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.
