Parametrizing growth in dark energy and modified gravity models

It is well known that an extremely accurate parametrization of the growth function of matter density perturbations in $\mathrm{\Lambda }\mathrm{CDM}$ cosmology, with errors below 0.25%, is given by $f(a)={\mathrm{\Omega }}_m^{\gamma }(a)$ with $\gamma \simeq 0.55$. In this work, we show that a simple modification of this expression also provides a good description of growth in modified gravity theories. We consider the model-independent approach to modified gravity in terms of an effective Newton constant written as $\mu (a,k)=G_{\mathrm{eff}}/G$ and show that $f(a)=\beta (a){\mathrm{\Omega }}_m^{\gamma }(a)$ provides fits to the numerical solutions with similar accuracy to that of $\mathrm{\Lambda }\mathrm{CDM}$. In the time-independent case with $\mu =\mu (k)$, simple analytic expressions for $\beta (\mu )$ and $\gamma (\mu )$ are presented. In the time-dependent (but scale-independent) case $\mu =\mu (a)$, we show that $\beta (a)$ has the same time dependence as $\mu (a)$. As an example, explicit formulas are provided in the Dvali-Gabadadze-Porrati (DGP) model. In the general case, for theories with $\mu (a,k)$, we obtain a perturbative expansion for $\beta (\mu )$ around the general relativity case $\mu =1$ which, for $f(R)$ theories, reaches an accuracy below 1%. Finally, as an example we apply the obtained fitting functions in order to forecast the precision with which future galaxy surveys will be able to measure the $\mu$ parameter.

Figure 1

From left to right: $\gamma$, $\beta$ and the relative error in the growth function as a function of $\mu$. The dashed red line corresponds to the analytical approximation in (20) with $\gamma ={\gamma }_{*}$ and $\beta$ in (19). The continuous blue line corresponds to the numerical fitting of $\beta$ and $\gamma$ to expression (22).

Figure 2

Errors for $f(z)$, ${\delta }_m(z)$ and $f(z){\delta }_m(z)$ in $\mathrm{\Lambda }\mathrm{CDM}$ using ${\gamma }_{*}=0.554$ (left) and ${\gamma }_{*}=0.550$ (right). We see that for ${\gamma }_{*}=0.554$ the error in $f(z)$ is below 0.25% but the error in ${\delta }_m(z)$ reaches 0.15%, whereas for ${\gamma }_{*}=0.550$, the error in ${\delta }_m(z)$ can be reduced below 0.05%, but the corresponding error for $f(z)$ grows to 0.7%.

Figure 3

From left to right, functions $\gamma (\mu )$ (31), $\beta (\mu )$ (19) and the growth function relative error, for time-independent $\mu$ for different values of ${\omega }_0$ with ${\omega }_1=0$. We have assumed that ${\mathrm{\Omega }}_m=0.271$.

Figure 4

From left to right, functions $\gamma (\mu )$ (31), $\beta (\mu )$ (19) and the growth function relative error, for time-independent $\mu$ for different values of ${\omega }_1$ with ${\omega }_0=-1$. We have assumed that ${\mathrm{\Omega }}_m=0.271$.

Figure 5

Relative difference between the numerical solution and the fitting function (34) for the model $\mu (a)=1+ba$. As we see, the maximum error is reached at $z=0$.

Figure 6

Numerical solutions in the model $\mu (a)=1+ba$ for different values of $b$. The fit according to Eq. (34) is not represented because it differs by less than 0.6% with respect to the numerical solution.

Figure 7

From left to right, the functions $\gamma ({\mathrm{\Omega }}_0)$, ${\mathrm{\Omega }}_0^{*}({\mathrm{\Omega }}_0)$ and the relative error of the growth function for the DGP model. The error decreases as we increase the value of ${\mathrm{\Omega }}_0$. The maximum error is less than 0.5% for typical values of ${\mathrm{\Omega }}_0$.

Figure 8

From left to right, the functions $\gamma ({\mu }_0)$, ${\mu }_0^{*}({\mu }_0)$ and the relative error of the growth function for the phenomenological model (43) for different values of ${\omega }_0$, setting ${\omega }_1=0$. The case with ${\omega }_0=-1$ is parametrized in (46) and (47).

Figure 9

From left to right, the functions $\gamma ({\mu }_0)$, ${\mu }_0^{*}({\mu }_0)$ and the relative error of the growth function for the phenomenological model (43) for different values of ${\omega }_1$, setting ${\omega }_0=-1$. The case with ${\omega }_1=0$ is parametrized in (46) and (47).

Figure 10

Left panel: $\mu (a,k)$ with $a=1$ as a function of $b(k/H_0{)}^2$ for the $f(R)$ theory in (55). We compare the exact expression in (56) with the approximation in Eq. (60). Right panel: The growth function relative error for the approximation (60) for the $\mathrm{\Lambda }\mathrm{CDM}$ background.

Figure 11

Errors in $f(z)$ for the phenomenological parametrization (63) for the $\mathrm{\Lambda }\mathrm{CDM}$ background.

Figure 12

1-$\sigma$ and $2\text{−}\sigma$ contours for $({\mathrm{\Omega }}_m,\mu )$ using an example of future galaxy redshift survey with $z$ in [0.5, 2.1], $15500{\mathrm{deg}}^2$, $\delta z=0.001$ and a mean galaxy density of $n(z)=1.12\times {10}^{-3}(\mathrm{h}/\mathrm{Mpc}{)}^3$.