One of the basic concepts in cosmology as a science describing the Universe (the mega-world surrounding us) as a whole, ignoring details of no significance in this respect.
The mathematical form of a cosmological model depends on which physical theory is adopted as basis for the description of moving matter: accordingly, one distinguishes between general-relativistic models, Newtonian models, steady-state models, models with variable gravitational constant, etc. The most important of these are the general-relativistic models. Astronomical systems may also be categorized as cosmological models: the Ptolemaic system, the Copernican system, etc. Modern cosmological models enable one to concentrate attention on essential details by introducing the concept of averaging physical properties over a physically large volume. The averaged values are assumed to be continuous and (usually) many times differentiable. That this averaging operation is possible is not self-evident. One can imagine a hierarchical model of the Universe, in which there exist qualitatively distinct objects of ever-increasing scales. However, the available observational data do not match a model of this type.
As yet, the averaging procedure for general-relativistic cosmological models lacks an adequate mathematical basis. The difficulty is here that different "microstates" , which yield the same cosmological model when averaged, constitute distinct pseudo-Riemannian manifolds, possibly even possessing distinct topological structures (see also Geometro-dynamics).
The physical basis for general-relativistic cosmological models is Einstein's general relativity theory (sometimes including the version with a cosmological constant; see Relativity theory). The mathematical form of general-relativistic cosmological models is the global geometry of pseudo-Riemannian manifolds. It is assumed that the topological structure of the manifold must be predicted theoretically. The choice of a specific topological structure for a cosmological model is complicated by the fact that models having different topologies and different global properties may be locally isometric. One method for solving this problem is to advance additional postulates, which either follow from general theoretical considerations (such as the causality principle), or are experimental facts (e.g. the postulate in [1], Vol. 2, Chapt. 24, follows from
where
Many other cosmological models have been proposed, in particular anisotropic homogeneous models (see [1], Vol. 2, Chapts. 18-22, [8]).
Prior to the appearance of general-relativistic cosmological models, it was implicitly assumed that the distribution of matter is isotropic, homogeneous and static. However, this assumption leads to what is known as the gravitational, photometrical and other paradoxes (infinitely large gravitational potential, infinitely large illuminance, etc.). General-relativistic models avoid these paradoxes (see [2]). With respect to mass distributions, good Newtonian approximations analogous to those valid in general-relativistic cosmological models have been obtained for certain general-relativistic cosmological models (see [7]). These cosmological models are also free of the above-mentioned paradoxes.
[1] | Ya.B. Zel'dovich, I.D. Novikov, "Relativistic astrophysics" , 1 - Stars and relativity; 2 - Structure and evolution of the Universe , Chicago (1971–1983) (Translated from Russian) |
[2] | A.Z. Petrov, "Einstein spaces" , Pergamon (1969) (Translated from Russian) |
[3] | A. Einstein, Sitzungsber. K. Preuss. Akad. Wissenschaft. (1917) pp. 142–152 |
[4] | A.A. Friedmann, Z. Phys. , 10 (1922) pp. 377–386 |
[5] | E.P. Hubble, Proc. Nat. Acad. Sci. , 15 (1929) pp. 168–173 |
[6] | A.A. Penzias, R.W. Wilson, Astrophys. J. , 142 (1965) pp. 419–421 |
[7] | O. Heckmann, E. Schücking, , Handbuch der Physik , 53 , Berlin (1959) pp. 489–519 |
[8] | V.A. Belinskii, E.M. Lifshits, I.M. Khalatnikov, Soviet Physics Uspekhi , 13 (1971) pp. 745–765 Uspekhi Fiz. Nauk , 102 : 3 (1970) pp. 463–500 |
[9] | R. Penrose, "Structure of space-time" C.M. DeWitt (ed.) J.A. Wheeler (ed.) , Batelle Rencontres 1967 Lectures in Math. Physics , Benjamin (1968) pp. 121–235 (Chapt. VII) |
Exact spherical symmetry around every point implies that the Universe is spatially homogeneous and that it admits a six-parameter group of isometries, whose surfaces of transitivity are space-like three-surfaces with constant curvature (see [a1], [a2]). This Universe has the above mentioned Robertson–Walker metric. The unexpectedly strong requirement of spherical symmetry around every point comes from the fact that here an observational symmetry (e.g. microwave background) is intended. The observations refer to the past null-cone and not to the three-surface of transitivity. All expanding Friedmann models with zero cosmological constant and non-negative density and pressure contain a singularity in the past (the "big bang" ). The past null-cone stops on this singularity without covering all material in the Universe. It is only possible to see material inside our "horizon" . The observed isotropy does not say anything about matter outside the horizon.
It has long been thought, that the initial singularity is a result of the imposed symmetry, but a number of recent theorems by S.W. Hawking and R. Penrose indicate that some initial singularity must exist in many models of the Universe irrespective of symmetry (see [a2], Chapt. 8). Because of this singularity and the resulting horizon the microwave background from two well separated points on the sky has no causal connection (i.e. do not have overlapping past null-cones). This makes it a surprise that the temperature ends up the same everywhere. A solution of this "horizon problem" is given in the inflationary model. This is a Friedmann model with a physically special equation of state for matter and vacuum at ultra-high temperature
Many other cosmological models have been considered in the literature. For a discussion of the Bianchi type I–IX models see [a5], Chapt. 11. Nearly all known models can be found in [a6].
[a1] | A.G. Walker, "Completely symmetric spaces" J. London Math. Soc. , 19 (1944) pp. 219–226 |
[a2] | S.W. Hawking, G.F.R. Ellis, "The large scale structure of space-time" , Cambridge Univ. Press (1973) |
[a3] | A.H. Guth, P.J. Steinhardt, Scientific American , May (1984) pp. 90–102 |
[a4] | G.W. Gibbons (ed.) S.W. Hawking (ed.) S.T.C. Siklos (ed.) , The very early universe , Cambridge Univ. Press (1983) |
[a5] | S.W. Hawking (ed.) W. Israel (ed.) , General relativity , Cambridge Univ. Press (1979) |
[a6] | D. Kramer, H. Stephani, M. MacCallum, E. Herlt, "Exact solutions of Einstein's field equations" , Cambridge Univ. Press (1980) |
[a7] | S. Weinberg, "Gravitation and cosmology" , Wiley (1972) pp. Chapt. 3 |
[a8] | P.T. Landsberg, D.A. Evans, "Mathematical cosmology" , Oxford Univ. Press (1977) |