  We study the prospect to detect the cosmic background of sterile neutrinos in
the Tritium $\beta$-decay, such as PTOLEMY-like experiments. The sterile
neutrino with mass between 1 eV - 10 keV may contribute to the local density as
warm or cold DM component. In this study, we investigate the possibility for
searching them in the models with different production in the early Universe,
without assuming sterile neutrino as full dark matter component. In these
models, especially with low-reheating temperature or phase transition, the
capture rate per year can be greatly enhanced to be $\mathcal O(10)$ without
violating other astrophysical and cosmological observations.
