New parametrization for the scale dependent growth function in general relativity

We study the scale-dependent evolution of the growth function delta(a,k) of cosmological perturbations in dark energy models based on general relativity. This scale dependence is more prominent on cosmological scales of 100h(-1) Mpc or larger. We derive a new scale-dependent parametrization which generalizes the well-known Newtonian approximation result f(0)(a)equivalent to dln delta(0)dlna=Omega(a)(gamma) (gamma=611 for Lambda CDM) which is a good approximation on scales less than 50h(-1) Mpc. Our generalized parametrization is of the form f(a)=f(0)(a)1+xi(a,k), where xi(a,k)=(3H(0)(2)Omega(0m))/(ak(2)). We demonstrate that this parametrization fits the exact result of a full general relativistic evaluation of the growth function up to horizon scales for both Lambda CDM and dynamical dark energy. In contrast, the scale independent parametrization does not provide a good fit on scales beyond 5% of the horizon scale (k similar or equal to 0.01h(-1) Mpc).