Predicting Supernova Neutrino Spectra after Slow Collective Flavor Conversion

Prepared by Ian Padilla-Gay

Neutrinos carry away most of the energy released in a core-collapse supernova. They cool the newly formed proto-neutron star, help heat material behind the stalled shock, and influence heavy-element production in the outflow. In the vicinity of stellar core, neutrinos are so abundant that they can affect one another’s flavor evolution in a collective fashion. We studied one such phenomenon, slow flavor conversion (SFC), using multi-energy, multi-angle simulations for three representative stages of an $18,M_\odot$ supernova: early accretion, late accretion, and cooling.

The initial electron- and heavy-lepton-flavor spectra cross in both the neutrino and antineutrino sectors (Figure 1). Below a crossing (see panels Fig 1a-1c), electron-flavor neutrinos are generally more abundant; above it, heavy-lepton flavors dominate. Because supernova heating is especially sensitive to energetic neutrinos, conversion above these crossings can have an important effect on the supernova environment.

Figure 1. Four graphs: early accretion, late accretion, cooling, and lepton-number spectra. — **Figure 1.** Initial spectra for early accretion, late accretion, and cooling. Panels (a)-(c) compare electron, antielectron, and heavy-lepton flavors; panel (d) shows the associated lepton-number spectra. Their crossings make the conversion strongly energy dependent.

We follow the flavor content with a density matrix $\rho(t,z,\omega,v_z)$ , where $\omega$ labels neutrino energy and $v_z$ its direction of propagation. It evolves according to $(\partial_t+v_z\partial_z)\rho=-i[H_{\rm vac}+H_{\nu\nu},\rho]$ , with the two terms on the right describing ordinary vacuum oscillations and coherent interactions among neutrinos. A stability analysis showed that the conversion is triggered mainly by spatially inhomogeneous disturbances. After roughly $t\simeq1000$ in simulation units (of order a few microseconds) the spatially averaged system settles into a nearly stationary state.

Figure 2. Three graphs: early accretion, late accretion, and cooling — **Figure 2.** Evolution of the electron-neutrino (solid) and electron-antineutrino (dashed) flavor fractions. Blue and orange denote the normal and inverted mass orderings; the dotted line marks flavor equipartition.

As Figure 2 shows, the flavor fractions remain close to their initial values, change rapidly when the instability saturates, and then level off. Accretion-phase electron-flavor fractions decrease, while cooling-phase fractions increase. The mass ordering changes the final, energy-integrated flavor fractions by no more than about 0.03. At the same time, the conversion probability for a given neutrino depends strongly on its energy, direction, and mass ordering.

Figure 3. Three columns, with two graphs in each: early accretion, late accretion, and cooling. — **Figure 3.** Initial spectra (solid), simulated post-SFC spectra (dashed), and our analytical estimates (dotted) in the normal mass ordering. High-energy modes generally convert most strongly.

Figure 3 shows the central spectral effect: above the crossing energy, the final $\nu_e$ and $\nu_x$ spectra move toward flavor equipartition, while conversion is more limited at lower energies. There, the final spectra lie between their initial state and complete flavor equipartition. Since charged-current heating is an energy-dependent integral of the spectrum and absorption cross section, this high-energy conversion matters a lot. The $\nu_e$ heating rate increases by 84% (62%) in early accretion, 12% (10%) in late accretion, and 47% (53%) during cooling for the normal (inverted) ordering. The $\bar\nu_e$ rate rises during early accretion and cooling, but falls by about 10% during late accretion because the initial high-energy $\bar\nu_e$ and $\bar\nu_x$ spectra are already similar.

The same changes make the simulated outflow more proton rich. The equilibrium electron fraction, set by the competition between $\nu_e$ absorption on neutrons and $\bar\nu_e$ absorption on protons, increases by 0.027 (0.017), 0.035 (0.029), and 0.036 (0.034) in the three phases for the normal (inverted) ordering. This could influence the production of proton-rich nuclei in neutrino-driven winds.

Most importantly, we found a practical way to estimate the final spectra after flavor evolution. Our “box” prescription reads the positive and negative regions of the initial lepton-number spectrum, places the smaller region near flavor equipartition, and adjusts the other to approximately conserve lepton number. The dotted curves in Figure 3 show the prediction. To test it, we measured the area between each predicted and simulated spectrum and normalized that difference by the simulated spectrum. Combining all flavors, the mismatch is only 1.74-8.55% across the six phase-ordering cases (below 10% in every case). Individual-flavor differences are also usually within 5-10%, with the largest, 13.2%, occurring for $\nu_x$ during early accretion.

SFC is therefore not a uniform reshuffling of neutrino identities: it leaves a distinct energy-dependent imprint that can affect supernova heating and composition. The simple “box” estimation could provide a practical way to carry that imprint into larger neutrino-transport calculations, allowing its consequences to be explored in more realistic CCSN simulations.

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