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Matter density is formally infinite at the location of caustic surfaces, where dark matter sheet folds in phase-space. The caustics separate regions with different number of streams and the volume elements change the parity by turning inside out when passing through the caustic stage. Being measure-zero structures, identification of caustics via matter density fields is usually restricted to fine-grained simulations. Here we employ a generic algorithm to identify caustics directly using the triangulation of the Lagrangian sub-manifold x(q,t) obtained in N-body simulations. In our approach the caustic surfaces are approximated by a set of triangles whose vertices are particles of the simulation. The major obstacle we encountered was insufficient sampling of small scale perturbations. We overcame it by a brute force approach. We continued to raise the scale of the cutoff in the initial power spectrum until obtained the reliable resolution of the caustics shells up to seven layers. Although quite modest, our result is the first reliable direct construction of caustic surfaces in N-body simulation. It reveals a number of unexpected geometrical features. In particular shapes of some of them are contrastingly different from the known shapes of the caustics formed in the Zeldovich approximation., Comment: 10 pages, 13 figures

Mysterious dark matter constitutes about 85% of all mass in the Universe. Clustering of dark matter plays the dominant role in the formation of all observed structures on scales from a fraction to a few hundreds of Mega-parsecs. Galaxies play a role of lights illuminating these structures so they can be observed. The observations in the last several decades have unveiled opulent geometry of these structures currently known as the cosmic web. Haloes are the highest concentrations of dark matter and host luminous galaxies. Currently the most accurate modeling of dark matter haloes is achieved in cosmological N-body simulations. Identifying the haloes from the distribution of particles in N-body simulations is one of the problems attracting both considerable interest and efforts. We propose a novel framework for detecting potential dark matter haloes using the field unique for dark matter -- multistream field. The multistream field emerges at the nonlinear stage of the growth of perturbations because the dark matter is collisionless. Counting the number of velocity streams in gravitational collapses supplements our knowledge of spatial clustering. We assume that the virialized haloes have convex boundaries. Closed and convex regions of the multistream field are hence isolated by imposing a positivity condition on all three eigenvalues of the Hessian estimated on the smoothed multistream field. In a single-scale analysis of high multistream field resolution and low softening length, the halo substructures with local multistream maxima are isolated as individual halo sites., Comment: 16 pages, 15 figures. Accepted for publication in MNRAS. Portions of this work appeared as arXiv:1608.05469v1, which was divided during the refereeing process of the journal, and published separately

Understanding of the observed structure in the universe can be reached only in the theoretical framework of dark matter. N-body simulations are indispensable for the analysis of the formation and evolution of the dark matter web. Two primary fields - density and velocity fields - are used in most of studies. However dark matter provides two additional fields which are unique for collisionless media only. These are the multi- stream field in Eulerian space and flip-flop field in Lagrangian space. The flip-flop field represents the number of sign reversals of an elementary volume of each collisionless fluid element. This field can be estimated by counting the sign reversals of the Jacobian at each particle at every time step of the simulation. The Jacobian is evaluated by numerical differentiation of the Lagrangian submanifold, i.e., the three-dimensional dark matter sheet in the six-dimensional space formed by three Lagrangian and three Eulerian coordinates. We present the results of the statistical study of the evolution of the flip-flop field from z = 50 to the present time z = 0. A number of statistical characteristics show that the pattern of the flip-flop field remains remarkably stable from z = 30 to the present time. As a result the flip-flop field evaluated at z = 0 stores a wealth of information about the dynamical history of the dark matter web. In particular one of the most intriguing properties of the flip-flop is a unique capability to preserve the information about the merging history of dark matter haloes., Comment: 19 pages, 25 figures, submitted to MNRAS

Topological connections in the single-streaming voids and multistreaming filaments and walls reveal a cosmic web structure different from traditional mass density fields. A single void structure not only percolates the multistream field in all the directions, but also occupies over 99 per cent of all the single-streaming regions. Sub-grid analyses on scales smaller than simulation resolution reveal tiny pockets of voids that are isolated by membranes of the structure. For the multistreaming excursion sets, the percolating structure is significantly thinner than the filaments in over-density excursion approach. Hessian eigenvalues of the multistream field are used as local geometrical indicators of dark matter structures. Single-streaming regions have most of the zero eigenvalues. Parameter-free conditions on the eigenvalues in the multistream region may be used to delineate primitive geometries with concavities corresponding to filaments, walls and haloes., Comment: 16 pages, 13 figures. Matches version accepted by MNRAS

We report the results of the first study of the multi-stream environment of dark matter haloes in cosmological N-body simulations in the LCDM cosmology. The full dynamical state of dark matter can be described as a three-dimensional sub-manifold in six dimensional phase space - the dark matter sheet. In our study we use a Lagrangian sub-manifold x = x(q,t) (where x and q are co-moving Eulerian and Lagrangian coordinates respectively), which is dynamically equivalent to the dark matter sheet but is more convenient for numerical analysis. Our major results can be summarized as follows. At the resolution of the simulation i.e. without additional smoothing, the cosmic web represents a hierarchical structure: each halo is embedded in the filamentary framework of the web predominantly at the filament crossings, and each filament is embedded in the wall like fabric of the web at the wall crossings. Locally, each halo or sub-halo is a peak in the number of streams field. The number of streams in the neighbouring filaments is higher than in the neighbouring walls. The walls are regions where number of streams is equal to three or a few. Voids are uniquely defined by the local condition requiring to be a single-stream flow region. The shells of streams around haloes are quite thin and the closest void region is typically within one and a half FOF radius from the center of the halo., Comment: 12 pages, 25 figures. Matches version accepted by MNRAS

A new computational paradigm for the analysis of substructure of the Cosmic Web in cosmological cold dark matter simulations is proposed. We introduce a new data-field --- the flip-flop field ---which carries wealth of information about the history and dynamics of the structure formation in the universe. The flip-flop field is an ordered data set in Lagrangian space representing the number of turns inside out sign reversals of an elementary volume of each collisionless fluid element represented by a computational particle in a N-body simulation. This field is computed using the Lagrangian submanifold, i.e. the three-dimensional dark matter sheet in the six-dimensional space formed by three Lagrangian and three Eulerian coordinates of the simulation particles. It is demonstrated that the very rich substructure of dark matter haloes and the void regions can be reliably and unambiguously recovered from the flip-flop field., Comment: 5 pages, 6 figures

We describe how the dynamics of cosmic structure formation defines the intricate geometric structure of the spine of the cosmic web. The Zeldovich approximation is used to model the backbone of the cosmic web in terms of its singularity structure. The description by Arnold et al. (1982) in terms of catastrophe theory forms the basis of our analysis. This two-dimensional analysis involves a profound assessment of the Lagrangian and Eulerian projections of the gravitationally evolving four-dimensional phase-space manifold. It involves the identification of the complete family of singularity classes, and the corresponding caustics that we see emerging as structure in Eulerian space evolves. In particular, as it is instrumental in outlining the spatial network of the cosmic web, we investigate the nature of spatial connections between these singularities. The major finding of our study is that all singularities are located on a set of lines in Lagrangian space. All dynamical processes related to the caustics are concentrated near these lines. We demonstrate and discuss extensively how all 2D singularities are to be found on these lines. When mapping this spatial pattern of lines to Eulerian space, we find a growing connectedness between initially disjoint lines, resulting in a percolating network. In other words, the lines form the blueprint for the global geometric evolution of the cosmic web., Comment: 37 pages, 21 figures, accepted for publication in MNRAS

Tessellations are valuable both conceptually and for analysis in the study of the large-scale structure of the universe. They provide a conceptual model for the 'cosmic web,' and are of great use to analyze cosmological data. Here we describe tessellations in another set of coordinates, of the initially flat sheet of dark matter that gravity folds up in rough analogy to origami. The folds that develop are called caustics, and they tessellate space into stream regions. Tessellations of the dark-matter sheet are also useful in simulation analysis, for instance for density measurement, and to identify structures where streams overlap., Comment: To appear in the proceedings of "The World a Jigsaw: Tessellations in the Sciences." 15 pages, 11 figures. Origami-galaxy crease patterns available at http://skysrv.pha.jhu.edu/~neyrinck/origalaxies.html

A new numerical technique to identify the cosmic web is proposed. It is based on locating multi-stream flows, i.e. the places where the velocity field is multi-valued. The method is local in Eulerian space, simple and computaionally efficient. This technique uses the velocities of particles and thus takes into account the dynamical information. This is in contrast with the majority of standard methods that use the coordinates of particles only. Two quantities are computed in every mesh cell: the mean and variance of the velocity field. In the cells where the velocity is single-valued the variance must be equal to zero exactly, therefore the cells with non-zero variance are identified as multi-stream flows. The technique has been tested in a N-body simulation of the \L CDM model. The preliminary analysis has shown that numerical noise does not pose a significant problem. The web identified by the new method has been compared with the web identified by the standard technique using only the particle coordinates. The comparison has shown overall similarity of two webs as expected, however they by no means are identical. For example, the isocontours of the corresponding fields have significantly different shapes and some density peaks of similar heights exhibit significant differences in the velocity variance and vice versa. This suggest that the density and velocity variance have a significant degree of independence. The shape of the two-dimensional pdf of density and velocity variance confirms this proposition. Thus, we conclude that the dynamical information probed by this technique introduces an additional dimension into analysis of the web., Comment: 19 pages, 10 figures

We present a novel method for computing the Minkowski Functionals from isodensity surfaces extracted directly from the Delaunay tessellation of a point distribution. This is an important step forward compared to the previous cosmological studies when the isodensity surface was built in the field on a uniform cubic grid and therefore having a uniform spatial resolution. The density field representing a particular interest in cosmology is the density of galaxies which is obtained from the highly nonuniform distribution of the galaxy positions. Therefore, the constraints caused by the spatially uniform grid put severe limitations on the studies of the geometry and shapes of the large-scale objects: superclusters and voids of galaxies. Our technique potentially is able to eliminate most of these limitations. The method is tested with some simple geometric models and an application to the density field from an N-body simulation is shown., Comment: Accepted paper ISVD2010, Quebec, Canada. IEEE CPS, ed. M.A. Mostafavi

The adhesion approximation is a simple analytical model suggested for explanation of the major geometrical features of the observed structure in the galaxy distribution on scales from 1 to (a few)x100/h Mpc. It is based on Burgers' equation and therefore allows analysis in considerable detail. A particular version of the model that assumes the infinitesimal viscosity naturally results in irregular tessellation of the universe. Generic elements of the tessellation: vertices, edges, faces and three-dimensional tiles can be associated with astronomical objects of different kinds: clusters, superclusters and voids of galaxies. Point-like vertices contain the most of the mass and one-dimensional edges (filaments) are the second massive elements. The least massive are the two-dimensional faces and tiles (voids). The evolution of the large-scale structure can be viewed as a continuous process that transports mass predominantly from the high- to low-dimensional elements of the tessellation. For instance, the mass from the cells flows into faces, edges and vertices, in turn the mass from faces flows into edges and vertices, etc. At the same time, the elements of the tessellation themselves are in continuous motion resulting in mergers of some vertices, growth of some tiles and shrinking and disappearance of the others as well as other metamorphoses., Comment: 18 pages, 11 figures

We have made a comparative study of morphological evolution in simulated DM halos and X-ray brightness distribution, and in optical clusters. Samples of simulated clusters include star formation with supernovae feedback, radiative cooling, and simulation in the adiabatic limit at three different redshifts, z = 0.0, 0.10, and 0.25. The optical sample contains 208 ACO clusters within redshift, $z \leq 0.25$. Cluster morphology, within 0.5 and 1.0 h$^{-1}$ Mpc from cluster center, is quantified by multiplicity and ellipticity. We find that the distribution of the dark matter halos in the adiabatic simulation appear to be more elongated than the galaxy clusters. Radiative cooling brings halo shapes in excellent agreement with observed clusters, however, cooling along with feedback mechanism make the halos more flattened. Our results indicate relatively stronger structural evolution and more clumpy distributions in observed clusters than in the structure of simulated clusters, and slower increase in simulated cluster shapes compared to those in the observed one. Within $z \leq 0.1$, we notice an interesting agreement in the shapes of clusters obtained from the cooling simulations and observation. We also notice that the different samples of observed clusters differ significantly in morphological evolution with redshift. We highlight a few possibilities responsible for the discrepancy in morphological evolution of simulated and observed clusters., Comment: Accepted for publication in MNRAS, 2006; 15 pages, 13 postscript figures

We report the {\em first} systematic study of the supercluster-void network in the $\Lambda$CDM concordance cosmology treating voids and superclusters on an equal footing. We study the dark matter density field in real space smoothed with the $L_s = 5$ \hm1 Mpc Gaussian window. Superclusters and voids are defined as individual members of over-dense and under-dense excursion sets respectively. We determine the morphological properties of the cosmic web at a large number of dark matter density levels by computing Minkowski functionals for every supercluster and void. At the adopted smoothing scale individual superclusters totally occupy no more than about 5% of the total volume and contain no more than 20% of mass if the largest supercluster is excluded. Likewise, individual voids totally occupy no more than 14% of volume and contain no more than 4% of mass if the largest void is excluded. The genus of individual superclusters can be $\sim 5$ while the genus of individual voids reaches $\sim 55$, implying significant amount of substructure in superclusters and especially in voids. Large voids are typically distinctly non-spherical., Comment: 6 pages, 4 figures, uses iaus.cls, Invited talk at IAU Colloquium 195 "Outskirts of galaxy clusters: intense life in the suburbs", Torino, Italy, March 12-16, 2004

We have studied morphological evolution in clusters simulated in the adiabatic limit and with radiative cooling. Cluster morphology in the redshift range, $0 < z < 0.5$, is quantified by multiplicity and ellipticity. In terms of ellipticity, our result indicates slow evolution in cluster shapes compared to those observed in the X-ray and optical wavelengths. The result is consistent with Floor, Melott & Motl (2003). In terms of multiplicity, however, the result indicate relatively stronger evolution (compared to ellipticity but still weaker than observation) in the structure of simulated clusters suggesting that for comparative studies of simulation and observation, sub-structure measures are more sensitive than the shape measures. We highlight a few possibilities responsible for the discrepancy in the shape evolution of simulated and real clusters., Comment: Submitted to MNRAS; 11 pages, 11 postscript figures

We study a sample of 112 galaxies of various Hubble types imaged in the Two Micron All Sky Survey (2MASS) in the Near-Infra Red (NIR; 1-2 $\mu$m) $J$, $H$, and $K_s$ bands. The sample contains (optically classified) 32 elliptical, 16 lenticulars, and 64 spirals acquired from the 2MASS Extended Source Catalogue. We use a set of non-parametric shape measures constructed from the Minkowski Functionals (MFs) for galaxy shape analysis. We use ellipticity ($\epsilon$) and orientation angle ($\Phi$) as shape diagnostics. With these parameters as functions of area within the isophotal contour, we note that the NIR elliptical galaxies with $\epsilon > 0.2$ show a trend of being centrally spherical and increasingly flattened towards the edge, a trend similar to images in optical wavelengths. The highly flattened elliptical galaxies show strong change in ellipticity between the center and the edge. The lenticular galaxies show morphological properties resembling either ellipticals or disk galaxies. Our analysis shows that almost half of the spiral galaxies appear to have bar like features while the rest are likely to be non-barred. Our results also indicate that almost one-third of spiral galaxies have optically hidden bars. The isophotal twist noted in the orientations of elliptical galaxies decreases with the flattening of these galaxies indicating that twist and flattening are also anti-correlated in the NIR, as found in optical wavelengths. The orientations of NIR lenticular and spiral galaxies show a wide range of twists., Comment: Accepted for publication in MNRAS. Revised version contains 17 pages, 23 PostScript figures

We have used N-body simulations to develop two independent methods to quantify redshift distortions known as the Bull's Eye effect (large scale infall plus small scale virial motion). This effect depends upon the mass density, $\Omega_0$, so measuring it can in principle give an estimate of this important cosmological parameter. We are able to measure the effect and distinguish between its strength for high and low values of $\Omega_0$. Unlike other techniques which utilize redshift distortions, one of our methods is relatively insensitive to bias. In one approach, we use path lengths between contour crossings of the density field. The other is based upon percolation. We have found both methods to be successful in quantifying the effect and distinguishing between values of $\Omega_0$. However, only the path lengths method exhibits low sensitivity to bias., Comment: 21 pages, 5 figures, 3 tables; Replaced version - minor corrections, replaced figure 2; To appear in ApJ, Jan. 20, 2004

We suggest a set of morphological measures that we believe can help in quantifying the shapes of two-dimensional cosmological images such as galaxies, clusters, and superclusters of galaxies. The method employs non-parametric morphological descriptors known as the Minkowski functionals in combination with geometric moments widely used in the image analysis. For the purpose of visualization of the morphological properties of image contour lines we introduce three auxiliary ellipses representing the vector and tensor Minkowski functionals. We study the discreteness, seeing, and noise effects on elliptic contours as well as their morphological characteristics such as the ellipticity and orientation. In order to reduce the effect of noise we employ a technique of contour smoothing. We test the method by studying simulated elliptic profiles of toy spheroidal galaxies ranging in ellipticity from E0 to E7. We then apply the method to real galaxies, including eight spheroidals, three disk spirals and one peculiar galaxy, as imaged in the near-infrared $K_s$-band (2.2 microns) with the Two Micron All Sky Survey (2MASS). The method is numerically very efficient and can be used in the study of hundreds of thousands images obtained in modern surveys., Comment: Accepted for publication in MNRAS. Revised version contains 20 pages, 17 PostScript figures. Results unchanged; high-resolution figures # 1,6,7,11,13,16 can be obtained from authors

We present 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, which generates a triangulated surface from a discrete data set representing (say) the distribution of galaxies in real (or redshift) space. Four Minkowski functionals -- surface area, volume, extrinsic curvature and genus -- describe the geometry and topology of the supercluster-void network. Ratio's of Minkowski functionals -- Shapefinders -- provide us with an excellent diagnostic of three dimensional shapes of clusters, superclusters and voids. Minkowski functionals and Shapefinders are determined for a triangulated iso-density surface using SURFGEN. SURFGEN is tested against both simply and multiply connected eikonal surfaces such as triaxial ellipsoids and tori. Remarkably, the first three Minkowski functionals are computed to better than 1% accuracy while the fourth (genus) is known exactly. SURFGEN also gives excellent results when applied to Gaussian random fields. Our results indicate that the surface modelling scheme SURFGEN is accurate and robust and can successfully be used to quantify the topology and morphology of the supercluster-void network in the universe. We apply SURFGEN to three cosmological models, $\L$CDM, $\T$CDM and SCDM and obtain interesting new results pertaining to the geometry, morphology and topology of large scale structure., Comment: 28 MNRAS-style pages, 24 figures -- Revised with enhanced discussion and added references. An appendix included which shows that SURFGEN gives excellent results when applied to Gaussian Random Fields. Accepted for publication in MNRAS

The study of shapes of the images of objects is an important issue not only because it reveals its dynamical state but also it helps to understand the object's evolutionary history. We discuss a new technique in cosmological image analysis which is based on a set of non-parametric shape descriptors known as the Minkowski Functionals (MFs). These functionals are extremely versatile and under some conditions give a complete description of the geometrical properties of objects. We believe that MFs could be a useful tool to extract information about the shapes of galaxies, clusters of galaxies and superclusters. The information revealed by MFs can be utilized along with the knowledge obtained from currently popular methods and thus could improve our understanding of the true shapes of cosmological objects., Comment: 3 pages, 1 figure, to appear in "The IGM/Galaxy Connection - The Distribution of Baryons at z=0" Conference Proceedings

We show how the morphological analysis of the maps of the secondary CMB anisotropies can detect an extended period of ``smoldering'' reionization, during which the universe remains partially ionized. Neither radio observations of the redshifted 21cm line nor IR observations of the redshifted Lyman-alpha forest will be able to detect such a period. The most sensitive to this kind of non-gaussianity parameters are the number of regions in the excursion set, the perimeter of the excursion set, and the genus of the largest (by area) region. For example, if the universe reionized fully at z=6, but maintained about 1/3 mean ionized fraction since z=20, then a 2 arcmin map with 500x500 pixel resolution and a signal-to-noise ratio S/N=1/2 allows to detect the non-gaussianity due to reionization with better than 99% confidence level., Comment: submitted to MNRAS

The assumption of Gaussianity of the primordial perturbations plays an important role in modern cosmology. The most direct test of this hypothesis consists in testing Gaussianity of the CMB maps. Counting the pixels with the temperatures in given ranges and thus estimating the one point probability function of the field is the simplest of all the tests. Other usually more complex tests of Gaussianity generally use a great deal of the information already contained in the probability function. However, the most interesting outcome of such a test would be the signal of non-Gaussianity independent of the probability function. It is shown that the independent information has purely morphological character i.e. it depends on the geometry and topology of the level contours only. As an example we discuss in detail the quadratic model $v=u+\alpha (u^2-1)$ ($u$ is a Gaussian field with $\bar{u}=0$ and $=1$, $\alpha$ is a parameter) which may arise in slow-roll or two-field inflation models. We show that in the limit of small amplitude $\alpha$ the full information about the non-Gaussianity is contained in the probability function. If other tests are performed on this model they simply recycle the same information. A simple procedure allowing to assess the sensitivity of any statistics to the morphological information is suggested. We provide an analytic estimate of the statistical limit for detecting the quadratic non-Gaussianity $\a_c$ as a function of the map size in the ideal situation when the scale of the field is resolved. This estimate is in a good agreement with the results of the Monte Carlo simulations of $256^2$ and $1024^2$ maps. The effect of resolution on the detection quadratic non-Gaussianity is also briefly discussed., Comment: 26 pages, 9 figures

We discuss the tests of nonGaussianity in CMB maps using morphological statistics known as Minkowski functionals. As an example we test degree-scale cosmic microwave background (CMB) anisotropy for Gaussianity by studying the \qmask map that was obtained from combining the QMAP and Saskatoon data. We compute seven morphological functions $M_i(\dt)$, $i=1,...,7$: six \mf and the number of regions $N_c$ at a hundred $\dt$ levels. We also introduce a new parameterization of the morphological functions $M_i(A)$ in terms of the total area $A$ of the excursion set. We show that the latter considerably decorrelates the morphological statistics and makes them more robust because they are less sensitive to the measurements at extreme levels. We compare these results with those from 1000 Gaussian Monte Carlo maps with the same sky coverage, noise properties and power spectrum, and conclude that the \qmask map is neither a very typical nor a very exceptional realization of a Gaussian field. At least about 20% of the 1000 Gaussian Monte Carlo maps differ more than the \qmask map from the mean morphological parameters of the Gaussian fields., Comment: 21 pages, 8 figures, 2 tables. 2002, ApJS, 141, accepted

We present a comparison between two observational and three theoretical mass functions for eight cosmological models suggested by the data from the recently completed BOOMERANG-98 and MAXIMA-1 cosmic microwave background (CMB) anisotropy experiments as well as peculiar velocities (PVs) and type Ia supernovae (SN) observations. The cosmological models have been proposed as the best fit models by several groups. We show that no model is in agreement with the abundances of X-ray clusters at $\sim 10^{14.7} h^{-1}M_{\odot}$.On the other hand, we find that the BOOM+MAX+{\sl COBE}:I, Refined Concordance and $\Lambda$MDM are in a good agreement with the abundances of optical clusters. The P11 and especially Concordance models predict a slightly lower abundances than observed at $\sim 10^{14.6} h^{-1}M_{\odot}$. The BOOM+MAX+{\sl COBE}:II and PV+CMB+SN models predict a slightly higher abundances than observed at $\sim 10^{14.9} h^{-1}M_{\odot}$. The nonflat MAXIMA-1 is in a fatal conflict with the observational cluster abundances and can be safely ruled out., Comment: 17 pages, 2 figures, reference added, figures changes, substantial revision made

The Press-Schechter theory of the cosmological mass function and its modifications allow to constraint cosmological scenarios of the structure formation. Recently a few new models have been suggested that explored the influence of anisotropic collapse on the shape of the mass function. I discuss in more detail a particular model that assumes that a fluid particle becomes a part of a gravitationally bound halo when the smallest eigenvalue of the deformation tensor of the filtered initial density field reaches a certain threshold (like the filtered density contrast reaches the threshold in the Press-Schechter formalism). Choosing the smallest eigenvalue guarantees that the fluid particle in question experiences collapse along all three axes. The model shows a better agreement with the N-body simulations than the standard Press-Schechter model., Comment: Talk at 15th Florida Workshop in Nonlinear Astronomy and Physics "The Onset of Nonlinearity", 17-19 February 2000, Gainesville, FL (12 pages, 5 figures)

We apply Shapefinders, statistical measures of `shape' constructed from two dimensional partial Minkowski functionals, to study the degree of filamentarity in the Las Campanas Redshift Survey (LCRS). In two dimensions, three Minkowski functionals characterise the morphology of an object, they are: its perimeter (L), area (S), and genus. Out of L and S a single dimensionless Shapefinder Statistic, F can be constructed (0 <=F <=1). F acquires extreme values on a circle (F = 0) and a filament (F = 1). Using F, we quantify the extent of filamentarity in the LCRS by comparing our results with a Poisson distribution with similar geometrical properties and having the same selection function as the survey. Our results unambiguously demonstrate that the LCRS displays a high degree of filamentarity both in the Northern and Southern galactic sections a result that is in general agreement with the visual appearance of the catalogue. It is well known that gravitational clustering from Gaussian initial conditions gives rise to the development of non-Gaussianity reflected in the formation of a network-like filamentary structure on supercluster scales. Consequently the fact that the smoothed LCRS catalogue shows properties consistent with those of a Gaussian random field (Colley 1997) whereas the unsmoothed catalogue demonstrates the presence of filamentarity lends strong support to the conjecture that the large scale clustering of galaxies is driven by gravitational instability., Comment: Accepted for publication in ApJ

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., Comment: 26 pages including 12 figures. Accepted for publication in ApJ

We examine several aspects of redshift distortions by expressing the redshift-space density in terms of the eigenvalues and orientation of the local Lagrangian deformation tensor. We explore the importance of multiple-streaming using the Zel'dovich approximation (ZA), and compute the average number of streams in real and redshift-space. It is found that multiple-streaming can be significant in redshift-space but negligible in real-space, even at moderate values of the linear fluctuation amplitude ($\sigma < 1$). Moreover, unlike their real-space counter-parts, redshift-space multiple-streams can flow past each other with minimal interactions. Such nonlinear redshift-space effects, which operate even when the real-space density field is quite linear, could suppress the classic compression of redshift-structures predicted by linear theory (Kaiser 1987). We also compute using the ZA the probability distribution function (PDF) of density, as well as $S_3$, in real and redshift-space, and compare it with the PDF measured from N-body simulations. The role of caustics in defining the character of the high density tail is examined. It is found that (non-Lagrangian) smoothing, due to both finite resolution or discreteness and small-scale velocity dispersions, is very effective in erasing caustic structures, unless the initial power spectrum is sufficiently truncated., Comment: 23 pages, 5 ps figures, submitted to ApJ

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) \propto k^{n} with spectral indices n=-1, 0 and the standard cold dark matter (SCDM) model with Omega =1 and h=0.5. For the power-law models, we use numerical data averaged over several different output times: ten output times from two N-body realizations for the n=-1 power-law model; four outputs from one realization for the n=0 model. While 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 simulations. 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., Comment: Revised version. Accepted by ApJ Letter. Significantly improved. Numerical test for the SCDM model included. 14 pages, Latex file, 2 psfigs

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, it allows 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., Comment: 28 pages latex using aasms4 and epsfig. 11 postscript figures embedded. Minor Changes, to reflect the published version

We analyse shapes of overdense regions (clusters and superclusters) in controlled N-body simulations of gravitational clustering with power law initial spectra P(k) \propto k^n, n = -3, -2, -1, 0. At values of the density just above the percolation transition the number of distinct (isolated) clusters peaks and we use this `natural threshold' to study the shapes and multiplicity function of clusters and superclusters. We find that the extent of both filamentarity and pancakeness increases as the simulation evolves, the former being appreciably larger than the latter at virtually all epochs and for all spectra considered by us. Our results also show that high density regions within very massive clusters/superclusters are likely to be noticeably filamentary or pancake/ribbon-like when compared to the less dense regions within these objects. We make a detailed study of two, moment-based `shape statistics' proposed, respectively, by Babul & Starkman (BS) and Luo & Vishniac (LV) and find that both LV and BS have certain built-in limitations: LV is biased towards oblate structures and tends to overemphasise this property; neither BS nor LV correctly describe the shape of strongly curved or topologically nontrivial objects. For instance, a thin filamentary torus and a ribbon are both described by BS and LV as being pancakes ! By contrast Shapefinders, a new shape diagnostic not constructed from density moments but from Minskowski functionals, does not suffer from these limitations and appears to faithfully reproduce the shapes of both simple and topologically complex objects., Comment: Latex, 25 pages with aaspp4.sty including 26 figures formatted with graphicx

We explore shapes of clusters and superclusters in the IRAS 1.2 Jy redshift survey with three reconstructions spanning the range \beta = 0.1, 0.5, 1.0, where \beta = \Omega^{0.6}/b, b is the bias factor and \Omega the present value of the dimensionless matter density. Comparing our results to Gaussian randomized reconstructions of the IRAS catalogue, we find structures having both planar and filamentary properties. For \beta = 0.5, 1.0 the largest structures in the survey have a distinct tendency to be filament-like} in general agreement with the results of N-body simulations., Comment: Latex (7 pages with aas2pp4.sty)

We modified the Press-Schechter (PS) formalism and then analytically derived a constrained mass distribution function n(M|varphi) for the regions having some specified value of the primordial gravitational potential, varphi. The resulting modified PS theory predicts that gravitationally bound clumps with masses corresponding to rich clusters are significantly biased toward the regions of negative primordial potential - the troughs of the potential. The prediction is quantitative, depending on the mass and the depth of the troughs, which can be tested in large N-body simulations. As an illustration of the magnitude of the effect we calculate the constrained mass function for the CDM model with Gamma = Omega h = 0.25 normalized to sigma_{8} = 1. In particular, we show that the probability of finding a clump of mass 10^{14} - 10^{15}h^{-1}M_{odot} in the region of negative initial potential is 1.3 - 3 times greater (depending on the mass) than that in the region of positiveinitial potential. The scale of the potential fluctuations R_{varphi}=sqrt{3} sigma_{varphi}/sigma_{varphi'} is shown to be approx 120 h^{-1} Mpc for the spectrum in question. The rms mass density contrast on this scale is only about sigma_{delta}(R_{varphi}) approx 0.03. Assuming that the modified PS theory is statistically correct, we conclude that clusters are significantly biased (b ge 10, b is a bias factor defined by Delta n_{cl}/ n_{cl} = b Delta rho_m/rho_m) toward the regions having negative initial potential., Comment: Latex file,15 pages,2 postscript figures, accepted by ApJ Letter

We construct a set of shape-finders which determine shapes of compact surfaces (iso-density surfaces in galaxy surveys or N-body simulations) without fitting them to ellipsoidal configurations as done earlier. The new indicators arise from simple, geometrical considerations and are derived from fundamental properties of a surface such as its volume, surface area, integrated mean curvature and connectivity characterized by the Genus. These `Shapefinders' could be used to diagnose the presence of filaments, pancakes and ribbons in large scale structure. Their lower-dimensional generalization may be useful for the study of two-dimensional distributions such as temperature maps of the Cosmic Microwave Background., Comment: Latex (8 pages with aaspp.sty) requires \usepackage{graphicx}). To appear in ApJ Lett

We report on a series of tests of agreement between three types of N-body simulations: PM, P$^3$M, and Tree codes. We find good agreement in both the individual and the statistical properties only on scales larger than the mean interparticle separation. As a result, we question most numerical results at and below below galaxy scales, either concerning primordial dark matter or baryonic matter coupled to it by gravitation. Mass resolution (specifically, sufficient to have the mean interparticle separation comparable to the smoothing scales) appears be necessary for results to converge., Comment: Latex, 12 pages, with 3 figures

An analytic approximation to the mass function for gravitationally bound objects is presented. We base on the Zel'dovich approximation to 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 objects - clumps occur in the regions where the lowest eigenvalue of the deformation tensor,lambda_{3}, is positive. We derive the conditional probability of lambda_{3}>0 as a function of the linearly extrapolated density contrast, delta, and the conditional probability distribution of delta provided that lambda_{3}>0. These two conditional probability distributions show that the most probable density of the bound regions (lambda_{3}>0) is roughly 1.5 at the characteristic mass scale, and that the probability of lambda_{3}>0 is almost unity in the highly overdense regions (delta>3*sigma). Finally an analytic mass function of clumps is derived with a help of one simple ansatz which is employed to treat the multistream regions beyond the validity of the Zel'dovich 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., Comment: A solution to the normalization problem is added. Latex file, 33 pages, 5 PostScript figures

We explore some of the effects that discreteness and two-body scattering may have on N-body simulations with ``realistic'' cosmological initial conditions. We use an identical subset of particles from the initial conditions for a $128^3$ Particle-Mesh (PM) calculation as the initial conditions for a variety P$^3$M and Tree code runs. We investigate the effect of mass resolution (the mean interparticle separation) since most ``high resolution'' codes only have high resolution in gravitational force. The phase-insensitive two--point statistics, such as the power spectrum (autocorrelation) are somewhat affected by these variations, but phase-sensitive statistics show greater differences. Results converge at the mean interparticle separation scale of the lowest mass-resolution code. As more particles are added, but the force resolution is held constant, the P$^3$M and the Tree runs agree more and more strongly with each other and with the PM run which had the same initial conditions. This shows high particle density is necessary for correct time evolution, since many different results cannot all be correct. However, they do not so converge to a PM run which continued the fluctuations to small scales. Our results show that ignoring them is a major source of error on comoving scales of the missing wavelengths. This can be resolved by putting in a high particle density. Since the codes never agree well on scales below the mean comoving interparticle separation, we find little justification for quantitative predictions on this scale. Some measures vary by 50%, but others can be off by a factor of three or more. Our results suggest possible problems with the density of galaxy halos, formation of early generation objects such as QSO absorber clouds, etc., Comment: Revised version to be published in Astrophysical Journal. One figure changed; expanded discussion, more information on code parameters. Latex, 44 pages, including 19 figures. Higher resolution versions of Figures 10-15 available at: ftp://kusmos.phsx.ukans.edu/preprints/nbody

We employ a percolation technique developed for pointwise distributions to analyze two-dimensional projections of the three northern and three southern slices in the Las Campanas Redshift Survey. One of the goals of this paper is to compare the visual impressions of the structure within distributions with objective statistical analysis. We track the growth of the largest cluster as an indicator of the network structure. We restrict our analysis to volume limited subsamples in the regions from 200 to 400 Mpc/h where the number density of galaxies is the highest. As a major result, we report a measurement of an unambiguous signal, with high signal-to-noise ratio (at least at the level of a few $\sigma$), indicating significant connectivity of the galaxy distribution which in two dimensions is indicative of a filamentary distribution. This is in general agreement with the visual impression and typical for the standard theory of the large-scale structure formation based on gravitational instability of initially Gaussian density fluctuations., Comment: The main body of the text (17 pages) is followed by the captions file plus 4 postscript figures. Submitted to ApJ

The large-scale structure in the Universe is believed to arise out of small random density perturbations generated in the very early Universe, that are amplified by gravity. Large and usually intricate N-body simulations are typically employed to model the complex nonlinear dynamics in a self gravitating medium. We suggest a very simple model which predicts, on large scales, the correct density and velocity distributions. The model does not involve an explicit computation of the gravitational force. It is based on a simple transformation of the variables and local conservation laws of mass and {\it generalized} linear momentum. The model demonstrates that the overall appearance of large-scale structure in the Universe is explicitly determined by the initial velocity field and it reveals the most significant large-scale effects of gravity on the formation of structure., Comment: Latex file (needs, epsf.sty, aaspp4.sty); postscript figures; 10 pages (including figures); To appear in ApJ Letters

We present percolation analyses of Wiener Reconstructions of the IRAS 1.2 Jy Redshift Survey. There are ten reconstructions of galaxy density fields in real space spanning the range $\beta= 0.1$ to $1.0$, where ${\beta}={\Omega^{0.6}}/b$, $\Omega$ is the present dimensionless density and $b$ is the bias factor. Our method uses the growth of the largest cluster statistic to characterize the topology of a density field, where Gaussian randomized versions of the reconstructions are used as standards for analysis. For the reconstruction volume of radius, $R {\approx} 100 h^{-1} $ Mpc, percolation analysis reveals a slight `meatball' topology for the real space, galaxy distribution of the IRAS survey. cosmology-galaxies:clustering-methods:numerical, Comment: Revised version accepted for publication in The Astrophysical Journal, January 10, 1997 issue, Vol.474

Using percolation statistics we, for the first time, demonstrate the universal character of a network pattern in the real space, mass distributions resulting from nonlinear gravitational instability of initial Gaussian fluctuations. Percolation analysis of five stages of the nonlinear evolution of five power law models reveals that all models show a shift toward a network topology if seen with high enough resolution. However, quantitatively, the shift is significantly different in different models: the smaller the spectral index ,n, the stronger the shift. On the contrary, the shift toward the "bubble" topology is characteristic only for the n <= -1 models. We find that the mean density of the percolating structures in the nonlinear density distributions generally is very different from the density threshold used to identify them and corresponds much better to a visual impression. We also find that the maximum of the number of structures (connected regions above or below a specified density threshold) in the evolved, nonlinear distributions is always smaller than in Gaussian fields with the same spectrum, and is determined by the effective slope at the cutoff frequency., Comment: The paper is 26 pages long. The latex file uses aasms.sty as a style file. There are 5 figures and 2 tables included.

The history and the major results of the study of the topology of the large-scale structure are briefly reviewed. Two techniques based on percolation theory and the genus curve are discussed. The preliminary results of the percolation analysis of the Wiener reconstruction of the IRAS $1.2 Jy$ redshift catalog are reported., Comment: Latex file with figures in postscript format, 8 pages

As galaxy redshift surveys probe deeper into the universe, they uncover ever more dramatic structures in the large-scale distribution of galaxies. In particular, the CfA2 and SSRS2 surveys to an apparent magnitude limit of 15.5 exhibit an impressive complex of sheets, filaments, and clusters. The formation of the large-scale structure in the universe results from the gravitational amplification of the primordial small perturbations of density. The primordial density perturbations, as quantum fluctuations at the very early stage, may reveal important information about the early universe and the laws of fundamental physics. One of the major obstacles to understanding the formation of the large-scale structure is the complexity of the evolution of the density inhomogeneities at the nonlinear stage when the observable structures form. One way of addressing this problem is to run three-dimensional numerical simulations. Here we review another approach based on the approximate analytic model of the nonlinear gravitational instability utilizing Burgers' equation of the nonlinear diffusion., Comment: Paper with 1 figure in postscript format, 14 pages. For additional figures contact the author

We examine the effect of a threshold bias on the power spectrum and the bispectrum in an ensemble of numerical simulations (Gaussian initial perturbations with power law spectra P(k) \sim k^n, n=+1, 0, -1, -2) and compare our results with theoretical predictions. Our simulations are evolved sufficiently that on the scale where we apply the threshold the rms fluctuation has developed significantly into the nonlinear regime. Thus, predictions based on perturbation theory do not necessarily apply. Nevertheless, we find our results for the power spectrum, biased power simply amplified by a numerical factor, follow predicted trends, far beyond the regime where perturbation theory is expected to be valid. We find that the biased bispectrum continues to follow the so-called hierarchical form, with reduced three-point amplitude Q \approx 1 in the strongly nonlinear regime, independent of initial spectrum. In the quasi-linear perturbative regime the three-point amplitude depends on configuration shape, a behavior that is found to give useful information about the amount of bias without information about the unbiased matter distribution., Comment: 20 pages of uuencoded compressed Postscript, also available on request

We apply the adhesion approximation to study the formation and evolution of voids in the Universe. Our simulations -- carried out using 128$^3$ 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 COBE-normalised Cold Dark Matter (hereafter CDM) model with $h_{50} = 1,$ scales approximately as $\bar D(z) = {\bar D_0\over \sqrt {1+z}},$ where $\bar D_0 \simeq 10.5$ 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 towards reconstructing the form of the primordial potential 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 merge 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 gravitational potential. A central result of this paper is that, Comment: Plain TeX, 38 pages Plus 16 Figures (available on request from the first author), IUCAA-28 To appear in The Astrophysical Journal, July 1994