Quasi-Isotropic Cycles and Nonsingular Bounces in a Mixmaster Cosmology

A Bianchi type-IX mixmaster spacetime is the most general spatially homogeneous solution of Einstein’s equations and it can represent the space-averaged universe. We introduce two novel mechanisms resulting in a mixmaster universe with nonsingular bounces that are quasi-isotropic. Matter with a nonlinear equation of state allows these bounces. Using a negative anisotropic stress successfully isotropizes this universe and mitigates the well-known mixmaster chaotic behavior. The Universe can be an eternal mixmaster spacetime, going through an infinite series of quasi-isotropic cycles separated by bounces.

Figure 1

Blue, orange, and green lines denote the scale factors $a$, $b$, and $c$, normalized to the curvature scale. The model is sourced by the nonlinear equation of state $\frac{1}{3}\rho -{\rho }^2$, with no anisotropic stress. Time is expressed in units of ${\rho }_c$.

Figure 2

Plot of the normalized dimensionless shear ${\sigma }^2/\rho$. The model is sourced by the nonlinear equation of state $\frac{1}{3}\rho -{\rho }^2$, with no anisotropic stress.

Figure 3

Blue, orange, and green lines denote the scale factors $a$, $b$, and $c$, normalized to the curvature scale. Anisotropic stress ${\pi }_{ab}=\kappa \sqrt{\rho }{\sigma }_{ab}$ has been added to the fluid, in addition to the nonlinear equation of state for the isotropic pressure that was sourcing the model before. $\kappa =-4$ in this computation. Time is expressed in units of ${\rho }_c$.

Figure 4

Normalized dimensionless shear ${\sigma }^2/\rho$ for a model sourced by quadratic equation of state $P=\frac{1}{3}\rho -{\rho }^2$ with the inclusion of anisotropic stress ${\pi }_{ab}=\kappa {\rho }^{1/2}{\sigma }_{ab}$. $\kappa =-4$ in this computation.