Parametrising non-linear dark energy perturbations

In this paper, we quantify the non-linear effects from k-essence dark energy through an effective parameter mu that encodes the additional contribution of a dark energy fluid or a modification of gravity to the Poisson equation. This is a first step toward quantifying non-linear effects of dark energy/modified gravity models in a more general approach. We compare our N-body simulation results from k-evolution with predictions from the linear Boltzmann code CLASS, and we show that for the k-essence model one can safely neglect the difference between the two potentials, Phi-psi, and short wave corrections appearing as higher order terms in the Poisson equation, which allows us to use single parameter mu for characterizing this model. We also show that for a large k-essence speed of sound the CLASS results are sufficiently accurate, while for a low speed of sound non-linearities in matter and in the k-essence field are non-negligible. We propose a tanh-based parameterisation for mu, motivated by the results for two cases with low (c(s)(2) = 10-7) and high (c(s)(2) = 10-4) speed of sound, to include the non-linear effects based on the simulation results. This parametric form of mu can be used to improve Fisher forecasts or Newtonian N-body simulations for k-essence models.