3 FIG. 3: Impact of increasing the solar mass splitting Δ m 2 ⊙ on the neutrino spectra at 500 km. Large Δ m 2 ⊙ makes the evo- lution more adiabatic, leading to the formation of a (broad) spectral split. FIG. 2: Impact of decreasing the solar mass splitting, Δ m 2 $_{⊙}$, on the neutrino spectra at 1000 km. Notice that, while a strictly vanishing Δ m 2 ⊙ gives the two-flavor result, even a tiny nonzero value of Δ m 2 ⊙ qualitatively changes the answer. two-flavor calculation, we set the solar mixing angle θ$_{12}$ to zero and drop the state that in vacuum is separated from the predominately ν$_{e}$ (¯ ν$_{e}$ ) state by Δ m 2 $_{⊙}$. It looks paradoxical at first that the solar splitting completely changes the evolution of neutrinos at high en- ergies, even though it is only ∼ 3% of the atmospheric splitting. Clearly, at this level Δ m 2 ⊙ cannot “overpower” Δ m 2 $_{atm}$; what it can do, however, is change the nature of the collective motion, allowing it to extend from the 2-flavor subspace into the full 3-flavor space. Evidently, this results in a radical change, not a small correction. Let us investigate how this change happens as the value of the solar splitting is turned on. The results are shown in Fig. 2. These at first may be even more surprising: the two-flavor spectrum is reproduced only when Δ m 2 ⊙ = 0; as soon as it is nonzero, even very small, the high-energy split disappears. Since for Δ m 2 ⊙ = 7 . 7 × 10 − 7 eV 2 (1% of its true value) the corresponding oscillation length is 10 4 km – much longer than the scales in the problem – one might think the two-flavor limit should be reached. Instead, the spectrum in this case is closer to the realistic three-flavor one than to the two-flavor one. To understand this result, it proves useful to examine the evolution as a function of radius in the matter mass basis [49, 51, 52]. Neutrinos, initially in the mass eigen- states, develop an instability which leads to large collec- tive oscillations[27, 37, 38]. What is interesting in our case is that, shortly after the collective oscillations de- velop between the “atmospheric” eigenstates ( ν$_{2}$$_{,}$$_{3}$ ), the third state ( ν$_{1}$ ) joins in. Therefore, not only is the initial configuration unstable, but the two-flavor trajectory is also unstable in the three-flavor space. A small nonzero Δ m 2 ⊙ is enough to displace the system from the two- flavor subspace; the instability makes it then run away from it (driven primarily by Δ m 2 $_{atm}$). The role of small Δ m 2 ⊙ is thus similar to that of small θ$_{13}$ in the develop- ment of the Kostelecky-Samuel instability [27, 37, 38]. Since the evolution is no longer constrained to the two- flavor subspace, its final state may be expected to be The results of the two-flavor calculations appear to be in very good agreement with the inverted hierarchy cal- culations of [48]. The three-flavor calculation results are visibly different: (i) the high-energy split in the ν$_{e}$ chan- nel is gone; (ii) in the ¯ ν$_{e}$ channel, the flavor swap prob- ability is neither zero, nor one, but increases gradually with neutrino energy. The swap is incomplete in this case. We perform multienergy, single-angle calculations of the evolution, starting at 40 km and ending at 1000 km. 4. Results: comparison of two- and three-flavor runs . – The resulting spectra at 1000 km for the IH case (Δ m 2 atm < 0) are presented in Fig. 1. We concentrate here on this case, because it most clearly illustrates the effect; both hierarchies are treated and contrasted in [49]. The top panels show the two-flavor calculations, the bot- tom ones, the corresponding three-flavor runs. The ν$_{e}$ spectra are on the left, those for ¯ ν$_{e}$ are on the right. The dashed curves and the dotted curves show the corre- sponding initial spectra (see legend). Animations show- ing the evolution of the spectra as a function of the radius for both IH and NH are available as Supplemental Ma- terial online [51, 52]. 5. Discussion . – Both of these results deserve further investigation. First we can establish that the physics driving flavor conversion before 1000 km are the neutrino- neutrino interactions, with the conventional MSW effect being unimportant. The atmospheric level crossing does occur here, but for the chosen parameters it is strongly nonadiabatic (flavor preserving). Moreover, it occurs when r ≳ 600 km, by which point the neutrino self- refraction effects have ceased. The small MSW effects are seen in the ¯ ν$_{e}$ channel as small wiggles. 0 10 20 30 40 50 60 0 0.5 1. 1.5 2. 2.5 3. neutrino energy @ MeV D n$_{e}$ at 1000 km, different D m$_{GLYPH<159>}$ 2 initial n$_{x}$ initial n$_{e}$ D m$_{GLYPH<159>}$ $^{2}$=$_{0.5}$H$_{std. val.}$ L D m$_{GLYPH<159>}$ $^{2}$=$_{0.2}$H$_{std. val.}$ L D m$_{GLYPH<159>}$ $^{2}$=$_{0.01}$H$_{std. val.}$ L D m$_{GLYPH<159>}$ $^{2}$= 0 0 10 20 30 40 50 60 0 0.5 1. 1.5 neutrino energy @ MeV D n$_{e}$ at 500 km, different D m$_{GLYPH<159>}$ 2 initial n$_{x}$ initial n$_{e}$ D m$_{GLYPH<159>}$ $^{2}$=$_{10}$H$_{std. val.}$ L D m$_{GLYPH<159>}$ $^{2}$=$_{5}$H$_{std. val.}$ L D m$_{GLYPH<159>}$ $^{2}$= H$_{std. val.}$L