  In the standard approach to studying cosmological structure formation, the
overall expansion of the Universe is assumed to be homogeneous, with the
gravitational effect of inhomogeneities encoded entirely in a Newtonian
potential. A topic of ongoing debate is to what degree this fully captures the
dynamics dictated by general relativity, especially in the era of precision
cosmology. To quantitatively assess this, we directly compare standard N-body
Newtonian calculations to full numerical solutions of the Einstein equations,
for cold matter with various magnitude initial inhomogeneities on scales
comparable to the Hubble horizon. We analyze the differences in the evolution
of density, luminosity distance, and other quantities defined with respect to
fiducial observers. This is carried out by reconstructing the effective
spacetime and matter fields dictated by the Newtonian quantities, and by taking
care to distinguish effects of numerical resolution. We find that the fully
general relativistic and Newtonian calculations show excellent agreement, even
well into the nonlinear regime. They only notably differ in regions where the
weak gravity assumption breaks down, which arise when considering extreme cases
with perturbations exceeding standard values.
