Newton to Einstein - dust to dust
  We investigate the relation between the standard Newtonian equations for a
pressureless fluid (dust) and the Einstein equations in a double expansion in
small scales and small metric perturbations. We find that parts of the Einstein
equations can be rewritten as a closed system of two coupled differential
equations for the scalar and transverse vector metric perturbations in Poisson
gauge. It is then shown that this system is equivalent to the Newtonian system
of continuity and Euler equations. Brustein and Riotto (2011) conjectured the
equivalence of these systems in the special case where vector perturbations
were neglected. We show that this approach does not lead to the Euler equation
but to a physically different one with large deviations already in the 1-loop
power spectrum. We show that it is also possible to consistently set to zero
the vector perturbations which strongly constrains the allowed initial
conditions, in particular excluding Gaussian ones such that inclusion of vector
perturbations is inevitable in the cosmological context. In addition we derive
nonlinear equations for the gravitational slip and tensor perturbations,
thereby extending Newtonian gravity of a dust fluid to account for nonlinear
light propagation effects and dust-induced gravitational waves.
