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 Poisson’s equation with this Helmholtz equation, involving a trivial change in the Green’s function kernel. Our results also motivate a simple, approximate (but very accurate) gauge transformation—${\delta }_{\mathrm{N}}(\mathbf{k})\approx {\delta }_{\mathrm{sim}}(\mathbf{k})\times (k^2+{\ell }^{-2})/k^2$—to convert the density field ${\delta }_{\mathrm{sim}}$ of standard collisionless $N$-body simulations (initialized in the comoving synchronous gauge) into the Newtonian gauge density ${\delta }_{\mathrm{N}}$ 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 superhorizon modes.

• Received 24 February 2016

DOI:https://doi.org/10.1103/PhysRevD.94.083511