Reconciling dark energy models with $f(R)$ theories

Higher-order theories of gravity have recently attracted a lot of interest as alternative candidates to explain the observed cosmic acceleration without the need of introducing any scalar field. A critical ingredient is the choice of the function $f(R)$ of the Ricci scalar curvature entering the gravity Lagrangian and determining the dynamics of the Universe. We describe an efficient procedure to reconstruct $f(R)$ from the Hubble parameter $H$ depending on the redshift $z$. Using the metric formulation of $f(R)$ theories, we derive a third order linear differential equation for $f[R(z)]$ which can be numerically solved after setting the boundary conditions on the basis of physical considerations. Since $H(z)$ can be reconstructed from the astrophysical data, the method we present makes it possible to determine, in principle, what is the $f(R)$ theory which best reproduces the observed cosmological dynamics. Moreover, the method allows to reconcile dark energy models with $f(R)$ theories finding out what is the expression of $f(R)$ which leads to the same $H(z)$ of the given quintessence model. As interesting examples, we consider “quiessence” (dark energy with constant equation of state) and the Chaplygin gas.

Figure 1

Reconstructed $f(z)$ for quiessence models with ${\Omega }_M=0.3$ and three different values of the dark energy barotropic factor, namely $w=-0.5$ (short dashed line), $w=-1$ (solid line), and $w=-1.5$ (long dashed line). We report $lf=\ln (-f)$ rather than $f$ to better show the results.

Figure 2

Reconstructed $f(R)$ for quiessence models with ${\Omega }_M=0.3$ and three different values of the dark energy barotropic factor, namely $w=-0.5$ (short dashed line), $w=-1$ (solid line), and $w=-1.5$ (long dashed line). We report $lf=\ln (-f)$ as function of $lR=\ln (-R)$ rather than $f(R)$ to better show the results.

Figure 3

Reconstructed $f(z)$ for Chaplygin gas with ${\Omega }_M=0.3$ and three different values of the normalization constant, namely $A_s=0.25$ (short dashed line), $A_s=0.50$ (solid line), and $A_s=0.75$ (long dashed line). We report $lf=\ln (-f)$ rather than $f$ to better show the results.

Figure 4

Reconstructed $f(R)$ for Chaplygin gas with ${\Omega }_M=0.3$ and three different values of the normalization constant, namely $A_s=0.25$ (short dashed line), $A_s=0.50$ (solid line), and $A_s=0.75$ (long dashed line). We report $lf=\ln (-f)$ as function of $lR=\ln (-R)$ rather than $f(R)$ to better show the results.