- the expansion of the Universe as verified by the redshifts in galaxy spectra and quantified by measurements of the Hubble constant , where is the Hubble constant;
- the deceleration parameter observed in distant galaxy spectra (although uncertainties about galactic evolution, intrinsic luminosities, and standard candles prevent an accurate estimate);
- the large scale isotropy and homogeneity of the Universe based on temperature anisotropy measurements of the microwave background radiation and peculiar velocity fields of galaxies (although the light distribution from bright galaxies is somewhat contradictory);
- the age of the Universe which yields roughly consistent estimates between the look-back time to the Big Bang in the FLRW model and observed data such as the oldest stars, radioactive elements, and cooling of white dwarf stars;
- the cosmic microwave background radiation suggests that the Universe began from a hot Big Bang and the data is consistent with a mostly isotropic model and a black body at temperature 2.7 K;
- CMBR precision measurements suggest best fit cosmological parameters in accord with the critical density standard model;
- the abundance of light elements such as
^{2}H,^{3}He,^{4}He, and^{7}Li, as predicted from the FLRW model, is consistent with observations, provides a bound on the baryon density and baryon-to-photon ratio, and is the earliest confirmation of the standard model; - the present mass density, as determined from measurements of luminous matter and galactic rotation curves, can be accounted for by the FLRW model with a single density parameter () to specify the metric topology;
- the distribution of galaxies and larger scale structures can be reproduced by numerical simulations in the context of inhomogeneous perturbations of the FLRW models;
- the detection of dark energy from observations of supernovae is generally consistent with accepted FLRW model parameters and cold dark matter + cosmological constant numerical structure formation models.

Because of these remarkable agreements between observation and theory, most work in the field of physical cosmology (see Section 4) has utilized the standard model as the background spacetime in which the large scale structure evolves, with the ambition to further constrain parameters and structure formation scenarios through numerical simulations. The most widely accepted form of the model is described by a set of dimensionless density parameters which sum to

where the different components measure the present mean baryon density , the dark matter density , the radiation energy , and the dark energy . The relative contributions of each source and their sum (which determines the topological curvature of the model) remains one of the most important issues in modern computational and observational cosmology. The reader is referred to [104] for a more in-depth review of the standard model, and to [128, 154] for a summary of observed cosmological parameter constraints and best fit “concordance” models. Peebles and Ratra [132] provide a comprehensive literature survey and an excellent review of the standard model, cosmological tests, and the evidence for dark energy and the cosmological constant.However, some important unanswered questions about the standard model concern the nature of the special conditions that produced an essentially geometrically flat Universe that is also homogeneous and isotropic to a high degree over large scales. In an affort to address these questions, it should be noted that many other cosmological models can be constructed with a late time behavior similar enough to the standard model that it is difficult to exclude them with absolute certainty. Consider, for example, the collection of homogeneous but arbitrarily anisotropic vacuum spacetimes known as the Bianchi models [141, 69]. There are nine unique models in this family of cosmologies, ranging from simple Bianchi I models representing the Kasner class of spacetimes (the flat FLRW model, sometimes referred to as Type I-homogeneous, belongs to this group), to the more complex and chaotic Bianchi IX or Mixmaster model (which also includes the closed FLRW model, Type IX-homogeneous). Several of these models will be discussed in the first section on relativistic cosmology (Section 3) dealing pre-dominately with the early Universe, where the models differ the most.

Anisotropic solutions, as well as more general (and in some cases exact) inhomogeneous cosmological models with initial singularities, can isotropize through anisotropic damping mechanisms and adiabatic cooling by the expansion of the Universe to resemble the standard FLRW model at late times. Furthermore, if matter is included in these spacetimes, the effective energy of anisotropy, which generally dominates matter energy at early times, tends to become less important over time as the Universe expands. The geometry in these matter-filled anisotropic spacetimes thus evolves towards an isotropic state. Quantum mechanical effects have also been proposed as a possible anisotropy damping mechanism that takes place in the early Universe to convert vacuum geometric energy to radiation energy and create particles. All of this suggests that the early time behavior and effects of local and global geometry are highly uncertain, despite the fact that the standard FLRW model is generally accepted as accurate enough for the late time description of our Universe.

Further detailed information on homogeneous (including Bianchi) universes, as well as more general classes of inhomogeneous cosmological models can be found in [105, 158, 70].

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