Revisiting Dark Matter Freeze-in and Freeze-out through Phase-Space Distribution

We revisit dark-matter production through freeze-in and freeze-out by solving the Boltzmann equations at the level of the phase-space distribution $f(p,t)$. Using the $2\to2$ annihilation and the $1\to2$ decay processes for illustration, we compare the resulting dark-matter relic abundance with that from the number-density approach. In the transition regime between freeze-in and freeze-out, we find the difference can be quite significant, or even by orders of magnitude if the annihilation of dark-matter particles or the decaying mediator is neglected. The freeze-in production in the $2\to2$ and the $1\to 2$ processes can also result in non-thermal phase-space distributions, or even multi-modal ones with out-of-equilibrium decay, which can potentially affect structure formation at late times. We also investigate how elastic scatterings can distort such non-thermal distributions.