We study the dynamical properties and space distribution of dark energy in the weak and strong gravitational fields caused by inhomogeneities of matter in the static world of galaxies and clusters. We show that the dark energy in the weak gravitational fields of matter density perturbations can condense or dilute, but amplitudes of its perturbations remain very small on all scales. We illustrate also how the "accretion" of the phantom dark energy onto the matter overdensity forms the dark energy underdensity. We analyze the behavior of dark energy in the gravitational fields of stars and black holes with the Schwarzschild metric. It is shown that, in the case of stars, the static solution of the differential equations for energy-momentum conservation exists and describes the distribution of density of dark energy inside and outside of a star. We have found that for stars and galaxies its value differs slightly from the average and is a bit higher for the quintessential scalar field as dark energy and a bit lower for the phantom one. The difference grows with the decrease of the effective sound speed of dark energy and is large in the neighborhood of neutron stars. We obtain and analyze also the solutions of equations that describe the stationary accretion of the dark energy as a test component onto the Schwarzschild black hole. It is shown that the rate of change of mass of the dark energy is positive in the case of quintessential dark energy and is negative in the case of the phantom one. PACS numbers: 95.36.+x,

We perform an analysis of the Cosmic Web as a complex network, which is built on a ΛCDM cosmological simulation. For each of nodes, which are in this case dark matter halos formed in the simulation, we compute 10 network metrics, which characterize the role and position of a node in the network. The relation of these metrics to topological affiliation of the halo, i.e. to the type of large scale structure, which it belongs to, is then investigated. In particular, the correlation coefficients between network metrics and topology classes are computed. We have applied different machine learning methods to test the predictive power of obtained network metrics and to check if one could use network analysis as a tool for establishing topology of the large scale structure of the Universe. Results of such predictions, combined in the confusion matrix, show that it is not possible to give a good prediction of the topology of Cosmic Web (score is ≈ 70 % in average) based only on coordinates and velocities of nodes (halos), yet network metrics can give a hint about the topological landscape of matter distribution.

We analyse the evolution of scalar field dark energy in the spherical halos of dark matter at the late stages of formation of gravitationally bound systems in the expanding Universe. The dynamics of quintessential dark energy at the center of dark matter halo strongly depends on the value of effective sound speed cs (in units of speed of light). If cs ∼ 1 (classical scalar field) then the dark energy in the gravitationally bound systems is only slightly perturbed and its density is practically the same as in cosmological background. The dark energy with small value of sound speed (cs < 0.1), on the contrary, is important dynamical component of halo at all stages of their evolution: linear, non-linear, turnaround, collapse, virialization and later up to current epoch. These properties of dark energy can be used for constraining the value of effective sound speed cs by comparison the theoretical predictions with observational data related to the large scale gravitationally bound systems.PACS numbers: 95.36.+x, 98.80.-k

Formation of halos in the Dark Ages from initial spherical perturbations is analyzed in a four component Universe (dark matter, dark energy, baryonic matter and radiation) in the approximation of relativistic hydrodynamics. Evolution of density and velocity perturbations of each component is obtained by integration of a system of nine differential equations from z = 10 8 up to virialization, which is described phenomenologically. It is shown that the number density of dark matter halos with masses M ∼ 10 8 − 10 9 M ⊙ virialized at z ∼ 10 is close to the number density of galaxies in comoving coordinates. The dynamical dark energy of classical scalar field type does not significantly influence the evolution of the other components, but dark energy with a small value of effective sound speed can affect the final halo state. Simultaneously, the formation/dissociation of the first molecules have been analyzed in the halos which are forming. The results show that number densities of molecules H 2 and HD at the moment of halo virialization are ∼ 10 3 and ∼ 400 times larger, respectively, than on a uniformly expanding background. It is caused by increased density and rates of reactions at quasilinear and nonlinear evolution stages of density and velocity of the baryonic component of halos.It is shown also that the temperature history of the halo is important for calculating the concentration of molecular ions with low binding energy. So, in a halo with virial temperature ∼ 10 5 K the number density of the molecular ion HeH + is approximately 100 times smaller than that on the cosmological background.

We analyse the evolution of cosmological perturbations which leads to the formation of large isolated voids in the Universe. We assume that initial perturbations are spherical and all components of the Universe (radiation, matter and dark energy) are continuous media with perfect fluid energy-momentum tensors, which interact only gravitationally. Equations of the evolution of perturbations for every component in the comoving to cosmological background reference frame are obtained from equations of energy and momentum conservation and Einstein's ones and are integrated numerically. Initial conditions are set at the early stage of evolution in the radiation-dominated epoch, when the scale of perturbation is much larger than the particle horizon. Results show how the profiles of density and velocity of matter and dark energy are formed and how they depend on parameters of dark energy and initial conditions. In particular, it is shown that final matter density and velocity amplitudes change within range ∼4-7% when the value of equation-of-state parameter of dark energy w vary in the range from -0.8 to -1.2, and change within ∼1% only when the value of effective sound speed of dark energy vary over all allowable range of its values.