  We point out that, for Dirac neutrinos, in addition to the standard thermal
cosmic neutrino background (C$\nu$B) there could also exist a non-thermal
neutrino background with comparable number density. As the right-handed
components are essentially decoupled from the thermal bath of standard model
particles, relic neutrinos with a non-thermal distribution may exist until
today. The relic density of the non-thermal (nt) background can be constrained
by the usual observational bounds on the effective number of massless degrees
of freedom $N_\mathrm{eff}$, and can be as large as
$n_{\nu_{\mathrm{nt}}}\lesssim 0.5\,n_\gamma$. In particular, $N_\mathrm{eff}$
can be larger than 3.046 in the absence of any exotic states. Non-thermal relic
neutrinos constitute an irreducible contribution to the detection of the
C$\nu$B, and, hence, may be discovered by future experiments such as PTOLEMY.
We also present a scenario of chaotic inflation in which a non-thermal
background can naturally be generated by inflationary preheating. The
non-thermal relic neutrinos, thus, may constitute a novel window into the very
early universe.
