Dark matter phase-in: producing feebly-interacting particles after a first-order phase transition

The freeze-in mechanism describes the out-of-equilibrium production of dark matter (DM) particles via feeble couplings or non-renormalisable interactions with large suppression scales. In the latter case, predictions suffer from a strong sensitivity to the initial conditions of the universe, such as the details of reheating. In this work, we investigate how this sensitivity is altered in the presence of a cosmological first-order phase transition. We show that freeze-in via non-renormalisable interactions is not always dominated by the highest temperatures of the Standard Model (SM) thermal bath, but instead may be governed by the period immediately after the phase transition, during which the decaying scalar field transfers its energy density to the SM radiation. We refer to this alternative production regime as DM $\textit{phase-in}$. Using numerical and approximate analytical solutions of the relevant Boltzmann equations, we determine the conditions that under which phase-in or conventional freeze-in production dominates the final DM abundance in terms of the type of interaction between the DM and SM particles, the amount of supercooling before and the evolution of the scalar field after the phase transition. In the phase-in regime, the DM abundance is correlated with the peak frequency of the gravitational wave signal associated with the phase transition, opening up new observational possibilities.