The Primordial Black Holes (PBHs) are a well-established probe for new physics in the very early Universe. We discuss here the possibility of PBH agglomeration into clusters that may have several prominent observable features. The clusters can form due to closed domain walls appearance in the natural and hybrid inflation models whose subsequent evolution leads to PBH formation. The dynamical evolution of such clusters discussed here is of crucial importance. Such a model inherits all the advantages of uniformly distributed PBHs, like possible explanation of supermassive black holes existence (origin of the early quasars), the binary black hole mergers registered by LIGO/Virgo through gravitational waves, which could provide ways to test the model in future, the contribution to reionization of the Universe. If PBHs form clusters, they could alleviate or completely avoid existing constraints on the abundance of uniformly distributed PBHs, thus allowing PBH to be a viable dark matter candidate. Most of the existing constraints on uniform PBH density should be re-considered to the case of PBH clustering. Furthermore, unidentified cosmic gamma-ray point-like sources could be (partially) accounted. We conclude that models leading to PBH clustering are favored compared to models predicting the uniform distribution of PBHs.

The nonbaryonic dark matter of the Universe is assumed to consist of new stable forms of matter. Their stability reflects symmetry of micro world and mechanisms of its symmetry breaking. In the early Universe heavy metastable particles can dominate, leaving primordial black holes (PBHs) after their decay, as well as the structure of particle symmetry breaking gives rise to cosmological phase transitions, from which massive black holes and/or their clusters can originate. PBHs can be formed in such transitions within a narrow interval of masses about 10 17 g and, avoiding severe observational constraints on PBHs, can be a candidate for the dominant form of dark matter. PBHs in this range of mass can give solution of the problem of reionization in the Universe at the redshift z ∼ 5 . . . 10. Clusters of massive PBHs can serve as a nonlinear seeds for galaxy formation, while PBHs evaporating in such clusters can provide an interesting interpretation for the observations of point-like gamma-ray sources. Analysis of possible PBH signatures represents a universal probe for super-high energy physics in the early Universe in studies of indirect effects of the dark matter.

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The simplistic model of the classical spacetime foam is considered, which consists of static wormholes embedded in Minkowski spacetime. We explicitly demonstrate that such a foam structure leads to a topological bias of point-like sources which can equally be interpreted as the presence of a dark halo around any point source. It is shown that a nontrivial halo appears on scales where the topological structure possesses local inhomogeneity, while the homogeneous structure reduces to a constant renormalization of the intensity of sources. We also show that in general dark halos possess both (positive and negative) signs depending on scales and specific properties of the topological structure of space.

Stochastic properties of the homogeneous Bianchi type-VIII and -IX ͑the mixmaster͒ models near the cosmological singularity are more distinctive in the Hamiltonian formalism in the Misner-Chitré parametrization. We show how the simplest analysis of the dynamical evolution leads, in a natural way, to the construction of a stationary invariant measure distribution which provides the complete statistical description of the stochastic behavior of these systems. We also establish the difference between the statistical description in the framework of the Misner-Chitré approach and that one based on the BKL ͑Belinski-Khalatnikov-Lifshitz͒ map by means of an explicit reduction of the invariant measure in the continuous case to the measure on the map. It turns out that the invariant measure in the continuous case contains an explicit information about durations of Kasner eras, while the measure in the case of the BKL map does not. ͓S0556-2821͑97͒00420-7͔PACS number͑s͒: 98.80. Hw, 05.45.ϩb The homogeneous Bianchi type-VIII and -IX cosmological models are known to have a chaotic behavior near the singularity when described in terms of the synchronous reference coordinates ͓1,2͔ and clear indications in favor of the covariance of this properties appeared, after a long debate, in a recent general analysis ͓3͔.The statistical nature of their dynamics turns out to be the most distinctive within framework of two approaches. The first one is the Belinski-Lifshitz-Khalatnikov ͑BKL͒ approach, when the evolution of basic variables is reduced to a discrete map ͑the BKL or Kasner map͒ ͓1͔. The second approach based on the use of the so-called Misner-Chitre variables in terms of which the evolution of the models is reduced to a billiard on the Lobachevsky plane, see Refs. ͓4-8͔. It is a well-known fact that the both approaches are equivalent, e.g., the construction of a Poincaré section for the billiard gives exactly the BKL map, see Ref. ͓8͔. However, there still remains an uncertainty in the relation between statistical properties of the models in the continuous case and the properties of the discrete BKL map ͑e.g., see papers in Ref. ͓2͔͒. Complete measure theoretic distributions, describing statistical properties of the BKL map, were constructed in Refs. ͓9-11͔ ͑see also references therein͒ and the invariant measure distribution in the continuous case ͑the case of the billiard͒ is known to be given by the volume element of the Lobashevsky plane ͓7,8,12͔. In this paper we display the correspondence between the both statistical descriptions by means of an explicit reduction of the invariant measure on the billiard to that one on the BKL map. The difference is the fact that the measure distribution in the continuous case contains explicitly information about durations of Kasner eras, while the invariant measure on the BKL map does not. In this sense we can claim that the invariant distribution in the continuous case corresponds to the more complete statistical description of the evolution of the Bianchi type-VIII and -IX models.Let...