  If neutrinos are Dirac particles and their right-handed component can be
copiously produced in the early universe, then they could influence a direct
observation of the cosmic neutrino background, which, most likely, will come
about with the recently proposed PTOLEMY experiment. For the production
mechanism of right-handed neutrinos we use a state-of-the-art version of gauge
field theory deformed by the spacetime noncommutativity, to disclose by it not
only the decoupling temperature for the said neutrino component, but also the
otherwise hidden coupling temperature. Considering two relevant processes, the
plasmon decay and the neutrino elastic scattering, we study the interplay
between the structure of the noncommutativity parameter $\theta^{\mu \nu}$
(type of noncommutativity) and the reheating temperature after inflation to
obtain otherwise elusive upper bound on the scale of noncommutativity
$\Lambda_{\rm NC}$. If PTOLEMY enhanced capture rate is due to spacetime
noncommutativity, we verify that a nontrivial maximum upper bound on
$\Lambda_{\rm NC}$ (a way below the Planck scale) emerges for a space-like
$\theta^{\mu \nu}$ and sufficiently high reheating temperature.
