On The Equivalence of the FRW Field Equations and those of Newtonian   Cosmology
  We present a simple argument to explain why the field equations of the Friedmann-Robertson-Walker metric are equivalent to those of Newtonian cosmology. By passing to the infinite limit of a family of conformally rescaled FRW metrics in suitable coordinates, we reveal Newtonian space and time. The limiting process preserves the Einstein equations and these may be elucidated directly from the Newtonian limit up to the determination of the scalar curvature parameter. Consideration of the conformally invariant scalar field equation on the FRW spacetime is used to recover the Einstein equations efficiently from the Newtonian theory. We proceed to examine the limiting procedure in connection with the Cartan formulation of Newtonian gravity. 
Clive G. Wells
gr-qc astro-ph.CO