  The Cosmic Neutrino Background (CNB) is a definite prediction of the standard
cosmological model and its direct discovery would represent a milestone in
cosmology and neutrino physics. In this work, we consider the capture of relic
neutrinos on a tritium target as a possible way to detect the CNB, as aimed for
by the PTOLEMY project. Crucial parameters for this measurement are the
absolute neutrino mass $m_\nu$ and the local neutrino number density
$n_\nu^{\rm loc}$. Within the $\Lambda$CDM model, cosmology provides a
stringent upper limit on the sum of neutrino masses of $\sum m_\nu < 0.12\,{\rm
eV}$, with further improvements expected soon from galaxy surveys by DESI and
EUCLID. This makes the prospects for a CNB detection and a neutrino mass
measurement in the laboratory very difficult. In this context, we consider a
set of non-standard cosmological models that allow for large neutrino masses
($m_\nu \sim 1\,{\rm eV}$), potentially in reach of the KATRIN neutrino mass
experiment or upcoming neutrinoless double-beta decay searches. We show that
the CNB detection prospects could be much higher in some of these models
compared to those in $\Lambda$CDM, and discuss the potential for such a
detection to discriminate between cosmological scenarios. Moreover, we provide
a simple rule to estimate the required values of energy resolution, exposure,
and background rate for a PTOLEMY-like experiment to cover a certain region in
the $(m_\nu,\, n_\nu^{\rm loc})$ parameter space. Alongside this paper, we
publicly release a code to calculate the CNB sensitivity in a given
cosmological model.
