This review is intended to provide a flavor of the variety of numerical cosmological calculations performed of different events occurring throughout the history of our Universe. The topics discussed range from the strong field dynamical behavior of spacetime geometry at early times near the Big Bang singularity and the epoch of inflation, to the late time evolution of large scale matter fluctuations and the formation of clusters of galaxies. For the most part, the nature of the calculations dealing with the early or late Universe can be distinguished by their basic motivations. For example, calculations of early Universe phenomena are designed to explore alternative cosmological models or topologies, and in some cases, different theories of gravity. They also tend to study the nature of topological singularities, geometric effects, and the problem of initial conditions or the origin of matter distributions. Calculations of the late Universe are generally focused to establish bounds on cosmological parameters in the context of the standard model, to resolve the correct structure formation scenario, to model the complex multi-physics interactions operating at vastly different scales, and to systematically compare invariant measures against observed data for both model validation and interpreting observations.

Although a complete, self-consistent, and accurate description of our Universe is impractical considering the complex multi-scale and multi-physics requirements, a number of enlightening results have been demonstrated through computations. For example, both monotonic AVTD and chaotic oscillatory BLK behavior have been found in the asymptotic approach to the initial singularity in a number of inhomogeneous cosmological models, though some issues remain concerning the generic nature of the singularity, including the effect of nonlinear mode coupling of spatial gradients to the oscillatory history, and the behavior in non-vacuum spacetimes with arbitrary global topology. Numerical calculations suggest that scalar fields play an important complicated role in the nonlinear or chaotic evolution of cosmological models with consequences for the triggering (or not) of inflation and the subsequent dynamics of structure formation. It is possible, for example, that these fields can influence the details of inflation and have observable ramifications as fractal patterns in the density spectrum, gravitational waves, galaxy distribution, and cosmic microwave background anisotropies. All these effects require further studies. Numerical simulations have also been used to place limits on curvature anisotropies and cosmological parameters at early times by considering primordial nucleosynthesis reactions in anisotropic and inhomogeneous cosmologies.

Finally, the large collection of calculations performed of the post-recombination epoch related to large scale structure formation (for example, cosmic microwave background, gravitational lensing, Ly absorption, and galaxy cluster simulations) have placed strong constraints on the standard model parameters and structure formation scenarios when compared to observations. Considering the range of models consistent with inflation, the preponderance of observational, theoretical and computational data suggest a best fit model of the late structure-forming Universe that is spatially flat with a cosmological constant and a small tilt in the power spectrum. These best fit model parameters, and in particular the introduction of a cosmological constant, are generally consistent with recent evidence of dark energy from supernovae and high precision CMBR observations.

Obviously many fundamental issues remain unresolved, including even the overall shape or topology of the cosmological model which best describes our Universe throughout its entire history. However, the field of numerical cosmology has matured in the development of computational techniques, the modeling of microphysics, and in taking advantage of current trends in computing technologies, to the point that it is now possible to perform high resolution multiphysics simulations and carry out reliable comparisons of numerical models with observational data.

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