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Dispelling a myth on dense neutrino media: fast pairwise conversions depend on energy

Pairwise conversions of neutrinos have complicated the challenging goal of quantifying the relevance of neutrino microphysics in compact astrophysical objects, limiting the ability to perform numerical simulations and encouraging the employment of semi-analytical tools, such as the linear stability analysis. Given the high neutrino density, the dependence of pairwise conversions on the neutrino energy has been neglected in the literature. We show that this naive simplification has limited our understanding of the flavor conversion phenomenology: fast pairwise conversions are strongly affected by the neutrino energy. An earlier onset of flavor conversions and a higher oscillation frequency are found as the vacuum frequency increases (i.e., the neutrino energy decreases), and the oscillation periodicity, otherwise present, is lost. Such effects are, however, not further exacerbated by the inclusion of spectral energy distributions for neutrinos, contrary to what was found for"slow"neutrino-neutrino conversions. Our findings highlight the limitations intrinsic to widely adopted approximations. In order to gauge their possible impact on the source physics, a more sophisticated modeling of pairwise conversions is necessary.


Introduction
Neutrino flavor conversions should play a significant, but yet poorly understood, role in dense astrophysical objects, such as core-collapse supernovae, compact binary mergers, as well as in the early Universe [1][2][3][4][5][6]. Matter-enhanced flavor conversions are generated by the forward scattering of neutrinos on the background matter [7,8]. In addition, the flavor evolution is also affected by the coherent forward scattering of neutrinos among themselves [3,[9][10][11][12][13][14][15]. The self-interactions of neutrinos induce non-linear effects on the flavor evolution, leading to a rich phenomenology, still vastly unexplored. A peculiar feature of neutrino-neutrino interactions is that they strongly depend on the angle between the momenta of the interacting neutrinos.
The impact of neutrino-neutrino interactions on the flavor evolution is challenging to predict because of the non-linear nature of the interaction. Therefore, numerical implementations have been adopted to investigate the flavor conversion phenomenology, building on the first attempts proposed in Refs. [12,15] and involving different levels of approximations. In addition, semi-analytical techniques, such as the linear stability analysis [16], have been massively employed and revealed to be especially useful in the cases most difficult to tackle numerically.
Recently, the possibility that pairwise conversions of neutrinos occur when the neutrino density is high has been raised [17][18][19]. In this case, "fast" pairwise conversions would develop at a rate much higher, √ 2G F (n ν − nν) few m −1 (with n ν,(ν) the (anti)neutrino density and G F the Fermi constant), than the typical rate of neutrino oscillations in vacuum, ∆m 2 /2E 0.1 km −1 (where ∆m 2 is the atmospheric mass difference and E O(10) MeV is the typical neutrino energy). Fast pairwise conversions have received a lot of attention lately as they could potentially lead to flavor decoherence, e.g. in the proximity of the neutrino decoupling region, with major implications on the source physics [5,[20][21][22][23][24][25][26][27][28][29].
Unlike the "slow" conversions due to neutrino-neutrino interactions, fast pairwise conversions have been postulated to be exclusively triggered by the neutrino-neutrino potential even in the absence of the vacuum frequency, if seed perturbations exist [17][18][19]. For this reason, the vacuum oscillation frequency has been neglected in the literature on fast pairwise conversions.
Fast pairwise conversions have been deemed to be driven by the angular distributions of neutrinos and antineutrinos. One of the necessary conditions leading to the development of fast pairwise conversions is the existence of electron neutrino lepton number (ELN) crossings [19]. The existence of ELN crossings means that, in order for fast neutrino conversions to occur at a given point, the number of electron neutrinos has to be larger than the one of electron antineutrinos in some direction and smaller in another direction. To explore whether favorable conditions for fast pairwise conversions exist, several methods have been proposed to search for ELN crossings in hydrodynamical simulations [30][31][32].
Due to the numerical complexity of the problem, most of the existing work on fast pairwise conversions is based on the linear stability analysis [19,[22][23][24][25][26][33][34][35]. Although very helpful, the oversimplifications due to the computational challenges and the wide employment of the linear stability analysis have led to the obfuscation of some relevant features of fast pairwise conversions. For example, recent work highlighted the dynamical role played by neutrino advection in hindering the growth of flavor instabilities [36], an effect that cannot be reproduced by means of the stability analysis.
In this paper, we attempt a direct comparison between the linear stability analysis and the numerical solution of the neutrino flavor evolution within a simplified setup and dispel some myths surrounding fast pairwise conversions. In particular, although the occurrence of ELN crossings is expected in the proximity of the decoupling region where the neutrinoneutrino potential is much larger than the vacuum frequency [19][20][21]32], the vacuum mixing parameters cannot be ignored, as currently done in the literature.
Our work focuses on the linear and non-linear regimes of pairwise conversions of neutrinos and aims to highlight the modifications induced in the flavor conversion phenomenology when the neutrino energy dependence is taken into account. In particular, we find that simple formal analogies widely adopted to intuitively explain the ν-ν interaction physics, such as the pendulum one proposed in Ref. [10], are not applicable to the case of fast pairwise conversions, as also pointed out in [29]. Given the highly non-linear nature of the system studied here, we refrain from providing an analytical understanding of our findings and mostly focus on discussing insightful examples of how the conversion physics is modified. This paper is organized as follows. We briefly review the neutrino equations of motion and the physics of fast pairwise conversions in Sec. 2. The dependence of the linear stability analysis on the vacuum frequency is discussed in Sec. 3. In Sec. 4 we focus on the nonlinear regime and explore the dependence of fast neutrino conversions on the vacuum mixing parameter and on the mass ordering. Our results are extended from a single-energy to a multi-energy framework in Sec. 5. Section 6 focuses on how our findings are further affected when the reflection symmetry is broken. Finally, conclusions are reported in Sec. 7.

Neutrino equations of motion
The equations of motion describing the neutrino flavor evolution can be written using the Wigner transformed density matrices to encode the flavor of neutrinos. For simplicity, we work in the two flavor approximation (ν e , ν x ) and thus we have two density matrices ρ and ρ for neutrinos and antineutrinos of momentum p, respectively. Each one is a 2 × 2 matrix and evolves in accordance with the Heisenberg equation of motion: The Hamiltonian, that governs the temporal evolution of the density matrix, has three contributions: the vacuum term, the matter term, and the neutrino-neutrino term: 3) In the two flavor approximation, the three contributions of the Hamiltonian are defined as follows where ω = ∆m 2 /2E is the vacuum frequency (with the squared mass splitting ∆m 2 > 0 for normal mass ordering and ∆m 2 < 0 for inverted ordering and E being the neutrino energy), θ V is the vacuum mixing angle, G F is the Fermi constant, n e is the effective number density of electrons in the medium, and µ = √ 2G F n ν is the strength of the self-interaction potential which is proportional to the neutrino number density n ν . The first two terms of the Hamiltonian, H vac and H mat , are linear in nature. On the other hand, the third term, H νν , is dependent on the density matrices and thus makes the equations of motion non-linear.
Throughout the paper, we assume µ = 10 5 km −1 and θ V = 10 −6 to mimic the effective suppression of mixing due to matter [37], and n e = 0. Moreover, in spherical coordinates (with θ being the polar angle and φ the azimuthal one) and under the assumption of azimuthal symmetry, the neutrino-neutrino term of the Hamiltonian becomes the angles θ and θ are the polar angular coordinates of neutrinos, and the integration over the azimuthal angle gives a 2π factor. Equation 2.8 is formally identical to the one used in, e.g., [15]; however, in this paper, we are interested in exploring the flavor conversion physics in a simple toy model and, consequently, we do not connect θ and θ to the geometry of the emission surface of an astrophysical source. The angular coordinates adopted in this paper should be interpreted as the local ones in the vicinity of the region where we solve the equations of motion. Neutrino-neutrino interactions are a consequence of the coherent forward scattering of neutrinos with other neutrinos: the incident neutrino exchanges momentum with the target neutrino. When the neutrino number density is large enough, it has been postulated that significant neutrino flavor evolution may occur as it is possible for a pair of neutrinos to coherently scatter of each other and change the flavor [17,18]: where α and β denote the neutrino flavor. Such pairwise conversions have been found to occur coherently when there is an ELN crossing [19].
The equations of motion that govern the flavor evolution in the presence of fast pairwise conversions of neutrinos are identical to the ones describing "slow" neutrino self-interactions [9]. However, whereas the slow neutrino self-interactions effectively occur when the neutrino selfinteraction potential is roughly comparable to the vacuum frequency, fast pairwise conversions can lead to significant flavor evolution for arbitrarily large neutrino-neutrino potential.
A necessary condition for significant flavor evolution is that the off-diagonal component of the Hamiltonian is not very small in comparison with the diagonal elements. This can be either due to a large off-diagonal term in the linear part of the Hamiltonian or to a dynamical increase in the off-diagonal term of H νν caused by the non-linear nature of the evolution. A dynamical rise in the off-diagonal term of the density matrix is referred to as "flavor instability." In the limit where we can ignore the momentum-changing scatterings, the occurrence of a flavor instability has been considered to be solely determined by the (anti)neutrino angular distributions, i.e., their relative shape and normalization [19]. In the case of core-collapse supernovae, conditions favoring the development of ELN crossings have been found in the proximity of the neutrino decoupling region where neutrinos gradually transition from being trapped to the free-streaming regime and in the pre-shock region [5,[20][21][22][23][24][25][26][27][28]. It should be noted that the cross-section of ν e with nucleons is larger than the one ofν e with nucleons, resulting in a larger decoupling radius for ν e in comparison toν e , see e.g. [20,21]; thus, there is a region near the decoupling region where ν e 's are more isotropically distributed thanν e 's whose distribution is slightly forward peaked. This scenario naturally provides a benchmark scenario where ELN crossings can occur and is a convenient case of study for our purposes.

Energy dependent linear stability analysis
In this section, we introduce the linear stability analysis tool commonly adopted to gauge whether favorable conditions for fast pairwise conversions exist. However, the linear stability analysis for exploring the existence of flavor instabilities due to pairwise conversions has been derived for vanishing vacuum frequency [19]. This simplification was naively justified because µ ∆m 2 /2E. Since our main goal is to investigate the dependence of fast pairwise conversions on ω = ∆m 2 /2E, we now include the vacuum term of the Hamiltonian.

Growth rate of the flavor instability
For the sake of simplicity, we consider an ensemble of neutrinos such that they are approximately in the electron flavor eigenstates and assume that the off-diagonal components are much smaller than the diagonal components. Hence, we can write the density matrices in the following form: with ρ ex ρ ee andρ ex ρ ee . If the flavor evolution is collective in nature, all momentum modes evolve with the same frequency denoted by Ω, Plugging this ansatz in Eq. 2.2, we get the following characteristic equations: where we adopt Q θ instead of Q p in the single-energy approximation. Moreover, the term H ee includes the vacuum term and it therefore depends on the neutrino energy E. An inspection of Eq. 3.3 implies that Q θ andQ θ are with a and c being undetermined constants depending on the neutrino energy. By substituting Eq. 3.4 in Eq. 3.3, we obtain where and It is possible to have multiple values of Ω for which Eq. 3.5 is satisfied, but only the solution with Im(Ω) > 0 implies a flavor instability.

Results of the linear stability analysis
In order to investigate the dependence of the flavor instability on the initial angular distribution of ν e andν e , we assume the following distributions: where the ν e density matrix is normalized such that π 0 2π 0 ρ ee (θ)dφ sin θdθ = 4π. The growth rate (see Eq. 3.2), Im(Ω), increases as a function of y 1 . The latter is assumed to be y 1 = 0.05, we also consider one energy mode for neutrinos and antineutrinos with E = 10 MeV.
The angular distributions of ν e 's andν e 's introduced above are plotted as functions of θ in the left panel of Fig. 1 and they exhibit one ELN crossing. For this initial configuration,  the linear stability analysis predicts a non-zero growth rate, as shown in the right panel of Fig. 1 (dashed lines) for different values of ∆m 2 (note that exploring the ω-dependence of fast pairwise conversions by varying ∆m 2 is equivalent to change E for our purposes). The growth rates of the off-diagonal term of the density matrix for different values of ∆m 2 do not drastically differ among each other in the linear regime; however, as we will discuss in Sec. 4, the evolution of the off-diagonal term of the neutrino density matrix shows a marked dependence on ∆m 2 in the non-linear regime.
4 Impact of the vacuum mixing parameter: single-energy configuration In order to explore the effect of the vacuum frequency, ω = ∆m 2 /2E, on the evolution of the neutrino flavor due to fast pairwise conversions, we now focus on the numerical computation of the flavor evolution for a neutrino ensemble with one energy mode for neutrinos and antineutrinos (E = 10 MeV). We assume the angular distributions introduced in Eq. 3.8. First, we focus on exploring the flavor evolution as a function of ∆m 2 for ∆m 2 > 0. Then, we discuss the dependence of the flavor evolution on the mass ordering.

Impact of the vacuum mixing parameter
The right panel of Fig. 1 shows the growth rate of the angle-integrated off-diagonal term of the neutrino density matrix for different values of ∆m 2 and its following evolution in the non-linear regime. The onset of the non-linear regime occurs earlier for larger ∆m 2 ; this is due to the fact that a larger ∆m 2 induces a larger off-diagonal seed in the density matrix that leads the instability to grow faster. In addition, quasi-periodic modulations of ρ ex dθ appear in the non-linear regime, and the oscillation frequency is higher for larger values of ∆m 2 . In order to better grasp the role of ω in fast pairwise conversions, Fig. 2 shows the initial and final (at t = 3 × 10 −6 sec) angular distributions of ν e andν e as functions of θ. The flavor instability starts to develop in the angular bins where the ELN crossing occurs and proceeds periodically, spreading in the neighboring angular bins. This is clearly shown in the animations provided as Supplemental Material. However, while we find a behavior remarkably similar to the one of "slow" bipolar oscillations [10] in the limit of ∆m 2 → 0, where the only dimension-full quantity in the equations of motion is µ, two different frequencies (∆m 2 /2E and µ) affect the flavor evolution when ∆m 2 0. As seen in the bottom panels of Fig. 2, the angular distributions acquire a more complex structure for larger ∆m 2 (e.g., for ∆m 2 corresponding to the measured atmospheric mass difference), and the periodicity of the flavor evolution is slowly lost as shown by the red line in the right panel of Fig. 1.
The animations highlight that, as time increases, the angular distributions keep developing fine structures. Consequently, it is extremely difficult to assess whether the angleaveraged flavor distribution ever reaches a steady state. Comparing the top and bottom panels of Fig. 2, and as also shown in the right panel of Fig. 1, it is clear that the time scale required to achieve a steady-state configuration-if ever reached-depends on the vacuum fre- quency. Figures 1 and 2 highlight that ∆m 2 = 0 plays a fundamental role in determining the frequency and the development of flavor conversions despite being much smaller than µ.
The temporal evolution of flavor can be intuitively explained by looking at the animations (see Supplemental Material). First ∆m 2 = 0 drives the polarization vectors to precess at a certain inclination angle with respect to the z axis, as it was found in the "slow" ν-ν interaction case, and this triggers the growth of ρ ex . However, the polarization vectors are strongly attracted towards the z axis again where they return in the t = 0 configuration, and the whole process repeats with a certain periodicity. This trend immediately suggests that the "pendulum analogy" [10] widely adopted to explain the ν-ν interaction physics in the slow regime does not hold in the "fast" regime. Figure 3 shows the temporal evolution of the angle-averaged transition probability: for different values of ∆m 2 . A rapid oscillatory behavior is clearly evident as typical of fast pairwise conversions. However, the period of conversions does depend on ∆m 2 as well as the oscillation amplitude. This behavior is not captured by the linear stability analysis and highlights the importance of numerical simulations in the non-linear regime. Notably, despite the growth rate being large as shown in the right panel of Fig. 1, the effective transition probability in Fig. 3 is small. Figure 3 highlights that, as the vacuum frequency increases, not only does the frequency of the flavor transition probability increases, but the deviation from the bipolar oscillation pattern is larger as ω increases.
Our results clearly show that, even if the ν-ν potential is much larger than the vacuum frequency, the vacuum frequency cannot be ignored, and it strongly affects the flavor evolution. The non-linear evolution of fast-pairwise flavor conversions was also explored in Ref. [38]. Despite the large ELN crossings, a modulation in the amplitude of the flavor conversions was found there, suggesting that the vacuum frequency may play a role, although this was not the focus of that paper.

Impact of the mass ordering
The counter-intuitive role played by the vacuum frequency on fast pairwise conversions naturally leads to investigate the impact of the neutrino mass ordering. In this section, we present the results of our numerical simulations with ∆m 2 < 0, while keeping all other parameters unchanged. The left panel of Fig. 4 shows the evolution of the off-diagonal terms of the density matrix as a function of time for normal (continuous line, ∆m 2 = 2.5 × 10 −3 eV 2 ) and inverted (dashed line, ∆m 2 = −2.5 × 10 −3 eV 2 ) ordering. One can see that the growth rate is steeper in normal ordering and the non-linear regime is reached earlier, as also visible in the right panel of Fig. 4. The latter shows the temporal evolution of P ex (see Eq. 4.1) for both orderings. In the case of inverted ordering, the oscillation amplitude is larger. In fact, as one can see from the animations provided as Supplemental Material, the system is more unstable in inverted ordering and it departs from its initial configuration a bit more at each precession, than in the normal ordering case. Our findings also suggest that the time required to reach the steady state configuration depends on the mass ordering (see Supplemental Material).
Importantly, our findings suggest that, although the flavor evolution depends on the mass ordering, the final flavor configuration is not easily predictable. Given the highly nonlinear nature of the system, it may be possible that for different angular distributions of neutrinos and antineutrinos, flavor conversions are enhanced in normal ordering instead than in inverted ordering, as shown in Fig. 4. This is another difference with respect to what found for "slow" ν-ν conversion where no flavor conversion is expected in normal ordering, see e.g. [10,12].

Impact of the vacuum mixing parameter: multi-energy configuration
In this section, we explore how the development of fast flavor conversions is further affected by a multi-energy distribution for the (anti)neutrino ensemble. For the sake of simplicity, we consider an initial ensemble of ν e andν e (without non-electron flavor neutrinos) with the angular distributions introduced in Eq. 3.8 and with Fermi-Dirac energy distributions: with temperatures Tν e = 4 MeV and T νe = 3 MeV, see dashed lines in the top panels of Fig. 5. The top panel of Fig. 5 shows the evolution of the spectral energy distributions of neutrinos (on the left) and antineutrinos (on the right) for ∆m 2 = 2.5 × 10 −3 eV 2 , the corresponding energy-averaged angular distributions are represented in the bottom panels. One can clearly see that the flavor conversions are triggered, as it was found in Sec. 4, by the ELN crossing present in the angular distributions. In turn, this induces modifications in the energy spectra.
In order to facilitate a comparison with the single-energy configuration discussed in Sec. 4 and better gauge the role played by the energy distribution in the final flavor configuration, Fig. 6 compares our multi-energy findings to the ones obtained for neutrinos and antineutrinos having one energy respectively, each correspondent to the average energy of the spectral distribution of Eq. 5.1 [39], i.e., E νe = 3.15 × T νe = 9.45 MeV and Eν e = 3.15 × Tν e = 12.60 MeV (average-energy configuration). For both scenarios, we assume the angular distributions introduced in Eq. 3.8. The angle-averaged transition probability P ex and its energy-integrated equivalent, P ex (obtained by integrating over energy each term in Eq. 4.1), are shown in Fig. 6 for ∆m 2 = 2.5 × 10 −6 eV 2 (on the left) and ∆m 2 = 2.5 × 10 −3 eV 2 (on the right). One can see that, in the absence of crossings in the energy distributions, the average-energy configurations give results very similar to the multi-energy ones, with no qualitative differences. Unsurprisingly, when ∆m 2 is small (∆m 2 = 2.5 × 10 −6 eV 2 ), we see a bipolar evolution of the neutrino flavor, as discussed in Sec. 5. The typical time-scale between two bipolar modes (time between two successive peaks of P ex ) is comparable to the single-energy configuration, see Sec. 5. Moreover, the oscillation frequency is strongly affected by ∆m 2 as the latter increases.
6 Breaking of the reflection symmetry Throughout this paper, we have assumed azimuthal symmetry for the sake of simplicity. The symmetry breaking adds a new layer of complexity to the problem that goes beyond adding another term in the Hamiltonian and leads to the spatial symmetry breaking of the system [36,40,41]. In its most general form, the ν-ν term of the Hamiltonian is The additional term, with respect to Eq. 2.8, that depends on sin θ may qualitatively change the neutrino flavor evolution and, thus, it should be included; however, at the same time, this additional term makes the numerical simulations unfeasible. The effect of the additional sin θ term in H νν can be mimicked without numerically solving the equations of motion by including the φ variable, but by evolving the neutrino  Figure 7. Angle-and energy-averaged transition probability P ex as a function of time for ∆m 2 = 2.5 × 10 −6 eV 2 (on the left), 2.5 × 10 −6 eV 2 (on the right), when the reflection symmetry is broken (solid lines, ρ(θ) = ρ(−θ)) and when it is not broken (dashed lines, ρ(θ) = ρ(−θ)). The dependence of the transition probability on ∆m 2 is still strong, and the neutrino flavor evolution is no longer bipolar even for small ∆m 2 .
flavor in a two dimensional system with θ ∈ [0, 2π] (instead than θ ∈ [0, π]) and with the phase space integration factor being dθ instead of sin θdθdφ. Then the resultant ν-ν term of the Hamiltonian is This approximation, although restricted to a two-dimensional space, allows to investigate the possible effect of the term proportional to sin θ in H νν on the flavor evolution without introducing the azimuthal coordinate, when the assumption of reflection symmetry (ρ(θ) = ρ(−θ) andρ(θ) =ρ(−θ)) is relaxed.
In the numerical solution, we include a small asymmetric perturbation of O(10 −6 ) in the initialρ ee . Moreover, we assume the energy (angular) distribution as in Eq. 5.1 (Eq. 3.8). Figure 7 shows the energy and angle averaged transition probability P ex as a function of time for the cases where the reflection symmetry is imposed (dashed lines) and when it is broken (solid lines) for ∆m 2 = 2.5 × 10 −6 eV 2 on the left and ∆m 2 = 2.5 × 10 −3 eV 2 on the right. One can see that there is still a strong dependence on ∆m 2 . However, as it was found for "slow" conversions [42], the inclusion of the asymmetric mode makes the system more unstable and the neutrino flavor evolution is no longer bipolar, even for small ∆m 2 .

Conclusions
Pairwise flavor conversions of neutrinos are expected to occur in a dense neutrino gas and have been considered to be exclusively driven by the angular distributions of ν e andν e . In particular, the existing literature and the massive employment of the linear stability analysis have lent support to the fact that fast pairwise conversions can be triggered by the existence of crossings in the angular distributions of ν e andν e (ELN crossings) and that pairwise conversions can also occur in the absence of the neutrino mass splitting.
In this paper, we focus on the rich and enigmatic phenomenology of fast pairwise conversions of neutrinos and dispel some naive generalizations on their nature. In particular, although the neutrino mass splitting has been traditionally neglected under the assumption that the vacuum frequency is a few orders of magnitude smaller than the ν-ν potential, we show that fast pairwise conversions are strongly affected by the neutrino energy and are sensitive to the neutrino mass ordering. Such a strong dependence on the neutrino vacuum frequency is surprising and counter-intuitive, but it highlights the highly non-linear nature of the dense neutrino gas.
Even if the energy-dependent linear stability analysis well reproduces the numerical solutions of the flavor evolution in the linear regime, the vacuum term affects the non-linear regime and the onset of fast flavor conversions, inducing higher frequency modulations in the conversion probability as the squared mass difference ∆m 2 increases (or, equivalently, the neutrino energy decreases). Interestingly, a regime very similar to the bipolar one [10] is observed for ∆m 2 → 0, but any oscillation periodicity disappears as ∆m 2 increases approaching the atmospheric mass splitting. This makes the pendulum analogy, widely employed to model the physics of slow ν-ν conversions, not suitable to grasp the basics of fast pairwise conversions.
Contrary to what was found for "slow" neutrino self-interactions, the spectral energy distributions of neutrinos and antineutrinos do not affect the flavor evolution dramatically. In fact, a qualitatively similar behavior is found when the multi-energy configuration for neutrinos and antineutrinos is approximated by two energy modes, each corresponding to the average energy of the neutrino and antineutrino spectral distributions. In the context of "slow" neutrino self-interactions, it was also shown that the relaxation of the symmetries imposed on the system make the latter more unstable. Similarly, for fast pairwise conversions, we find that the inclusion of an asymmetric mode breaks any periodicity in the oscillation patterns and favors the growth of the instability.
In conclusion, our findings highlight the highly counter-intuitive phenomenology of fast pairwise conversions and further corroborate the need to move beyond simple semi-analytical treatments. Only in this way will we be able to grasp the neutrino flavor evolution physics and finally gauge its relevance in astrophysical sources.