Sergei F. Shandarin

The Aspen–Amsterdam void finder comparison project

Despite a history that dates back at least a quarter of a century studies of voids in the large–scale structure of the Universe are bedevilled by a major problem: there exist a large number of quite different void–finding algorithms, a fact that has so far got in the way of groups comparing their results without worrying about whether such a comparison in fact makes sense. Because of the recent increased interest in voids, both in very large galaxy surveys and in detailed simulations of cosmic structure formation, this situation is very unfortunate. We here present the first systematic comparison study of thirteen different void finders constructed using particles, haloes, and semi– analytical model galaxies extracted from a subvolume of the Millennium simulation. The study includes many groups that have studied voids over the past decade. We show their results and discuss their differences and agreements. As it turns out, the basic results of the various methods agree very well with each other in that they all locate a major void near the centre of our volume. Voids have very underdense centres, reaching below 10 percent of the mean cosmic density. In addition, those void finders that allow for void galaxies show that those galaxies follow similar trends. For example, the overdensity of void galaxies brighter than mB = 20 is found to be smaller than about 0.8 by all our void finding algorithms.

The Cosmological mass distribution function in the Zeldovich approximation

An analytic approximation of the mass function for gravitationally bound objects is presented. Based on the Zeldovich approximation, we extend the Press-Schechter formalism to a nonspherical dynamical model. A simple extrapolation of that approximation suggests that the gravitational collapse along all three directions, which eventually leads to the formation of real virialized object clumps, occurs in the regions where the lowest eigenvalue of the deformation tensor, λ3, is positive. We derive the conditional probability of λ3 > 0 as a function of the linearly extrapolated density contrast δ and the conditional probability distribution of δ, provided that λ3 > 0. These two conditional probability distributions show that the most probable density of the bound regions (λ3 > 0) is roughly 1.5 on the characteristic mass scale M* and that the probability of λ3 > 0 is almost unity in the highly overdense regions (δ > 3σ). Finally, an analytic mass function of clumps is derived with the help of one simple Ansatz, which is employed to treat the multistream regime beyond the validity of the Zeldovich approximation. The resulting mass function is renormalized by a factor of 12.5, which we justify with a sharp k-space filter by means of the modified Jedamzik analysis. Our mass function is shown to be different from the Press-Schechter one, having a lower peak and predicting more small-mass objects.

The Cosmic Web, Multi-Stream Flows, and Tessellations

Understanding the structure of the matter distribution in the Universe due to the action of the gravitational instability – the cosmic web – is complicated by lack of direct analytic access to the nonlinear domain of structure formation. Here, we suggest and apply a novel tessellation method designed for cold dark matter (CDM) N-body cosmological simulations. The method is based on the fact that the initial CDM state can be described by a 3-D manifold (in a 6-D phase space) that remains continuous under evolution. Our technique uses the full phase space information and has no free parameters; it can be used to compute multi-stream and density fields, the main focus

Morphology of the supercluster-void network in ΛCDM cosmology

We report here the first systematic study of the supercluster-void network in theCDM concordance cosmology in which voids and superclusters are treated on an equal footing. We study the dark matter density field in real space smoothed on a scale of 5 h −1 Mpc. Superclusters are defined as individual members of an overdense excursion set, and voids are defined as individual members of a complementary underdense excursion set at the same density threshold. We determine the geometric, topological and morphological properties of the cosmic web at a large set of density levels by computing Minkowski functionals for every supercluster and void using SURFGEN (described recently by Sheth et al.). The properties of the largest (percolating) supercluster and the complementary void are found to be very different from those of the individual superclusters and voids. In total, the individual superclusters occupy no more than about 5 per cent of the volume and contain no more than 20 per cent of the mass if the largest supercluster is excluded. Likewise, in total, individual voids occupy no more than 14 per cent of the volume and contain no more than 4 per cent of the mass if the largest void is excluded. Although superclusters are more massive and voids are more voluminous, the difference in maximum volumes is no greater than an order of magnitude. The genus value of individual superclusters can be ∼5, while the genus of individual voids can reach ∼50, implying a significant amount of substructure in superclusters and especially in voids. One of our main results is that large voids, as defined through the dark matter density field in real space, are distinctly non-spherical. Ke yw ords: methods: numerical - galaxies: statistics - cosmology: theory - large-scale struc- ture of Universe.

Disentangling the Cosmic Web. I. Morphology of Isodensity Contours

We apply Minkowski functionals and various derived measures to decipher the morphological properties of large-scale structure seen in simulations of gravitational evolution. Minkowski functionals of isodensity contours serve as tools to test global properties of the density field. Furthermore, we identify coherent objects at various threshold levels and calculate their partial Minkowski functionals. We propose a set of two derived dimensionless quantities, planarity and filamentarity, which reduce the morphological information in a simple and intuitive way. Several simulations of the gravitational evolution of initial power-law spectra provide a framework for systematic tests of our method.

The Evolution of Voids in the Adhesion Approximation

We apply the adhesion approximation to study the formation and evolution of voids in the universe. Our simulations-carried out using 128(exp 3) particles in a cubical box with side 128 Mpc-indicate that the void spectrum evolves with time and that the mean void size in the standard Cosmic Background Explorer Satellite (COBE)-normalized cold dark matter (CDM) model with H(sub 50) = 1 scals approximately as bar D(z) = bar D(sub zero)/(1+2)(exp 1/2), where bar D(sub zero) approximately = 10.5 Mpc. Interestingly, we find a strong correlation between the sizes of voids and the value of the primordial gravitational potential at void centers. This observation could in principle, pave the way toward reconstructing the form of the primordialpotential from a knowledge of the observed void spectrum. Studying the void spectrum at different cosmological epochs, for spectra with a built in k-space cutoff we find that the number of voids in a representative volume evolves with time. The mean number of voids first increases until a maximum value is reached (indicating that the formation of cellular structure is complete), and then begins to decrease as clumps and filaments erge leading to hierarchical clustering and the subsequent elimination of small voids. The cosmological epoch characterizing the completion of cellular structure occurs when the length scale going nonlinear approaches the mean distance between peaks of the gravitaional potential. A central result of this paper is that voids can be populated by substructure such as mini-sheets and filaments, which run through voids. The number of such mini-pancakes that pass through a given void can be measured by the genus characteristic of an individual void which is an indicator of the topology of a given void in intial (Lagrangian) space. Large voids have on an average a larger measure than smaller voids indicating more substructure within larger voids relative to smaller ones. We find that the topology of individual voids is strongly epoch dependent, with void topologies generally simplifying with time. This means that as voids grow older they become progressively more empty and have less structure within them. We evaluate the genus measure both for individual voids as well as for the entire ensemble of voids predicted by CDM model. As a result we find that the topology of voids when taken together with the void spectrum is a very useful statistical indicator of the evolution of the structure of the universe on large scales.

Coherent structures in the universe and the adhesion model

An adhesion model is used to study the formation process of large-scale structures due to nonlinear gravitational growth of small initial fluctuations in the universe dominated by dark matter. The model is compared with 2D N-body simulations with initial power-law spectral indices n = -2, 0, +2, and various cutoffs. It is found that the adhesion model imitates the skeleton of the structure extremely well for the parameters of the initial spectra until the stage when the nonlinear scale reaches the correlation length R(phi) of the initial gravitational potential. The model explains the origin of large-scale coherent sructures, such as superpancakes and superfilaments, as a result of coherent motion of clumps due to large-scale inhomogeneities in the initial gravitational potential. It is found that clumps of mass identified in the N-body simulations correspond to several knots in the adhesion model, which influences the way of calculating the mass distribution function. The distribution functions of velocities and masses of clumps and areas of cells in the adhesion model satisfy self-similar scaling laws of the n = 2 model.

Measuring the geometry and topology of large-scale structure using SURFGEN: methodology and preliminary results

Observations of the universe reveal that matter within it clusters on a variety of scales. On scales between 10 - 100 Mpc, the universe is spanned by a percolating network of superclusters interspersed with large and almost empty regions – voids. This paper, the first in a series, presents a new ansatz which can successfully be used to determine the morphological properties of the supercluster-void network. The ansatz is based on a surface modelling scheme (SURFGEN), developed explicitly for the purpose, which generates a triangulated surface from a discrete data set representing (say) the distribution of galaxies in real (or redshift) space. The triangulated surface describes, at progressively lower density thresholds, clusters of galaxies, superclusters of galaxies and voids. Four Minkowski functionals (MFs) – surface area, volume, extrinsic curvature and genus – describe the geometry and topology of the supercluster-void network. On a discretised and closed triangulated surface the MFs are determined using SURFGEN. Ratio’s of the Minkowski functionals provide us with an excellent diagnostic of three dimensional shapes of clusters, superclusters and voids. Minkowski functionals can be studied at different levels of the density contrast and therefore probe the morphology of large scale structure on a variety of length scales. Our method for determining the Minkowski functionals of a triangulated iso-density surface is tested against both simply and multiply connected eikonal surfaces such as triaxial ellipsoids and tori. The performance of our code is thereby evaluated using density distributions which are pancake-like, filamentary, ribbon-like and spherical. Remarkably, the first three Minkowski functionals are computed to better than 1% accuracy while the fourth (genus) is known exactly. SURFGEN also gives very accurate results when applied to Gaussian random fields. We apply SURFGEN to study morphology in three cosmological models, �CDM, �CDM and SCDM, at the present epoch. Geometrical properties of the supercluster-void network are found to be sensitive to the underlying

Comparison of Analytical Mass Functions with Numerical Simulations

We present numerical testing results of our mass function derived in our previous paper and compare the testing results with those of the popular Press-Schechter (PS) mass function. Two fiducial models are considered for the test: the scale-free power-law spectra P(k)∝kn with spectral indices n=-1, 0 and the standard cold dark matter (SCDM) model with Ω=1 and h=0.5. For the power-law models, we use numerical data averaged over several different output times: 10 output times from two N-body realizations for the n=-1 power-law model; four outputs from one realization for the n=0 model. For the SCDM model, we consider four outputs separately at redshifts z=0, 0.43, 1.14, and 1.86 from one large N-body simulation. The comparison results show that our mass function fits the numerical data in a much improved way over the PS one. Thus, we expect that our mass function can be a viable alternative of the PS mass function in applications to various areas.

MINKOWSKI FUNCTIONALS AND CLUSTER ANALYSIS FOR CMB MAPS

We suggest novel statistics for the CMB maps that are sensitive to non-Gaussian features. These statistics are natural generalizations of the geometrical and topological methods that have been already used in cosmology such as the cumulative distribution function and genus. We compute the distribution functions of the Partial Minkowski Functionals for the excursion set above or bellow a constant temperature threshold. Minkowski Functionals are additive and are translationally and rotationally invariant. Thus, they can be used for patchy and/or incomplete coverage. The technique is highly efficient computationally (it requires only O(N) operations, where N is the number of pixels per one threshold level). Further, the procedure makes it possible to split large data sets into smaller subsets. The full advantage of these statistics can be obtained only on very large data sets. We apply it to the 4-year DMR COBE data corrected for the Galaxy contamination as an illustration of the technique.