Using (4+1) split and energy conditions to study the induced matter in 5D Ricc-flat cosmology

We use $(4+1)$ split to derive the 4D induced energy density $\rho$ and pressure $p$ of the $5D$ Ricci-flat cosmological solutions which are characterized by having a bounce instead of a bang. The solutions contain two arbitrary functions of time $t$ and, therefore, are mathematically rich in giving various cosmological models. By using four known energy conditions (null, weak, strong, and dominant) to pick out and study physically meaningful solutions, we find that the 4D part of the $5D$ solutions asymptotically approach to the standard 4D FRW models and the expansion of the universe is decelerating for normal induced matter for which all the four energy conditions are satisfied. We also find that quintessence might be normal or abnormal, depending on the parameter $w$ of the equation of state. If $-1\leqslant w<-1/3$, the expansion of the universe is accelerating and the quintessence is abnormal because the strong energy condition is violated while other three are satisfied. For phantom, all the four energy conditions are violated. Before the bounce all the four energy conditions are violated, implying that the cosmic matter before the bounce could be explained as a phantom which has a large negative pressure and makes the universe bouncing. In the early times after the bounce, the dominant energy condition is violated while the other three are satisfied, and so the cosmic matter could be explained as a super-luminal acoustic matter.