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Newtonian simulations are routinely used to examine the matter dynamics onnon-linear scales. However, even on these scales, Newtonian gravity is not acomplete description of gravitational effects. A post-Friedmann approach showsthat the leading order correction to Newtonian theory is a vector potential inthe metric. This vector potential can be calculated from N-body simulations,requiring a method for extracting the velocity field. Here, we present the fulldetails of our calculation of the post-Friedmann vector potential, using theDelauney Tesselation Field Estimator (DTFE) code. We include a detailedexamination of the robustness of our numerical result, including the effects ofbox size and mass resolution on the extracted fields. We present the powerspectrum of the vector potential and find that the power spectrum of the vectorpotential is $\sim 10^5$ times smaller than the power spectrum of the fullynon-linear scalar gravitational potential at redshift zero. Comparing ournumerical results to perturbative estimates, we find that the fully non-linearresult can be more than an order of magnitude larger than the perturbativeestimate on small scales. We extend the analysis of the vector potential tomultiple redshifts, showing that this ratio persists over a range of scales andredshifts. We also comment on the implications of our results for the validityand interpretation of Newtonian simulations.

The fully non-linear post-Friedmann frame-dragging vector potential: Magnitude and time evolution from N-body simulations