Freeze-in sterile neutrino dark matter in a class of U$(1)^prime$ models with inverse seesaw

We study a class of general U$(1)^prime$ models to explain the observed dark matter relic abundance and light neutrino masses. The model contains three right handed neutrinos and three gauge singlet Majorana fermions to generate the light neutrino mass via the inverse seesaw mechanism. We assign one pair of degenerate sterile neutrinos to be the dark matter candidate whose relic density is generated by the freeze-in mechanism. We consider different regimes of the masses of the dark matter particle and the ${Z^prime}$ gauge boson. The production of the dark matter can occur at different reheating temperatures in various scenarios depending on the masses of the ${Z^prime}$ boson and the dark matter candidate. We also note that if the mass of the sterile neutrino dark matter is $gtrsim 1 rm{MeV}$ and if the $Z^prime$ is heavier than the dark matter, the decay of the dark matter candidate into positrons can explain the long standing puzzle of the galactic $511rm{keV}$ line in the Milky Way center observed by the INTEGRAL satellite. We constrain the model parameters from the dark matter analysis, vacuum stability and the collider searches of heavy ${Z^prime}$ at the LHC. For the case with light $Z^prime$, we also compare how far the parameter space allowed from dark matter relic density can be probed by the future lifetime frontier experiments SHiP and FASERs in the special case of $U(1)_{B-L}$ model.