Equivalent off-diagonal cosmological models and ekpyrotic scenarios in
  f(R)-modified, massive and Einstein gravity
  We re-investigate how generic off-diagonal cosmological solutions depending,
in general, on all spacetime coordinates can be constructed in massive and
f-modified gravity using the anholonomic frame deformation method. There are
constructed new classes of locally anisotropic and (in) homogeneous
cosmological metrics with open and closed spatial geometries. By resorting such
solutions, we show that they describe the late time acceleration due to
effective cosmological terms induced by nonlinear off-diagonal interactions,
possible modifications of the gravitational action and graviton mass. The
cosmological metrics and related St\" uckelberg fields are constructed in
explicit form up to nonholonomic frame transforms of the
Friedmann-Lama\^{\i}tre-Robertson-Walker (FLRW) coordinates. The solutions
include matter, graviton mass and other effective sources modelling nonlinear
gravitational and matter fields interactions with polarization of physical
constants and deformations of metrics, which may explain dark energy and dark
matter effects. However, we argue that it is not obligatory always to modify
gravity if we consider effective generalized Einstein equations with nontrivial
vacuum and/or non-minimal coupling with matter. Indeed, we state certain
conditions when such configurations mimic interesting solutions in general
relativity and modifications, for instance, when we can extract the general
Painlev\' e-Gullstrand and FLRW metrics. In a more general context, we
elaborate on a reconstruction procedure for off-diagonal cosmological solutions
which describe cyclic and ekpyrotic universes. Finally, there are discussed
open issues and further perspectives.
