  The metric of an FRW universe filled with pressureless dust is shown to
agree, close to any spacetime point, with a curved Newtonian-type metric where
Einstein's equations simplify to those of Newtonian gravity. The agreement is
shown to quadratic order in the local coordinates, so the curvatures agree.
This result is established by expressing both metrics in Riemann normal form.
This approach gives a local Newtonian understanding of cosmology that avoids
the paradoxes of global Newtonian cosmology.
