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 $100h^{-1}\mathrm{Mpc}$ or larger. We derive a new scale-dependent parametrization which generalizes the well-known Newtonian approximation result $f_0(a)\equiv \frac{d\ln {\delta }_0}{d\ln a}=\Omega (a{)}^{\gamma }$ ($\gamma =\frac{6}{11}$ for $\Lambda \mathrm{CDM}$) which is a good approximation on scales less than $50h^{-1}\mathrm{Mpc}$. Our generalized parametrization is of the form $f(a)=\frac{f_0(a)}{1+\xi (a,k)}$, where $\xi (a,k)=(3H_0^2{\Omega }_{0\mathrm{m}})/(a{k}^2)$. 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 $\Lambda \mathrm{CDM}$ and dynamical dark energy. In contrast, the scale independent parametrization does not provide a good fit on scales beyond 5% of the horizon scale ($k\simeq 0.01h^{-1}\mathrm{Mpc}$).

Figure 1

(a) The growth factor $f$ obtained from the solution of the general relativistic system ($k=0.01h^{-1}\mathrm{Mpc}$, ${\Omega }_{0\mathrm{m}}=0.3$, $\Lambda \mathrm{CDM}$, dotted line) compared with the scale independent parametrization (continuous line) and the corresponding generalized scale-dependent parametrization (thick dashed line) (b) Similar to (a) for $k=0.004h^{-1}\mathrm{Mpc}$. (c) Similar to (a) for $k=0.001h^{-1}\mathrm{Mpc}$.

Figure 2

(a) The accuracy of the scale independent parametrization in terms of $\Delta$ as defined in Eq. (2.16). (b) Likewise for the scale-dependent parametrization.

Figure 3

(a) The values of the parameters $(\gamma ,\alpha )$ that fit best the general relativistic growth solution as functions of the dark energy equation of state parameter $w_0$ ($w_1$ was fixed to 0). Blue disks correspond to $\alpha$, black triangles to $\gamma$. The error bars in $\alpha$ describe the $1\sigma$ variation of $\alpha$ as the scale $k$ is varied in the range $k\in [0.001,0.1]h^{-1}\mathrm{Mpc}$. (b) Similar to (a) for the dark energy equation of state parameter $w_1$ ($w_0$ was fixed to $-1.4$).

Figure 4

$\Delta$ for the scale independent and scale-dependent parametrizations at a scale of $k=0.004h{\mathrm{Mpc}}^{-1}$ for the ($w_0$, $w_1$) choices shown.