Abstract
We use gauge-invariant cosmological perturbation theory to calculate the displacement field that sets the initial conditions for $N$-body simulations. Using first and second-order fully relativistic perturbation theory in the synchronous-comoving gauge, allows us to go beyond the Newtonian predictions and to calculate relativistic corrections to it. We use an Einstein--de Sitter model, including both growing and decaying modes in our solutions. The impact of our results should be assessed through the implementation of the featured displacement in cosmological $N$-body simulations.
Highlights
Recent years have seen tremendous progress in both observational and theoretical cosmology
For simplicity we restricted our derivation to an Einstein–de Sitter universe, the calculations can be readily extended to other models of the Universe
We have focused on determining all the terms in the solutions, including the decaying modes of the displacement field, which were so far unknown and could become important at early times, where the influence of the cosmological constant should be negligible
Summary
Recent years have seen tremendous progress in both observational and theoretical cosmology. Relativistic corrections can be important in simulations even using Newtonian physics and cold dark matter particles, when the size of the simulated box is the same as the horizon size at the time that the initial conditions are set. Some recent work has focused on the effects of incorporating relativistic corrections into N -body simulations [6,7,8,9], or interpreting Newtonian simulations in terms of General Relativity [10, 11]. We compute the second-order corrections to the initial conditions using cosmological perturbation theory and keeping the decaying mode at all orders. The paper is structured as follows: we present the governing equations for the system for background, linear and second-order perturbations.
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