The frame-dragging vector potential on galaxy scales from Dark-Matter-only Newtonian $N$-body simulations

Effects of General Relativity are usually neglected in the non-linear evolution of structures, where Newtonian $N$-body simulations are traditionally employed. In the post-Friedmann expansion framework, a weak-field relativistic approximation purpose-built for cosmology, a frame-dragging gravito-magnetic vector potential arises at leading order, sourced by momentum currents. At this order, the vector potential contributes to the metric while leaving the dynamics of the matter fields unaffected, as it does not appear in the Euler equation. It can therefore be extracted a posteriori from standard N-body simulations, where the dynamics is purely Newtonian. Using the Delaunay Tessellation Field Estimator code on the IllustrisTNG simulations, here we extend previous work in order to compute the power spectrum of this vector potential down to galactic scales. The magnitude of the vector potential is two orders of magnitude larger than predicted by perturbation theory, and is a $1\% \sim 0.1\%$ effect compared to the non-linear Newtonian scalar gravitational potential. In the redshift range considered here, the gravito-magnetic effect remains subdominant, without showing any enhancement during a particular phase in the evolution of structures, aside from the continuous growth of non-linearity at low redshift. Although this seems to suggest that, within the $\Lambda$CDM model, no significant gravito-magnetic effects contribute to the non-linear evolution of cosmic structures, i.e. to the dynamics of massive particles, possible observational consequences, e.g. in lensing, deserve further exploration.