Does the finite size of the proto-neutron star preclude supernova neutrino flavor scintillation due to turbulence?

Abstract

The transition probabilities describing the evolution of a neutrino with a given energy along some ray through a turbulent supernova profile are random variates unique to each ray. If the proto--neutron-star source of the neutrinos were a point, then one might expect the evolution of the turbulence would cause the flavor composition of the neutrinos to vary in time i.e. the flavor would scintillate. But in reality the proto--neutron star is not a point source---it has a size of order $\ensuremath{\sim}10\text{ }\text{ }\mathrm{km}$, so the neutrinos emitted from different points at the source will each have seen different turbulence. The finite source size will reduce the correlation of the flavor transition probabilities along different trajectories and reduce the magnitude of the flavor scintillation. To determine whether the finite size of the proto--neutron star will preclude flavor scintillation, we calculate the correlation of the neutrino flavor transition probabilities through turbulent supernova profiles as a function of the separation $\ensuremath{\delta}x$ between the emission points. The correlation will depend upon the power spectrum used for the turbulence, and we consider two cases: when the power spectrum is isotropic, and the more realistic case of a power spectrum which is anisotropic on large scales and isotropic on small. Although it is dependent on a number of uncalibrated parameters, we show the supernova neutrino source is not of sufficient size to significantly blur flavor scintillation in all mixing channels when using an isotropic spectrum, and this same result holds when using an anisotropic spectrum, except when we greatly reduce the similarity of the turbulence along parallel trajectories separated by $\ensuremath{\sim}10\text{ }\text{ }\mathrm{km}$ or less.