Relativistic modeling of cosmological structures with Bianchi IX spacetimes

We develop a relativistic framework to investigate the evolution of cosmological structures from the initial density perturbations to the highly nonlinear regime. Our approach involves proposing a procedure to match “best-fit,” exact Bianchi IX (BIX) spacetimes to finite regions within the perturbed Friedmann-Lemaître-Robertson-Walker universe, characterized by a positive averaged spatial curvature. This method enables us to approximately track the nonlinear evolution of the initial perturbation using an exact solution. Unlike standard perturbation theory and exact solutions with a high degree of symmetry (such as spherical symmetry), our approach is applicable to generic initial data, with the only requirement being positive spatial curvature. By employing the BIX symmetries, we can systematically incorporate the approximate effects of shear and curvature into the process of collapse. Our approach addresses the limitations of both standard perturbation theory and highly symmetric exact solutions, providing valuable insights into the nonlinear evolution of cosmological structures.

Figure 1

The anisotropy potential $V_{IX}({\beta }_{\pm })$ of the Bianchi type IX model.

Figure 2

The gauge-fixing conditions $\delta {\mathcal{C}}^0=0$ and $\delta {\mathcal{C}}^{\overrightarrow{k}}=0$ fix, respectively, the time coordinate and the spatial coordinates in the perturbed spacetime model.

Figure 3

The three Bianchi IX vector fields are transported to a domain in the matching hypersurface where they are expressed in the Cartesian coordinates.

Figure 4

The behavior of the anisotropic and isotropic Bianchi IX models in comparison to the background evolution for the first initial dataset.

Figure 5

The behavior of the anisotropic and isotropic Bianchi IX models in comparison to the background evolution the second initial dataset.