  We continue the investigation of a recent proposal on alternative matching
conditions for self-gravitating defects which generalize the standard matching
conditions. The reasoning for this study is the need for consistency of the
various codimension defects and the existence of a meaningful equation of
motion at the probe limit, things that seem to lack from the standard approach.
These matching conditions arise by varying the brane-bulk action with respect
to the brane embedding fields (and not with respect to the bulk metric at the
brane position) in a way that takes into account the gravitational
back-reaction of the brane to the bulk. They always possess a Nambu-Goto probe
limit and any codimension defect is seemingly consistent for any second order
bulk gravity theory. Here, we consider in detail the case of a codimension-1
brane in five-dimensional Einstein gravity, derive the generic alternative
junction conditions and find the Z_{2}-symmetric braneworld cosmology, as well
as its bulk extension. Compared to the standard braneworld cosmology, the new
one has an extra integration constant which accounts for the today matter and
dark energy contents, therefore, there is more freedom for accommodating the
observed cosmic features. One branch of the solution possesses the asymptotic
linearized LFRW regime. We have constrained the parameters so that to have a
recent passage from a long deceleration era to a small today acceleration epoch
and we have computed the age of the universe, consistent with current data, and
the time-varying dark energy equation of state. For a range of the parameters
it is possible for the presented cosmology to provide a large acceleration in
the high energy regime.
