Supernova neutrinos and antineutrinos: ternary luminosity diagram and spectral split patterns

In core-collapse supernovae, the nu(e) and nu(e) species may experience collective flavor swaps to non-electron species nu(x), within energy intervals limited by relatively sharp boundaries (splits). These phenomena appear to depend sensitively upon the initial energy spectra and luminosities. We investigate the effect of generic variations of the fractional luminosities (l(e), l((e) over bar), l(x)) with respect to the usual energy equipartition case (1/6, 1/6, 1/ 6), within an early-time supernova scenario with fixed thermal spectra and total luminosity. We represent the constraint l(e) + l((e) over bar) + 4l(x) = 1 in a ternary diagram, which is explored via numerical experiments ( in single-angle approximation) over an evenly-spaced grid of points. In inverted hierarchy, single splits arise in most cases, but an abrupt transition to double splits is observed for a few points surrounding the equipartition one. In normal hierarchy, collective effects turn out to be unobservable at all grid points but one, where single splits occur. Admissible deviations from equipartition may thus induce dramatic changes in the shape of supernova (anti) neutrino spectra. The observed patterns are interpreted in terms of initial flavor polarization vectors (defining boundaries for the single/double split transitions), lepton number conservation, and minimization of potential energy.