Tilted cosmological models of Bianchi type V

Cosmological models of Bianchi types V and I containing a perfect fluid with a linear equation of state plus cosmological constant are investigated. In general, these spacetimes are tilted and describe fluids with expansion, shear, and vorticity. We use a tetrad approach where our variables are the Riemann tensor, the Ricci rotation coefficients, and a subset of the tetrad vector components. This set, called $S$, describes a spacetime when its elements are constrained by certain integrability conditions and due to a theorem by Cartan $S$ gives a complete local description of the spacetime. With the help of the Lie algebra, the full line element is constructed up to quadratures in terms of the elements in $S$. The system obtained by imposing the integrability conditions and Einstein’s equations on the elements in $S$ can be reduced to an integrable system of five coupled first order ordinary differential equations. In general, exact solutions to this system are hard to find, but the linearized equations around the open Friedmann models are easily integrated. The full system is also studied numerically and the perturbative solutions agree well with the numerical ones in the appropriate domains. We also give some examples of numerical solutions in the nonperturbative regime.

Figure 1

Comparison between numerical calculation (solid curve) and perturbations to first (dashed) and second order (dotted) for $v$ in the case of dust and $\Lambda =0$. Initial values: ${\gamma }_{131}=1$, $v=0.03$, $B=0.03$, ${\sigma }_{12}=0.03$, and ${\sigma }_{22}=0.03$.

Figure 2

Comparison between numerical calculation (solid) and perturbations to first (dashed) and second order (dotted) for $v$ in the case of radiation and $\Lambda =0$. Initial values: ${\gamma }_{131}=1$, $v=0.03$, $B=0.03$, ${\sigma }_{12}=0.03$, and ${\sigma }_{22}=0.03$.

Figure 3

${\sigma }_{ij}/\theta \to 0$ as $\rho \to 0$ for dust and $\Lambda =1$. Initial values: $\rho =0.8$, $v=-0.09$, ${\gamma }_{131}=0.08$, $B=0.1$, ${\sigma }_{12}=0.11$, ${\sigma }_{22}=0.12$. From top to bottom (at $\rho =0.8$): ${\sigma }_{11}/\theta$, ${\sigma }_{13}/\theta$, ${\sigma }_{12}/\theta$, ${\sigma }_{22}/\theta$, ${\sigma }_{23}/\theta$, and $v$.

Figure 4

${\omega }_{13}/\theta$ (upper curve) and $v$ as $\rho \to 0$ for radiation and $\Lambda =0$. Initial values: $\rho =0.8$, $v=-0.09$, ${\gamma }_{131}=0.08$, $B=0.1$, ${\sigma }_{12}=0.11$, and ${\sigma }_{22}=0.12$.

Figure 5

$v$, ${\sigma }_{11}/\theta$, and ${\sigma }_{22}/\theta$ for radiation and $\Lambda =0$ when $\rho \to \infty$. Initial values: $\rho =1$, $v=0.2$, ${\gamma }_{131}=1$, $B=-0.2$, ${\sigma }_{12}=0$, and ${\sigma }_{22}=0.7$. From top to bottom: ${\sigma }_{11}/\theta$, $v$, and ${\sigma }_{22}/\theta$.

Figure 6

$v$ for dust and $\Lambda =0$ when $\rho \to \infty$. Initial values: $\rho =1$, $v=0.2$, ${\gamma }_{131}=1$, $B=-0.2$, ${\sigma }_{12}=0.01$, and ${\sigma }_{22}=0.7$.