Recently, the cosmological deceleration-acceleration transition redshift in f (R) gravity has been considered in order to address consistently the problem of cosmic evolution. It is possible to show that the deceleration parameter changes sign at a given redshift according to observational data. Furthermore, a f (R) gravity cosmological model can be constructed in brane-antibrane system starting from the very early universe and accounting for the cosmological redshift at all phases of cosmic history, from inflation to late time acceleration. Here we propose a f (R) model where transition redshifts correspond to inflation-deceleration and deceleration-late time acceleration transitions starting from a BIon system. At the point where the universe was born, due to the transition of k black fundamental strings to the BIon configuration, the redshift is approximately infinity and decreases with reducing temperature (z ∼ T ^2). The BIon is a configuration in flat space of a universe-brane and a parallel anti-universebrane connected by a wormhole. This wormhole is a channel for flowing energy from extra dimensions into our universe, occurring at inflation and decreasing with redshift as z ∼ T 4+1/7. Dynamics consists with the fact that the wormhole misses its energy and vanishes as soon as inflation ends and deceleration begins. Approaching two universe branes together, a tachyon is originated, it grows up and causes the formation of a wormhole. We show that, in the framework of f (R) gravity, the cosmological redshift depends on the tachyonic potential and has a significant decrease at deceleration-late time acceleration transition point (z ∼ T ^2/3). As soon as today acceleration approaches, the redshift tends to zero and the cosmological model reduces to the standard CDM cosmology.
