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This paper establishes asymptotic time dependences of characteristic sizes of astrophysical and cosmological objects. These dependences are obtained on the basis of uncertainty principle applied in cosmic scales in approximation of spherical symmetry in Euclidean geometry. The proposed analytical approach makes it possible to determine spatial boundaries of the uniformity of matter distribution in the Universe, and a size of cosmic sphere which contains numerous groups of interacting universes.

At present the increasing number of scientific articles is devoted to different problems of astrophysics and cosmology: from formation and evolution of stars and galaxies to formation of the large-scale structure of the Universe under influences of dark matter and dark energy (see, for example, [

The most significant cosmological problem is determination of spatial boundaries corresponded to uniformity and isotropy of the Universe, i.e., to Cosmological principle [

In accordance with conception [

In this paper, an attempt of qualitative analysis of the above problems is undertaken. With this purpose, extended interpretation of an uncertainty principle applied to the sizes of objects in cosmic scales is proposed. The proposed analytical approach makes it possible to obtain asymptotic time dependences for characteristic sizes of astrophysical and cosmological objects in approximation of spherical symmetry in Euclidean geometry. Globular star clusters, superclusters of galaxies and the Universe itself are considered as such objects. The existence of groups of interacting universes is postulated, and the issue of a size of cosmic sphere involving the great number of such groups is considered.

By analogy with quantum mechanics which considers discrete and continuous energy spectra of microcosm objects it is possible to single out relatively small-scaled regions for cosmic structures, where, as a result of gravitation, mass distribution is non-uniform (analogy is with discrete energy spectrum). In more extended structures, where gravitation “gets smeared” (spreads) in space, mass distribution becomes quasi-uniform (analogy is with continuous energy spectrum).

Such analogy provides precondition for an attempt to extend the uncertainty principle to cosmic scales. The essence of the proposed uncertainty principle in cosmic scales is in the fact that during the time of a single act of gravitational interaction between cosmic structures, their sizes cannot be determined exactly. It is connected with the fact that unless the elementary act is finished, it is impossible to determine to which object each of the interacting surface elements the most closely located to each other is assigned. Stars are elements for globular clusters, galaxies are elements for super-clusters, clusters of galaxies are elements for the Universe.

In accordance with the generally accepted concept about formation of large cosmological objects from smaller ones (embryos) [

On the basis of universal relation for half-width of wave packet and half-width of spectral line (resulted from Fourier theorem)

Here _{c} is 2π-reduced cosmological constant of action;

Here _{0} is Hubble constant, G is gravitation constant. Hence it follows that each type of the objects under consideration has own cosmological constant of action: the larger a structure embryo is, the larger K_{c} is.

Formally, it follows from ratio (1) that

That means that formation of a large-scale Universe structure is of pulsed nature, i.e. there is development of initial perturbations in matter distribution as a result of instability of cosmological flows leading to appearance of non-uniformities. If perturbations are not large, it is reasonably to consider the mean size

In case of intensive process of object interaction (at close distances or at large flows of dark mass-energy) the value

Here ∆E is energy level width, corresponding to a specific process. Since in relations (1) and (3) we are dealing with object sizes that vary continuously with time, these relations can be considered as differential. From the relations (1) and (3) we obtain the following differential relation:

From here it is obvious that in intensive processes the action constant becomes “virtual”.

We consider globular clusters consisting of red giants having an average mass of m_{g} = 4M_{S}, where M_{S} is the mass of the Sun [

Here m_{bcl} is the mass of globular cluster,

With the Universe life time

Thus, the uncertainty principle works well enough within the range of astrophysical scales. This suggests that it might be used in cosmological scales. As

Here clusters are embryos for superclusters, and superclusters are embryos for the Universe.

Solving Equation (7) in quadratures with regard to the initial condition

This dependence is valid up to the “moment” of time t_{1}, from which dark energy begins to prevail in matter density in the Universe. In accordance with generally accepted ideas [

Let us calculate consecutively theoretical sizes of the considered structures according with Equation (8) based on the data [_{c}/ρ = 33,_{0} of embryo of a larger isotropic structure we find out that the calculated characteristic size of the Universe at the time t_{1} is equal approximately to

It should be noted that the size R_{un} is significantly less than the “light” radius_{light}. By analogy with microcosm and mesocosm the exponent 1/3 attached to t in Equation (8) corresponds to Brownian motion of colliding objects, and the final result of collisions consists in irreversible aggregation of objects [_{1} the time dependence of characteristic size of the Universe takes the following form:

The second term in Equation (9) expresses the ultimate ultrarelativistic stage of Universe expansion with the constant velocity

Let us consider an issue about a size of closed cosmic sphere which contains a great number of groups of interacting universes. We will determine the ultimate value of cosmological action constant

Here, _{p} is proton mass; N_{b} ~ 10^{80} is number of baryons in the Universe [_{c} corresponds approximately to the cosmic mass 3.986 × 10^{54} kg, given in [

The exponent 2/5 attached to t corresponds to the analogy with free-molecular flux of Brownian particles, i.e., with irreversible growth of object as a result of successive addition of structure elements. With ρ = 0.03ρ_{c} [_{light} by almost two orders of magnitude. Hence one can conclude that all interactions within the cosmic sphere can occur only between “neighboring” universes with sizes and distances between them which not exceed the distance travelled by light. The interacting universes are formed independently from each other, for example, as a result of great number of “Big bangs” distributed in space and time randomly inside the sphere (11). Number of universes formed inde-

pendently inside the cosmic sphere can be evaluated as

A distinctive feature of this phenomenological approach is in assumption of the existence of many interacting universes in space with spherical Euclidian geometry and in introduction of the concept of cosmological action constant, the value of which depends on a scale of a cosmological object, which is an embryo for a larger structure under consideration. The given results do not contradict to well known ideas [

To describe an intermediate stage of expansion affected by dark energy one can accept the value of mass in Equation (10) for cosmological constant of action as equal to the ultimately possible value

Here

(see Equation (9)), we obtain the following equation for the Universe size-time relation:

This equation reflects accelerated expansion, and the type of time dependence of characteristic size of the Universe is in compliance with de Sitter cosmological model for Euclidean geometry at Λ > 0. The type of pre-exponential factor corresponds to Dirac cosmological model for Euclidean geometry at the atomic time scale when Λ = 0 [_{c} and ρ is rearranged to the following equation:

Equation (14) corresponds approximately to the Hubble law.

Thus, uncertainty principle and Cosmological principle do not contradict each other and are mutually complementary. The presented results correspond to fundamental concept [

From Equation (13) we obtain that the universe size at present stage is

The above results are obtained in one-dimensional approximation. With more complicated space-time geometry proposed in [_{cB} is not essential, since both values are much larger than the light radius. If one substitutes theoretical formation time of large-scale structure of the Universe (which is equal to 2.2×10^{10} years [_{cB} becomes insignificant.

It should be noted that the proposed approach does not assume any prohibitions on existence of great number of independent cosmic spheres (11) in infinite space.

Extended interpretation of uncertainty principle applied to the sizes of objects in cosmic scales in approximation of spherical symmetry in Euclidean geometry was proposed. Evidently first phenomenological outer-space constant of action was determined with regard to the sizes and masses of embryos of astrophysical and cosmological objects. The obtained adequate asymptotic time dependences of characteristic sizes of objects under consideration demonstrate that the “ordinary” space-time relations are valid for determination of cosmic boundaries. The presented results indicate the existence of the great number of independent groups of interacting universes. The proposed approach does not contradict to well-known ideas about stochastic quantum nature of cosmological phenomena and completes the existent notions on large numbers in quantized cosmos.