Abstract. In this paper, we consider the Universe deep inside of the cell of uniformity.At these scales, the Universe is filled with inhomogeneously distributed discrete structures (galaxies, groups and clusters of galaxies), which disturb the background Friedmann model. We propose mathematical models with conformally flat, hyperbolic and spherical spaces. For these models, we obtain the gravitational potential for an arbitrary number of randomly distributed inhomogeneities. In the cases of flat and hyperbolic spaces, the potential is finite at any point, including spatial infinity, and valid for an arbitrary number of gravitating sources. For both of these models, we investigate the motion of test masses (e.g., dwarf galaxies) in the vicinity of one of the inhomogeneities. We show that there is a distance from the inhomogeneity, at which the cosmological expansion prevails over the gravitational attraction and where test masses form the Hubble flow. For our group of galaxies, it happens at a few Mpc and the radius of the zero-acceleration sphere is of the order of 1 Mpc, which is very close to observations. Outside of this sphere, the dragging effect of the gravitational attraction goes very fast to zero.

In the framework of the concordance cosmological model the first-order scalar and vector perturbations of the homogeneous background are derived in the weak gravitational field limit without any supplementary approximations. The sources of these perturbations (inhomogeneities) are presented in the discrete form of a system of separate point-like gravitating masses. The found expressions for the metric corrections are valid at all (sub-horizon and super-horizon) scales and converge at all points except at locations of the sources. The average values of these metric corrections are zero (thus, first-order backreaction effects are absent). Both the Minkowski background limit and the Newtonian cosmological approximation are reached under certain well-defined conditions. An important feature of the velocity-independent part of the scalar perturbation is revealed: up to an additive constant this part represents a sum of Yukawa potentials produced by inhomogeneities with the same finite time-dependent Yukawa interaction range. The suggested connection between this range and the homogeneity scale is briefly discussed along with other possible physical implications.

We consider the Universe deep inside the cell of uniformity. At these scales, the Universe is filled with inhomogeneously distributed discrete structures (galaxies, groups and clusters of galaxies), which perturb the background Friedmann model. Here, the mechanical approach (Eingorn & Zhuk, 2012) is the most appropriate to describe the dynamics of the inhomogeneities which is defined, on the one hand, by gravitational potentials of inhomogeneities and, on the other hand, by the cosmological expansion of the Universe. In this paper, we present additional arguments in favor of this approach. First, we estimate the size of the cell of uniformity. With the help of the standard methods of statistical physics and for the galaxies of the type of the Milky Way and Andromeda, we get that it is of the order of 190 Mpc which is rather close to observations. Then, we show that the nonrelativistic approximation (with respect to the peculiar velocities) is valid for z 10, i.e. approximately for 13 billion years from the present moment. We consider scalar perturbations and, within the ΛCDM model, justify the main equations. Moreover, we demonstrate that radiation can be naturally incorporated into our scheme. This emphasizes the viability of our approach. This approach gives a possibility to analyze different cosmological models and compare them with the observable Universe. For example, we indicate some problematic aspects of the spatially flat models. Such models require a rather specific distribution of the inhomogeneities to get a finite potential at any points outside gravitating masses. We also criticize the application of the Schwarzschild-de Sitter solution to the description of the motion of test bodies on the cosmological background.

In Kaluza-Klein model with toroidal extra dimensions, we obtain the metric coefficients in a weak field approximation for delta-shaped matter sources. These metric coefficients are applied to calculate the formulas for frequency shift, perihelion shift, deflection of light and parameterized post-Newtonian (PPN) parameters. In the leading order of approximation, the formula for frequency shift coincides with well known general relativity expression. However, for perihelion shift, light deflection and PPN parameter γ we obtain formulas Dπrg/[(D − 2)a(1 − e 2 )], (D − 1)rg/[(D − 2)ρ] and 1/(D − 2) respectively, where D is a total number of spatial dimensions. These expressions demonstrate good agreement with experimental data only in the case of ordinary three-dimensional (D = 3) space. This result does not depend on the size of the extra dimensions. Therefore, in considered multidimensional Kaluza-Klein models the point-like masses cannot produce gravitational field which corresponds to the classical gravitational tests.

In Kaluza-Klein models with toroidal compactification of the extra dimensions, we investigate soliton solutions of Einstein equation. The nonrelativistic gravitational potential of these solitons exactly coincides with the Newtonian one. We obtain the formulas for perihelion shift, deflection of light, time delay of radar echoes and post-Newtonian (PPN) parameters. Using the constraint on PPN parameter , we find that the solitonic parameter k should be very big: jkj ! 2:3 Â 10 4 . We define a soliton solution which corresponds to a pointlike mass source. In this case the soliton parameter k ¼ 2, which is clearly contrary to this restriction. A similar problem with the observations takes place for static spherically symmetric perfect fluid with the dustlike equation of state in all dimensions. The common for both of these models is the same (dustlike) equations of state in our three dimensions and in the extra dimensions. All dimensions are treated at equal footing. This is the crucial point. To be in agreement with observations, it is necessary to break the symmetry (in terms of equations of state) between the external/our and internal spaces. It takes place for black strings which are particular examples of solitons with k ! 1. For such k, black strings are in concordance with the observations. Moreover, we show that they are the only solitons which are at the same level of agreement with the observations as in general relativity. Black strings can be treated as perfect fluid with dustlike equation of state p 0 ¼ 0 in the external/our space and very specific equation of state p 1 ¼ Àð1=2Þ" in the internal space. The latter equation is due to negative tension in the extra dimension. We also demonstrate that dimension 3 for the external space is a special one. Only in this case we get the latter equation of state. We show that the black string equations of state satisfy the necessary condition of the internal space stabilization. Therefore, black strings are good candidates for a viable model of astrophysical objects (e.g., Sun) if we can provide a satisfactory explanation of negative tension for particles constituting these objects.

In Kaluza-Klein models with an arbitrary number of toroidal internal spaces, we investigate soliton solutions which describe the gravitational field of a massive compact object. Each d i -dimensional torus has its own scale factor C i , i ¼ 1; . . . ; N, which is characterized by a parameter i . We single out the physically interesting solution corresponding to a pointlike mass. For the general solution we obtain equations of state in the external and internal spaces. These equations demonstrate that the pointlike mass soliton has dustlike equations of state in all spaces. We also obtain the parametrized post-Newtonian parameters, which give the possibility to obtain the formulas for perihelion shift, deflection of light and time-delay of radar echoes. Additionally, the gravitational experiments lead to a strong restriction on the parameters of the model: ¼ P N i¼1 d i i ¼ Àð2:1 AE 2:3Þ Â 10 À5 . The pointlike mass solution withcontradicts this restriction. The condition ¼ 0 satisfies the experimental limitation and defines a new class of solutions which are indistinguishable from general relativity. We call such solutions latent solitons. Black strings and black branes with i ¼ 0 belong to this class. Moreover, the condition of stability of the internal spaces singles out black strings/branes from the latent solitons and leads uniquely to the black string/brane equations of state p i ¼ À"=2, i ¼ 1; . . . ; N, in the internal spaces and to the number of the external dimensions d 0 ¼ 3. The investigation of multidimensional static spherically symmetric perfect fluid with a dustlike equation of state in the external space confirms the above results.

In this paper, we consider the Universe at the late stage of its evolution and deep inside the cell of uniformity. At these scales, the Universe is filled with inhomogeneously distributed discrete structures (galaxies, groups and clusters of galaxies). Supposing that the Universe contains also the cosmological constant and a perfect fluid with a negative constant equation of state (EoS) parameter (e.g., quintessence, phantom or frustrated network of topological defects), we investigate scalar perturbations of the Friedmann–Robertson–Walker metrics due to inhomogeneities. Our analysis shows that, to be compatible with the theory of scalar perturbations, this perfect fluid, first, should be clustered and, second, should have the EoS parameter . In particular, this value corresponds to the frustrated network of cosmic strings. Therefore, the frustrated network of domain walls with is ruled out. A perfect fluid with neither accelerates nor decelerates the Universe. We also obtain the equation for the nonrelativistic gravitational potential created by a system of inhomogeneities. Due to the perfect fluid with , the physically reasonable solutions take place for flat, open and closed Universes. This perfect fluid is concentrated around the inhomogeneities and results in screening of the gravitational potential.

We investigate the classical gravitational tests for the six-dimensional Kaluza-Klein model with spherical (of a radius a) compactification of the internal space. The model contains also a bare multidimensional cosmological constant Λ6. The matter, which corresponds to this ansatz, can be simulated by a perfect fluid with the vacuum equation of state in the external space and an arbitrary equation of state with the parameter ω1 in the internal space. For example, ω1 = 1 and ω1 = 2 correspond to the monopole two-forms and the Casimir effect, respectively. In the particular case Λ6 = 0, the parameter ω1 is also absent: ω1 = 0. In the weak-field approximation, we perturb the background ansatz by a point-like mass. We demonstrate that in the case ω1 > 0 the perturbed metric coefficients have the Yukawa type corrections with respect to the usual Newtonian gravitational potential. The inverse square law experiments restrict the parameters of the model: a/ √ ω1 6 × 10 −3 cm. Therefore, in the Solar system the parameterized post-Newtonian parameter γ is equal to 1 with very high accuracy. Thus, our model satisfies the gravitational experiments (the deflection of light and the time delay of radar echoes) at the same level of accuracy as General Relativity. We demonstrate also that our background matter provides the stable compactification of the internal space in the case ω1 > 0. However, if ω1 = 0, then the parameterized post-Newtonian parameter γ = 1/3, which strongly contradicts the observations.