Theoretical cosmology began, within the framework of the general theory of relativity (GRT), when Einstein proposed the first mathematical model of the stationary Universe (1916). Before using the GRT equations in his model, Einstein makes a number of prophetic assumptions: “the relative velocities of stars are very small compared to the speed of light.” Therefore, long before other scientists realized this fact, Einstein clearly noted that, firstly, in cosmological models it is quite sufficient to use the Newtonian approximation; secondly, “matter in a large area of space is evenly distributed”, that is, the density of matter in the Universe can be considered constant: “matter is at rest for a long time”; and, finally, thirdly, if we introduce into the Poisson equation for the gravitational potential φ a term describing antigravity (-Λφ), then “The universe can be considered as a closed a continuum having a finite spatial (three-dimensional) volume.” Following these assumptions, Einstein modifies his GRT equations, similar to the generalized Poisson equation, and obtains a model of the stationary Universe in the form:

The model meets all the criteria of correctness of both physics and mathematics, but it turns out, as Friedman showed, it is unstable. To make our universe stable, Einstein’s theoretical model lacked another “prophetic” assumption. The authors of the Steady State Theory model came very close to this assumption Bondi, Gold, Hoyle.

In Einstein’s model, as in Friedman’s model, there is a so-called singularity problem. A cosmological singularity can be avoided by introducing a hypothetical field with a negative energy density. This goal was pursued by the Hoyle-Narlikar model (HN model), as the theory of the stationary Universe. For two decades, starting in 1948, a significant part of the theoretical and observational work of cosmologists was devoted to the verification, development and criticism of the HN model. Initially, the idea of a stationary global universe with continuous birth of baryonic matter was expressed simultaneously in the work of Bondi-Gold and in the work of Hoyle. But this model was developed in detail by Khoylomi Narlikar on the basis of GRT equations with a Λ-term and an additional term describing a certain C-field that creates matter. The main point of the HN model is the hypothesis of the constant birth of baryonic matter in our Universe from a hypothetical C-field. Therefore, this model is also called the theory of continuous creation of matter:

By selecting a suitable A, one can identically satisfy the stationarity condition. In 1963, Hoyle and Narlikar suggested that the birth of matter occurs mainly where the density of matter is already high (!). Unfortunately, according to the totality of observational data and theoretical calculations, the HN model is considered to be refuted. At the same time, Zeldovich and Novikov “pay tribute to the intellectual courage of the authors of this theory, since the discussions around it were useful and contributed to the general rise of cosmology” (!). It should be added that the idea of the continuous birth of “matter” in our Universe has not exhausted itself with the HN model, since in addition to baryonic matter, certain hopes can currently be placed on the mysterious non-baryonic dark matter, the structure and birth of which remains a big mystery for now.

### Observational data

Since 1994, Karachentsev’s group has obtained very interesting observational data:

1) the galaxies of our local group (numbering about 40), together with the dark matter halo (TM), form a system centered near the two largest galaxies (the Milky Way and the Andromeda Nebula). Gravitational attraction, mainly due to the TM of the local group, neutralizes the anti-gravitational influence of dark energy (TE);

2) a small number of dwarf galaxies (numbering about 20), under the influence of TE, manage to overcome the gravitational attraction of the local group and form the so-called “Hubble flow”, obeying the general law of the scattering of galaxies in the Universe;

3) the mass of TM in the local group is many times (5-6) greater than the mass of baryonic matter. This makes it possible, when analyzing the behavior of a local group, to consider galaxies as “test particles” in the TE iTM field;

4) in some local groups (including our local group), there is a separation of part of the TM from the general halo. It is assumed that a large number of TM can be located in the inputs;

5) it is noticed that the older the galaxy, the greater the density of TM associated with this galaxy (!).

We do not know what dark matter consists of, but theoretically we imagine how it forms gravitational potential pits where baryonic matter “falls”, forming the structure of the Universe “necessary for the birth of living matter”. Several neighboring groups of galaxies with a halo from TM form superclusters in the form of “pancakes” of Zeldovich with a size of about 30 Mpc (Local universe). And finally, superclusters form chains, filaments, which include (5-20) superclusters. It should be noted that the Shine group (part of the Cosmikflous-2 project” has built a detailed computer simulation of the movement of galaxies in the Local universe, having modeled the trajectories of 1382 galaxies over the past 13 billion years.

### A model of the Local universe

A natural question arises: for what reason should the Local universe be stationary? Such a universe, in the first approximation, is stabilized by mutually compensating forces of gravitational attraction of galaxies and forces of antigravity repulsion of TE (Einstein’s model). Additional stability of the system should be provided by “invisible baryonic matter” (the Hoyle-Narlikar model). But, following the lessons of the analysis of the HN model, now instead of the continuous birth of baryonic matter (which does not exist), it remains to assume that dark matter is continuously being born somewhere: exactly where the density of matter is maximum, that is, inside galaxies. The mathematical model of the stationary Local universe is presented below, where the ideas of the Einstein model and the Hoyle-Narlikar model are used. In particular, similar to the hypothesis of the HN model, the hypothesis of continuous exponential birth of dark matter in the depths of galaxies of Local universes is accepted. It is this dark matter that forms the halo.

Consider a stationary group of galaxies (neglecting the mass of galaxies) immersed in the halo TM. Let’s assume that during the evolution of the Local universe, the mass of TM (M) grew exponentially:

where α is the relative growth rate of TM.

Hence, using the well-known procedure for obtaining the Friedman equations in the Newtonian approximation, for the density of TM (PTM)

we get the first equation:

where H is the Hubble constant

Further, using the law of gravitational interaction TM (Newton’s law of gravitation) taking into account TE (in the form of a Λ-term):

we get the second equation:

here: γ is the gravitational constant of the interaction TM, c is

the speed of light, Λ is the cosmological constant TE.

Combining equations (5) and (7), we obtain a nonlinear equation describing the dynamics of the PTM of the global universe:

Taking into account the mass of the baryonic substance will not change the general form

of equation (8).

Turning to the Local universe, we can neglect the second term in (8). The stationarity of the Local universe is determined by the condition:

that is, the third term in (8) disappears and the equation of the dynamics of the PTM of the Local universe takes on a simple form:

The solution of this equation is the Weierstrass function (elliptic binary periodic function of a complex argument):