We revisit the issue of interpreting the results of large volume cosmological simulations in the context of large scale general relativistic effects. We look for simple modifications to the nonlinear evolution of the gravitational potential $psi$ that lead on large scales to the correct, fully relativistic description of density perturbations in the Newtonian gauge. We note that the relativistic constraint equation for $psi$ can be cast as a diffusion equation, with a diffusion length scale determined by the expansion of the Universe. Exploiting the weak time evolution of $psi$ in all regimes of interest, this equation can be further accurately approximated as a Helmholtz equation, with an effective relativistic screening scale $ell$ related to the Hubble radius. We demonstrate that it is thus possible to carry out N-body simulations in the Newtonian gauge by replacing Poissons equation with this Helmholtz equation, involving a trivial change in the Greens function kernel. Our results also motivate a simple, approximate (but very accurate) gauge transformation - $delta_{rm N}(mathbf{k}) approx delta_{rm sim}(mathbf{k})times (k^2+ell^{-2})/k^2$ - to convert the density field $delta_{rm sim}$ of standard collisionless N-body simulations (initialised in the comoving synchronous gauge) into the Newtonian gauge density $delta_{rm N}$ at arbitrary times. A similar conversion can also be written in terms of particle positions. Our results can be interpreted in terms of a Jeans stability criterion induced by the expansion of the Universe. The appearance of the screening scale $ell$ in the evolution of $psi$, in particular, leads to a natural resolution of the Jeans swindle in the presence of super-horizon modes.
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