Phantom crossing, equation-of-state singularities, and local gravity constraints in f (R) models

We identify the class of f (R) dark energy models which have a viable cosmology, i.e. a matter dominated epoch followed by a late-time acceleration. The deviation from a ΛCDM model (f = R - Λ) is quantified by the function m = R f_{, R R} / f_{, R}. The matter epoch corresponds to m (r = - 1) ≃ + 0 (where r = - R f_{, R} / f) while the accelerated attractor exists in the region 0 ≤ m < 1. We find that the equation of state w_DE of all such "viable" f (R) models exhibits two features: w_DE diverges at some redshift z_c and crosses the cosmological constant boundary ("phantom crossing") at a redshift z_b smaller than z_c. Using the observational data of Supernova Ia and Cosmic Microwave Background, we obtain the constraint m < O (0.1) and we find that the phantom crossing could occur at z_b ≳ 1, i.e., within reach of observations. If we add local gravity constraints, the bound on m becomes very stringent, with m several orders of magnitude smaller than unity in the region whose density is much larger than the present cosmological density. The representative models that satisfy both cosmological and local gravity constraints take the asymptotic form m (r) = C (- r - 1)^p with p > 1 as r approaches -1.