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

TL;DR

This paper investigates the gravito-magnetic frame-dragging vector potential on galaxy scales within a PF cosmology, using DM-only IllustrisTNG simulations and the DTFE method to extract momentum-density sources. By solving the PF equations in Fourier space and cross-checking against second-order perturbation theory, it shows the vector potential is amplified in the non-linear regime yet remains a small correction compared to the scalar potential, with a corrected vector-to-scalar ratio of order 10^-5 across scales and redshifts. The study demonstrates non-linear growth of the vector potential, reveals vortical patterns in the vector field, and provides a global Fourier-space representation of the gravito-magnetic potential from N-body data, highlighting finite-box effects and a need for power-missing corrections. The results support the view that Newtonian N-body simulations are robust for most cosmological observables within ΛCDM, while indicating potential observational windows (e.g., lensing) to probe GR contributions and motivating future hydrodynamical and fully relativistic analyses at galactic scales.

Abstract

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, contributing to the metric even if the dynamics of matter fields at this order is Newtonian, and can thus be extracted from $N$-body simulations. 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 red-shift 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 $Λ$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.

Figures (21)

• Figure 1: Matter power spectra for the TNG300-2-Dark (red), TNG100-2-Dark (blue), and TNG50-2-Dark (green) simulations. The linear matter power spectrum is plotted as reference (grey solid line), together with the non-linear HaloFit prediction (grey dashed line).
• Figure 2: Power spectra for the velocity and momentum fields, plotted with dot-dashed and dashed lines respectively, for the TNG300-2-Dark (red), TNG100-2-Dark (blue), and TNG50-2-Dark (green) simulations, both normalized by $(\mathcal{H} f)^2$ . The linear matter power spectrum is plotted as reference (grey solid line), together with the matter power spectra of the three simulations (coloured dotted lines). All matter power spectra are normalized by $k^2$.
• Figure 3: Power spectra of the gradients of the velocity and momentum density fields for the TNG300-2-Dark (red), TNG100-2-Dark (blue), and TNG50-2-Dark (green) simulations. The divergence and vorticity of the velocity are represented as solid and dashed lines, respectively, while the divergence and vorticity of the momentum are shown with dotted and dash-dotted lines. The linear matter power spectrum is plotted as a reference (grey solid line).
• Figure 4: Power spectra of the main sources of momentum vorticity, i.e. $\delta \nabla \times \mathbf{v}$ (dotted lines) and $\nabla \delta \times \mathbf{v}$ (dashed lines), for the TNG300-2-Dark (red), TNG100-2-Dark (blue), and TNG50-2-Dark (green) simulations.
• Figure 5: Top: power spectra of the vector potential for the TNG300-2-Dark (red), TNG100-2-Dark (blue), and TNG50-2-Dark (green) simulations. The power spectrum of the vector potential predicted with the second-order perturbation theory from Equation (\ref{['eq:2nd_order']}) is shown as reference (grey solid line), along with the non-linear prediction computed from the same equation with the non-linear matter power spectrum of HaloFit (grey dashed line). Bottom: same plot but normalised for the power spectrum of the linear scalar potential $\Delta_{\phi^{\mathrm{N}}}^\mathrm{lin}$.