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We consider the impact of neutrino self-interactions described by an four-fermion coupling on cosmological observations. Implementing the Boltzmann hierarchy for interacting neutrinos first derived in: 1409. 1577 into the Boltzmann solver CLASS, we perform a detailed analysis of the effects of the interaction on the cosmic microwave (CMB) anisotropies, and compare our results with known in the literature. While we find good agreement between our approach and the relaxation time approximation used in some recent, the popular \ (c_², c_²\) -parameterisation fails to reproduce the correct scale dependence of CMB temperature power spectrum. We then proceed to derive constraints on effective coupling constant G_ using currently available data via an MCMC analysis. Interestingly, our results reveal a posterior distribution, where one mode represents the standard\CDM limit with G\ ₄₅₅ \ 10⁸ \\, G\ ₅, and the other scenario in which neutrinos self-interact with an effective coupling constantG\ ₄₅₅ \ 3 \ 10⁹ \\, G\ ₅.
Oldengott et al. (Fri,) studied this question.