  We study the formation of a superfluid condensate of neutrinos inside a
neutrino cluster. The attractive interaction between neutrinos is mediated by a
scalar boson which is lighter than a neutrino. We consider the appearance of
neutrino bound states consisting of particles with oppositely directed spins.
The gap equation for such a system is derived. Based on numerical simulations
of the neutrino distribution in a cluster, we find the phase transition
temperature and the coherence length inside such a cluster for various
parameters of the system. The constraints on the parameters of the Yukawa
interaction, resulting in the neutrino superfluidity, are derived. We obtain
that the cosmic neutrino background can contribute to the superfluid condensate
inside a neutrino cluster having realistic characteristics. The mechanism of
the neutrino cluster cooling in the early universe, based on the plasmons
\v{C}erenkov radiation, is proposed.
