Linear stability analysis of collective neutrino oscillations without spurious modes

Collective neutrino oscillations are induced by the presence of neutrinos themselves. As such they are intrinsically nonlinear phenomena and are much more complex than linear counterparts such as the vacuum or MSW oscillations. They obey integro-differential equations, numerical solutions of which are also very challenging. If one focuses on the onset of the collective oscillations, on the other hand, the equations can be linearized and the technique of linear analysis can be employed. Unfortunately, however, it is well known that such an analysis, when applied with discretizations of continuous angular distributions, suffers from the appearance of so-called spurious modes, unphysical eigenmodes of the discretized linear equations. In this paper, we analyze in detail the origin of these unphysical modes and present a simple solution to this annoying problem. We have found that the spurious modes originate from the artificial production of pole singularities instead of a branch cut in the Riemann surface by the discretizations. The branching point singularities in the Riemann surface for the original undiscretized equations can be recovered by approximating the angular distributions with polynomials and then performing the integrals analytically. We demonstrate for some examples that this simple prescription removes the spurious modes indeed. We also propose an even simpler method: a piecewise linear approximation to the angular distribution. It is shown that the same methodology is applicable to the multi-energy case as well as to the dispersion relation approach that was proposed very recently.