Constraints on $N_{\rm{eff}}$ of high energy non-thermal neutrino injections upto $z\sim 10^8$ from CMB spectral distortions and abundance of light elements

High energy neutrinos and anti-neutrinos ($\gtrsim$ 100 GeV) can inject energetic electromagnetic particles into the baryon-photon plasma in the high redshift universe through electroweak showers from electroweak bremsstrahlung, inelastic scattering with the background electrons and nucleons, and by pair-production of standard model particles on background neutrinos and anti-neutrinos. In this paper, we evolve the particle cascades of high energy non-thermal neutrinos injections, using dark matter decay as a specific example, including relevant collision processes of these neutrinos with the background particles and taking into account the expansion of the universe. We study the effect of these non-thermal neutrino injections on the CMB spectral shape and abundance of light elements produced in the big bang nucleosynthesis. We show that CMB spectral distortions and abundance of light elements can constrain neutrino energy density at the recombination, parameterized as contribution to $N_{\rm{eff}}$, from high energy neutrino injection. These constraints are stronger by several orders of magnitudes compared to the CMB anisotropy constraints. We also show that CMB spectral distortions can probe neutrino injections to significantly higher redshifts ($z>2\times 10^6$) as compared to pure electromagnetic energy injection.