# Evaluating approximate asymptotic distributions for fast neutrino flavor
conversions in a periodic 1D box

Zewei Xiong  0000-0002-2385-6771<EMAIL_ADDRESS>GSI Helmholtzzentrum für
Schwerionenforschung, Planckstraße 1, 64291 Darmstadt, Germany  Meng-Ru Wu
0000-0003-4960-8706 Institute of Physics, Academia Sinica, Taipei 11529,
Taiwan  Institute of Astronomy and Astrophysics, Academia Sinica, Taipei
10617, Taiwan  Physics Division, National Center for Theoretical Sciences,
Taipei 10617, Taiwan  Sajad Abbar  0000-0001-8276-997X Max-Planck-Institut für
Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, D-80805 München,
Germany  Soumya Bhattacharyya  0000-0001-5331-4597 Institute of Physics,
Academia Sinica, Taipei 11529, Taiwan  Manu George  0000-0002-8974-5986
Institute of Physics, Academia Sinica, Taipei 11529, Taiwan  Chun-Yu Lin
0000-0002-7489-7418 National Center for High-performance Computing, National
Applied Research Laboratories, Hsinchu 30076, Taiwan

###### Abstract

The fast flavor conversions (FFCs) of neutrinos generally exist in core-
collapse supernovae and binary neutron-star merger remnants, and can
significantly change the flavor composition and affect the dynamics and
nucleosynthesis processes. Several analytical prescriptions were proposed
recently to approximately explain or predict the asymptotic outcome of FFCs
for systems with different initial or boundary conditions, with the aim for
providing better understandings of FFCs and for practical implementation of
FFCs in hydrodynamic modeling. In this work, we obtain the asymptotic survival
probability distributions of FFCs in a survey over thousands of randomly
sampled initial angular distributions by means of numerical simulations in
one-dimensional boxes with the periodic boundary condition. We also propose
improved prescriptions that guarantee the continuity of the angular
distributions after FFCs. Detailed comparisons and evaluation of all these
prescriptions with our numerical survey results are performed. The survey
dataset is made publicly available to inspire the exploration and design for
more effective methods applicable to realistic hydrodynamic simulations.

##  I Introduction

A great amount of neutrinos are produced in dense astrophysical environments
such as core-collapse supernovae (CCSNe) and the remnants of binary neutron
star mergers (BNSMs). Their fluxes are so intense that the coherent forward
scattering among those neutrinos can lead to significant changes in their
flavor content through the collective flavor instabilities, particularly the
fast flavor conversion (FFC; see e.g., [1, 2, 3, 4, 5, 6] for reviews) at the
vicinity of the cores of CCSNe and accretion disks of BNSMs, which can play
important roles in the dynamics and the nucleosynthesis of those environments
[7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].

FFC happens when the angular distribution of the neutrino lepton number
between any two distinct flavors takes both positive and negative values
simultaneously [18] with the transition points often dubbed as “zero-
crossings”. Given that the multi-dimensional simulations usually provide only
the angular moments instead of the full distributional information, various
approximate or parametric methods were adopted [19, 20, 21, 22, 23, 24, 25,
26] and found the existence of FFC near or even inside the neutrinosphere in
CCSNe [27, 28, 29, 30, 31, 32, 22, 33, 34, 35, 36] as well as ubiquitously in
the post-merger remnants of BNSMs [37, 25].

The spatial and temporal scales associated with the development of the fast
flavor instability can be in sub-centimeter and sub-nanosecond, much shorter
than the typical scales considered in the hydrodynamical simulations for CCSNe
and BNSMs. This naturally brings up a challenge to incorporate FFC into the
hydrodynamical simulations. To overcome this challenge, one possible solution
is to decompose this problem into two scale hierarchies: performing the local
dynamical simulations at a small scale and summarizing with useful parametric
prescriptions that can be applied to the hydrodynamic simulation more
efficiently.

The outcome of FFCs has been extensively studied based on local dynamical
simulations in tiny boxes with a periodic boundary condition [38, 39, 40, 41,
42, 43, 44, 45, 46, 47, 48, 49] and may be affected by adopting a different
boundary condition [50]. These studies suggest that the flavor conversions
undergo the kinematic decoherence in general [51, 20, 52, 40, 53] and reach
asymptotically to quasi-stationary states achieving complete or partially
flavor equilibration as allowed by the conservation of neutrino lepton number
[51, 54], when coarse-grained over the box size.

An early attempt to obtain an analytical description on the asymptotic
distribution was made in a homogeneous neutrino gas [55], and a growing number
of schemes have been recently proposed for the local simulations allowing the
advection of neutrinos inside the box [40, 45, 49]. Those methods often
contain an artificial discontinuity for the survival probability distribution
near the zero-crossing, which is further passed to the asymptotic angular
distributions of neutrino number densities. In this paper, we propose new
prescriptions by imposing a continuous transition at the zero-crossing and
performed numerical FFC simulations for $\sim\mathcal{O}(8000)$ systems with
randomly sampled initial angular distributions that covers the major parameter
space for FFCs to occur near the neutrino decoupling regions. For each
prescription, we compare the predicted asymptotic angular distributions after
FFC with those obtained by numerical simulations, and evaluate in detail their
performance using different metrics of errors. Our improved prescriptions of
the asymptotic distributions not only predict the angular moments in the
asymptotic state more accurately, but also can be directly implemented in the
discrete-ordinate neutrino transport with the advection on a large scale [56,
57, 58, 59, 60].

We describe the simulation setup over thousands of parameter sets in Sec. II.
We present various analytical prescriptions to determine the asymptotic
distributions with and without continuous transitions at the zero-crossing in
Sec. III. The results of the performance evaluation for those asymptotic
prescriptions are presented in Sec. IV. Finally, we provide our discussions
and conclusions in Sec. V. We adopt natural units ($\hbar=c=1$) throughout the
paper.

##  II Survey of simulations

###  II.1 Equation of motion

We use the code COSE$\nu$ [61] to evolve the FFCs in a similar setup of one-
dimensional (1D) box as described in Ref. [41] assuming translation symmetry
in the $x$ and $y$ directions, axial symmetry around the $z$-axis, and
periodic boundary condition in the $z$ direction. We consider in the
simulation that the oscillations start from electron flavor $\nu_{e}$
($\bar{\nu}_{e}$) initially and can be converted to one heavy-lepton flavor
$\nu_{x}$ ($\bar{\nu}_{x}$)111The effects of heavy-lepton flavor neutrinos in
the initial condition will be discussed in Sec. IV.3.. We neglect the vacuum
mixing and neutrino-matter forward scattering. Although it is reported that
the so-called collisional flavor instability induced by neutrino emission and
absorption may interplay with the FFC [62, 63, 64, 65, 66, 67, 68], we neglect
all collisional processes in the 1D-box setup that we consider. The equation
of motion (EOM) for the normalized neutrino (antineutrino) density matrix
$\varrho$ ($\bar{\varrho}$) is given by

$\displaystyle(\partial_{t}+v_{z}\partial_{z})\varrho(t,z,v_{z})$
$\displaystyle=-i[H(t,z,v_{z}),\varrho(t,z,v_{z})],$
$\displaystyle(\partial_{t}+v_{z}\partial_{z})\bar{\varrho}(t,z,v_{z})$
$\displaystyle=i[H^{*}(t,z,v_{z}),\bar{\varrho}(t,z,v_{z})],$ (1)

with the Hamiltonian of coherent forward-scattering at specific $t$ and $z$

$\displaystyle H(v_{z})=$
$\displaystyle\mu\int_{-1}^{1}dv_{z}^{\prime}(1-v_{z}v_{z}^{\prime})[g_{\nu}(v_{z}^{\prime})\varrho(v_{z}^{\prime})-g_{\bar{\nu}}(v_{z}^{\prime})\bar{\varrho}^{*}(v_{z}^{\prime})]$
(2)

where $\mu=\sqrt{2}G_{F}n_{\nu_{e}}$, $G_{F}$ is the Fermi constant,
$n_{\nu_{e}}$ is the number density for $\nu_{e}$, and $g_{\nu}$
($g_{\bar{\nu}}$) is the initial angular distribution for $\nu_{e}$
($\bar{\nu}_{e}$) as a function of the projected velocity $v_{z}$. The
distribution $g_{\nu}$ for $\nu_{e}$ is normalized with the zeroth moment
$I_{\nu}=\int_{-1}^{1}dv_{z}g_{\nu}(v_{z})=1$. The zeroth moment for
$\bar{\nu}_{e}$,
$I_{\bar{\nu}}=\int_{-1}^{1}dv_{z}g_{\bar{\nu}}(v_{z})=n_{\bar{\nu}_{e}}/n_{\nu_{e}}$,
indicates whether the condition is $\nu_{e}$\- or $\bar{\nu}_{e}$-dominant.
The angular distribution of neutrino electron lepton number ($\nu$ELN)

$G(v_{z})=g_{\nu}(v_{z})-g_{\bar{\nu}}(v_{z}),$ (3)

determines the existence of fast flavor instability.

Without explicitly including the vacuum mixing in the EOM, we trigger the FFC
by seeding random perturbations in the initial condition

$\displaystyle\varrho_{ee}(z,v_{z})=\bar{\varrho}_{ee}(z,v_{z})=$
$\displaystyle[1+\sqrt{1-\epsilon^{2}(z)}]/2,$
$\displaystyle\varrho_{xx}(z,v_{z})=\bar{\varrho}_{xx}(z,v_{z})=$
$\displaystyle[1-\sqrt{1-\epsilon^{2}(z)}]/2,$
$\displaystyle\varrho_{ex}(z,v_{z})=\bar{\varrho}_{ex}(z,v_{z})=$
$\displaystyle\epsilon(z)/2,$ (4)

where a real number $\epsilon$ is randomly assigned for each $z$ and follows a
uniform distribution between 0 and $10^{-2}$.

###  II.2 Setup and parameters

We consider two types of initial angular distributions. The first one is
described by a Gaussian function [69]

$g_{\nu(\bar{\nu})}(v_{z})\propto\exp[-(v_{z}-1)^{2}/(2\sigma^{2}_{\nu(\bar{\nu})})],$
(5)

and the second one is obtained from maximum entropy closure [25]

$g_{\nu(\bar{\nu})}(v_{z})\propto\exp[v_{z}/\sigma_{\nu(\bar{\nu})}].$ (6)

For both types, $\sigma_{\nu(\bar{\nu})}$ is a parameter associated with the
neutrino (antineutrino) flux factor, i.e., the ratio between the first and
zeroth angular moments

$F_{\nu(\bar{\nu})}=\frac{J_{\nu(\bar{\nu})}}{I_{\nu(\bar{\nu})}}$ (7)

where $J_{\nu(\bar{\nu})}=\int_{-1}^{1}dv_{z}v_{z}g_{\nu(\bar{\nu})}(v_{z})$.
For both types, since the zeroth moment of $\nu_{e}$ are normalized, the
initial angular distributions can be uniquely determined by three parameters
$I_{\bar{\nu}}$, $F_{\nu}$, and $F_{\bar{\nu}}$.

We choose these three parameters in the following way. First we randomly
assign $I_{\bar{\nu}}$ and $F_{\bar{\nu}}$ following uniform distributions
ranging from 0.5 to 1.6 and from 0.3 to 0.9, respectively. We take the lower
limit for $F_{\bar{\nu}}$ as 0.3 for our survey. This is because a smaller
flux factor implies more isotropic neutrino angular distribution generally
obtained near the neutrino optically thick region associated with higher
density and temperature. Under those conditions, the highly degenerate
electrons leads to a large neutrino chemical potential of electron flavor so
that $g_{\nu}$ dominates over $g_{\bar{\nu}}$ in the whole $v_{z}$ range,
i.e., the fast flavor instability is less likely to occur for
$F_{\bar{\nu}}\lesssim 0.3$. We also do not consider flux factor higher than
0.9 because neutrinos become highly collimated towards one direction and may
not be well captured with the angular resolution in the current setup. We then
randomly assign $F_{\nu}$ in a uniform distribution from $\sim
0.65F_{\bar{\nu}}$ to $\min(0.9,1.6F_{\bar{\nu}})$ where “min” stands for the
minimum function. This constraint is to avoid the situation where either
$\nu_{e}$ or $\bar{\nu}_{e}$ has much larger flux factor than the other, which
usually does not occur in realistic systems because the decoupling regions of
$\nu_{e}$ or $\bar{\nu}_{e}$ are not very far apart.

If a zero-crossing at $v_{c}$ where $G(v_{c})=0$ exists, we adopt this
parameter set and perform the simulation in a periodic 1D box using the
corresponding initial angular distributions. Otherwise, this parameter set is
rejected, and we continue to generate new parameters. In all simulations, the
number of spatial grids is $N_{z}$=6000. The size of the 1D box is
$L_{z}=1200\mu^{-1}$. We adopt the finite volume method as well as the seventh
order weighted essentially nonoscillatory scheme.

We repeat the same procedure above until 8000 parameter sets are adopted for
the Gaussian-type distributions with zero-crossings. In each parameter set we
take two different angular resolutions with $N_{v_{z}}=50$ and
$N_{v_{z}}=100$. For the maximum-entropy distributions we use the same
parameter sets of $I_{\bar{\nu}}$, $F_{\nu}$, and $F_{\bar{\nu}}$ as in the
Gaussian type. We further exclude those not having any zero-crossings in the
maximum-entropy type, which reduces the size of the sample to 7668 parameter
sets for this case.

###  II.3 Determination of the asymptotic distributions

During the simulation of each set of initial angular distributions, the space-
averaged survival probabilities for $\nu_{e}$ and $\bar{\nu}_{e}$ at each $t$
are defined as

$\displaystyle\langle\varrho_{ee}\rangle_{z}(v_{z})$
$\displaystyle=\frac{1}{L}\int dz\varrho_{ee}(z,v_{z}),$
$\displaystyle\langle\bar{\varrho}_{ee}\rangle_{z}(v_{z})$
$\displaystyle=\frac{1}{L}\int dz\bar{\varrho}_{ee}(z,v_{z}),$ (8)

respectively. The overall space-averaged survival probabilities are

$\displaystyle\langle
P_{ee}\rangle=\int_{-1}^{1}dv_{z}g_{\nu}(v_{z})\langle\varrho_{ee}\rangle_{z}(v_{z})\Big{/}\int_{-1}^{1}dv_{z}g_{\nu}(v_{z}),$
$\displaystyle\langle
P_{\bar{e}\bar{e}}\rangle=\int_{-1}^{1}dv_{z}g_{\bar{\nu}}(v_{z})\langle\bar{\varrho}_{ee}\rangle_{z}(v_{z})\Big{/}\int_{-1}^{1}dv_{z}g_{\bar{\nu}}(v_{z}),$
(9)

respectively.

In the presence of the initial perturbation, both $\langle P_{ee}\rangle$ and
$\langle P_{\bar{e}\bar{e}}\rangle$ start from values very close to 1 and
decrease under the fast flavor instability until reaching the first minimum
point. They bounce back but do not return to 1. Instead, they enter into a
ringdown phase with gradually damped oscillation amplitude and eventually
approach asymptotic values [see, e.g., Fig. 5(b) in Ref. [41]].

We take a practical approach to determine whether the system has reached the
asymptotic state as follows. For each simulation, we record the times when
$\langle P_{ee}\rangle$ reach the first and second minima as $t_{1}$ and
$t_{2}$, respectively and define $\Delta T=t_{2}-t_{1}$. Then, we end the
simulation at $t_{f}=t_{2}+N_{t}\Delta T$ with $N_{t}=20$ to cover roughly
$N_{t}$ more periods during the ringdown phase. Because
$\langle\varrho_{ee}\rangle_{z}(v_{z})$ and
$\langle\bar{\varrho}_{ee}\rangle_{z}(v_{z})$ may still fluctuate in time at
the end of the simulation, we compute the time-averaged survival probabilities

$\displaystyle P_{ee}(v_{z})$ $\displaystyle=\frac{1}{\Delta
T}\int_{t_{f}-\Delta T}^{t_{f}}dt\langle\varrho_{ee}\rangle_{z}(t,v_{z}),$
$\displaystyle P_{\bar{e}\bar{e}}(v_{z})$ $\displaystyle=\frac{1}{\Delta
T}\int_{t_{f}-\Delta
T}^{t_{f}}dt\langle\bar{\varrho}_{ee}\rangle_{z}(t,v_{z}),$ (10)

over the last time interval of $\Delta T$ as our final data outputs.

Since Eqs. (1) and (2) imply the relation that
$\varrho_{ee}(t,z,v_{z})=\bar{\varrho}_{ee}(t,z,v_{z})$ and
$P_{ee}(v_{z})=P_{\bar{e}\bar{e}}(v_{z})$, the time- and space-averaged
angular distributions after the FFCs can be computed as
$\tilde{g}_{\nu_{e}}(v_{z})=g_{\nu}(v_{z})P_{ee}(v_{z})$ and
$\tilde{g}_{\bar{\nu}_{e}}(v_{z})=g_{\bar{\nu}}(v_{z})P_{ee}(v_{z})$
accordingly. It follows that the zeroth and first moments for $\nu_{e}$
($\bar{\nu}_{e}$) after the FFCs are
$\tilde{I}_{\nu_{e}(\bar{\nu}_{e})}=\int_{-1}^{1}dv_{z}\tilde{g}_{\nu_{e}(\bar{\nu}_{e})}$
and
$\tilde{J}_{\nu_{e}(\bar{\nu}_{e})}=\int_{-1}^{1}dv_{z}v_{z}\tilde{g}_{\nu_{e}(\bar{\nu}_{e})}$
respectively. We store the time-averaged final distributions for the survival
probability as well as the first two angular moments for $\nu_{e}$ and
$\bar{\nu}_{e}$ described above for the entire sets with 8000 and 7668
different initial conditions for the Gaussian and maximum-entropy types,
respectively. The full dataset is available in [70].

##  III Analytical prescriptions

For both Gaussian and maximum-entropy types that we consider, the initial
$\nu$ELN distribution $G(v_{z})$ is ensured to allow at most one zero-
crossing. Thanks to this feature, we can divide the $v_{z}$ range into two
parts separated by the zero-crossing $v_{c}$. The integrals over both parts
are

$\displaystyle I_{+}$
$\displaystyle=\left|\int_{-1}^{1}dv_{z}G(v_{z})\Theta[G(v_{z})]\right|,$
$\displaystyle I_{-}$
$\displaystyle=\left|\int_{-1}^{1}dv_{z}G(v_{z})\Theta[-G(v_{z})]\right|,$
(11)

respectively, where $\Theta$ is the Heaviside theta function. For the sake of
convenience, in the rest of the paper we call the $v_{z}$ range over which the
above integral is smaller (larger) as the “small” (“large”) side, and use
$v_{z}^{<}$ ($v_{z}^{>}$) to denote that range.

Based on the observation from the numerical simulations, a complete flavor
equilibration is approximately achieved on the small side on a coarse-grained
sense. A general description on the asymptotic survival probability within
two-flavor oscillations222In the three-flavor case where $\nu_{\mu}$ and
$\nu_{\tau}$ are indistinguishable, the expression is $P_{ee}^{\rm
3f}(v_{z})=1-4[1-P_{ee}^{\rm 2f}(v_{z})]/3$. can be given as

$P_{ee}^{\rm 2f}(v_{z})=\begin{cases}\frac{1}{2}&{\rm for~{}}v_{z}^{<},\\\
P_{ee}(v_{z})&{\rm for~{}}v_{z}^{>},\end{cases}$ (12)

where the distributions on the large side $P_{ee}(v_{z})$ can be formulated by
a chosen analytical prescription. Below, based on the same assumption on the
small side, we will describe both previously formulated prescriptions and the
improved ones proposed in this paper.

###  III.1 Prescriptions with abrupt transition

It was suggested in Refs. [49, 40] to use a box-like expression, whose
spatially-averaged survival probabilities $P_{ee}(v_{z})$ is constant in
$v_{z}^{>}$, to describe the asymptotic distribution for neutrino survival
probabilities. Assuming that the small side undergoes a complete flavor
equilibration, the conservation of total $\nu$ELN requires on the large side
that

$P_{ee}(v_{z})=1-\frac{I_{<}}{2I_{>}},$ (13)

where $I_{<}=\min(I_{-},I_{+})$ and $I_{>}=\max(I_{-},I_{+})$.

Another prescription assumes that $P_{ee}(v_{z})$ is linear in $v_{z}$ on the
large side [40, 45]:

$P_{ee}(v_{z})=\frac{1}{2}+\frac{I_{>}-I_{<}}{4I_{>}}[1\mp
v_{c}\mp\frac{3}{2}(v_{c}^{2}-1)v_{z}],$ (14)

where the sign $-$ ($+$) denotes that $v_{z}=1$ is on the large (small) side.
Furthermore, one can follow the same method of Refs. [40, 45] to include the
second-order quadratic Legendre polynomial so that $P_{ee}(v_{z})$ is
quadratic in $v_{z}$. This adds an additional term to Eq. (14) and result in

$\displaystyle P_{ee}(v_{z})=\frac{1}{2}+\frac{I_{>}-I_{<}}{4I_{>}}[$
$\displaystyle 1\mp v_{c}\mp\frac{3}{2}(v_{c}^{2}-1)v_{z}$
$\displaystyle\mp\frac{5}{4}(v_{c}^{3}-v_{c})(3v_{z}^{2}-1)].$ (15)

Both linear and quadratic prescriptions ensure $P_{ee}(v_{z})\geq 1/2$ on the
large side to avoid introducing an additional zero-crossing in the asymptotic
state. However, it is important to note that Eq. (14) and Eq. (15) both do not
guarantee the conservation of total $\nu$ELN or the constraint that
$P_{ee}(v_{z})\leq 1$ on the large side.

###  III.2 Prescriptions with continuous transition

None of the above prescriptions ensures a continuous transition at the zero-
crossing $v_{c}$, which can lead to an artificial discontinuity in the final
asymptotic angular distributions $\tilde{g}_{\nu_{e}}$ and
$\tilde{g}_{\bar{\nu}_{e}}$. To avoid this, we propose a new prescription for
the large side as

$P_{ee}(v_{z})=1-\frac{1}{2}h(|v_{z}-v_{c}|/a),$ (16)

where $h(x)$ is a $v_{z}$-dependent function that monotonically decreases from
1 to 0 when $x$ increases from 0 to infinity. We try three different double-
power laws for $h(x)$ as $(x^{2}+1)^{-1/2}$, $(x^{2}+1)^{-1}$, and
$(x+1)^{-2}$, denoted as power-1/2, -1, and -2, respectively. In addition, we
take one more exponential function $h(x)=\exp(-x)$. For any choice of $h(x)$,
the coefficient $a$ can be numerically solved using the Newton-Raphson method
for the following equation

$I_{<}=\int_{v_{z}^{>}}dv_{z}G(v_{z})h(|v_{z}-v_{c}|/a),$ (17)

which can be derived based on the $\nu$ELN conservation.

Because the right-hand side of Eq. (17) is monotonic in $a$, an interpolation
method can also be used to effectively solve the coefficient $a$ in practice.
For a set of $a$, $\\{a_{i}\\}$, including several finite positive $a_{i}$
values as well as $a_{i}=0^{+}$ and $a_{i}=\infty$, the corresponding values
for
$\Gamma(a_{i})=(I_{>})^{-1}\int_{v_{z}^{>}}dv_{z}G(v_{z})h(|v_{z}-v_{c}|/a_{i})$
can be calculated covering a range from 0 to 1. For a given $I_{<}/I_{>}$, one
can find the interval defined by a pair of adjacent values $a_{i}$ and
$a_{i+1}$ where $I_{<}/I_{>}$ is sandwiched by $\Gamma(a_{i})$ and
$\Gamma(a_{i+1})$. Then, the asymptotic distribution can be interpolated as

$\displaystyle P_{ee}(v_{z})=1-\frac{1}{2}\left[\right.$ $\displaystyle\gamma
h(|v_{z}-v_{c}|/a_{i})$
$\displaystyle\left.+(1-\gamma)h(|v_{z}-v_{c}|/a_{i+1})\right]$ (18)

with $\gamma=|\Gamma(a_{i+1})-I_{<}/I_{>}|/|\Gamma(a_{i+1})-\Gamma(a_{i})|$.
We note that the total $\nu$ELN is also conserved in this practical scheme.
Although this procedure can be applied to any of the four above schemes with
continuous transitions, we demonstrate its practicability in this work by
taking $a_{i}=\\{0^{+},0.04,0.2,1,\infty\\}$ based on the power-1/2
prescription and denote this practical scheme as power-1/2-i.

We elaborate further here on the above choice of the monotonic function $h(x)$
on the large side. Based on the observation from simulation results, the
unstable eigenmode that grows the fastest in the linear regime usually has
larger amplitude at the small side in $v_{z}$. When evolving into the
nonlinear regime, it often results in more flavor conversion closer to the
small side. Since the unstable eigenmode has a continuous distribution in
$v_{z}$, this implies that for a $v_{z}$ that is farther away from the small
side, it generally experiences less flavor conversion. As a result, when the
system relaxes to the quasi-stationary state through kinematic decoherence,
the spatially averaged survival probability keeps the memory of the continuous
transition, which lead to a typically larger value of $P_{ee}(v_{z})$ closer
to the small side.

##  IV Results

In this section, we assess the performance of various analytical prescriptions
in predicting the asymptotic distributions of survival probabilities and the
relevant angular moments. We use superscripts “sim” and “pre” to distinguish
those quantities from the simulations and prescriptions, respectively. We will
compare eight prescriptions including box-like, linear, quadratic, power-1/2,
power-1/2-i, power-1, power-2, and exponential ones described in Sec. III in
the following analysis.

###  IV.1 Two representative conditions

To illustrate some general behaviors of those prescriptions, we show in Fig. 1
the angular distributions and asymptotic survival probabilities
$P_{ee}(v_{z})$ for two representative conditions obtained with simulations as
well as those from the analytical prescriptions. For $P_{ee}(v_{z})$, all
eight different prescriptions are shown in the plots while for the angular
distribution, we only show analytical results derived using the box-like [Eq.
(13), quadratic, [Eq. (15)], and the power-1/2 of Eq. (16). The first typical
condition represents a system dominated by electron neutrinos with
$(I_{\bar{\nu}},F_{\nu},F_{\bar{\nu}})\approx(0.87,0.47,0.65)$ while the
second one is dominated by antineutrinos with
$(I_{\bar{\nu}},F_{\nu},F_{\bar{\nu}})\approx(1.27,0.79,0.86)$.333Those two
conditions are provided as ID 58 and 32, respectively, in the dataset [70].
Panels (a)-(c) [(d)-(f)] show results obtained in the $\nu_{e}$
($\bar{\nu}_{e}$) dominant condition with the initial angular distributions
$g_{\nu}$ and $g_{\bar{\nu}}$ parametrized by the Gaussian-type distribution.
For the $\nu_{e}$ dominant case, we also show in Panels (g)-(i) results
obtained with $g_{\nu}$ and $g_{\bar{\nu}}$ parametrized by the maximum-
entropy distribution.

For both condition, $G(v_{z})>0$ for $v<v_{c}$ because
$F_{\nu}<F_{\bar{\nu}}$, i.e., $g_{\nu}$ is less forward-peaked than
$g_{\bar{\nu}}$. This range is the large side when $I_{\bar{\nu}}<1$
($\nu_{e}$-dominated case) and only undergoes incomplete conversions towards
flavor equilibration as shown in Figs. 1(a)-(c). When $I_{\bar{\nu}}>1$, this
range becomes the small side and reaches approximate flavor equilibration
shown in Figs. 1(d)-(f). Obviously, approximate flavor equilibration and
incomplete flavor conversions are obtained in the range $v_{z}<v_{c}$ for
cases with $I_{\bar{\nu}}<1$ and $I_{\bar{\nu}}>1$, correspondingly. In
addition, we also show that very similar results are obtained with
$N_{v_{z}}=50$ when compared to those obtained with $N_{z}=100$ in panels (b)
and (e).

(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)

Figure 1: Angular distributions (upper panel) and survival probabilities
(lower two panels) with the initial distributions parametrized by the Gaussian
type. The left (middle) panels are with smaller (larger) initial
$\bar{\nu}_{e}$ zeroth moment than $\nu_{e}$ as described in Sec. IV.1. The
right panels are with the same $\nu_{e}$-dominant moments as in the left
panels but with distributions characterized by the maximum-entropy type. The
gray vertical line marks the zero-crossing at $v_{c}$.

For the analytical prescriptions, these panels show that large differences on
the large side exist between results obtained using Eqs. (13)-(16). However,
using Eq. (16) with different double-power or exponential functions $h(x)$
generally gives rise to similar outcomes. For $P_{ee}(v_{z})$ with the
$\nu_{e}$-dominant condition ($I_{\bar{\nu}_{e}}<1$), both the box-like and
linear prescriptions contains relatively large deviations $\sim 0.1-0.3$ from
simulation outcome near the zero-crossing $v_{c}$. The quadratic prescription
matches better with the simulation result in the range of $v_{z}\lesssim 0.1$.
However, the deviation near $v_{c}$ is similarly large as with the linear
prescription because the second-order Legendre polynomial has a small
contribution near $v_{z}=\sqrt{3}/3\approx 0.57$. As mentioned already in Sec.
III, the large deviations around $v_{c}$ for are related to the inherit
discontinuities in these prescriptions.

Significant improvements are obtained near $v_{c}$ when using prescriptions
with continuous transitions. For all continuously transitioning cases with
either the double-power law or the exponential function, the survival
probabilities increase from 0.5 at the small side to larger values as $v_{z}$
decreases without any discontinuity. More specifically, the power-2 and
exponential prescriptions contain first-order discontinuity at $v_{c}$, while
the other two prescriptions are first-order continuous at $v_{c}$, which leads
to slightly more flat $P_{ee}$ at $v_{z}\lesssim v_{c}$ and therefore more
flavor conversions. All these prescriptions have small deviations of $\lesssim
0.04$ from the simulation result in the range of $0<v_{z}<v_{c}$, with both
power-1/2 and power-1/2-i schemes showing the best agreement. For $v_{z}<0$,
larger deviations up to $\sim 0.15$ appear for all of them, particularly for
the power-1 and exponential prescriptions. However, in terms of the angular
distributions, because the initial $g_{\nu(\bar{\nu})}$ are forward-peaked,
the deviations at negative $v_{z}$ only result in negligible differences in
the asymptotic angular distribution $\tilde{g}_{\nu_{e}}(v_{z})$, as shown in
Fig. 1(a). On the other hand, the abrupt transition of $P_{ee}$ at $v_{c}$ for
all discontinuous prescriptions lead to artificial peaks in
$\tilde{g}_{\nu_{e}}(v_{z})$ with an obvious discontinuity at $v_{c}$. We note
here that if one plots the oscillated $\nu$ELN distributions as in Ref. [49],
these discontinuities at $v_{c}$ will disappear because $G(v_{c})=0$ and the
deviations around $v_{c}$ will appear small. Nevertheless, the physical
angular distributions $g_{\nu}$ and $g_{\bar{\nu}}$ are generally nonzero
there so that this feature of discontinuity can hardly be avoided.

With the $\bar{\nu}_{e}$-dominant condition ($I_{\bar{\nu}}>1$), the large
side ranges from $v_{z}=v_{c}$ to $v_{z}=1$ with the zero-crossing
$v_{c}\approx 0.72$ as shown in Fig. 1(d). Similar to the $\nu_{e}$-dominant
case, taking the box-like, linear, or quadratic prescription also results in
larger deviations from simulation outcome than taking the continuous
prescriptions. Interestingly, $P_{ee}(v_{z})$ with the quadratic prescription
appears nearly linear in this case. This is because the additional quadratic
Legendre contribution is derived based on the whole $v_{z}$ range, which
results in in a large linear term compared to the quadratic term for the
narrower $v_{z}>v_{c}$ range where we apply the prescription. For all the
continuous prescriptions, the agreements in $P_{ee}$ with the simulation
results appear to be even better than the $\nu_{e}$-dominant case.

For the $\nu_{e}$-dominant case with the same parameter set
$(I_{\bar{\nu}},F_{\nu},F_{\bar{\nu}})\approx(0.87,0.47,0.65)$ but taking the
initial $g_{\nu(\bar{\nu})}$ given by the maximum-entropy distributions, Fig.
1(g)-(i) show that the resulting asymptotic distributions are qualitatively
similar to those obtained with Gaussian function discussed above. Comparing to
the Gaussian case, the zero-crossing $v_{c}$ is shifted from $\approx 0.6$ to
0.7, but the small side remains at $v_{z}>v_{c}$. Here, different analytical
prescriptions for the large side with abrupt transitions at $v_{c}$ also show
similarly large differences in $P_{ee}(v_{z})$ while those formulated with
continuous transitions result in similar $P_{ee}(v_{z})$ that match better
with the numerical result. Note that here the asymptotic $P_{ee}(v_{z})$
obtained with simulations do contain some noticeable differences from the
Gaussian case shown in panel (b). This can affect the comparison of different
analytical prescriptions to simulation outcome. For instance, the linear
prescription now performs better than the quadratic one in $v_{z}<0$ as shown
in Panel (h). Also, the region where approximate flavor equilibration is
achieved is extended to $v_{z}\approx 0.5$ below $v_{c}$, around which the
power-1 prescription fits the simulation result better.

Looking at the small side, although all schemes assume the same flavor
equilibration with $P_{ee}=0.5$ as an approximation, we observe an interesting
phenomenon that slight overconversions are possible for some parameter sets
with either types of initial distributions. For example, $P_{ee}$ can be $\sim
0.4$ for $v_{z}\gtrsim 0.8$ for the $\nu_{e}$-dominant case with the initial
maximum-entropy distribution, and for $0.5\lesssim v_{z}\lesssim 0.6$ for the
$\bar{\nu}_{e}$-dominant case with the initial Gaussian distribution,
independent of the choice of angular resolutions. The overconversions may
result from some specific unstable eigenmodes when fast instability develops
from the linear to nonlinear regime (see, e.g., [41, 45]). Although it leads
to nonzero asymptotic $\nu$ELN on the small side, it does not lead to an
additional spectrum crossing and fast flavor instability in the asymptotic
state, because the nonzero $\nu$ELN there has the same sign as in the large
side. However, the presence of the overconversions can result in systematic
biases in the predictions of the box-like prescription as well as those
continuous ones due to the imposed constraint from the conservation of
$\nu$ELN. Since there are more flavor conversion than the equilibration on the
small side due to the overconversions, the $\nu$ELN conservation then implies
that there will be also more flavor conversion on the large side obtained by
simulations than results derived with those analytical prescriptions, as shown
in Fig. 1.

###  IV.2 Overall performance

Going beyond the explicit comparisons based on only a few examples, we further
evaluate the overall performance of each analytical prescription for all
parameter sets by calculating several useful error quantities, including the
root mean square errors for $\tilde{g}_{\nu_{e}}(v_{z})$ over the entire
$v_{z}$ range as well as over the large side only, and the differences for the
first two angular moments. We write those error quantities in terms of
$\nu_{e}$ explicitly as

$\displaystyle E(\tilde{g}_{\nu_{e}})$
$\displaystyle=\frac{1}{2}\left[\int_{-1}^{1}dv_{z}\left|\tilde{g}^{\rm
pre}_{\nu_{e}}(v_{z})-\tilde{g}^{\rm
sim}_{\nu_{e}}(v_{z})\right|^{2}\right]^{1/2},$ $\displaystyle
E(\tilde{g}_{\nu_{e}}^{>})$
$\displaystyle=\frac{\left[\int_{v_{z}^{>}}dv_{z}\left|\tilde{g}^{\rm
pre}_{\nu_{e}}(v_{z})-\tilde{g}^{\rm
sim}_{\nu_{e}}(v_{z})\right|^{2}\right]^{1/2}}{\int_{v_{z}^{>}}dv_{z}},$
$\displaystyle E(\tilde{I}_{\nu_{e}})$ $\displaystyle=|\tilde{I}^{\rm
pre}_{\nu_{e}}-\tilde{I}^{\rm sim}_{\nu_{e}}|,$ $\displaystyle
E(\tilde{J}_{\nu_{e}})$ $\displaystyle=|\tilde{J}^{\rm
pre}_{\nu_{e}}-\tilde{J}^{\rm sim}_{\nu_{e}}|.$ (19)

The error $E(\tilde{g}_{\nu_{e}}^{>})$ is evaluated excluding the contribution
from the small side because the same flavor equilibration is assumed in all
prescriptions. One can replace all subscripts of $\nu_{e}$ by $\bar{\nu}_{e}$
for the corresponding errors in the antineutrino sector.

We notice that some $\nu$ELN distributions in our sample have very “shallow”
zero crossings, i.e., small ratios of $(I_{<}/I_{>})\ll 1$. For these cases,
their zero-crossings are close to $v_{z}=-1$ or 1. As a result, nearly no
flavor conversion occurs on the large side due to the $\nu$ELN conservation,
similar to the conditions found in large radii of a CCSN [47]. These cases can
be empirically classified as with no flavor conversion, and hence not included
in the performance comparison. After excluding these shallow distributions
with the ratio $(I_{<}/I_{>})<10^{-2}$, the numbers of parameter sets are
reduced to $N^{\rm set}=7479$ and 7162 for the Gaussian and maximum-entropy
types, respectively.

Figure 2 shows the error quantities for all non-shallow distributions of the
Gaussian type with the box-like, quadratic, and power-1/2 prescriptions for
all parameter sets. The indices for each panel are numbered such that the the
corresponding error quantities obtained with the power-1/2 prescription
decrease with increasing index numbers for the sake of clearer presentation.
The top panels show that the power-1/2 prescriptions clearly outperform the
box-like and quadratic predictions for the distributional errors
$E(\tilde{g}_{\nu_{e}})$, $E(\tilde{g}_{\bar{\nu}_{e}})$ and
$E(\tilde{g}_{\nu_{e}}^{>})$. Most of the $E(\tilde{g}_{\nu_{e}})$ and
$E(\tilde{g}_{\bar{\nu}_{e}})$ with the box-like prescription sit around $\sim
0.08$. Comparatively, the power-1/2 scheme provide improvement for these two
errors by up to $\sim 75\%$ for parameter sets with indices $\sim 5000$. When
taking the quadratic prescription, although it generally gives rise to smaller
error compared to the box-like scheme, these two errors have larger variations
with errors as large as 0.5 for some parameter sets. For
$E(\tilde{g}_{\nu_{e}}^{>})$, similar features hold except that now there
exist large variations in errors for all three prescriptions. The underlying
reason is that the common contribution from the small side is not included
when computing $E(\tilde{g}_{\nu_{e}}^{>})$.

(a)
(b)
(c)
(d)
(e)
(f)

Figure 2: Error quantities defined in Eqs. (19) for the box-like (black dots),
quadratic (cyan dots), and power-1/2 (red dots) prescriptions. For each panel,
the indices of the parameter sets are sorted so that the errors obtained with
the power-1/2 scheme follow a descending order, so that the red dots form a
red curve. The Gaussian type is adopted for the initial angular distributions.
The horizontal lines show the arithmetic mean errors $\mathcal{E}$ over the
whole sets as defined in Eq. (20) for the box-like (black dashed), linear
(magenta dashed), quadratic (blue dashed), power-1/2 (red solid), power-1/2-i
(magenta solid), power-1 (green solid), power-2 (yellow solid), and
exponential (blue solid) prescriptions, respectively. Note that additional
cases where the truncation of $P_{ee}=1$ is implemented for the linear and
quadratic prescriptions are shown in the upper panel by the yellow and green
dashed lines, respectively.

The distributions of errors in moments $E(\tilde{I}_{\nu_{e}})$,
$E(\tilde{J}_{\nu_{e}})$, and $E(\tilde{J}_{\bar{\nu}_{e}})$ show somewhat
different and interesting behaviors. Both the box-like and quadratic
prescriptions show large variations. For some cases, they can perform better
than the power-1/2 one in moments despite their worse performance in
distributional errors discussed above. The seemly contradictory result is
related to the cancellation of positive ad negative contributions of errors in
Eq. (19) as well as the overconversions and systematic biases discussed in
Sec. IV.1. For example, for the asymptotic zeroth moments
$\tilde{I}_{\nu_{e}}$ in the antineutrino-dominant case presented in Figs.
1(d)-(f), are $\approx 0.673$, 0.681, and 0.707 for the simulation, quadratic,
and power-1/2 prescriptions, respectively. Although clearly the
$\tilde{g}(\nu_{e})$ obtained with the power-1/2 prescription resembles better
the $g(\nu_{e})$ from the simulation than with the quadratic scheme, the
latter results in smaller $E(\tilde{I}_{\nu_{e}})$ due to the cancellation of
contributions from the integration range of $0.7<v_{z}<0.85$ and from
$v_{z}>0.85$. Such a cancellation does not happen with the power-1/2
prescription where the $\nu$ELN conservation is imposed, because it predicts
larger values of $P_{ee}(v_{z})$ and $\tilde{g}_{\nu_{e}}$ than simulation
values on both the small side and the large side due to the overconversions on
the small side (see discussions in Sec. IV.1). If we have taken the amount of
overconversion contribution $\int_{v_{z}^{<}}dv_{z}g_{\nu}(1/2-P_{ee}^{\rm
sim})\approx 0.014$ into account for the power-1/2 scheme, it will lead to a
reduced $\tilde{I}_{\nu_{e}}=0.679$, which will be better than 0.681 obtained
with the quadratic scheme.

We further calculate the arithmetic mean error for each prescription

$\mathcal{E}=\frac{1}{N^{\rm set}}\sum_{i}^{N^{\rm set}}E_{i},$ (20)

where $i$ sums over the whole ensemble of parameter sets. Those mean errors
are shown by the horizontal lines in Fig. 2 with values listed in Table 1. The
prescriptions with abrupt and continuous transitions at $v_{c}$ are
represented by the dashed and solid lines, respectively. In addition, to
prevent $P_{ee}(v_{z})$ from exceeding the unity in the linear and quadratic
prescriptions as discussed in Sec. III.1, both Eqs. (14) and (15) are replaced
by $\min[1,P_{ee}(v_{z})]$, and the corresponding arithmetic errors are shown
in Figs. 2(a)-(c) as the yellow and green dashed lines. Statistically, the
truncation at $P_{ee}=1$ presents negligible improvement on the mean error.
Both the box-like and linear prescriptions yield the largest errors for all
measures. Comparatively, the quadratic prescription provides visible
improvements while all other prescriptions with continuous transition at
$v_{c}$ further reduce the mean errors and show similar performances in
general. It is noteworthy that the power-1/2-i scheme interpolating with 5
points for the coefficient $a$ in Eq. (17) has errors merely $\sim 5\%$–20%
greater than those with the power-1/2 prescription.

Table 1: Arithmetic mean errors $\mathcal{E}/10^{-2}$ for all eight prescriptions. The Gaussian type is adopted for the initial angular distributions. Prescriptions | $\tilde{g}_{\nu_{e}}$ | $\tilde{g}_{\bar{\nu}_{e}}$ | $\tilde{g}_{\nu_{e}}^{>}$ | $\tilde{g}_{\bar{\nu}_{e}}^{>}$ | $\tilde{I}_{\nu_{e}}$ | $\tilde{I}_{\bar{\nu}_{e}}$ | $\tilde{J}_{\nu_{e}}$ | $\tilde{J}_{\bar{\nu}_{e}}$
---|---|---|---|---|---|---|---|---
Box-like | 6.89 | 7.30 | 11.4 | 12.4 | 4.14 | 4.14 | 2.34 | 2.38
Linear | 6.97 | 7.86 | 13.2 | 15.3 | 3.01 | 3.46 | 2.74 | 3.18
Quadratic | 4.65 | 4.99 | 7.81 | 8.70 | 2.15 | 2.12 | 1.39 | 1.46
Power-1/2 | 2.44 | 2.55 | 3.04 | 3.46 | 1.52 | 1.52 | 0.87 | 0.91
Power-1/2-i | 2.62 | 2.72 | 3.47 | 3.87 | 1.82 | 1.82 | 1.04 | 1.07
Power-1 | 2.83 | 3.01 | 3.56 | 4.11 | 1.25 | 1.25 | 0.81 | 0.86
Power-2 | 2.46 | 2.55 | 3.21 | 3.57 | 1.71 | 1.71 | 0.98 | 1.01
Exponential | 2.57 | 2.65 | 3.15 | 3.50 | 1.55 | 1.55 | 0.95 | 0.99

To further check whether the ranking of the arithmetic averages may be
affected by specific outlier parameter sets with large error, we calculate one
more metric $\mathcal{R}$, which is the fraction of the best performance for
each type of errors $E$ in a subset of prescriptions. As the average errors
from prescriptions with abrupt and continuous transitions are clearly
separated into two groups, we use two subsets: subset A includes box-like,
linear, quadratic, and power-1/2 prescriptions, and subset B includes four
prescriptions with continuous transition at $v_{c}$. They are compared in
Table 2 for cases with initial Gaussian angular distributions. Consistent with
the previous analysis, the power-1/2 prescription has the best performance
predominantly in $\approx 98\%$ of all samples among the prescription subset
A. In terms of the predictions for first two moments, the linear and quadratic
schemes can perform better for $\approx 10$–30% of the parameter sets, while
the power-1/2 prescription performs better for $\approx 45$–60% of all
samples. In subset B, the power-1/2 and power-2 prescriptions share similar
best performance percentages $\approx 30\%$ for $\tilde{g}_{\nu_{e}}$ and
$\tilde{g}_{\bar{\nu}_{e}}$, while the power-1/2 and exponential ones are
slightly worse. With regards to the first two moments, the power-1
prescription dominates and give rises to best values of $\mathcal{R}\approx
63$–73%.

Table 2:  Best performance fraction $\mathcal{R}$ in percentage in two subsets of prescriptions. The Gaussian type is adopted for the initial angular distributions. The total number of parameter sets is $N^{\rm set}=7479$. Subset A | $\tilde{g}_{\nu_{e}}$ | $\tilde{g}_{\bar{\nu}_{e}}$ | $\tilde{I}_{\nu_{e}}$ | $\tilde{I}_{\bar{\nu}_{e}}$ | $\tilde{J}_{\nu_{e}}$ | $\tilde{J}_{\bar{\nu}_{e}}$
---|---|---|---|---|---|---
Box-like | 0.5 | 0.5 | 6.3 | 6.4 | 7.4 | 7.8
Linear | 0.5 | 0.4 | 28.0 | 25.0 | 10.3 | 9.7
Quadratic | 1.4 | 1.6 | 22.2 | 24.3 | 22.2 | 22.9
Power-1/2 | 97.7 | 97.5 | 43.5 | 44.3 | 60.1 | 59.6
Subset B | $\tilde{g}_{\nu_{e}}$ | $\tilde{g}_{\bar{\nu}_{e}}$ | $\tilde{I}_{\nu_{e}}$ | $\tilde{I}_{\bar{\nu}_{e}}$ | $\tilde{J}_{\nu_{e}}$ | $\tilde{J}_{\bar{\nu}_{e}}$
Power-1/2 | 30.5 | 29.2 | 5.5 | 5.5 | 10.3 | 12.1
Power-1 | 15.8 | 14.2 | 72.9 | 72.9 | 66.2 | 63.3
Power-2 | 30.6 | 31.9 | 15.2 | 15.2 | 17.6 | 17.9
Exponential | 23.1 | 24.8 | 6.4 | 6.4 | 6.0 | 6.7

Most of the results discussed above does not change when taking the maximum
entropy type of presumed angular distributions. To avoid repetition, we only
show in Table 3 the corresponding best performance fraction $\mathcal{R}$
obtained here for the maximum entropy type. There, the power-1/2 prescription
still performs best in subset A, and the power-1 prescription has the largest
$\mathcal{R}\sim 80\%$ for the angular moments in subset B. The minor
difference is that the power-1 prescription also outperforms the power-1/2 and
power-2 schemes in $\tilde{g}_{\nu_{e}}$ and $\tilde{g}_{\bar{\nu}_{e}}$.

Table 3: Same as in Table 2 except that the maximum-entropy type is adopted for the initial angular distributions. The total number of parameter sets is $N^{\rm set}=7162$. Subset A | $\tilde{g}_{\nu_{e}}$ | $\tilde{g}_{\bar{\nu}_{e}}$ | $\tilde{I}_{\nu_{e}}$ | $\tilde{I}_{\bar{\nu}_{e}}$ | $\tilde{J}_{\nu_{e}}$ | $\tilde{J}_{\bar{\nu}_{e}}$
---|---|---|---|---|---|---
Box-like | 0.3 | 0.3 | 3.5 | 3.4 | 3.9 | 3.8
Linear | 0.2 | 0.3 | 19.0 | 16.4 | 7.1 | 6.5
Quadratic | 1.0 | 0.8 | 16.8 | 17.0 | 18.1 | 17.2
Power-1/2 | 98.5 | 98.7 | 60.6 | 63.1 | 70.8 | 72.5
Subset B | $\tilde{g}_{\nu_{e}}$ | $\tilde{g}_{\bar{\nu}_{e}}$ | $\tilde{I}_{\nu_{e}}$ | $\tilde{I}_{\bar{\nu}_{e}}$ | $\tilde{J}_{\nu_{e}}$ | $\tilde{J}_{\bar{\nu}_{e}}$
Power-1/2 | 25.2 | 25.1 | 1.1 | 1.1 | 2.2 | 2.3
Power-1 | 55.0 | 54.6 | 80.0 | 80.0 | 78.9 | 78.4
Power-2 | 9.8 | 10.4 | 7.3 | 7.3 | 7.6 | 7.7
Exponential | 10.0 | 9.9 | 11.6 | 11.6 | 11.3 | 11.6

###  IV.3 Dependence in parameter space

The errors in Fig. 2 are ranked regardless of the shape of the distributions
or the moments. To gain a better understanding on how a specific prescription
works better in certain range of the the explored parameter space, we examine
the dependence of the simulation outcome and the errors associated with each
analytical prescriptions in this section. For this purpose, we show the
qualitative features of the asymptotic values of moments from the simulations,
and the evaluated errors with different prescriptions in the parameter space
of $I_{\bar{\nu}}$, $F_{\nu}$, and $F_{\bar{\nu}}$ in Fig. 3 and Fig. 4,
respectively. In both figures, the empty diagonal region in each panel
indicates where the fast flavor instability does not exist. It is more likely
to have fast instabilities when the flux factors are significantly different
from each other or when $I_{\bar{\nu}}$ is closer to $I_{\nu}=1$.

(a)
(b)
(c)
(d)
(e)
(f)

Figure 3: Parameter dependence of the asymptotic values of the first two
moments obtained by simulations. The regions between the two red lines are
where we generate our parameter sets. The Gaussian type initial angular
distributions are adopted for this figure.

(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)

Figure 4: Parameter dependence of the best performed prescriptions for
$E(\tilde{g}_{\nu_{e}})$ with the prescription subset A in the first row,
$E(\tilde{g}_{\nu_{e}})$ with the prescription subset B in the second row,
$E(\tilde{I}_{\nu_{e}})$ with subset A in the third row, and
$E(\tilde{J}_{\nu_{e}})$ with subset A in the fourth row. Colors of the dots
represent different types of prescriptions including the box-like (black),
linear (magenta), quadratic (cyan), power-1/2 (gray), power-1 (green), power-2
(yellow), and exponential (blue) ones. The two red lines delineates the region
where we generate our parameter sets as in Fig. 4. The Gaussian type is
adopted for the initial angular distributions.

When $I_{\bar{\nu}}\approx 1$, and $F_{\nu}$ greatly deviates from
$F_{\bar{\nu}}$, the zero-crossing typically appears in rather central part of
the $v_{z}$ range. As a result, nearly flavor equilibration
($\tilde{I}_{\nu_{e}}^{\rm sim}\simeq\tilde{J}_{\nu_{e}}^{\rm
sim}/J_{\nu}\simeq 0.5$) for both zeroth and first moments are achieved in
most parameter sets as shown by the darker regions in Figs. 3(b) and (e). When
$F_{\nu}\approx F_{\bar{\nu}}$, the initial shape of distributions for
$\nu_{e}$ and $\bar{\nu}_{e}$ are so similar that the zero-crossing of the ELN
is close to either $v_{z}=-1$ or $v_{z}=1$, which leads to incomplete flavor
conversion on the large side even with $I_{\bar{\nu}}\approx 1$. Such a
similar trend also applies to the cases with a large asymmetry of zeroth
moments between $\nu_{e}$ and $\bar{\nu}_{e}$ for larger or smaller
$I_{\bar{\nu}_{e}}$ shown in Figs. 3(a), (c), (d) and (f). Closer to the
central blank regions of these panels, both the $\tilde{I}_{\nu_{e}}^{\rm
sim}$ and $\tilde{J}_{\nu_{e}}^{\rm sim}/J_{\nu}$ deviate from 0.5
systematically. Nearly complete flavor conversion can only happen when
$F_{\nu}$ and $F_{\bar{\nu}}$ differ significantly.

For most of parameter sets, the changes of the zeroth and first moments are
correlated. However, unlike the zeroth moment of electron flavor neutrinos
that can only decrease assuming no $\nu_{x}$ in the initial state, the first
moment after the FFC can be larger than the initial value. For example, at the
bottom-left corner of panel (f) [$1.3\leq I_{\bar{\nu}}\leq 1.6$] where
$F_{\nu}\sim 0.25$ and $F_{\bar{\nu}}\sim 0.35$, the ratio
$\tilde{J}_{\nu_{e}}^{\rm sim}/J_{\nu}$ can be $\approx 1.2$. This is because
the flavor conversion occurs mostly in the range of the back-propagating
neutrinos with $v_{z}<0$, which contribute a nonnegligible amount to the
moments.

Let us now look at how different analytical prescriptions work in different
regions of the moment space. Fig. 4(a)-(c) show that, independent of
$I_{\bar{\nu}}$, $F_{\nu}$, and $F_{\bar{\nu}}$, the power-1/2 prescription
has universally the best performance in the subset A in predicting the
asymptotic distribution $\tilde{g}_{\nu_{e}}$; see also Table 2. When
considering the subset B shown in Fig. 4(d)-(f), different prescriptions
occupy visibly different parameter space for $0.5\leq I_{\bar{\nu}}\leq 0.8$
and $1.3\leq I_{\bar{\nu}}\leq 1.6$ for providing least errors in
$E(\tilde{g}_{\nu_{e}})$. For instance, panel (f) shows that with the
antineutrino-dominant condition $1.3\leq I_{\bar{\nu}}\leq 1.6$ and
$F_{\nu}<F_{\bar{\nu}}$, the best prescription gradually transitions from the
power-2 to the exponential followed by the power-1 and then power-1/2 types,
as the flux factors increases. In the same plot but at the corner with
$F_{\nu}>F_{\bar{\nu}}$, the best prescription transitions from the power-1/2
type to the power-2 type followed by the exponential one. Interestingly, there
does not appear to be any specific prescription that predominately provides
the least distributional error $E(\tilde{g}_{\nu_{e}})$ in any part of the
parameter space with $0.9\leq I_{\bar{\nu}}\leq 1.2$, indicated by the mixed
colors in panel (e).

The bottom two panels display the best performance prescriptions for
$E(\tilde{I}_{\nu_{e}})$ and $E(\tilde{J}_{\nu_{e}})$ within the subset A.
Panels (g)-(i) show somewhat similar domain-like patterns regarding the best
performing prescription for $E(\tilde{I}_{\nu_{e}})$. Although the power-1/2
scheme still outperforms other abrupt prescriptions in a large fraction of the
parameter space as indicated by Table 2, the linear and quadratic
prescriptions can outperform the power-1/2 in some particular parameter
regions. For example, the linear prescription has the best performance when
$0.5\leq I_{\bar{\nu}}\leq 0.8$, $F_{\nu}\sim 0.6$, and $F_{\bar{\nu}}\sim
0.5$, or when $1.3\leq I_{\bar{\nu}}\leq 1.6$, $F_{\nu}\sim 0.4$–0.6, and
$F_{\bar{\nu}}\sim 0.7$–0.8. The quadratic prescription has the best
performance when $0.5\leq I_{\bar{\nu}}\leq 0.8$, $F_{\nu}\sim 0.8$, and
$F_{\bar{\nu}}\sim 0.6$, or when $1.3\leq I_{\bar{\nu}}\leq 1.6$, $F_{\nu}\sim
0.7$, and $F_{\bar{\nu}}\sim 0.8$. As for the box-like prescription, it only
provides better performance for certain parameter sets that are sparsely
distributed in panel (h) with $0.9\leq I_{\bar{\nu}}\leq 1.2$.

Comparing panels (j)-(l) showing the best prescription distribution for
$E(\tilde{J}_{\nu_{e}})$ to panels (g)-(i), it shows that the distribution of
the best-performance domains can vary when considering different moments. For
instance, although for the quadratic prescription, it performs the best in
similar parameter space for both $E(\tilde{I}_{\nu_{e}})$ and
$E(\tilde{J}_{\nu_{e}})$, in the region with $1.3\leq I_{\bar{\nu}}\leq 1.6$,
$F_{\nu}\sim 0.4$–0.6, and $F_{\bar{\nu}}\sim 0.7$–0.8, the power-1/2 replaces
the linear prescription as the best prescription in subset A for
$E(\tilde{J}_{\nu_{e}})$.

###  IV.4 Effects of heavy lepton flavor neutrinos

In the previous discussions the heavy lepton flavor neutrinos are not taken
into consideration in the initial angular distributions because the $\nu$ELN
distribution is unchanged if the same amount of $\nu_{x}$ and $\bar{\nu}_{x}$
distributions are assumed. As a result, the evolution and asymptotic
distribution for survival probabilities are expected to be the same, although
the asymptotic angular distributions $\tilde{g}_{\nu_{e}}$ and
$\tilde{g}_{\bar{\nu}_{e}}$ can be affected.

Because the inclusion of heavy lepton flavor neutrinos introduces more
dimensions in the parameter space, we do not perform detail analysis here for
the performance evaluation. Instead, we provide a specific example below to
illustrate how to evaluate its impact on the errors obtained in earlier
sections, which can be generally applied by post-processing the data set that
we released. Assuming both $\nu_{x}$ and $\bar{\nu}_{x}$ have the same flux
factor as $\bar{\nu}_{e}$ initially for simplicity, their angular
distributions can be characterized by the zeroth moment $I_{\nu_{x}}$ as

$g_{\nu_{x}}(v_{z})=g_{\bar{\nu}_{x}}(v_{z})=\frac{I_{\nu_{x}}}{I_{\bar{\nu}}}g_{\bar{\nu}}(v_{z}).$
(21)

With the asymptotic survival probability unaffected, the final distribution
for $\nu_{e}$ now becomes

$\tilde{g}_{\nu_{e}}(v_{z})=g_{\nu}(v_{z})P_{ee}(v_{z})+\frac{I_{\nu_{x}}}{I_{\bar{\nu}}}g_{\bar{\nu}}(v_{z})[1-P_{ee}(v_{z})],$
(22)

for both the simulated $\tilde{g}_{\nu_{e}}^{\rm sim}$ and predicted
$\tilde{g}_{\nu_{e}}^{\rm pre}$. In Figure 5, we show the distributional
errors $E(\tilde{g}_{\nu_{e}})$ as a function of $I_{\nu_{x}}/I_{\bar{\nu}}$
computed based on Eq. (22) for the two conditions considered in Sec. IV.1 with
the initial distributions parameterized by the Gaussian and maximum-entropy
functions.

Figure 5: Distributional errors $E(\tilde{g}_{\nu_{e}})$ as functions of
$I_{\nu_{x}}/I_{\bar{\nu}}$ for eight prescriptions.

The errors $E(\tilde{g}_{\nu_{e}})$ for all prescriptions start to decrease as
$I_{\nu_{x}}$ increases at the beginning. They reach the minimum at
$I_{\nu_{x}}/I_{\bar{\nu}}\sim 0.6$–1 and eventually increase again. The
reduction of errors for $I_{\nu_{x}}\sim I_{\bar{\nu}}$ is because for more
similar distributions of $\nu_{e}$ and $\nu_{x}$, less changes to the
$\nu_{e}$ distribution can occur due to the conversion of $\nu_{x}$ to
$\nu_{e}$. Specifically, the minimum locates at $I_{\nu_{x}}/I_{\bar{\nu}}$
slightly less than 1 where the crossing between angular distributions of
$g_{\nu}$ and $g_{\nu_{x}}$ happens at the small side, as this allows to
minimize the systematic error contribution due to the observed overconversions
seen in simulations discussed earlier.

Comparing different analytical prescriptions, all three panels show that those
with the continuous transitions at $v_{c}$ again provide similar errors with
different $I_{\nu_{x}}/I_{\bar{\nu}}$ and generally perform better than the
abrupt prescriptions. For the abrupt ones, the quadratic prescription still
has better performance than the box-like and linear ones, but can be slightly
outperformed, e.g., by the linear scheme at $I_{\nu_{x}}/I_{\bar{\nu}}\sim
0.6$ or the box-like scheme at $I_{\nu_{x}}/I_{\bar{\nu}}\sim 0.9$ in the
antineutrino-dominant condition with the initial Gaussian type shown in panel
(b).

##  V Discussion and conclusions

In this paper, we conducted a comprehensive survey over a large sample of
initial neutrino angular distributions to investigate the outcome of the
asymptotic state of FFC in the periodic 1D-box setup. Several thousands of
simulations for initial $\nu_{e}$ and $\bar{\nu}_{e}$ angular distributions
parametrized by the Gaussian and the maximum-entropy functions that can also
be specified by the initial zeroth moment of $\bar{\nu}_{e}$, $I_{\bar{\nu}}$,
and flux factors, $F_{\nu}$, and $F_{\bar{\nu}}$ were performed to times when
the systems reach close to the asymptotic states. These results provide a
database for the design of effective treatments so that FFCs can be
approximately incorporated into realistic hydrodynamic simulations that
include classical neutrino transport.

We found that in the asymptotic state, flavor conversions on one side of the
$\nu$ELN (defined as the “small” side in this work) happen in a way that the
system evolves towards flavor equilibration to eliminate the zero crossing
when averaging over the entire box, as pointed out in several earlier works.
Interestingly, we also found that slight overconversions on the small side in
the asymptotic state can happen as a general final outcome of the system,
which however, does not introduce new zero crossings.

Assuming flavor equilibration on the small side, we formulated several new
analytical prescriptions that aim to improve the existing formulations
including the box-like and the linear prescriptions proposed in Refs. [40, 45,
49], which provided analytical formula to characterize the asymptotic state on
the “large” side of the $\nu$ELN. One of our new proposals extends these
existing ones and includes the second-order Legendre polynomial correction,
resulting in a quadratic velocity dependence on the large side. More
importantly, to overcome the artificial discontinuity encountered at the zero-
crossing when using the box-like, linear, and quadratic expressions, we
provided several new prescriptions that continuously connect the flavor
conversion probabilities on the small and the large sides while respect the
$\nu$ELN conservation.

Based on our simulation data, we first compared in detail the asymptotic
states predicted by these analytical prescriptions with those obtained
numerically for two representative examples. We then evaluated the overall
performance for the entire data sets using several error measures, including
the distributional errors, net differences of first two angular moments, the
fraction of best performance. We found that despite all prescriptions provide
reasonable predictions with small distributional errors $\lesssim 0.15$ and
angular moment differences $\lesssim 0.05$, the prescriptions with continuous
transitions at zero crossings systematically outperform those with abrupt
transitions. Specifically, the quadratic prescription reduces the average
errors by $\sim 30$–50% from the box-like and linear schemes, while all the
continuous prescriptions give rise to another factor of $\sim 30$–60%
improvement from the quadratic scheme mainly due to the imposed condition
continuity around $v_{c}$.

There exist certain advantages and disadvantages associated with these
prescriptions. The evaluation the box-like, linear, and quadratic schemes can
be directly done with the explicit formulae given Sec. III.1. For the linear
and the extended quadratic scheme, they were derived by adding corrections
upon the box-like prescription, neither the $\nu$ELN conservation nor non-
negative transition probability is ensured unless some additional truncation
is imposed. For the continuous prescriptions, it in principle requires extra
computational efforts to solve the width coefficient $a$ iteratively. However,
we also demonstrated that one can use the interpolation method to obtain the
asymptotic distributions efficiently without sacrificing much the accuracy of
their predictive power. Moreover, if one wants to directly implement these
prescriptions with neutrino transport solvers that adopt the discrete-ordinate
schemes, taking the abrupt prescriptions will introduce large errors
associated with angular distribution discontinuity when numerical evaluating
the angular advection, which can be avoided with continuous prescriptions.

We have also discussed the dependence of the outcome on the parameter space
and the impact when including non-negligible heavy lepton neutrinos in the
initial condition. We found that similar conclusions discussed above generally
hold for cases including the heavy lepton flavors – the continuous
prescriptions perform better than those abrupt schemes. However, we also noted
for some parameter space, the linear and quadratic schemes (without $\nu$ELN
conservation constraint) in fact give rise to smaller errors in individual
angular moment differences than all the continuous prescriptions, due to the
accidental cancellation effect when integrating over the distributions. For
the continuous prescriptions with $\nu$ELN conservation being imposed, the
generally obtained flavor overconversions prevent the accidental cancellation
to occur and causes a larger systematic bias.

Several questions remain to be addressed beyond this work and we list a few
below. Do the overconversions on the small side depend on the periodic
boundary conditions? Will they be suppressed in the presence of collisions? If
not, how can we improve the formulation of the asymptotic state to account for
the overconversions? Can these prescriptions be applicable to more general
scenarios, e.g., cases where the azimuthal symmetry is broken? Answering all
those questions certainly requires many following-up studies and will help
achieve the ultimate goal of implementing flavor conversions of neutrinos in
hydrodynamical simulations of supernovae and neutron star mergers.

###### Acknowledgements.

We thank Oliver Just, Gabriel Martínez-Pinedo, and Yong-Zhong Qian for
fruitful discussions. Z. X., M.-R. W., and S. A. are grateful to the Mainz
Institute for Theoretical Physics (MITP) of the Cluster of Excellence PRISMA+
(Project ID 39083149), for its hospitality and its partial support during the
completion of this work. Z. X. acknowledges support of the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (ERC Advanced Grant KILONOVA No. 885281). M.-R. W., S. B., and M. G.
acknowledge supports from the National Science and Technology Council, Taiwan
under Grant No. 110-2112-M-001-050 and 111-2628-M-001-003-MY4, the Academia
Sinica under Project No. AS-CDA-109-M11, and Physics Division, National Center
for Theoretical Sciences, Taiwan. S. A. was supported by the German Research
Foundation (DFG) through the Collaborative Research Centre “Neutrinos and Dark
Matter in Astro- and Particle Physics (NDM),” Grant SFB-1258, and under
Germany’s Excellence Strategy through the Cluster of Excellence ORIGINS
EXC-2094-390783311. Software: Numpy [71], Matplotlib [72], and Scipy [73].

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