Self-gravity stabilises topological vortices in ultralight scalar field dark matter (universe)

Self-gravity stabilises topological vortices in ultralight scalar field dark matter (universe) This is Version 2 of the paper, significantly expanded from the original Zenodo release (v1.0). The new version includes: Full 2D simulations in an expanding Universe (comoving coordinates, matter-dominated expansion a(t)∝t2/3a(t)∝t2/3), demonstrating that self-gravity stabilises topological vortices above a critical effective gravitational coupling G~c≈0.3G~c≈0.3–0.40.4. First 3D simulations with the same expanding background, confirming the stabilisation effect in three dimensions and showing the formation of a network of vortex lines. Corrected physics: fixed scale_rho normalisation, proper FFT energy normalisation, adaptive vortex detection threshold, and inclusion of Hubble friction. New observational connection: a dedicated discussion of polar ring galaxies as direct observational evidence for the wave nature of the cosmic scalar field. Main results 2D: vortex number jumps from zero to ~15–78 at G≥0.4G≥0.4; energy ratio ∣Egrav∣/Ekin∣Egrav∣/Ekin decays from ~16 to ~4, indicating virialisation. 3D: total vortex line length shows a similar phase transition, rising to ~30–40 at G≥0.6G≥0.6; energy ratio decays from ~110 to ~20. Polar ring galaxies: the anisotropic interference pattern of the scalar field naturally produces flattened, non-spherical gravitational potentials, consistent with observed halo flattening in NGC 4650A (axis ratio c/a≈0.3c/a≈0.3–0.40.4). Code and data All simulation scripts, analysis notebooks, and averaged data files are publicly available at the GitHub repository. The code reproduces all figures in the paper. Citation Please cite this work as: K. Kashcheyevu, DeepSeek AI, Self-gravity stabilises topological vortices in ultralight scalar field dark matter (universe), Zenodo (2026), DOI: [to be inserted].