# Dark energy perturbations in $N$-body simulations

## TL;DR

This paper develops fully relativistic $N$-body simulations incorporating dark energy perturbations, demonstrating their significance at large scales and ensuring consistency with linear theory predictions from the CLASS Boltzmann solver.

## Contribution

It introduces a method to include dark energy perturbations in $N$-body simulations compatible with general relativity, tested with both fluid and PPF formalisms.

## Key findings

- Dark energy perturbations can cause up to tens of percent differences at large scales.
- Simulations match linear theory results from CLASS for all linear scales.
- Including relativistic effects is crucial for upcoming large scale structure surveys.

## Abstract

We present $N$-body simulations which are fully compatible with general relativity, with dark energy consistently included at both the background and perturbation level. We test our approach for dark energy parameterised as both a fluid, and using the parameterised post-Friedmann (PPF) formalism. In most cases, dark energy is very smooth relative to dark matter so that its leading effect on structure formation is the change to the background expansion rate. This can be easily incorporated into Newtonian $N$-body simulations by changing the Friedmann equation. However, dark energy perturbations and relativistic corrections can lead to differences relative to Newtonian $N$-body simulations at the tens of percent level for scales $k < (10^{-3} \unicode{x2013} 10^{-2})\,\mathrm{Mpc}^{-1}$, and given the accuracy of upcoming large scale structure surveys such effects must be included. In this paper we will study both effects in detail and highlight the conditions under which they are important. We also show that our $N$-body simulations exactly reproduce the results of the Boltzmann solver CLASS for all scales which remain linear.