Linear perturbations in viable $f(R)$ theories

We describe the cosmological evolution predicted by three distinct $f(R)$ theories, with emphasis on the evolution of linear perturbations. The most promising observational tools for distinguishing $f(R)$ theories from $\Lambda \mathrm{CDM}$ are those intrinsically related to the growth of structure, such as weak lensing. At the linear level, the enhancement in the gravitational potential provided by the additional $f(R)$ “fifth force” can separate the theories, whereas at the background level they can be indistinguishable. Under the stringent constraints imposed on the models by Solar System tests and galaxy-formation criteria, we show that the relative difference between the models’ linear evolution of the lensing potential will be extremely hard to detect even with future space-based experiments such as Euclid, with a maximum value of approximately 4% for small scales. We also show the evolution of the gravitational potentials under more relaxed local constraint conditions, where the relative difference between these models and $\Lambda \mathrm{CDM}$ could prove discriminating.

Figure 1

The Hubble parameter, Ricci scalar, and $f(R)$ effective equation of state as a function of redshift, $z$, for the Starobinsky model, compared to the $\Lambda \mathrm{CDM}$ model.

Figure 2

The $V(R)$ potential as a function of the Ricci scalar for the Starobinsky model.

Figure 3

The lensing potential ${\Phi }_{+}$ and $\chi$ as a function of the scale factor, $a$, for the Starobinsky model, for the first set of parameters.

Figure 4

The lensing potential ${\Phi }_{+}$ and $\chi$ as a function of $a$ for the Starobinsky model, for the second set of parameters.

Figure 5

The form of $f_{RR}$ as a function of redshift for the two cases considered in the Starobinsky model.

Figure 6

The Hubble parameter, Ricci scalar, and $f(R)$ effective equation of state for the Hu-Sawicki model, compared to the $\Lambda \mathrm{CDM}$ model.

Figure 7

The potential $V(R)$ for the two different cases considered in the Hu-Sawicki model.

Figure 8

The lensing potential ${\Phi }_{+}$ and $\chi$ as a function of $a$ for the Hu-Sawicki model, for the first set of parameters.

Figure 9

The lensing potential ${\Phi }_{+}$ and $\chi$ as a function of $a$ for the Hu-Sawicki model, for the second set of parameters.

Figure 10

The form of $f_{RR}$ as a function of redshift for the two cases considered in the Hu-Sawicki model.

Figure 11

The Hubble parameter, Ricci scalar, and $f(R)$ effective equation of state for the exponential model, compared to the $\Lambda \mathrm{CDM}$ model.

Figure 12

The potential $V(R)$ for the two cases considered in the exponential model.

Figure 13

The lensing potential ${\Phi }_{+}$ and $\chi$ as a function of $a$ for the exponential model, for the first set of parameters.

Figure 14

The lensing potential ${\Phi }_{+}$ and $\chi$ as a function of $a$ for the exponential model, for the second set of parameters.

Figure 15

The form of $f_{RR}$ as a function of redshift for the two different cases considered in the exponential model.

Figure 16

The evolution of the linear perturbations as a function of the scale factor, $a$, for the $f(R)$ model with $w_{\mathrm{eff}}=-1$, against the $\Lambda \mathrm{CDM}$ model. The middle plot shows the relative difference between both models in the evolution of ${\Phi }_{+}$. The result for $k=0.01h/\mathrm{Mpc}$ was enhanced by a factor of 100 to allow its visualization.