Long-term 3+1 simulations of primordial black hole formation during radiation domination

We develop an efficient three-dimensional numerical-relativity framework for primordial black-hole (PBH) formation from superhorizon curvature perturbations in a radiation-dominated Universe. We implement flux-conservative relativistic hydrodynamics in the adaptive-mesh-refinement code \textsc{GRChombo} and introduce a cosmologically scaled Gamma-driver that allows the cosmic-time step to grow in proportion to the scale factor. For a representative long-term simulation, the scaled driver preserves the apparent-horizon mass evolution and constraint behavior while reducing the number of coarse-level advances by a factor of approximately $94$ relative to the standard driver. We also construct a conformal-time version of the moving-puncture gauge as an independent check. Applying the framework to a spherical Gaussian curvature profile, we find a collapse threshold $0.79578 < μ_c < 0.79580$ and a critical exponent $γ\simeq 0.3559$, consistent with previous spherically symmetric results. We further fit the late-time PBH mass growth to the Zel'dovich--Novikov accretion law, demonstrating that the code can follow both near-critical collapse and long-term post-formation evolution in three dimensions. The framework provides a foundation for future studies of PBH formation beyond spherical symmetry.