# 8.2. Multi-Zenith-Angle Instability¶

Multi-zenith-angle instability was first described in a paper [Banerjee2011].

In What Context

System with neutrino self-interactions.

Non-isotropic neutrino gas would have velocity (or momentum) related interactions, $$1-\vec v_p\cdot\vec v_q$$, which is in fact a $$1 -\frac{2\sqrt{\pi}}{\sqrt{3}} Y_1^0(\theta,\phi)$$ term.

A small anisotropy leads to a runaway flavor equipartition. [2]

 [2] Raffelt, G. & Smirnov, A. Self-induced spectral splits in supernova neutrino fluxes. Phys. Rev. D 76, (2007). In this paper the author adds a small perturbation to a perfectly isotropic neutrino antineutrino gas. The results show multi-angle instability.

Fig. 8.1 A figure from Raffelt & Simirnov (2007) shows the instability from anisotropic small perturbations. Potential energy grows expotentially, where $$-E_1 = \mu/4 \vec{D_1}^2$$ .

## 8.2.1. References & Notes¶

 [Banerjee2011] Banerjee, A., Dighe, A., & Raffelt, G. (2011). Linearized flavor-stability analysis of dense neutrino streams. Physical Review D, 84(5), 53013. https://doi.org/10.1103/PhysRevD.84.053013

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