Current universe (assumed here to be normal matter on the brane) is pressureless from observations. In this case the energy condition is $rho_0geq0$ and $p_0=0$. By using this condition, brane models can be distinguished. Then, assuming arbitrary com
ponent of matter in DGP model, we use four known energy conditions to study the matter on the brane. If there is nonnormal matter or energy (for example dark energy with $w<-1/3$) on the brane, the universe is accelerated.
An exact solution of brane universe is studied and the result indicates that Friedmann equations on the brane are modified with an extra term. This term can play the role of dark energy and make the universe accelerate. In order to derive the $Lambda
$CDM Universe from this global brane model, the new solutions are obtained to describe the $5D$ manifold.
By studying a tachyon field on the DGP brane model, in order to embed the 4D standard Friedmann equation with a brane tachyon field in 5D bulk, the metric of the 5D spacetime is presented. Then, adopting the inverse square potential of tachyon field,
we obtain an expanding universe with power-law on the brane and an exact 5D solution.
We discuss the exact solutions of brane universes and the results indicate the Friedmann equations on the branes are modified with a new density term. Then, we assume the new term as the density of dark energy. Using Wetterichs parametrization equati
on of state (EOS) of dark energy, we obtain the new term varies with the red-shift z. Finally, the evolutions of the mass density parameter $Omega_2$, dark energy density parameter $Omega_x$ and deceleration parameter q_2 are studied.
We discuss the correspondence between the DGP brane cosmology and 5D Ricci-flat cosmology by letting their metrics equal each other. By this correspondence, a specific geometrical property of the arbitrary integral constant I in DGP metric is given a
nd it is related to the curvature of 5D bulk. At the same time, the relation of arbitrary functions $mu$ and $ u$ in a class of Ricci-flat solutions is obtained from DGP brane metric.
