Cosmological reconstruction of realistic modified F(R) gravities
  The cosmological reconstruction scheme for modified $F(R)$ gravity is
developed in terms of e-folding (or, redshift). It is demonstrated how any FRW
cosmology may emerge from specific $F(R)$ theory. The specific examples of
well-known cosmological evolution are reconstructed, including $\Lambda$CDM
cosmology, deceleration with transition to phantom superacceleration era which
may develop singularity or be transient. The application of this scheme to
viable $F(R)$ gravities unifying inflation with dark energy era is proposed.
The additional reconstruction of such models leads to non-leading gravitational
correction mainly relevant at the early/late universe and helping to pass the
cosmological bounds (if necessary). It is also shown how cosmological
reconstruction scheme may be generalized in the presence of scalar field.
