Collective pair conversion νeνe ↔ νxνx by forward scattering, where x = µ or τ , may be generic for supernova neutrino transport. Depending on the local angular intensity of the electron lepton number carried by neutrinos, the conversion rate can be "fast," i.e., of the order of √ 2GF(nν e − nν e ) ∆m 2 atm /2E. We present a novel approach to understand these phenomena: a dispersion relation for the frequency and wave number (Ω, K) of disturbances in the mean field of νeνx flavor coherence. Run-away solutions occur in "dispersion gaps," i.e., in "forbidden" intervals of Ω and/or K where propagating plane waves do not exist. We stress that the actual solutions also depend on the initial and/or boundary conditions, which need to be further investigated.Introduction.-The physics of core-collapse supernova (SN) explosions and neutron-star (NS) mergers raises unique questions about flavor evolution in environments where neutrinos are dense. Their decoupling strongly depends on flavor because β reactions dominate for ν e and ν e . As a result, the ν eνe flux of the SN accretion phase exceeds the ν xνx fluxes [1], an effect that is even more pronounced in NS mergers [2,3]. Moreover, the SN ν e flux is larger than theν e one (deleptonization) and the other way round in NS mergers.The subsequent flavor evolution matters because SN neutrinos not only carry away energy, but also deposit some of it in the gain region below the stalled SN shock by ν e + n → p + e − andν e + p → n + e + , thus driving the delayed explosion. At later stages, neutrinos regulate the nucleosynthesis outcome in the neutrino-driven wind. The neutrino signal from the next nearby SN will also depend on the flavor ratio.In the SN region of interest, the matter density is large and suppresses conventional flavor conversion of the type ν e (p) → ν x (p), which is driven by neutrino masses and mixing. This effect becomes important only at larger radii where neutrinos undergo an MSW resonance [4]. Stochastic density variations from turbulence might stimulate flavor conversions [5], but have been found to be ineffective during the accretion phase [6].Neutrino-neutrino interactions can famously change this picture [1,[7][8][9][10][11][12][13][14][15] because flavor off-diagonal refraction by ν e ν x coherence spawns conversion [16][17][18]. In this way, neutrinos feed back upon themselves and can develop collective run-away modes. Neutral-current interactions preserve flavor, so we are dealing with flavor exchange of the type ν e (p) + ν x (k) ↔ ν x (p) + ν e (k) and especially ν e (p) +ν e (k) ↔ ν x (p) +ν x (k) by forward scattering. Such pairwise swaps preserve net flavor, but still modify subsequent charged-current interactions.The impact of refractive ν eνe ↔ ν xνx conversion has never been studied in SN simulations because such effects seemed to arise only beyond the shock wave [19]. Yet, Sawyer has long held that such conclusions result from overly simplified assumptions about neutrino distri-

Neutrino flavor evolution in core-collapse supernovae, neutron-star mergers, or the early universe is dominated by neutrino-neutrino refraction, often spawning "self-induced flavor conversion", i.e., shuffling of flavor among momentum modes. This effect is driven by collective run-away modes of the coupled "flavor oscillators" and can spontaneously break the initial symmetries such as axial symmetry, homogeneity, isotropy, and even stationarity. Moreover, the growth rates of unstable modes can be of the order of the neutrino-neutrino interaction energy instead of the much smaller vacuum oscillation frequency: self-induced flavor conversion does not always require neutrino masses. We illustrate these newly found phenomena in terms of simple toy models. What happens in realistic astrophysical settings is up to speculation at present

Neutrino-neutrino refraction in dense media can cause self-induced flavor conversion triggered by collective run-away modes of the interacting flavor oscillators. The growth rates were usually found to be of order a typical vacuum oscillation frequency ∆m 2 /2E. However, even in the simple case of a ν e beam interacting with an opposite-movingν e beam, and allowing for spatial inhomogeneities, the growth rate of the fastest-growing Fourier mode is of order µ = √ 2G F n ν , a typical ν-ν interaction energy. This growth rate is much larger than the vacuum oscillation frequency and gives rise to flavor conversion on a much shorter time scale. This phenomenon of "fast flavor conversion" occurs even for vanishing ∆m 2 /2E and thus does not depend on energy, but only on the angle distributions. Moreover, it does not require neutrinos to mix or to have masses, except perhaps for providing seed disturbances. We also construct a simple homogeneous example consisting of intersecting beams and study a schematic supernova model proposed by Ray Sawyer, where ν e andν e emerge with different zenith-angle distributions, the key ingredient for fast flavor conversion. What happens in realistic astrophysical scenarios remains to be understood.

Self-induced flavor conversion of supernova (SN) neutrinos is a generic feature of neutrino-neutrino dispersion. The corresponding run-away modes in flavor space can spontaneously break the original symmetries of the neutrino flux and in particular can spontaneously produce small-scale features as shown in recent schematic studies. However, the unavoidable "multi-angle matter effect" shifts these small-scale instabilities into regions of matter and neutrino density which are not encountered on the way out from a SN. The traditional modes which are uniform on the largest scales are most prone for instabilities and thus provide the most sensitive test for the appearance of self-induced flavor conversion. As a by-product we clarify the relation between the time evolution of an expanding neutrino gas and the radial evolution of a stationary SN neutrino flux. Our results depend on several simplifying assumptions, notably stationarity of the solution, the absence of a "backward" neutrino flux caused by residual scattering, and global spherical symmetry of emission.

IceCube has measured a diffuse astrophysical flux of TeV-PeV neutrinos. The most plausible sources are unique high energy cosmic ray accelerators like hypernova remnants (HNRs) and remnants from gamma ray bursts in star-burst galaxies, which can produce primary cosmic rays with the required energies and abundance. In this case, however, ordinary supernova remnants (SNRs), which are far more abundant than HNRs, produce a comparable or larger neutrino flux in the ranges up to 100-150 TeV energies, implying a spectral break in the IceCube signal around these energies. The SNRs contribution in the diffuse flux up to these hundred TeV energies provides a natural baseline and then constrains the expected PeV flux.