  Gravitational production of massive particles due to cosmic expansion can be
significant during the inflationary and reheating period of the Universe. If
the particle also has non-gravitational interactions that do not significantly
affect its production, numerous observational probes open up, including
cosmological probes. In this work, we focus on the gravitational production of
light vector bosons that couple feebly to the Standard Model (SM) particles.
Due to the very feeble coupling, the light vector bosons never reach thermal
equilibrium, and if the Hubble scale at the end of inflation is above $10^8$
GeV, the gravitational production can overwhelm the thermal production via the
freeze-in mechanism by many orders of magnitude. As a result, much stronger
constraints from the Big Bang Nucleosynthesis (BBN) can be placed on the
lifetime and mass of the vector bosons compared to the scenario where only
thermal production is considered. As an example, we study the sub-GeV scale
dark photons, which couple to the SM only through kinetic mixing, and derive
constraints on the mass and kinetic mixing parameter of the dark photon from
the photodisintegration effects on the light element abundances relevant at the
end of the BBN when the cosmic age was around $10^4$ s.
