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. 
S. Capozziello, V.F. Cardone, A. Troisi