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The generalization of Jeans equation in expanding and rotating Universe is given. We found the generalized frequency of baryonic substrate oscillations in the rotating Universe. In doing this, two cases were considered: the generalized wave vector coincides with the Jeans wave vector and second case, when the generalized wave vector tends to zero.

It is well known that discovery of cosmic vacuum radically changed main cosmological models [

Cosmic vacuum actively influences on the process of large-scale objects formation in the Universe, also (see, for example [

Moreover, cosmic vacuum (or dark energy) is the reason of Universe rotation. One of the first articles devoted to this problem searching was done by Ellis and Olive [

In our article [

The Universe rotation, in its turn, can plays significant role for describing physical processes that are elapsing in it. Talk about the Universe rotation’s influence on the process of galaxy creation.

It’s well-known that classical approach to study the origin of galaxies is basing on Jeans equations [

So, the Jeans equation generalization for the dark energy with nonstationary equation of state and its application to study the antigravitational instability of cosmic substrate in the Newtonian cosmology was done in article [

Further, in [

Moreover, the approach developed in this work admits direct generalizations for other modified Gauss-Bon- net theory, string-inspired gravity, etc. In these cases, the constrained Poisson equation may be even more complicated due to the presence of extra scalar(s) in non-local or string-inspired gravity.

Dark matter density profiles from the Jeans equation were considered in article [

In [

And last, stellar dynamics in the galactic centre―proper motions and anisotropy―was examined in [^{*}. They explicitly include velocity anisotropy in estimating the central mass distribution. They also show how Leonard-Merritt and Bahcall-Tremainemass estimates give systematic offsets in the inferred mass of the central object when applied to finite concentric rings for power law clusters. Corrected Leonard-Merritt projected mass estimators and Jeans equation modeling confirm previous conclusions (from isotropic models) that a compact central concentration (central density

As we said above the problem of stability the homogeneous distribution matter mathematically was formulated and solved in the framework of small perturbations theory at first by Jeans. He took into account two factors: gravity that tend to conserve a substance dividing into separate bunches, and pressure that have a tendency to smooth the appeared nonhomogeneities.

Recall the equations of hydrodynamics and gravitation in the Newtonian approximation for an ideal gas -

where

To obtain a solution for the perturbations it is necessary use the method of decomposing an arbitrary pertur- bation on the system of orthogonal functions, and examining the time evolution of the perturbations’ components.

Following [

where

Substituting these expressions into equations of hydrodynamics, we will consider only terms that are linear with respect to

In monograph [

where

The aim of our work is a further generalization of Equation (4) for the case of rotating Universe. For doing this, it is necessary take into account that rotation of the Universe changes Hubble “constant”.

In fact, this question was considered in our paper [

In expression (6),

This equation is the basis of our following searching. In doing this its second term plays the role of “friction” force. And this force, as it clear from physical consideration, increases “friction” at the centrifugal force presence. Beside, next searching of the generalized Jeans equation relates to the vacuum-dominated epoch that exactly allows consider the baryonic mass density as approximately invariable parameter.

Concerning third term of Equation (7) we mark that its standard physical interpretation is the Hook-like force.

For solving Equation (7) we set the standard designation

for oscillations’ frequency of baryonic substrate. So, equation (7) be accompanied with (8) is an ordinary diffe- rential equation of the second order with constant coefficients. Its general solution is as follows

where

For searching these frequencies we’ll impose the general condition existence

that allows simplify expression (10) by decomposing it into the Taylor series. Thus, the generalized oscillation frequency of baryonic substrate in the rotating Universe takes on the form

From (12) clearly that the first frequency is

Hence, from (13) we find the standard critical value of Jeans vector module

The second frequency becomes more complicated and equals to

Now let discuss the generalized length of Jeans basing on (15). For doing this, in accordance with [

It is clear that the generalized wave vector module

First case relates to condition when generalized wave vector’s module coincides with the Jeans wave vector module, i.e.

Hence, with the value of Hubble constant [

From this expression we see that angular velocity depends both on the baryonic mass density as well as vacuum mass density.

Now consider the case when generalized wave vector module tends to zero that corresponds to the absence of wave perturbations in baryonic matter density. In this case

Hence, we find the corresponding angular velocity of the Universe

This formula indicates that at matter dominated epoch

As the result we can make the following conclusions.

1) In the framework of Big Bang model, the inflationary stage of the Universe evolution was described by condition

2) In the case of cosmic vacuum neglecting

3) Rotation of the Universe, as it clears from general physical considerations, is an external factor that can produce the wave disturbances in protogalactic substrate. However, in some cases the rotation doesn’t leads to such wave existence (see expression (20)). Unfortunately, such types of astronomical observations based on the well-known deep sky surveys (HDF, CDF, GOODS, etc.) do not give the possibility of this effect detection.

The authors express their deep gratitude to JSC “National Centre of Space Research and Technology” for the support of this research in the budget program 055 subprogram 101 “Grant funding of scientific research”.

We also thank our reviewers whose remarks made our article physically more transparent.