Thermal Evolution and Hydrodynamic Filtering of Pseudoscalar Dark Matter

Filtered dark matter provides a mechanism for producing massive dark matter particles during a first order phase transition. The resulting abundance can be modified by plasma hydrodynamics. We investigate this effect in a complex singlet extension of the Standard Model, focusing on the deflagration regime. The entropy normalized dark matter abundance $Y_\chi$ is calculated using both analytic and numerical methods. As the dark matter mass increases, we identify three patterns of abundance evolution before nucleation, ranging from thermal equilibrium to thermal suppression and freeze-out. For $\xi_w=0.01$, we find that shock heating increases $Y_\chi$ by factors of approximately $4.3$, $5.8$, and $32$ for $m_\chi=1.78$, $2.30$, and $5.03~\mathrm{TeV}$, respectively. For the two lighter masses, the numerical and analytic results agree within $5\%$, and the difference is reduced to about $1\%$ or less when hydrodynamic effects are included. The heaviest case requires a numerical treatment because dark matter freeze-out before nucleation. Our results show that hydrodynamic reheating can enhance $Y_\chi$ even when dark matter has already frozen out before the phase transition. In this regime, $Y_\chi$ is determined by the dark matter abundance established before nucleation and hydrodynamic filtering at the bubble wall.