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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|Geometro-dynamics]]).

The physical basis for general-relativistic cosmological models is Einstein's general relativity theory (sometimes including the version with a [[Cosmological constant|cosmological constant]]; see [[Relativity theory|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 [[#References|[1]]], Vol. 2, Chapt. 24, follows from  $  CP $-
violation). The construction of a cosmological model usually begins with the assumption of some specific type of symmetry, in respect to which one distinguishes between homogeneous and isotropic cosmological models, anisotropic homogeneous cosmological models, and the like (see [[#References|[1]]], Vol. 2, [[#References|[2]]]). The first general-relativistic cosmological model was proposed by A. Einstein in 1917 (see [[#References|[3]]]); it was static, homogeneous and isotropic and included a  $  \Lambda $-
term, i.e. a cosmological constant. Subsequently, A.A. Friedmann developed a non-static homogeneous isotropic model, known as the Friedmann model [[#References|[4]]]. The non-static nature predicted by this model was observed in 1929 (see [[#References|[5]]]). The Friedmann model has different versions, depending on the values of the parameters that figure in it. If the density of matter  $  \rho $
is not greater than some critical value  $  \rho _ {0} $, 
one has what is called an open model; if  $  \rho > \rho _ {0} $
one has a closed model. In terms of suitable coordinates, the metric of the Friedmann cosmological model has the form

$$ 
ds  ^ {2}  = \ 
c  ^ {2}  dt  ^ {2} -
\left (

\frac{R ( t) }{R _ {0} }

\right )  ^ {2}
\left [

\frac{dr  ^ {2} }{1 - kr  ^ {2} /R _ {0}  ^ {2} }
 +
r  ^ {2} ( d \theta  ^ {2} + \sin  ^ {2}  \theta \ 
d \phi  ^ {2} ) \right ] ,
$$

where  $  t $
denotes time,  $  \rho $
and  $  \rho _ {0} $
are the average and so-called critical densities of matter at the moment of time in question,  $  c $
is the velocity of light, and  $  r, \theta $
and  $  \phi $
are coordinates. This metric is also known as the Robertson–Walker metric. The critical density  $  \rho _ {0} $
is a certain function of time, and it turns out that the magnitude  $  \rho - \rho _ {0} $
does not change sign. If  $  k < 0 $, 
the spatial cross-section  $  t = \textrm{ const } $
is Lobachevskii space; if  $  k = 0 $
it is Euclidean space (though the cosmological model itself is not flat); if  $  k > 0 $
one obtains spherical space. The function  $  R ( t) $(
the world radius) is determined from the Einstein equations and the equations of state; it vanishes at one ( $  k \leq  0 $) 
or two ( $  k > 0 $) 
values of  $  t $, 
and simultaneously the average density, curvature and other physical characteristics of the model become infinite. At such points the cosmological model is said to have a singularity. Depending on the equation of state, one speaks of cold (the pressure  $  p = 0 $) 
or hot ( $  p = \epsilon /3 $, 
where  $  \epsilon $
is the energy density) models. The discovery (see [[#References|[6]]]) of isotropic equilibrium radiation ( $  T \approx 3  ^ {n}  K  $) 
corroborates the hot model. Regardless of the crude nature of the Friedmann models, they already convey the main features of the structure of the Universe. For the further construction of cosmological models on the basis of Friedmann models see [[#References|[1]]], Vol 2, Chapt. 3. A theory has been developed for the evolution of small deviations of a cosmological model from a Friedmann model. A result of this evolution is apparently the formation of galactic clusters and other astronomical objects. The available observational data seem to imply that the real Universe is described to a good degree of accuracy by a Friedmann model. These data, however, do not permit the determination of the sign of  $  k $(
it seems somewhat more probable that  $  k < 0 $). 
There are other possible topological interpretations of Friedmann models, obtained by different factorizations by a spatial section (different ways of pasting it together). The observed data impose only weak restrictions on the nature of these factorizations (see [[#References|[1]]], Vol. 2). In a logically consistent theory, the construction of a cosmological model must begin with the selection of a manifold — the carrier of a pseudo-Riemannian metric. However, there is as yet no method for selecting manifolds. There are only a few restrictions on the possible global structure of the cosmological model, based on the causality principle and on  $   \mathop{\rm CP} $-
violation (see [[#References|[1]]], Vol. 2).

Many other cosmological models have been proposed, in particular anisotropic homogeneous models (see [[#References|[1]]], Vol. 2, Chapts. 18-22, [[#References|[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 [[#References|[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 [[#References|[7]]]). These cosmological models are also free of the above-mentioned paradoxes.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.Z. Petrov,   "Einstein spaces" , Pergamon  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Einstein,   ''Sitzungsber. K. Preuss. Akad. Wissenschaft.''  (1917)  pp. 142–152</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.A. Friedmann,   ''Z. Phys.'' , '''10'''  (1922)  pp. 377–386</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.P. Hubble,   ''Proc. Nat. Acad. Sci.'' , '''15'''  (1929)  pp. 168–173</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.A. Penzias,   R.W. Wilson,   ''Astrophys. J.'' , '''142'''  (1965)  pp. 419–421</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  O. Heckmann,   E. Schücking,  , ''Handbuch der Physik'' , '''53''' , Berlin  (1959)  pp. 489–519</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  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)</TD></TR></table>

====Comments====
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 [[#References|[a1]]], [[#References|[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 [[#References|[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  $  ( \sim 10  ^ {25}  K  ) $. 
In this regime a phase transition is expected in which the true vacuum appears as rapidly expanding bubbles in a surrounding of negative pressure. During this transition the Universe and the horizon expand exponentially (hence inflation). The presently visible Universe is only a tiny fraction of one bubble of the phase transition and fits inside the inflation horizon by a large margin. Also other long-standing problems find a natural solution in this model. For a physical introduction see [[#References|[a3]]], for a more rigorous treatment see [[#References|[a4]]].

Many other cosmological models have been considered in the literature. For a discussion of the Bianchi type I–IX models see [[#References|[a5]]], Chapt. 11. Nearly all known models can be found in [[#References|[a6]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.G. Walker,   "Completely symmetric spaces"  ''J. London Math. Soc.'' , '''19'''  (1944)  pp. 219–226</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.W. Hawking,   G.F.R. Ellis,   "The large scale structure of space-time" , Cambridge Univ. Press  (1973)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.H. Guth,   P.J. Steinhardt,   ''Scientific American'' , '''May'''  (1984)  pp. 90–102</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G.W. Gibbons (ed.)  S.W. Hawking (ed.)  S.T.C. Siklos (ed.) , ''The very early universe'' , Cambridge Univ. Press  (1983)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  S.W. Hawking (ed.)  W. Israel (ed.) , ''General relativity'' , Cambridge Univ. Press  (1979)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  D. Kramer,   H. Stephani,   M. MacCallum,   E. Herlt,   "Exact solutions of Einstein's field equations" , Cambridge Univ. Press  (1980)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  S. Weinberg,   "Gravitation and cosmology" , Wiley  (1972)  pp. Chapt. 3</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  P.T. Landsberg,   D.A. Evans,   "Mathematical cosmology" , Oxford Univ. Press  (1977)</TD></TR></table>