NS‑VGR: A Non-linear Saturated Velocity–Gravity Relation for Cosmic Velocity Field Reconstruction

The reconstruction of galaxy peculiar velocity fields is crucial for probing large-scale structure and testing cosmological models. This work introduces NS‑VGR, a non‑linear saturated velocity–gravity relation that applies different corrections in overdense clusters and underdense voids. Using the CosmicFlows‑4 catalog of 9,355 galaxies, NS‑VGR achieves a 55% improvement in correlation between predicted and observed peculiar velocities over standard linear theory while retaining fast O(N log N) performance on 128^3 grids.