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 $\texttt{CLASS}$, 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
$\texttt{CLASS}$ 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.
