Brane world models with bulk perfect fluid and broken 4D Poincaré invariance

We consider 5D brane world models with broken global 4D Poincaré invariance (4D part of the spacetime metric is not conformal to the Minkowski spacetime). The bulk is filled with the negative cosmological constant and may contain a perfect fluid. In the case of empty bulk (the perfect fluid is absent), it is shown that one brane solution always has either a physical or a coordinate singularity in the bulk. We cut off these singularities in the case of compact two brane model and obtain regular exact solutions for both 4D Poincaré broken and restored invariance. When the perfect fluid is present in the bulk, we get the master equation for the metric coefficients in the case of arbitrary bulk perfect fluid equation of state (EoS) parameters. In two particular cases of EoS, we obtain the analytic solutions for thin and thick branes. First one generalizes the well known Randall-Sundrum model with one brane to the case of the bulk anisotropic perfect fluid. In the second solution, the 4D Poincaré invariance is restored. Here, the spacetime goes asymptotically to the anti-de Sitter one far from the thick brane.

Figure 1

The schematic plot of the two-brane model. The red/upper left-right arrow indicates the points of identification. The black arrows show directions of normal vectors to the branes.

Figure 2

Models with restored 4D Poincaré invariance. Orange/top and blue/bottom lines represent $A(\xi )$ and $B(\xi )$ metric coefficients, respectively. Left graph: $\mu L=1$, $\mu /m=2$, $b_2=0$: $(\kappa /\mu ){\epsilon }_{(1)}=-(\kappa /\mu ){\epsilon }_{(2)}=2.25$, ${\mathrm{\Omega }}_{(1)}={\mathrm{\Omega }}_{(2)}=-1$. Right graph: $\mu L=1$, $\mu /m=1/3$, $b_2=-1$: $(\kappa /\mu ){\epsilon }_{(1)}=-(\kappa /\mu ){\epsilon }_{(2)}=-6$, ${\mathrm{\Omega }}_{(1)}={\mathrm{\Omega }}_{(2)}=-1$.

Figure 3

Models with broken 4D Poincaré invariance. Orange/top and blue/bottom lines represent $A(\xi )$ and $B(\xi )$ metric coefficients, respectively. Left graph: $\mu L=1$, $\mu /m=2$ and $b_2=-1/3$: $(\kappa /\mu ){\epsilon }_{(1)}=0.75$, $(\kappa /\mu ){\epsilon }_{(2)}\approx -1.96$, ${\mathrm{\Omega }}_{(1)}\approx -6.33$, ${\mathrm{\Omega }}_{(2)}\approx -1.21$. Right graph: $\mu L=1$, $\mu /m=1/2$ and $b_2=1$: $(\kappa /\mu ){\epsilon }_{(1)}=13.5$, $(\kappa /\mu ){\epsilon }_{(2)}\approx -5.15$, ${\mathrm{\Omega }}_{(1)}\approx -0.41$, ${\mathrm{\Omega }}_{(2)}\approx -0.84$.

Figure 4

The plot of the metric coefficients $A(\xi )$ (orange/top line) and $B(\xi )$ (blue/bottom line) in the case ${\omega }_1=0$ and ${\omega }_0=4/3$. Here, $(\kappa /m)\epsilon =-3\sqrt{2/5}$ and $\mathrm{\Omega }=-2$. 4D Poincaré invariance is broken.

Figure 5

The plot of the metric coefficient $B(\xi )=-A(\xi )$ (blue/bottom line) and the dimensionless perfect fluid energy density $(\kappa /m^2)ϵ(\xi )$ (green/top line) in the case of the thick brane.