A free-fall-based switching criterion for P^3 T N-body methods in collisional stellar systems

The P$^3$T scheme is a hybrid method for simulating gravitational $N$-body systems. It combines a fast particle-tree (PT) algorithm for long-range forces with a high-accuracy particle-particle (PP, direct $N$-body) solver for short-range interactions. Preserving both PT efficiency and PP accuracy requires a robust PT-PP switching criterion. We introduce a simple free-fall-based switching criterion for general stellar systems, alongside the commonly used velocity-dispersion-based ($\sigma$-based) criterion. Using the \textsc{petar} code with the P$^3$T scheme and slow-down algorithmic regularization for binaries and higher-order multiples, we perform extensive simulations of star clusters to evaluate how each criterion affects energy conservation and binary evolution. For systems in virial equilibrium, we find that the free-fall-based criterion is generally more accurate for low-$\sigma$ or loose clusters containing binaries, whereas the $\sigma$-based criterion is better suited for high-$\sigma$ systems. Under subvirial or fractal initial conditions, both criteria struggle to maintain high energy conservation; however, the free-fall-based criterion improves as the tree timestep is reduced, whereas the $\sigma$-based degrades due to its low-accuracy treatment of two-body encounters.