  We investigate the production of particle Dark Matter (DM) in a minimal
freeze-in model considering a non-instantaneous reheating phase after
inflation. We demonstrate that for low reheating temperatures, bosonic or
fermionic reheating from monomial potentials can lead to a different evolution
in the DM production and hence to distinct predictions for the parent particle
lifetime and mass, constrained by long-lived particle (LLP) searches. We
highlight that such scenario predicts parent particle decay lengths larger
compared to using the instantaneous reheating approximation. Moreover, we
demonstrate the importance of an accurate definition of the reheating
temperature and emphasize its relevance for the correct interpretation of
experimental constraints. We explore different models of inflation, which can
lead to the considered reheating potential. We find that the extent to which
the standard DM freeze-in production can be modified crucially depends on the
underlying inflationary model. Based on the latest CMB constraints, we derive
lower limits on the decay length of the parent particle and confront these
results with the corresponding reach of LLP searches. Our findings underscore
the impact of the specific dynamics of inflation on DM freeze-in production and
highlight their importance for the interpretation of collider signatures. At
the same time, our results indicate the potential for LLP searches to shed
light on the underlying dynamics of reheating.
