Although the new era of high precision cosmology of the cosmic microwave background (CMB) radiation improves our knowledge to understand the infant as well as the presentday Universe, it also leads us to question the main assumption of the exact isotropy of the CMB. There are two pieces of observational evidence demonstrating that there is no exact isotropy. These are first the existence of small anisotropy deviations from isotropy of the CMB radiation and second, the presence of large angle anomalies that are shown as real features by the Planck satellite results. These evidences are particularly important since isotropy is one of the two main postulates of the Copernican principle on which the FRW models are built. This almost isotropic CMB radiation implies that the universe is almost a FRW universe, as is proved by previous studies.Assuming the matter component forms the deviations from isotropy in the CMB density fluctuations when matter and radiation decouples, we here attempt to find possible constraints on these deviations by using the Bianchi type I (BI) anisotropic model which is asymptotically equivalent to the standard FRW. Then we obtain the separations from the FRW type scale factor and the Hubble parameter in the form of time dependent functions. Hence we put constraints on the anisotropy coefficients from the anisotropy upper limits of the recent Planck data and from the theoretical consistency relation. These constraints lead us to obtain a BI model which becomes to be an almost FRW in time that is consistent with observational data of the CMB.

In this study, we obtain the size distribution of voids as a 3-parameter redshift independent lognormal void probability function (VPF) directly from the Cosmic Void Catalog (CVC). Although many statistical models of void distributions are based on the counts in randomly placed cells, the log-normal VPF that we here obtain is independent of the shape of the voids due to the parameter-free void finder of the CVC. We use three void populations drawn from the CVC generated by the Halo Occupation Distribution (HOD) Mocks which are tuned to three mock SDSS samples to investigate the void distribution statistically and the effects of the environments on the size distribution. As a result, it is shown that void size distributions obtained from the HOD Mock samples are satisfied by the 3-parameter log-normal distribution. In addition, we find that there may be a relation between hierarchical formation, skewness and kurtosis of the log-normal distribution for each catalog. We also show that the shape of the 3-parameter distribution from the samples is strikingly similar to the galaxy log-normal mass distribution obtained from numerical studies. This similarity of void size and galaxy mass distributions may possibly indicate evidence of nonlinear mechanisms affecting both voids and galaxies, such as large scale accretion and tidal effects. Considering in this study all voids are generated by galaxy mocks and show hierarchical structures in different levels, it may be possible that the same nonlinear mechanisms of mass distribution affect the void size distribution.

Following up on previous studies, we here complete a full analysis of the void size distributions of the Cosmic Void Catalog (CVC) based on three different simulation and mock catalogs; dark matter, haloes and galaxies. Based on this analysis, we attempt to answer two questions: Is a 3-parameter log-normal distribution a good candidate to satisfy the void size distributions obtained from different types of environments? Is there a direct relation between the shape parameters of the void size distribution and the environmental effects? In an attempt to answer these questions, we here find that all void size distributions of these data samples satisfy the 3-parameter log-normal distribution whether the environment is dominated by dark matter, haloes or galaxies. In addition, the shape parameters of the 3-parameter log-normal void size distribution seem highly affected by environment, particularly existing substructures. Therefore, we show two quantitative relations given by linear equations between the skewness and the maximum tree depth, and variance of the void size distribution and the maximum tree depth directly from the simulated data. In addition to this, we find that the percentage of the voids with nonzero central density in the data sets has a critical importance. If the number of voids with nonzero central densities reaches ≥ %3.84 in a simulation/mock sample, then a second population is observed in the void size distributions. This second population emerges as a second peak in the log-normal void size distribution at larger radius.

Observational studies show that voids are prominent features of the large scale structure of the present day Universe. Even though their emerging from the primordial density perturbations and evolutionary patterns differ from dark matter halos, N -body simulations and theoretical models have shown that voids also merge together to form large void structures. In this study, following Sheth & van de Weygaert (2004), we formulate an analytical approximate description of the hierarchical void evolution of growing voids by adopting the halo merging algorithm given by Lacey & Cole (1993) in the Einstein de Sitter (EdS) Universe. To do this, we take into account the general volume distribution of voids which consists of two main void processes: merging and collapsing. We show that the volume distribution function can be reduced to a simple form, by neglecting the collapsing void contribution since the collapse process is negligible for large size voids. Therefore, the void volume fraction has a contribution only from growing voids. This algorithm becomes the analogue of the halo merging algorithm. Based on this growing void distribution, we obtain the void merging algorithm in which we define and formulate void merging and absorption rates, as well as void size and redshift survival probabilities and also failure rates in terms of the self similar and currently favored dark energy dominated cold dark matter models in the EdS Universe.

In hierarchical evolution, voids exhibit two different behaviors related with their surroundings and environments, they can merge or collapse. These two different types of void processes can be described by the two-barrier excursion set formalism based on Brownian random walks. In this study, the analytical approximate description of the growing void merging algorithm is extended by taking into account the contributions of voids that are embedded into overdense region(s) which are destined to vanish due to gravitational collapse. Following this, to construct a realistic void merging model that consists of both collapse and merging processes, the two-barrier excursion set formalism of the void population is used. Assuming spherical voids in the Einstein de Sitter Universe, the void merging algorithm which allows us to consider the two main processes of void hierarchy in one formalism is constructed. In addition to this, the merger rates, void survival probabilities, void size distributions in terms of the collapse barrier and finally, the void merging tree algorithm in the self-similar models are defined and derived.