There are several classical cosmological models built using general relativity, supplemented by the homogeneity and isotropy of space.
Einstein's closed universe has a constant positive curvature of space, which becomes static due to the introduction of the so-called cosmological parameter into the general relativity equations, which acts as an antigravitational field.
In the expanding with acceleration of de Sitter's universe with non-curved space, there is no ordinary matter, but it is also filled with an anti-gravitating field.
There are also the closed and open universes of Alexander Friedman; the boundary world of Einstein - de Sitter, which gradually decreases the expansion rate to zero over time, and finally, the Lemaitre universe, the progenitor of the Big Bang cosmology, growing from a supercompact initial state. All of them, and especially the Lemaitre model, became the predecessors of the modern standard model of our universe.
The space of the universe in different models has different curvatures, which can be negative (hyperbolic space), zero (flat Euclidean space, corresponding to our universe) or positive (elliptical space).
The first two models are open universes, expanding endlessly, the last one is closed, which will sooner or later collapse. The illustration from top to bottom shows two-dimensional analogs of such a space.
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There are, however, other universes, also generated by a very creative, as they say now, using the equations of general relativity. They correspond much less (or do not correspond at all) to the results of astronomical and astrophysical observations, but they are often very beautiful, and sometimes elegantly paradoxical.
True, mathematicians and astronomers have invented them in such quantities that we will have to limit ourselves to only a few of the most interesting examples of imaginary worlds.
From string to pancake
After the appearance (in 1917) of the fundamental work of Einstein and de Sitter, many scientists began to use the equations of general relativity to create cosmological models. One of the first to do this was the New York mathematician Edward Kasner, who published his solution in 1921.
His universe is very unusual. It lacks not only gravitating matter, but also an anti-gravitating field (in other words, there is no Einstein cosmological parameter). It would seem that in this ideally empty world nothing can happen at all.
However, Kasner assumed that his hypothetical universe evolved unevenly in different directions. It expands along two coordinate axes, but contracts along the third axis. Therefore, this space is obviously anisotropic and resembles an ellipsoid in its geometric outlines.
As such an ellipsoid stretches in two directions and contracts along the third, it gradually turns into a flat pancake. At the same time, Kasner's universe does not grow thin at all, its volume increases in proportion to age.
At the initial moment, this age is equal to zero - and, therefore, the volume is also zero. However, the Kasner universes are not born from a point singularity, like the world of Lemaitre, but from something like an infinitely thin spoke - its initial radius is equal to infinity along one axis and zero along the other two.
What is the secret of the evolution of this empty world? Since its space “shifts” in different ways along different directions, gravitational tidal forces arise, which determine its dynamics. It would seem that one can get rid of them if we equalize the expansion rates along all three axes and thereby eliminate the anisotropy, but mathematics does not allow such liberties.
True, one can set two of the three speeds equal to zero (in other words, fix the dimensions of the universe along two coordinate axes). In this case, the Kasner world will grow in only one direction, and strictly proportional to the time (this is easy to understand, since this is how its volume must increase), but this is all that we can achieve.
Kazner's universe can remain by itself only if it is completely empty. If you add a little matter to it, it will gradually begin to evolve like the isotropic universe of Einstein-de Sitter.
In the same way, when a nonzero Einstein parameter is added to its equations, it (with or without matter) will asymptotically enter the regime of exponential isotropic expansion and turn into de Sitter's universe.
However, such "additions" really change only the evolution of the already existing universe. At the moment of her birth, they practically do not play a role, and the universe evolves according to the same scenario.
Although the Kasner world is dynamically anisotropic, its curvature at any given time is the same along all coordinate axes. However, the equations of general relativity admit the existence of universes that not only evolve with anisotropic velocities, but also have anisotropic curvature.
Such models were built in the early 1950s by the American mathematician Abraham Taub. Its spaces can behave in some directions like open universes, and in others - like closed ones. Moreover, over time, they can change their sign from plus to minus and from minus to plus.
Their space not only pulsates, but literally turns inside out. Physically, these processes can be associated with gravitational waves, which deform space so strongly that they locally change its geometry from spherical to saddle and vice versa. In general, strange worlds, albeit mathematically possible.
Unlike our Universe, which expands isotropically (that is, at the same speed regardless of the chosen direction), Kasner's universe simultaneously expands (along two axes) and contracts (along the third).
Fluctuations of the Worlds
Soon after the publication of Kasner's work, articles by Alexander Fridman appeared, the first in 1922, the second in 1924. These papers presented surprisingly elegant solutions to the equations of general relativity, which had an extremely constructive effect on the development of cosmology.
Friedman's concept is based on the assumption that, on average, matter is distributed in outer space as symmetrically as possible, that is, completely homogeneous and isotropic.
This means that the geometry of space at each moment of a single cosmic time is the same at all its points and in all directions (strictly speaking, such a time still needs to be correctly determined, but in this case this problem is solvable).
It follows that the rate of expansion (or contraction) of the universe at any given moment is again independent of direction. Friedmann's universes are therefore quite unlike Kasner's model.
In the first article, Friedman built a model of a closed universe with a constant positive curvature of space. This world arises from an initial point state with an infinite density of matter, expands to a certain maximum radius (and, therefore, maximum volume), after which it collapses again into the same singular point (in mathematical language, a singularity).
However, Friedman did not stop there. In his opinion, the found cosmological solution does not have to be limited by the interval between the initial and final singularities; it can be continued in time both forward and backward.
The result is an endless cluster of universes strung on the time axis, which border each other at singularity points. In the language of physics, this means that Friedmann's closed universe can oscillate endlessly, dying after each contraction and reborn to new life in the subsequent expansion.
This is a strictly periodic process, since all oscillations continue for the same length of time. Therefore, each cycle of the existence of the universe is an exact copy of all other cycles.
This is how Friedman commented on this model in his book “The World as Space and Time”: “Further, there are cases when the radius of curvature changes periodically: the universe contracts to a point (into nothing), then again from a point brings its radius to a certain value, then again, decreasing the radius of its curvature, it turns into a point, etc.
One involuntarily recalls the legend of Hindu mythology about the periods of life; it is also possible to talk about "the creation of the world from nothing", but all this should be considered as curious facts that cannot be solidly confirmed by insufficient astronomical experimental material."
The graph of the potential of the Mixmaster universe looks so unusual - the potential pit has high walls, between which there are three "valleys". Below are the equipotential curves of such a “universe in a mixer”.
A few years after the publication of Friedman's articles, his models gained fame and recognition. Einstein became seriously interested in the idea of an oscillating universe, and he was not alone. In 1932, it was taken up by Richard Tolman, professor of mathematical physics and physical chemistry at Caltech.
He was neither a pure mathematician, like Friedman, nor an astronomer and astrophysicist, like de Sitter, Lemaitre and Eddington. Tolman was a recognized specialist in statistical physics and thermodynamics, which he first combined with cosmology.
The results were very nontrivial. Tolman came to the conclusion that the total entropy of the cosmos should increase from cycle to cycle. The accumulation of entropy leads to the fact that more and more of the energy of the universe is concentrated in electromagnetic radiation, which from cycle to cycle affects its dynamics more and more strongly.
Because of this, the length of the cycles increases, each next one becomes longer than the previous one. Oscillations persist, but cease to be periodic. Moreover, in each new cycle, the radius of Tolman's universe increases.
Consequently, at the stage of maximum expansion, it has the smallest curvature, and its geometry is more and more and for more and more long time approaches the Euclidean one.
Richard Tolman, while designing his model, missed an interesting opportunity, which John Barrow and Mariusz Dombrowski drew attention to in 1995. They showed that the oscillatory regime of the Tolman universe is irreversibly destroyed when an antigravitational cosmological parameter is introduced.
In this case, Tolman's universe on one of the cycles no longer contracts into a singularity, but expands with increasing acceleration and turns into de Sitter's universe, which in a similar situation is also done by the Kasner universe. Antigravity, like zeal, overcomes everything!
Universe in the Mixer
In 1967, American astrophysicists David Wilkinson and Bruce Partridge discovered that relic microwave radiation from any direction, discovered three years earlier, arrives on Earth with practically the same temperature.
With the help of a highly sensitive radiometer, invented by their compatriot Robert Dicke, they showed that temperature fluctuations of relict photons do not exceed a tenth of a percent (according to modern data, they are much less).
Since this radiation originated earlier than 4,00,000 years after the Big Bang, the results of Wilkinson and Partridge gave reason to believe that even if our universe was not nearly ideally isotropic at the time of birth, it acquired this property without much delay.
This hypothesis was a big problem for cosmology. In the first cosmological models, the isotropy of space was laid from the very beginning simply as a mathematical assumption. However, back in the middle of the last century it became known that the equations of general relativity make it possible to construct a set of non-isotropic universes. In the context of these results, the almost ideal isotropy of the CMB required an explanation.
This explanation appeared only in the early 1980s and turned out to be completely unexpected. It was built on a fundamentally new theoretical concept of a superfast (as they usually say, inflationary) expansion of the Universe in the first moments of its existence. In the second half of the 1960s, science was simply not ripe for such revolutionary ideas. But, as you know, in the absence of stamped paper, they write in plain paper.
The prominent American cosmologist Charles Misner, immediately after the publication of the article by Wilkinson and Partridge, tried to explain the isotropy of microwave radiation using quite traditional means.
According to his hypothesis, the inhomogeneities of the early Universe gradually disappeared due to the mutual "friction" of its parts due to the exchange of neutrino and light fluxes (in his first publication, Mizner called this supposed effect neutrino viscosity).
According to him, such viscosity can quickly smooth out the initial chaos and make the Universe almost perfectly homogeneous and isotropic.
Misner's research program looked beautiful, but did not bring practical results. The main reason for its failure was again revealed through microwave analysis.
Any processes involving friction generate heat, this is an elementary consequence of the laws of thermodynamics. If the primary inhomogeneities of the Universe were smoothed out due to neutrino or some other viscosity, the CMB energy density would differ significantly from the observed value.
As the American astrophysicist Richard Matzner and his aforementioned English colleague John Barrow showed in the late 1970s, viscous processes can eliminate only the smallest cosmological inhomogeneities. For the complete "smoothing" of the Universe, other mechanisms were required, and they were found within the framework of the inflationary theory.
Nevertheless, Mizner received many interesting results. In particular, in 1969 he published a new cosmological model, the name of which he borrowed … from a kitchen appliance, a home mixer made by Sunbeam Products! Mixmaster Universe is constantly beating in the strongest convulsions, which, according to Mizner, make the light circulate along closed paths, mixing and homogenizing its contents.
However, later analysis of this model showed that, although photons in Mizner's world do make long journeys, their mixing effect is very insignificant.
Nevertheless, the Mixmaster Universe is very interesting. Like Friedman's closed universe, it arises from zero volume, expands to a certain maximum and contracts again under the influence of its own gravity. But this evolution is not smooth, like Friedman's, but absolutely chaotic and therefore completely unpredictable in detail.
In youth, this universe oscillates intensively, expanding in two directions and contracting in a third - like in Kasner. However, the orientations of the expansions and contractions are not constant - they change places randomly.
Moreover, the frequency of oscillations depends on time and tends to infinity when approaching the initial instant. Such a universe undergoes chaotic deformations, like jelly trembling on a saucer. These deformations can again be interpreted as a manifestation of gravitational waves moving in different directions, much more violent than in the Kasner model.
The Mixmaster Universe entered the history of cosmology as the most complex of the imaginary universes created on the basis of "pure" general relativity. Since the early 1980s, the most interesting concepts of this kind began to use the ideas and mathematical apparatus of quantum field theory and the theory of elementary particles, and then, without much delay, and superstring theory.