{e}$ and $\nu_{x}$, in the two-flavor oscillation scenario considered
here.

We extend the analysis in [83] by studying the variation of the average
electron neutrino survival probability ($\langle P_{ee}\rangle$) in the
$\beta-F$ plane. The result, in the form of a density plot for the average
probability111The averaging is done over $66$ samples similar to Fig. 5a., is
shown in Fig. 5b. A rainbow color scheme is used to illustrate $\langle P_{\rm
ee}\rangle$, where the violet end of the spectrum implies $\langle P_{\rm
ee}\rangle\sim 0$ and the red end signifies $\langle P_{\rm ee}\rangle\sim 1$.
The area where complete flavor depolarization occurs is shown by a greenish-
yellow shade corresponding to $\langle P_{\rm ee}\rangle\sim 0.5$. For
$\beta\sim 0$, we find that $\langle P_{\rm ee}\rangle\sim 0.5$ for any given
value of the turbulence amplitude $F$. As $\beta$ decreases to $-1$, higher
amplitudes of turbulence lead to flavor depolarization. This generalizes the
results of [83] and confirms that for large values of $F$, flavor
depolarization holds for other values of $\beta$ as well.

There are distinct differences between the analysis in [83] and our current
work. We assume a more realistic matter density profile based on actual
simulations in contrast to the linear profile assumed for simplicity in the
former work. While the simple profile helps in getting some analytical
estimates, the effects of a realistic profile need to be addressed numerically
and lead to more realistic predictions for the upcoming detectors.
Furthermore, the seeding of the turbulence in this work uses a variant of the
Randomization method which is different from the simple random noise
turbulence used in the previous work.

(a) (a)

(b) (b)

Figure 5:  (a) The electron neutrino survival probability as a function of the
noise amplitude $F$ for $\beta=-5/6$, to match with the results of [83] (see
Fig. 1 there); (b) The electron neutrino survival probability as a function of
$\beta$ ranging from $-1$ to $0$ for various values of $F$. Each data point in
the numerics was repeated for $66$ times and the average value was used.

##  IV Events rates and statistical analysis

In previous sections, we discussed the effects of turbulence on the neutrino
signatures and survival probabilities. This section focuses on the prospects
and the capabilities of the upcoming neutrino detectors to constrain the
parameters associated with turbulence, in particular, the amplitude $(C_{*})$
and the spectral index $(\alpha)$. The detectors we consider for this work are
the Deep Underground Neutrino Experiment (DUNE) and Hyper-Kamiokande (HK).
While the former is largely sensitive to a flux of $\nu_{e}$, the latter is
better suited for detecting the $\bar{\nu}_{e}$ flux.

The supernova neutrino spectrum at the source (the neutrinosphere) is assumed
to be a pinched Fermi-Dirac spectrum [100],

$\frac{dN_{\nu}^{\rm src}(E_{\nu})}{dE_{\nu}}=\frac{L_{\nu}}{\langle
E_{\nu}\rangle^{2}}\frac{(1+\alpha_{\nu})^{(1+\alpha_{\nu})}}{\Gamma(1+\alpha_{\nu})}\bigg{(}\frac{E_{\nu}}{\langle
E_{\nu}\rangle}\bigg{)}^{\alpha_{\nu}}{\rm
exp}\bigg{(}-(1+\alpha_{\nu})\frac{E_{\nu}}{\langle
E_{\nu}\rangle}\bigg{)}\,,$ (20)

where $N_{\nu}^{\rm src}$ is the number of neutrinos at the source for any
given flavor with energy $E_{\nu}$, $L_{\nu}$ is the neutrino luminosity for
that particular flavor, $\langle E_{\nu}\rangle$ is the mean neutrino energy,
$\alpha_{\nu}$ is a pinching parameter associated with the given flavor.
Considering the cooling phase, we choose
$L_{\nu_{e}}=L_{\bar{\nu}_{e}}=5\times 10^{51}$ erg,
$L_{\nu_{x}}=L_{\bar{\nu}_{x}}=10^{52}$ erg, $\langle E_{\nu_{e}}\rangle=12$
MeV, $\langle E_{\bar{\nu}_{e}}\rangle=15$ MeV, $\langle
E_{\nu_{x}}\rangle=\langle E_{\bar{\nu}_{x}}\rangle=18$ MeV, and
$\alpha_{\nu_{e}}=\alpha_{\bar{\nu}_{e}}=\alpha_{\nu_{x}}=\alpha_{\bar{\nu}_{x}}=3$.
The neutrinos emitted from the neutrinosphere then propagate through the
matter profile encountering the forward and reverse shocks, along with the
turbulence which leads to neutrino oscillations. Let $P_{ee}$ be the survival
probability of $\nu_{e}$ while propagation. In this case, the number of
$\nu_{e}$ neutrinos at the end of the matter profile is given by

$\frac{dN_{\nu_{e}}^{\rm
SN}(E_{\nu})}{dE_{\nu}}=P_{ee}(E_{\nu})\frac{dN_{\nu_{e}}^{\rm
src}(E_{\nu})}{dE_{\nu}}+\big{(}1-P_{ee}(E_{\nu})\big{)}\frac{dN_{\nu_{x}}^{\rm
src}(E_{\nu})}{dE_{\nu}}\,,$ (21)

where $N_{\nu_{e}}^{\rm SN}$ is the number of $\nu_{e}$ after the initial
neutrino spectra from the neutrinosphere have propagated through the matter
profile. Similarly one can obtain the spectra for $\bar{\nu}_{e}$. Once the
neutrinos leave the supernova they propagate through the interstellar medium
to reach Earth. This can be effectively treated as propagation in
vacuum222Note that by this time, the neutrinos will have decohered and will
travel as an incoherent superposition of mass eigenstates.. The vacuum
survival probability for $\nu_{e}$ is given by

$P_{ee}^{\rm vac}(E_{\nu})=1-{\rm sin}^{2}2\theta\ {\rm
sin}^{2}\bigg{(}\frac{\Delta m^{2}R}{4E_{\nu}}\bigg{)}\simeq 1-\frac{1}{2}{\rm
sin}^{2}2\theta=0.58\,,$ (22)

where $\theta_{\rm vac}$ is the vacuum mixing angle and $\Delta m^{2}_{\rm
vac}$ is the vacuum mass-squared difference. We assume $\theta_{\rm
vac}=0.9967$ rads and $\Delta m^{2}_{\rm vac}=2.6\times 10^{-3}\rm eV^{2}$.
Thus the neutrino spectra at the Earth, neglecting Earth-matter effects, is
given by

$\frac{dN_{\nu_{e}}^{\rm earth}(E_{\nu})}{dE_{\nu}}=P_{ee}^{\rm
vac}(E_{\nu})\frac{dN_{\nu_{e}}^{\rm
SN}(E_{\nu})}{dE_{\nu}}+\big{(}1-P_{ee}^{\rm
vac}(E_{\nu})\big{)}\frac{dN_{\nu_{x}}^{\rm SN}(E_{\nu})}{dE_{\nu}}\,.$ (23)

A similar analysis can be repeated for obtaining the $\bar{\nu}_{e}$ spectra
on earth which will be relevant for HK. We do not consider earth matter
effects for this work.

The total number of neutrinos of a given species $\alpha$ ($N_{\nu_{\alpha}}$)
detected in a neutrino detector from a source at a distance $R$ is given by

$N_{\nu_{\alpha}}=\Delta t\frac{N_{\rm tar}}{4\pi R^{2}}\int dE_{r}\int
dE_{t}\ \frac{dN_{\nu_{\alpha}}^{\rm
earth}(E_{t})}{dE_{t}}\sigma_{\nu_{\alpha}}(E_{t})W(E_{r},E_{t})\,,$ (24)

where $N_{\rm tar}$ is the number of targets for a given channel in the
detector fiducial volume, $\sigma_{\nu_{\alpha}}$ is the neutrino interaction
cross-section with the targets for the relevant channel, $\Delta t$ is the
duration for which we assume the density profile to remain approximately fixed
and similar to what is considered for the matter profile, and $W$ is the
Gaussian energy resolution function which depends on the intrinsic properties
of the detector. The experimentally reconstructed energy in the detector is
given by $E_{r}$ and the actual neutrino energy is given by $E_{t}\equiv
E_{\nu}$. We fix the distance to the supernova at $R=10$ kpc for our current
work. The energy resolution function is defined as [101]

$W(E_{r},E_{t})=\frac{1}{\sqrt{2\pi}\sigma_{E}(E_{r})}\exp{\bigg{(}-\frac{1}{2}\frac{(E_{r}-E_{t})^{2}}{\sigma_{E}(E_{r})^{2}}\bigg{)}}\,.$
(25)

The sensitivity of the neutrino detectors for different values of $C_{*}$ and
$\alpha$ is estimated using a simple $\chi^{2}$ estimator

$\chi^{2}=\frac{\big{(}N_{\rm obs}-N_{\rm exp}\big{)}^{2}}{N_{\rm obs}}\,,$
(26)

where $N_{\rm obs}$ is the number of neutrino events observed in the neutrino
detector, $N_{\rm exp}$ is the expected number of neutrino events in the
detector. We fix $N_{\rm exp}$ to the case when $C_{*}=0.1$, and
$\alpha=1.65$, corresponding to the Kolmogorov turbulence, unless otherwise
specified. We consider $\Delta t=1\,{\rm s}$.

Figure 6:  Density plot to show the total number of $\nu_{e}$ events for DUNE
(_left_) and $\bar{\nu}_{e}$ events for HK (_right_) on the $C_{*}-\alpha$
plane. The source is assumed to be at a distance of $R=10$ kpc. See text for
details on the other parameters used.

###  IV.1 DUNE

The DUNE experiment [102] consists of a near and a far detector separated by
$1300$ km. The former is located at Fermilab, Illinois while the latter is
located about $1.5$ km below the ground at the Stanford Underground Research
Facility (SURF), South Dakota. The far detector is a liquid argon time
projection chamber (LArTPC) with $70$ kt (fiducial mass of $40$ kt) of liquid
argon. We will focus on the far detector in this work.

The relevant interaction channel for supernova neutrinos is the charged
current interaction of electron neutrinos with argon nuclei,
$\nu_{e}+^{40}{\rm Ar}\rightarrow^{40}{\rm K}^{*}+e^{-}$. We use $N_{\rm
tar}=6\times 10^{32}$. The interaction cross-section $\sigma_{\nu_{e}}$
associated with the $\nu_{e}-\rm Ar$ interaction cross section is taken from
[103]. The energy resolution for DUNE is taken to be $\sigma_{E}/{\rm
MeV}=0.11\sqrt{E_{r}/{\rm MeV}}+0.02E_{r}/{\rm MeV}$.

The result for the total number of electron neutrino events $N_{\nu_{e}}$ for
DUNE is shown in the left panel of Fig. 6 as a density plot in the
$C_{*}-\alpha$ plane. The color map uses a rainbow template to denote the
lowest number of events in violet and the highest in red. For a typical
galactic scale supernova ($R\sim 10$ kpc) we have $\sim 10^{3}$ events in DUNE
in the $\nu_{e}$ channel. There are two interesting aspects that we note right
away - (a) as $\alpha$ varies from $1.0$ to $2.0$, the number of events
$N_{\nu}$ does not change significantly, which implies that it would be
difficult to put constraints on the value of $\alpha$ using the DUNE detector;
(b) as $C_{*}$ is varied from $0.001$ to $1.0$ the number of events in the
DUNE detector has a significant variation and goes from $\sim 1350$ (red) for
lower values of $C_{*}\sim 0.001$ to $\sim 900$ (violet) for large values of
$C_{*}\sim 1.0$. Thus, DUNE can be used to place constraints on the value of
$C_{*}$ for a typical galactic supernova in the near future. We discuss the
two points below using a $\chi^{2}$ plot as discussed below.

The significant variation of the number of events with $C_{*}$ is confirmed in
Fig. 7a, which shows the variation of $\chi^{2}$ versus $C_{*}$ for
$\alpha=1.65$. The null hypothesis is considered to be $C_{*}=0.1$. We note
that for low values of $C_{*}<0.05$, the $\chi^{2}$ starts growing and it
increases as the amplitude of turbulence decreases333Note that for $C_{*}=0.1$
the value of $\chi^{2}$ goes to $0$ from Eq. (26).. However, for larger values
of $C_{*}>0.05$, constraints become weak. To get a better understanding of
this behaviour, we compare the event spectra for different values of $C_{*}$
in Fig. 7c. The null hypothesis $(C_{*}=0.1)$ is shown with a dotted dark blue
line. The cases corresponding to $C_{*}=0.005,0.02$, and $0.5$ are shown as
solid purple, orange, and red lines. On comparing the case of $C_{*}=0.005$
with the expected case we see a significant difference in the number of events
in the energy bins above $17$ MeV leading to the high value of $\chi^{2}$. As
$C_{*}$ increases the number of events in the energy bins becomes similar to
the expected case of $C_{*}=0.1$ explaining the trend of $\chi^{2}$ as seen in
Fig. 7a.

The variation over $\alpha$ does not have a specific trend for any value of
$C_{*}$ and the $\chi^{2}$ values obtained are extremely low and we do not
show them here. This is evident from Fig. 8a where we show the event spectra
for $\alpha=1.05$ (solid orange), $1.65$ (dashed dark blue), and $2.0$ (solid
purple) for $C_{*}=0.1$. The expected scenario for this case is again chosen
to be $C_{*}=0.1$ and $\alpha=1.65$ shown as the dashed dark blue line. We
note that the spectra for all the three cases shown look very similar to the
expected case in dotted dark blue, explaining the low values of $\chi^{2}$.
This highlights the ineffectiveness of DUNE to put constraints on the value of
$\alpha$.

(a) (a)

(b) (b)

(c) (c)

(d) (d)

Figure 7:  (a) Plot of $\chi^{2}$ estimator for different values of $C_{*}$
for DUNE assuming $\alpha=1.65$ (corresponding to Kolmogorov turbulence); (b)
same as (a) but for HK; (c) The expected number of $\nu_{e}$ events per energy
bin for DUNE assuming $\alpha=1.65$ and $C_{*}=0.005$ (solid purple), $0.02$
(solid orange), and $0.5$ (solid red). The $N_{\rm exp}$ for the $\chi^{2}$
estimator is considered to be the case when $C_{*}=0.1$ and $\alpha=1.65$,
which is shown as a dotted dark blue line; (d) same as (c) but for HK where
$\bar{\nu}_{e}$ events are considered.

(a) (a)

(b) (b)

Figure 8:  The expected number of $\nu_{e}$ and $\bar{\nu}_{e}$ events per
energy bin corresponding to DUNE (_left_) and HK (_right_) for a fixed value
of $C_{*}=0.1$ and different values of $\alpha=1.05$ (solid orange), $1.65$
(dotted dark blue), and $2.0$ (solid purple) respectively. The source is
assumed to be at a distance of $R=10$ kpc.

###  IV.2 Hyper-Kamiokande

Hyper-Kamiokande (HK) [104] is an upcoming $260$ kton (fiducial $187$ kton)
liquid scintillator detector, being built at the Kamioka Observatory in Japan.
It is an upgrade to the Super-Kamiokande detector but with around $10$ times
the fiducial volume. It will consist of two tanks of ultrapure water acting as
the medium for neutrino detection. The primary detection channel in HK is the
inverse-beta decay (IBD), $\bar{\nu}_{e}+p\rightarrow e^{+}+n$. We use the IBD
cross-section from [105]. Similar to Super-Kamiokande, the energy resolution
for HK is given by $\sigma_{E}/{\rm MeV}=0.6\sqrt{E_{r}/{\rm MeV}}$ [103]. The
number of targets for HK is assumed to be $1.2\times 10^{35}$.

The density plot for the number of $\bar{\nu}_{e}$ events for HK in the
$C_{*}-\alpha$ plane is shown in the right panel of Fig. 6. The details of the
colors used are discussed in Sec. IV.1. We have $\sim 7\times 10^{4}$
$\bar{\nu}_{e}$ events for a typical galactic scale supernova ($R=10$ kpc).
This number is much higher than DUNE owing to the larger number of targets.
This already hints at the possibility that HK would be a better detector than
DUNE at constraining the parameter range for turbulence. However, similar to
DUNE, the variation in $N_{\bar{\nu}_{e}}$ is insignificant across the
different values of $\alpha$ for a given value of $C_{*}$.

Using HK, relatively strong constraints can be placed on the amplitude of
turbulence as seen by the variation in $N_{\bar{\nu}_{e}}$ with $C_{*}$ for a
fixed value of $\alpha$ in the right panel of Fig. 6. This is also evident
from Fig. 7b where we have very large values of $\chi^{2}$ for $C_{*}\lesssim
0.05$, keeping $\alpha$ fixed at $1.65$. To confirm our results, we show the
event spectra for $\alpha=1.65$ and different values of $C_{*}$ in Fig. 7d.
Once again, the expected case is assumed to be $C_{*}=0.1$ and $\alpha=1.65$
and is shown in dotted dark blue. We note that similar to the case of DUNE,
for $E_{\nu}\leq 20$ MeV, the spectra are similar among all the cases, however
for $E_{\nu}>20$ MeV, there is a significant difference in the spectra between
$C_{*}=0.1$ and $C_{*}=0.005$ (solid purple line) and $C_{*}=0.02$ (solid
orange line) leading to the comparatively high values of $\chi^{2}$. However,
for $C_{*}=0.5$ (solid red line) the event spectra look similar to the
expected case giving a low value of $\chi^{2}$ as seen in Fig. 7b.

Finally in Fig. 8b, we show the event spectra for $C_{*}=0.1$ for different
values of $\alpha$. The spectra look similar to the expected case of
$C_{*}=0.1$, indicating that HK, while sensitive, would not be able to put
very strong constraints on $\alpha$.

##  V Conclusions and Implications

Neutrino propagation through a turbulent medium leads to high degrees of non-
adiabaticity. Hydrodynamical simulations of core-collapse supernova hint
towards the presence of various instabilities in the region behind the shock,
leading to the matter profile becoming turbulent. Such a turbulent medium can
leave its signature on the neutrino spectra, thereby allowing future neutrino
detectors to probe the strength of turbulence inside a core-collapse
supernova. In this work, we performed a detailed study of the effects of
turbulence on neutrino propagation and outlined the prospects for upcoming
neutrino detectors like DUNE and HK to constrain the parameter space
associated with turbulence.

We considered a usual supernova matter profile snapshot, characteristic of the
cooling phase, and used the variant C formalism of the Randomization method to
seed in the turbulence. This involves a stratified sampling of the modes to
have a minimal effect associated with the physical scales. The two main
parameters associated with turbulence are the amplitude ($C_{*}$) and the
spectral index associated with the power spectrum ($\alpha$). We numerically
evolved the neutrino oscillation probabilities using the slab approximation,
which is appropriate for a highly oscillatory profile as is the case in the
presence of turbulence. It is important to note that for getting reasonable
results, the discretization scale, which in this case is given by the width of
the slabs, should be smaller than the length scales associated with the
variation of the matter profile. To illustrate the difference between neutrino
propagation in the presence and absence of turbulence, we contrasted various
quantities like the electron neutrino survival probability, the effective
mixing angle in matter, and the adiabaticity parameter. We showed that these
quantities change significantly depending on whether the matter background has
turbulence or not.

Depending on the location of the forward and reverse shocks, the neutrino
spectra from a supernova can be very different due to the non-adiabaticity
associated with the shock fronts. In particular, the H-resonance associated
with $\Delta m^{2}_{\rm atm}$ can break adiabaticity at the forward and
reverse shocks. This can lead to a double-dip structure in the neutrino
survival probability, as was discussed in detail in [77]. We revisited this
issue and studied if the double-dip feature still survives in the presence of
turbulence between the shockfronts. We found that as the profile becomes more
turbulent, flavor depolarization kicks in. In fact, in our study, if the
strength of turbulence exceeds $30\%$, the double-dip structure is washed out,
leading to complete flavor depolarization. This implies that the double-dip
signature which can provide more information about the positions of the
forward and reverse shocks may fail to do so in the presence of strong
turbulence in between the shocks.

We considered the prospects of upcoming neutrino detectors like DUNE and HK to
detect neutrinos from a galactic core-collapse supernova and constrain the
parameters associated with turbulence in Sec. IV. Given the larger fiducial
volume, HK is better in constraining the turbulence parameters than DUNE. Most
importantly, we note that there is a significant variation of the number of
events as the amplitude increases from sub-percent to $\mathcal{O}(1)$ for a
fixed value of the spectral index. As a result, both DUNE and HK would be able
to place reasonable constraints on the amplitude of the turbulence. However,
the same cannot be said for the spectral index as the sensitivity is not too
large.

We conclude by discussing potential improvements in our results. With regard
to the turbulence, we assumed a simplified model of one-dimensional to
illustrate our results. A more realistic picture can be obtained using three-
dimensional computations where the turbulence samples $k_{x},k_{y}$, and
$k_{z}$ from given power spectral distributions. However, this would be
computationally intensive. Furthermore, we also assumed that the matter
density profile along with the turbulence is constant for the duration of
neutrino propagation we consider, which is set to $1$ second. This is a
reasonable approximation since the density profiles from numerical simulations
do not have significant changes over such durations in the cooling phase.

The results presented in this paper rely on an effective two-flavor framework.
This can be further improved to include three-flavor effects. Furthermore, we
also did not consider the effect of collective flavor oscillations on our
results. Particularly, if the neutrinos are susceptible to fast flavor
instabilities occurring near the core, then flavor depolarization might
already have happened by the time the neutrinos reach the shock-front [41,
62]. In that case, the turbulence signature becomes degenerate and cannot be
disentangled from the depolarization due to fast conversions.

Understanding the complex nature of turbulence along with its very existence
still remains a challenging problem. This work highlights the possibilities of
constraining turbulence in the era of large-scale neutrino detectors like DUNE
and HK. The statistics from a future galactic supernova at the upcoming
neutrino detectors should be good enough to gain invaluable insights into
turbulence. In fact, for a very nearby supernova (within $1$ kpc) [106, 107]
the complete spectrum can help place strong constraints on both the amplitude
and the spectral index associated with turbulence, and allow us to investigate
the double-dip in the neutrino spectra associated with shocks. A planned
$100$-kt neutrino detector THEIA [108] can also help place strong constraints
owing to its large fiducial volume. We conclude that with the next galactic
core-collapse supernova in the era of DUNE and HK, we can significantly
improve our understanding of the evolution of shocks inside the SN, and the
underlying hydrodynamic instabilities. This can help to answer the question
about whether the turbulence inside a core-collapse supernova follows a
Kolmogorov spectrum.

###### Acknowledgements.

We would like to thank Basudeb Dasgupta, Amol Dighe, Shigeo S. Kimura, Kohta
Murase, and George Zahariade for helpful discussions and invaluable
suggestions. M. M. wishes to thank the Astronomical Institute at Tohoku
University, the Yukawa Institute for Theoretical Physics (YITP), Kyoto
University, and the Institute for Advanced Study (IAS), Princeton, for their
hospitality where major parts of this work were completed. M. M. is supported
by NSF Grant No. AST-2108466. M. M. also acknowledges support from the
Institute for Gravitation and the Cosmos (IGC) Postdoctoral Fellowship.

##  Appendix A Algorithm

In this appendix, we outline the algorithm used for calculating the neutrino
survival probabilities. Computing the spatial evolution of $\bm{\Psi}_{e}$
from Eq. (7) is enough to give us the survival and disappearance probabilities
while the neutrinos propagate through some arbitrary matter profile. To do
this we implement the following:

  * •

Initial conditions: $L$, $n$ which specify $\Delta x$, $\theta$, $\Delta
m^{2}$, $E$, and the initial wavefunction in the flavor basis
$\bm{\Psi}_{e}(x_{0})$.

  * •

The information about the given matter density profile is taken into account
through Eq. (5).

  * •

We can then calculate $\theta_{M,i}$ and $\Delta m_{M,i}$ at the $i-$th slab
using Eqs. (10) and (4) respectively.

  * •

Now, using Eqs. (8) and (9) we can evaluate the term in the square brackets
which gives us the wavefunction $\bm{\Psi}(x_{i})$ at the $i-$th slab, where,
$i$ spans from $1,\dots,n$.

  * •

The survival or disappearance probability at every slab $i$ can then be
calculated from Eq. (12). The mass eigenstate probabilities can be obtained by
first getting the wavefunction in the mass basis using Eq. (11) and then
calculating the probability.

## References

  * Hirata  _et al._ [1987] K. Hirata _et al._ (Kamiokande-II), Observation of a Neutrino Burst from the Supernova SN 1987a, Phys. Rev. Lett. 58, 1490 (1987).
  * Hirata  _et al._ [1988] K. S. Hirata _et al._ , Observation in the Kamiokande-II Detector of the Neutrino Burst from Supernova SN 1987a, Phys. Rev. D 38, 448 (1988).
  * Bionta  _et al._ [1987] R. M. Bionta _et al._ , Observation of a Neutrino Burst in Coincidence with Supernova SN 1987a in the Large Magellanic Cloud, Phys. Rev. Lett. 58, 1494 (1987).
  * Alekseev  _et al._ [1987] E. N. Alekseev, L. N. Alekseeva, V. I. Volchenko, and I. V. Krivosheina, Possible Detection of a Neutrino Signal on 23 February 1987 at the Baksan Underground Scintillation Telescope of the Institute of Nuclear Research, JETP Lett. 45, 589 (1987).
  * Duan  _et al._ [2006] H. Duan, G. M. Fuller, and Y.-Z. Qian, Collective neutrino flavor transformation in supernovae, Phys. Rev. D 74, 123004 (2006), arXiv:astro-ph/0511275 .
  * Hannestad  _et al._ [2006] S. Hannestad, G. G. Raffelt, G. Sigl, and Y. Y. Y. Wong, Self-induced conversion in dense neutrino gases: Pendulum in flavour space, Phys. Rev. D74, 105010 (2006), [Erratum: Phys. Rev.D76,029901(2007)], arXiv:astro-ph/0608695 [astro-ph] .
  * Duan  _et al._ [2007] H. Duan, G. M. Fuller, J. Carlson, and Y.-Z. Qian, Analysis of Collective Neutrino Flavor Transformation in Supernovae, Phys. Rev. D 75, 125005 (2007), arXiv:astro-ph/0703776 .
  * Raffelt [2011] G. G. Raffelt, $n$-mode coherence in collective neutrino oscillations, Phys. Rev. D 83, 105022 (2011).
  * Fogli  _et al._ [2007] G. Fogli, E. Lisi, A. Marrone, and A. Mirizzi, Collective neutrino flavor transitions in supernovae and the role of trajectory averaging, Journal of Cosmology and Astroparticle Physics 2007 (12), 010.
  * Raffelt and Sigl [2007] G. G. Raffelt and G. Sigl, Self-induced decoherence in dense neutrino gases, Phys. Rev. D 75, 083002 (2007), arXiv:hep-ph/0701182 .
  * Esteban-Pretel  _et al._ [2007] A. Esteban-Pretel, S. Pastor, R. Tomas, G. G. Raffelt, and G. Sigl, Decoherence in supernova neutrino transformations suppressed by deleptonization, Phys. Rev. D 76, 125018 (2007), arXiv:0706.2498 [astro-ph] .
  * Dasgupta and Dighe [2008] B. Dasgupta and A. Dighe, Collective three-flavor oscillations of supernova neutrinos, Phys. Rev. D 77, 113002 (2008), arXiv:0712.3798 [hep-ph] .
  * Dasgupta  _et al._ [2008] B. Dasgupta, A. Dighe, and A. Mirizzi, Identifying neutrino mass hierarchy at extremely small theta(13) through Earth matter effects in a supernova signal, Phys. Rev. Lett. 101, 171801 (2008), arXiv:0802.1481 [hep-ph] .
  * Dasgupta  _et al._ [2010] B. Dasgupta, A. Mirizzi, I. Tamborra, and R. Tomas, Neutrino mass hierarchy and three-flavor spectral splits of supernova neutrinos, Phys. Rev. D 81, 093008 (2010), arXiv:1002.2943 [hep-ph] .
  * Raffelt and Smirnov [2007] G. G. Raffelt and A. Yu. Smirnov, Self-induced spectral splits in supernova neutrino fluxes, Phys. Rev. D76, 081301 (2007), [Erratum: Phys. Rev.D77,029903(2008)], arXiv:0705.1830 [hep-ph] .
  * Sarikas  _et al._ [2012] S. Sarikas, G. G. Raffelt, L. Hudepohl, and H.-T. Janka, Suppression of Self-Induced Flavor Conversion in the Supernova Accretion Phase, Phys. Rev. Lett. 108, 061101 (2012), arXiv:1109.3601 [astro-ph.SR] .
  * Banerjee  _et al._ [2011] A. Banerjee, A. Dighe, and G. G. Raffelt, Linearized flavor-stability analysis of dense neutrino streams, Phys. Rev. D84, 053013 (2011), arXiv:1107.2308 [hep-ph] .
  * Raffelt  _et al._ [2013] G. Raffelt, S. Sarikas, and D. de Sousa Seixas, Axial Symmetry Breaking in Self-Induced Flavor Conversion of Supernova Neutrino Fluxes, Phys. Rev. Lett. 111, 091101 (2013), [Erratum: Phys.Rev.Lett. 113, 239903 (2014)], arXiv:1305.7140 [hep-ph] .
  * Chakraborty  _et al._ [2016a] S. Chakraborty, R. S. Hansen, I. Izaguirre, and G. Raffelt, Self-induced flavor conversion of supernova neutrinos on small scales, JCAP 01, 028, arXiv:1507.07569 [hep-ph] .
  * Mirizzi and Serpico [2012] A. Mirizzi and P. D. Serpico, Flavor Stability Analysis of Dense Supernova Neutrinos with Flavor-Dependent Angular Distributions, Phys. Rev. D 86, 085010 (2012), arXiv:1208.0157 [hep-ph] .
  * Capozzi  _et al._ [2016] F. Capozzi, B. Dasgupta, and A. Mirizzi, Self-induced temporal instability from a neutrino antenna, JCAP 1604 (04), 043, arXiv:1603.03288 [hep-ph] .
  * Das  _et al._ [2017] A. Das, A. Dighe, and M. Sen, New effects of non-standard self-interactions of neutrinos in a supernova, JCAP 05, 051, arXiv:1705.00468 [hep-ph] .
  * Airen  _et al._ [2018] S. Airen, F. Capozzi, S. Chakraborty, B. Dasgupta, G. Raffelt, and T. Stirner, Normal-mode Analysis for Collective Neutrino Oscillations, JCAP 12, 019, arXiv:1809.09137 [hep-ph] .
  * Stapleford  _et al._ [2020] C. J. Stapleford, C. Fröhlich, and J. P. Kneller, Coupling Neutrino Oscillations and Simulations of Core-Collapse Supernovae, Phys. Rev. D 102, 081301 (2020), arXiv:1910.04172 [astro-ph.HE] .
  * Sawyer [2016] R. F. Sawyer, Neutrino cloud instabilities just above the neutrino sphere of a supernova, Phys. Rev. Lett. 116, 081101 (2016), arXiv:1509.03323 [astro-ph.HE] .
  * Chakraborty  _et al._ [2016b] S. Chakraborty, R. S. Hansen, I. Izaguirre, and G. G. Raffelt, Self-induced neutrino flavor conversion without flavor mixing, JCAP 1603 (03), 042, arXiv:1602.00698 [hep-ph] .
  * Dasgupta  _et al._ [2017] B. Dasgupta, A. Mirizzi, and M. Sen, Fast neutrino flavor conversions near the supernova core with realistic flavor-dependent angular distributions, JCAP 1702 (02), 019, arXiv:1609.00528 [hep-ph] .
  * Sawyer [2005] R. F. Sawyer, Speed-up of neutrino transformations in a supernova environment, Phys. Rev. D72, 045003 (2005), arXiv:hep-ph/0503013 [hep-ph] .
  * Sawyer [2009] R. F. Sawyer, The multi-angle instability in dense neutrino systems, Phys. Rev. D79, 105003 (2009), arXiv:0803.4319 [astro-ph] .
  * Izaguirre  _et al._ [2017] I. Izaguirre, G. G. Raffelt, and I. Tamborra, Fast Pairwise Conversion of Supernova Neutrinos: A Dispersion-Relation Approach, Phys. Rev. Lett. 118, 021101 (2017), arXiv:1610.01612 [hep-ph] .
  * Capozzi  _et al._ [2017] F. Capozzi, B. Dasgupta, E. Lisi, A. Marrone, and A. Mirizzi, Fast flavor conversions of supernova neutrinos: Classifying instabilities via dispersion relations, Phys. Rev. D96, 043016 (2017), arXiv:1706.03360 [hep-ph] .
  * Dighe and Sen [2018] A. Dighe and M. Sen, Nonstandard neutrino self-interactions in a supernova and fast flavor conversions, Phys. Rev. D 97, 043011 (2018), arXiv:1709.06858 [hep-ph] .
  * Dasgupta and Sen [2018] B. Dasgupta and M. Sen, Fast Neutrino Flavor Conversion as Oscillations in a Quartic Potential, Phys. Rev. D97, 023017 (2018), arXiv:1709.08671 [hep-ph] .
  * Dasgupta  _et al._ [2018] B. Dasgupta, A. Mirizzi, and M. Sen, Simple method of diagnosing fast flavor conversions of supernova neutrinos, Phys. Rev. D98, 103001 (2018), arXiv:1807.03322 [hep-ph] .
  * Abbar  _et al._ [2018] S. Abbar, H. Duan, K. Sumiyoshi, T. Takiwaki, and M. C. Volpe, On the occurrence of fast neutrino flavor conversions in multidimensional supernova models, (2018), arXiv:1812.06883 [astro-ph.HE] .
  * Azari  _et al._ [2019] M. D. Azari, S. Yamada, T. Morinaga, W. Iwakami, H. Nagakura, and K. Sumiyoshi, Linear Analysis of Fast-Pairwise Collective Neutrino Oscillations in Core-Collapse Supernovae based on the Results of Boltzmann Simulations, (2019), arXiv:1902.07467 [astro-ph.HE] .
  * Johns  _et al._ [2020a] L. Johns, H. Nagakura, G. M. Fuller, and A. Burrows, Neutrino oscillations in supernovae: angular moments and fast instabilities, Phys. Rev. D 101, 043009 (2020a), arXiv:1910.05682 [hep-ph] .
  * Glas  _et al._ [2020] R. Glas, H. T. Janka, F. Capozzi, M. Sen, B. Dasgupta, A. Mirizzi, and G. Sigl, Fast Neutrino Flavor Instability in the Neutron-star Convection Layer of Three-dimensional Supernova Models, Phys. Rev. D 101, 063001 (2020), arXiv:1912.00274 [astro-ph.HE] .
  * Shalgar  _et al._ [2019] S. Shalgar, I. Padilla-Gay, and I. Tamborra, Neutrino propagation hinders fast pairwise flavor conversions, (2019), arXiv:1911.09110 [astro-ph.HE] .
  * Abbar [2020] S. Abbar, Searching for Fast Neutrino Flavor Conversion Modes in Core-collapse Supernova Simulations, JCAP 05, 027, arXiv:2003.00969 [astro-ph.HE] .
  * Bhattacharyya and Dasgupta [2020] S. Bhattacharyya and B. Dasgupta, Fast Neutrino Flavor Conversion at Late Time, (2020), arXiv:2005.00459 [hep-ph] .
  * Bhattacharyya and Dasgupta [2021a] S. Bhattacharyya and B. Dasgupta, Fast Flavor Depolarization of Supernova Neutrinos, Phys. Rev. Lett. 126, 061302 (2021a), arXiv:2009.03337 [hep-ph] .
  * Li and Siegel [2021] X. Li and D. M. Siegel, Neutrino Fast Flavor Conversions in Neutron-Star Postmerger Accretion Disks, Phys. Rev. Lett. 126, 251101 (2021), arXiv:2103.02616 [astro-ph.HE] .
  * Padilla-Gay  _et al._ [2022] I. Padilla-Gay, I. Tamborra, and G. G. Raffelt, Neutrino Flavor Pendulum Reloaded: The Case of Fast Pairwise Conversion, Phys. Rev. Lett. 128, 121102 (2022), arXiv:2109.14627 [astro-ph.HE] .
  * Wu  _et al._ [2021] M.-R. Wu, M. George, C.-Y. Lin, and Z. Xiong, Collective fast neutrino flavor conversions in a 1D box: Initial conditions and long-term evolution, Phys. Rev. D 104, 103003 (2021), arXiv:2108.09886 [hep-ph] .
  * Richers  _et al._ [2021a] S. Richers, D. E. Willcox, N. M. Ford, and A. Myers, Particle-in-cell Simulation of the Neutrino Fast Flavor Instability, Phys. Rev. D 103, 083013 (2021a), arXiv:2101.02745 [astro-ph.HE] .
  * Richers  _et al._ [2021b] S. Richers, D. Willcox, and N. Ford, Neutrino fast flavor instability in three dimensions, Phys. Rev. D 104, 103023 (2021b), arXiv:2109.08631 [astro-ph.HE] .
  * Martin  _et al._ [2020] J. D. Martin, C. Yi, and H. Duan, Dynamic fast flavor oscillation waves in dense neutrino gases, Phys. Lett. B 800, 135088 (2020), arXiv:1909.05225 [hep-ph] .
  * Abbar and Volpe [2019] S. Abbar and M. C. Volpe, On Fast Neutrino Flavor Conversion Modes in the Nonlinear Regime, Phys. Lett. B790, 545 (2019), arXiv:1811.04215 [astro-ph.HE] .
  * Johns  _et al._ [2020b] L. Johns, H. Nagakura, G. M. Fuller, and A. Burrows, Fast oscillations, collisionless relaxation, and spurious evolution of supernova neutrino flavor, Phys. Rev. D 102, 103017 (2020b), arXiv:2009.09024 [hep-ph] .
  * Sigl [2022] G. Sigl, Simulations of fast neutrino flavor conversions with interactions in inhomogeneous media, Phys. Rev. D 105, 043005 (2022), arXiv:2109.00091 [hep-ph] .
  * Xiong and Qian [2021] Z. Xiong and Y.-Z. Qian, Stationary solutions for fast flavor oscillations of a homogeneous dense neutrino gas, Phys. Lett. B 820, 136550 (2021), arXiv:2104.05618 [astro-ph.HE] .
  * Abbar and Capozzi [2022] S. Abbar and F. Capozzi, Suppression of fast neutrino flavor conversions occurring at large distances in core-collapse supernovae, JCAP 03 (03), 051, arXiv:2111.14880 [astro-ph.HE] .
  * Delfan Azari  _et al._ [2020] M. Delfan Azari, S. Yamada, T. Morinaga, H. Nagakura, S. Furusawa, A. Harada, H. Okawa, W. Iwakami, and K. Sumiyoshi, Fast collective neutrino oscillations inside the neutrino sphere in core-collapse supernovae, Phys. Rev. D 101, 023018 (2020), arXiv:1910.06176 [astro-ph.HE] .
  * Chakraborty and Chakraborty [2020] M. Chakraborty and S. Chakraborty, Three flavor neutrino conversions in supernovae: slow & fast instabilities, JCAP 2001 (01), 005, arXiv:1909.10420 [hep-ph] .
  * Capozzi  _et al._ [2020] F. Capozzi, M. Chakraborty, S. Chakraborty, and M. Sen, Fast flavor conversions in supernovae: the rise of mu-tau neutrinos, Phys. Rev. Lett. 125, 251801 (2020), arXiv:2005.14204 [hep-ph] .
  * Bhattacharyya and Dasgupta [2021b] S. Bhattacharyya and B. Dasgupta, Fast flavor oscillations of astrophysical neutrinos with 1, 2, …, $\infty$ crossings, JCAP 07, 023, arXiv:2101.01226 [hep-ph] .
  * Nagakura and Johns [2021] H. Nagakura and L. Johns, New method for detecting fast neutrino flavor conversions in core-collapse supernova models with two-moment neutrino transport, Phys. Rev. D 104, 063014 (2021), arXiv:2106.02650 [astro-ph.HE] .
  * Nagakura  _et al._ [2021] H. Nagakura, L. Johns, A. Burrows, and G. M. Fuller, Where, when, and why: Occurrence of fast-pairwise collective neutrino oscillation in three-dimensional core-collapse supernova models, Phys. Rev. D 104, 083025 (2021), arXiv:2108.07281 [astro-ph.HE] .
  * Dasgupta [2022] B. Dasgupta, Collective Neutrino Flavor Instability Requires a Crossing, Phys. Rev. Lett. 128, 081102 (2022), arXiv:2110.00192 [hep-ph] .
  * Shalgar and Tamborra [2021] S. Shalgar and I. Tamborra, Three flavor revolution in fast pairwise neutrino conversion, Phys. Rev. D 104, 023011 (2021), arXiv:2103.12743 [hep-ph] .
  * Bhattacharyya and Dasgupta [2022] S. Bhattacharyya and B. Dasgupta, Elaborating the Ultimate Fate of Fast Collective Neutrino Flavor Oscillations, (2022), arXiv:2205.05129 [hep-ph] .
  * Wu  _et al._ [2017] M.-R. Wu, I. Tamborra, O. Just, and H.-T. Janka, Imprints of neutrino-pair flavor conversions on nucleosynthesis in ejecta from neutron-star merger remnants, Phys. Rev. D 96, 123015 (2017), arXiv:1711.00477 [astro-ph.HE] .
  * Zaizen  _et al._ [2020] M. Zaizen, J. F. Cherry, T. Takiwaki, S. Horiuchi, K. Kotake, H. Umeda, and T. Yoshida, Neutrino halo effect on collective neutrino oscillation in iron core-collapse supernova model of a 9.6 $M_{\odot}$ star, JCAP 06, 011, arXiv:1908.10594 [astro-ph.HE] .
  * Nagakura  _et al._ [2019] H. Nagakura, T. Morinaga, C. Kato, and S. Yamada, Fast-pairwise collective neutrino oscillations associated with asymmetric neutrino emissions in core-collapse supernova 10.3847/1538-4357/ab4cf2 (2019), arXiv:1910.04288 [astro-ph.HE] .
  * Duan  _et al._ [2010] H. Duan, G. M. Fuller, and Y.-Z. Qian, Collective neutrino oscillations, Ann. Rev. Nucl. Part. Sci. 60, 569 (2010), arXiv:1001.2799 [hep-ph] .
  * Mirizzi  _et al._ [2016] A. Mirizzi, I. Tamborra, H.-T. Janka, N. Saviano, K. Scholberg, R. Bollig, L. Hüdepohl, and S. Chakraborty, Supernova Neutrinos: Production, Oscillations and Detection, Riv. Nuovo Cim. 39, 1 (2016), arXiv:1508.00785 [astro-ph.HE] .
  * Tamborra and Shalgar [2021] I. Tamborra and S. Shalgar, New Developments in Flavor Evolution of a Dense Neutrino Gas, Ann. Rev. Nucl. Part. Sci. 71, 165 (2021), arXiv:2011.01948 [astro-ph.HE] .
  * Richers and Sen [2022] S. Richers and M. Sen, Fast Flavor Transformations, in  _Handbook of Nuclear Physics_, edited by I. Tanihata, H. Toki, and T. Kajino (2022) pp. 1–17, arXiv:2207.03561 [astro-ph.HE] .
  * Mikheev and Smirnov [1985] S. P. Mikheev and A. Yu. Smirnov, Resonance Amplification of Oscillations in Matter and Spectroscopy of Solar Neutrinos, Sov. J. Nucl. Phys. 42, 913 (1985), [,305(1986)].
  * Wolfenstein [1978] L. Wolfenstein, Neutrino oscillations in matter, Phys. Rev. D 17, 2369 (1978).
  * Schirato and Fuller [2002] R. C. Schirato and G. M. Fuller, Connection between supernova shocks, flavor transformation, and the neutrino signal, (2002), arXiv:astro-ph/0205390 [astro-ph] .
  * Takahashi  _et al._ [2003] K. Takahashi, K. Sato, H. E. Dalhed, and J. R. Wilson, Shock propagation and neutrino oscillation in supernova, Astropart. Phys. 20, 189 (2003), arXiv:astro-ph/0212195 .
  * Lunardini and Smirnov [2003] C. Lunardini and A. Y. Smirnov, Probing the neutrino mass hierarchy and the 13 mixing with supernovae, JCAP 06, 009, arXiv:hep-ph/0302033 .
  * Fogli  _et al._ [2003] G. L. Fogli, E. Lisi, D. Montanino, and A. Mirizzi, Analysis of energy and time dependence of supernova shock effects on neutrino crossing probabilities, Phys. Rev. D68, 033005 (2003), arXiv:hep-ph/0304056 [hep-ph] .
  * Fogli  _et al._ [2005] G. L. Fogli, E. Lisi, A. Mirizzi, and D. Montanino, Probing supernova shock waves and neutrino flavor transitions in next-generation water-Cerenkov detectors, JCAP 0504, 002, arXiv:hep-ph/0412046 [hep-ph] .
  * Tomas  _et al._ [2004] R. Tomas, M. Kachelriess, G. Raffelt, A. Dighe, H. T. Janka, and L. Scheck, Neutrino signatures of supernova shock and reverse shock propagation, JCAP 09, 015, arXiv:astro-ph/0407132 .
  * Dasgupta and Dighe [2007] B. Dasgupta and A. Dighe, Phase effects in neutrino conversions during a supernova shock wave, Phys. Rev. D 75, 093002 (2007), arXiv:hep-ph/0510219 .
  * Galais  _et al._ [2010] S. Galais, J. Kneller, C. Volpe, and J. Gava, Shockwaves in Supernovae: New Implications on the Diffuse Supernova Neutrino Background, Phys. Rev. D81, 053002 (2010), arXiv:0906.5294 [hep-ph] .
  * Friedland and Mukhopadhyay [2022] A. Friedland and P. Mukhopadhyay, Near-critical supernova outflows and their neutrino signatures, Phys. Lett. B 834, 137403 (2022), arXiv:2009.10059 [astro-ph.HE] .
  * Radice  _et al._ [2018] D. Radice, E. Abdikamalov, C. D. Ott, P. Mösta, S. M. Couch, and L. F. Roberts, Turbulence in Core-Collapse Supernovae, J. Phys. G 45, 053003 (2018), arXiv:1710.01282 [astro-ph.HE] .
  * Fogli  _et al._ [2006] G. L. Fogli, E. Lisi, A. Mirizzi, and D. Montanino, Damping of supernova neutrino transitions in stochastic shock-wave density profiles, JCAP 06, 012, arXiv:hep-ph/0603033 .
  * Friedland and Gruzinov [2006] A. Friedland and A. Gruzinov, Neutrino signatures of supernova turbulence, arXiv preprint astro-ph/0607244 (2006), astro-ph/0607244 .
  * Kneller and Volpe [2010] J. P. Kneller and C. Volpe, Turbulence effects on supernova neutrinos, Phys. Rev. D 82, 123004 (2010), arXiv:1006.0913 [hep-ph] .
  * Borriello  _et al._ [2014] E. Borriello, S. Chakraborty, H.-T. Janka, E. Lisi, and A. Mirizzi, Turbulence patterns and neutrino flavor transitions in high-resolution supernova models, JCAP 11, 030, arXiv:1310.7488 [astro-ph.SR] .
  * Lund and Kneller [2013] T. Lund and J. P. Kneller, Combining collective, MSW, and turbulence effects in supernova neutrino flavor evolution, Phys. Rev. D 88, 023008 (2013), arXiv:1304.6372 [astro-ph.HE] .
  * Kneller and Kabadi [2015] J. P. Kneller and N. V. Kabadi, Sensitivity of neutrinos to the supernova turbulence power spectrum: Point source statistics, Phys. Rev. D 92, 013009 (2015), arXiv:1410.5698 [hep-ph] .
  * Patton  _et al._ [2015] K. M. Patton, J. P. Kneller, and G. C. McLaughlin, Stimulated neutrino transformation through turbulence on a changing density profile and application to supernovae, Phys. Rev. D 91, 025001 (2015), arXiv:1407.7835 [hep-ph] .
  * Yang and Kneller [2018] Y. Yang and J. P. Kneller, Neutrino Flavour Evolution Through Fluctuating Matter, J. Phys. G 45, 045201 (2018), arXiv:1510.01998 [quant-ph] .
  * Kneller and de los Reyes [2017] J. P. Kneller and M. de los Reyes, The effect of core-collapse supernova accretion phase turbulence on neutrino flavor evolution, J. Phys. G 44, 084008 (2017), arXiv:1702.06951 [astro-ph.HE] .
  * Abbar [2021] S. Abbar, Turbulence Fingerprint on Collective Oscillations of Supernova Neutrinos, Phys. Rev. D 103, 045014 (2021), arXiv:2007.13655 [astro-ph.HE] .
  * Dolence  _et al._ [2013] J. C. Dolence, A. Burrows, J. W. Murphy, and J. Nordhaus, Dimensional Dependence of the Hydrodynamics of Core-Collapse Supernovae, Astrophys. J. 765, 110 (2013), arXiv:1210.5241 [astro-ph.SR] .
  * Couch and Ott [2015] S. M. Couch and C. D. Ott, The Role of Turbulence in Neutrino-Driven Core-Collapse Supernova Explosions, Astrophys. J. 799, 5 (2015), arXiv:1408.1399 [astro-ph.HE] .
  * Botella  _et al._ [1987] F. J. Botella, C. S. Lim, and W. J. Marciano, Radiative corrections to neutrino indices of refraction, Phys. Rev. D 35, 896 (1987).
  * Akhmedov  _et al._ [2002] E. K. Akhmedov, C. Lunardini, and A. Y. Smirnov, Supernova neutrinos: Difference of muon-neutrino - tau-neutrino fluxes and conversion effects, Nucl. Phys. B 643, 339 (2002), arXiv:hep-ph/0204091 .
  * Mirizzi  _et al._ [2009] A. Mirizzi, S. Pozzorini, G. G. Raffelt, and P. D. Serpico, Flavour-dependent radiative correction to neutrino-neutrino refraction, JHEP 10, 020, arXiv:0907.3674 [hep-ph] .
  * Pantaleone [1995] J. T. Pantaleone, Neutrino flavor evolution near a supernova’s core, Phys. Lett. B 342, 250 (1995), arXiv:astro-ph/9405008 .
  * Giunti and Kim [2007] C. Giunti and C. W. Kim,  _Fundamentals of Neutrino Physics and Astrophysics_ (2007).
  * Majda and Kramer [1999] A. J. Majda and P. R. Kramer, Simplified models for turbulent diffusion: Theory, numerical modelling, and physical phenomena, Physics Reports 314, 237 (1999).
  * Keil  _et al._ [2003] M. T. Keil, G. G. Raffelt, and H.-T. Janka, Monte Carlo study of supernova neutrino spectra formation, Astrophys. J. 590, 971 (2003), arXiv:astro-ph/0208035 .
  * Ohlsson [2001] T. Ohlsson, Equivalence between neutrino oscillations and neutrino decoherence, Phys. Lett. B 502, 159 (2001), arXiv:hep-ph/0012272 .
  * Abi  _et al._ [2020] B. Abi _et al._ (DUNE), Deep Underground Neutrino Experiment (DUNE), Far Detector Technical Design Report, Volume I Introduction to DUNE, JINST 15 (08), T08008, arXiv:2002.02967 [physics.ins-det] .
  * Jana  _et al._ [2022] S. Jana, Y. P. Porto-Silva, and M. Sen, Exploiting a future galactic supernova to probe neutrino magnetic moments, JCAP 09, 079, arXiv:2203.01950 [hep-ph] .
  * Bian  _et al._ [2022] J. Bian _et al._ (Hyper-Kamiokande), Hyper-Kamiokande Experiment: A Snowmass White Paper, in  _Snowmass 2021_ (2022) arXiv:2203.02029 [hep-ex] .
  * Strumia and Vissani [2003] A. Strumia and F. Vissani, Precise quasielastic neutrino/nucleon cross-section, Phys. Lett. B 564, 42 (2003), arXiv:astro-ph/0302055 .
  * Mukhopadhyay  _et al._ [2020] M. Mukhopadhyay, C. Lunardini, F. X. Timmes, and K. Zuber, Presupernova neutrinos: directional sensitivity and prospects for progenitor identification, Astrophys. J. 899, 153 (2020), arXiv:2004.02045 [astro-ph.HE] .
  * Healy  _et al._ [2023] S. Healy, S. Horiuchi, M. Colomer Molla, D. Milisavljevic, J. Tseng, F. Bergin, K. Weil, and M. Tanaka, Red Supergiant Candidates for Multimessenger Monitoring of the Next Galactic Supernova, (2023), arXiv:2307.08785 [astro-ph.SR] .
  * Askins  _et al._ [2020] M. Askins _et al._ (Theia), THEIA: an advanced optical neutrino detector, Eur. Phys. J. C 80, 416 (2020), arXiv:1911.03501 [physics.ins-det] .