Cosmological Simulation with Population III Stellar Feedback and Metal Enrichment I: Model Description And Convergence Test

We present SPARK (Simulation of Pop III in Arepo with Radiation and Kinetic feedback) framework, a new Pop III + Pop II subgrid framework implemented in the moving-mesh code {\sc arepo}, designed to study the impact of Pop III feedback on star formation in the early universe. The framework combines primordial non-equilibrium chemistry, metal-line cooling, IMF-sampled stellar evolution with SN feedback, and approximate Lyman-Werner (LW) and ionizing radiation transport. We run a suite of $1c{\rm Mpc}/h$ box simulations with different initial conditions and resolutions from $z=127$ to $z=10$. The highest gas mass and spatial resolution in the fiducial simulation reach $\sim10\,{\rm M_{\odot}}$ and $\sim4\,{\rm pc}$, respectively. The model successfully reproduces the Pop II star formation rate density (SFRD) consistent with previous theoretical works across all initial conditions, with minor variation mainly driven by local halo interactions and LW irradiation. We find that the volume filling factor of metal-enriched gas converges to $\sim0.5-2\%$ at $z=10$, with scatter driven by stochastic and mixing model variations. Convergence is achieved once subhalos with $M_{\rm subhalo}\gtrsim 10^{6.5}\,{\rm M_{\odot}}$ are resolved, and the total stellar mass at $z=10$ is largely insensitive to initial conditions or the resolution considered in this work. A fiducial simulation requires $\sim 10^4$ CPU hours, making the framework computationally tractable for larger box simulations and enabling future large parameter studies of stellar physics or environment effects such as Pop III IMF variations, X-ray radiation, or the streaming velocity at high redshift.