A bridge between unified cosmic history by $f(R)$-gravity and BIonic
  system
  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 froma 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 ($z\sim T^{2}$). The BIon is a configuration in flat space
of a universe-brane and a parallel anti-universe-brane 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 $z\sim
T^{4+1/7}$. 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 ($z\sim
T^{2/3}$). As soon as today acceleration approaches, the redshift tends to zero
and the cosmological model reduces to the standard $\Lambda$CDM cosmology.
