Freeze-in produced dark matter in the ultra-relativistic regime

When dark matter particles only feebly interact with plasma constituents in the early universe, they never reach thermal equilibrium. As opposed to the freeze-out mechanism, where the dark matter abundance is determined at $T \ll M$, the energy density of a feebly interacting state builds up and increases over $T \gtrsim M$. In this work, we address the impact of the high-temperature regime on the dark matter production rate, where the dark and Standard Model particles are ultra-relativistic and nearly light-like. In this setting, multiple soft scatterings, as well as $2 \to 2$ processes, are found to give a large contribution to the production rate. Within the model we consider in this work, namely a Majorana fermion dark matter of mass $M$ accompanied by a heavier scalar $-$ with mass splitting $\Delta M$ $-$ which shares interactions with the visible sector, the energy density can be dramatically underestimated when neglecting the high-temperature dynamics. We find that the overall effective $1 \leftrightarrow 2$ and $2 \to2$ high-temperature contributions to dark-matter production give $\mathcal{O}(10)$ (20\%) corrections for $\Delta M /M =0.1$ ($\Delta M /M =10$) to the Born production rate with in-vacuum masses and matrix elements. We also assess the impact of bound-state effects on the late-time annihilations of the heavier scalar, in the context of the super-WIMP mechanism.