Fourier Analysis of the Parametric Resonance of the Neutrino Oscillation in the Presence of Inhomogeneous Matter

We study the parametric resonance of the neutrino oscillation through the matter whose density varies spatially. The Fourier analysis of the matter effect enables us to clarify the parametric resonance condition, which is summarized in a frequency matching between the neutrino oscillation and the spatial variation of the matter density. As a result, the n-th Fourier mode of a matter density profile modifies the energy spectrum of the νμ → νe appearance probability at around the n-th dip. The oscillation probability of neutrinos passing through the interior of the Earth is strongly affected by the matter on the baseline. In such cases, the spatial variation of the matter density can make the probability quite different from that under the constant matter density [1, 2, 3]. The possibility of the parametric resonance of the neutrino oscillation with the matter density has been discussed [1]. We exploit the Fourier analysis of the matter density profile to study the effect of parametric resonance [3, 4]. Generally, the parametric resonance of an oscillation emerges when an oscillation parameter changes with a resonant frequency, which is typically twice the natural frequency of the system. Our formulation based on the Fourier analysis gives a clear view of this frequency matching, which is essential to the parametric resonance. We carry out our analysis in a two-flavor framework. This simplification does not spoil our point as can be seen in Fig. 1, in which we compare the νμ → νe appearance probability in the threeand two-flavor calculation, taking the baseline length L = 12000 km. The three-flavor result assumes the density profile of the Preliminary Reference Earth Model (solid curve), while two-flavor result uses the constant density case (dotted curve) and the cosine variation around the average density (broken curve). Renormalized by the factor of sin 2θ23, the two-flavor result with the cosine profile shows the similar feature as the three-flavor one. The evolution equation of two flavors of neutrinos νe and νμ in the matter density ρ(x) is i d dx (