Zeldovich Approximation Research Articles

Light Propagation in Inhomogeneous Universes. I. Methodology and Preliminary Results

We describe a numerical algorithm that simulates the propagation of light in inhomogeneous universes. This algorithm computes the trajectories of light rays between the observer, located at redshift z = 0, and distant sources located at high redshift using the multiple lens plane method. The deformation and deflection of light beams as they interact with each lens plane are computed using the filled-beam approximation. We use a particle-particle/particle-mesh (P3M) N-body numerical code to simulate the formation of large-scale structure in the universe. We extend the length resolution of the simulations to submegaparsec scales by using a Monte Carlo method for locating galaxies inside the computational volume according to the underlying distribution of background matter. The observed galaxy two-point correlation function is reproduced. This algorithm constitutes a major improvement over previous methods, which either neglected the presence of large-scale structure, neglected the presence of galaxies, neglected the contribution of distant matter (matter located far from the beam), or used the Zeldovich approximation for simulating the formation of large-scale structure. In addition, we take into account the observed morphology-density relation when assigning morphological types to galaxies, something that was ignored in all previous studies. To test this algorithm, we perform 1981 simulations for three different cosmological models: an Einstein-de Sitter model with density parameter Ω0 = 1, an open model with Ω0 = 0.2, and a flat, low-density model with Ω0 = 0.2 and a cosmological constant of λ0 = 0.8. In all models, the initial density fluctuations correspond to a cold dark matter power spectrum normalized to COBE. In each simulation, we compute the shear and magnification resulting from the presence of inhomogeneities. Our results are the following: (1) The magnification is totally dominated by the convergence, with the shear contributing less than one part in 104. (2) Most of the cumulative shear and magnification is contributed by matter located at intermediate redshifts, z = 1-2. (3) The actual value of the redshift at which the largest contribution to shear and magnification occurs depends on the cosmological model. In particular, the lens planes contributing the most are located at larger redshift for models with smaller Ω0. (4) The number of galaxies directly hit by the beam increases with redshift, while the contribution of lens planes to the shear and magnification decrease with increasing lens plane redshift for z > 2, which indicates that the bulk of the shear and magnification does not originate from direct hits, but rather from the tidal influence of nearby and more distant galaxies and background matter. (5) The average contributions of background matter and nearby galaxies to the shear is comparable for models with small Ω0. For the Einstein-de Sitter model, the contribution of the background matter exceeds the contribution of nearby galaxies by nearly 1 order of magnitude.

The large scale structure of the universe: Dynamical and statistical aspects

The description of gravitational clustering is essentially statistical but its origin is dynamical. Hence both aspects of clustering: dynamical and statistical, must be understood in order to arrive at a proper appreciation of the subject of gravitational instability and the formation and evolution of the large scale structure of the Universe. Key dynamical aspects of gravitational clustering such as the Zeldovich approximation and its extension — the adhesion model are reviewed. Statistical indicators of clustering such as the correlation function and percolation theory, as applied to the large scale structure of the Universe have also been focussed on.

Accretion of cold and hot dark matter onto cosmic string filaments.

The Zeldovich approximation is applied to study the accretion of hot and cold dark matter onto moving long strings. It is assumed that such defects carry a substantial amout of small-scale structure, thereby acting gravitationally as a Newtonian line source whose effects dominate the velocity perturbations. Analytical expressions for the turnaround surfaces are derived and the mass inside of these surfaces is calculated. Estimates are given for the redshift dependence of ${\ensuremath{\Omega}}_{\mathrm{nl}}$, the fraction of mass in nonlinear objects. Depending on parameters, it is possible to obtain ${\ensuremath{\Omega}}_{\mathrm{nl}}=1$ at the present time. Even with hot dark matter, the first nonlinear filamentary structures form at a redshift close to 100, and there is sufficient nonlinear mass to explain the observed abundance of high redshift quasars and damped Lyman $\ensuremath{\alpha}$ systems. These results imply that moving strings with small-scale structure are the most efficient seeds to produce massive nonlinear objects in the cosmic string model.

The Evolution of Correlation Functions in the Zeldovich Approximation and Its Implications for the Validity of Perturbation Theory

We investigate whether it is possible to study perturbatively the transition in cosmological clustering from a single-streamed flow to a multistreamed flow. We do this by considering a system whose dynamics is governed by the Zeldovich approximation (ZA) and calculating the evolution of the two-point correlation function using two methods, (1) distribution functions and (2) hydrodynamic equations without pressure and vorticity. The latter method breaks down once multistreaming occurs whereas the former does not. We find that the two methods yield the same results to all orders in a perturbative expansion of the two-point correlation function. We thus conclude that we cannot study the transition from a single-streamed flow to a multistreamed flow in a perturbative expansion. We expect this conclusion to hold even if full gravitational dynamics (GD) is used instead of ZA.

Adding Long‐Wavelength Modes to anN‐Body Simulation

We present a new method to add long-wavelength power to an evolved N-body simulation, making use of the Zeldovich approximation to change positions and velocities of particles. We describe the theoretical framework of our technique and apply it to a P3M cosmological performed on a cube of 100 Mpc on a side, obtaining a new simulation of 800 Mpc on a side. We study the effect of the power added by long waves by means of several statistics of the density and velocity field, and we suggest possible applications of our method to the study of the large-scale structure of the universe.

Nonlinear Evolution of the Genus Statistics in the Zeldovich Approximation

Evolution of genus density is calculated from Gaussian initial conditions using Zel'dovich approximation. A new approach is introduced which formulates the desired quantity in a rotationally invariant manner. It is shown that normalized genus density does not depend on the initial spectral shape but is a function of the fluctuation amplitude only.

A New Statistical Indicator to Study Nonlinear Gravitational Clustering and Structure Formation

In an expanding universe, velocity field and gravitational force field are proportional to each other in the linear regime. Neither of these quantities evolve in time and these can be scaled suitably so that the constant of proportionality is unity and velocity and force field are equal. The Zeldovich approximation extends this feature beyond the linear regime, until formation of pancakes. Nonlinear clustering which takes place {\it after} the breakdown of Zeldovich approximation, breaks this relation and the mismatch between these two vectors increases as the evolution proceeds. We suggest that the difference of these two vectors could form the basis for a powerful, new, statistical indicator of nonlinear clustering. We define an indicator called velocity contrast, study its behaviour using N-Body simulations and show that it can be used effectively to delineate the regions where nonlinear clustering has taken place. We discuss several features of this statistical indicator and provide simple analytic models to understand its behaviour. Particles with velocity contrast higher than a threshold have a correlation function which is biased with respect to the original sample. This bias factor is scale dependent and tends to unity at large scales.

The Peak-Patch Picture of Cosmic Catalogs. I. Algorithms

view Abstract Citations (341) References (49) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Peak-Patch Picture of Cosmic Catalogs. I. Algorithms Bond, J. R. ; Myers, S. T. Abstract We develop a picture of cosmic structure formation that identifies virialized cosmological objects with a point process of "peak patches" in the initial (Lagrangian) space. The strongly nonlinear internal dynamics of the collapsing patch is largely decoupled from the weakly nonlinear dynamics describing peak-patch flow. A modest set of local field measurements on the patch region suffices to approximately determine the fate of the peak. The candidate peak points are identified using a hierarchy of smoothing operations on the density field. Patch size and mass are found by requiring complete collapse of a homogeneous ellipsoid model for internal patch dynamics which includes the important influence of the external tidal field. Exclusion algorithms that prevent peak-patch overlaps trim the candidate list into the final "hierarchical peak" list. The Zeldovich approximation with locally adaptive filtering of the displacement field is used for external dynamics to move the surviving patches to their final state (Eulerian) positions, augmented by a quadratic nonlinear correction where necessary. Thus both statistical and dynamical clustering of the peaks are included. Our technique is the natural generalization of both the Press-Schechter method (to include nonlocal effects) and the BBKS single-filter peaks theory (to allow a mass spectrum and solve the cloud-in-cloud problem). It allows efficient Monte Carlo constructions of three-dimensional catalogs of objects such as groups and clusters of galaxies, and, with more selection criteria, of galaxies in their formation phases. Publication: The Astrophysical Journal Supplement Series Pub Date: March 1996 DOI: 10.1086/192267 Bibcode: 1996ApJS..103....1B Keywords: COSMOLOGY: THEORY; GALAXIES: FORMATION; GALAXIES: CLUSTERS: GENERAL; METHODS: NUMERICAL full text sources ADS | Related Materials (2) Part 2: 1996ApJS..103...41B Part 3: 1996ApJS..103...63B

Matter accretion by cosmic string loops and wakes.

We develop quantative methods for the study of structure formation with cosmic strings in an expanding universe. The gravitational effects of arbitrary string configurations are calculated using linearized gravity. The growth of density fluctuations is then studied using these gravitational forces as a source term in the Zeldovich approximation. We use either the adhesion modification or an N-body tree code to project this fluctuation growth into the nonlinear regime. These methods are applied to specific loop and long string solutions beginning at equal matter and radiation ${\mathit{t}}_{\mathrm{eq}}$ on scales corresponding to about 10 Mpc today (h=0.5). We reproduce analytic results for spherical and planar collapse. We show that these methods are applicable to accretion about closed oscillating loops and in the wakes of moving long strings which possess significant small-scale structure, quantitatively confirming the wiggly string approximation with a renormalized string energy density \ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{}. We demonstrate the efficiency of the fragmentation of wakes created by wiggly strings by the present day. These methods are sufficiently computationally efficient to employ in the study of an evolving string network. For the cosmic string scenario, we conclude that reliable quantitative predictions must take into account non-Gaussianity, vorticity generation, and nonlinear fragmentation effects.

Nonlinear approximations to gravitational instability: A comparison in the quasi-linear regime

We compare different nonlinear approximations to gravitational clustering in the weakly nonlinear regime, using as a comparative statistic the evolution of non-Gaussianity which can be characterised by a set of numbers $S_p$ describing connected moments of the density field at the lowest order in $ $: $ _c \simeq S_n ^{n-1}$. Generalizing earlier work by Bernardeau (1992) we develop an ansatz to evaluate all $S_p$ in a given approximation by means of a generating function which can be shown to satisfy the equations of motion of a homogeneous spherical density enhancement in that approximation. On the basis of the values of we show that approximations formulated in Lagrangian space (such as the Zeldovich approximation and its extensions) are considerably more accurate than those formulated in Eulerian space such as the Frozen Flow and Linear Potential approximations. In particular we find that the $n$th order Lagrangian perturbation approximation correctly reproduces the first $n+1$ parameters $S_n$. We also evaluate the density probability distribution function for the different approximations in the quasi-linear regime and compare our results with an exact analytic treatment in the case of the Zeldovich approximation.

Optimizing the Zel'dovich approximation

We have recently learned that the Zeldovich approximation can be successfully used for a far wider range of gravitational instability scenarios than formerly proposed; we study here how to extend this range. In previous work we studied the accuracy of several analytic approximations to gravitational clustering in the mildly nonlinear regime. We found that the ``truncated Zel'dovich approximation" (TZA) was better than any other (except in one case the ordinary Zeldovich approximation) over a wide range from linear to mildly nonlinear ($\sigma \sim 3$) regimes. TZA sets Fourier amplitudes equal to zero for {\it all} wavenumbers greater than $k_{n\ell}$, where $k_{n\ell}$ marks the transition to the nonlinear regime. Here, we study crosscorrelation of generalized TZA with a group of $n$--body simulations for three shapes of window function: sharp $k$--truncation (as in CMS), tophat in coordinate space, or a Gaussian. We also study the crosscorrelation as a function of initial scale within each window type. We find $k$--truncation, which was so much better than other things tried in CMS, is the {it worst} of these three window shapes. We find that a Gaussian window $e^{-k^2/2k_G^2}$ applied to the intial Fourier amplitudes is the best choice. It produces a greatly improved crosscorrelation all cases we studied. The optimum choice of $k_G$ for the Gaussian window is (spectrum-- dependent) 1--1.5 times $k_{n\ell}$, with $k_{n\ell}$ defined by (3). Although all three windows produce similar power spectra and density distribution functions after application of the Zeldovich approximation, phase agreement with the $n$--body simulation is better for the Gaussian window. We ascribe Gaussian window success to its superior treatment of phase evolution.

Nonlinear approximations to gravitational instability: A comparison in second-order perturbation theory

view Abstract Citations (28) References (10) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Nonlinear Approximations to Gravitational Instability: A Comparison in Second-Order Perturbation Theory Munshi, Dipak ; Starobinsky, Alexei A. Abstract Nonlinear approximation methods such as the Zeldovich approximation, and more recently the frozen flow and linear potential approximations, are sometimes used to simulate nonlinear gravitational instability in the expanding Universe. We investigate the relative accuracy of these approximations by comparing them with the exact solution using second order perturbation theory. We evaluate the density and velocity fields in these approximations to second order, and also determine the skewness parameter $S_3 = \langle \delta^3 \rangle / \langle (\delta^{(1)})^2\rangle ^2$ for each of the approximations again in second order. We find that $S_3 = 4, ~3, ~3.4$ for the Zeldovich approximation, the frozen flow and the linear potential approximations respectively as compared to $S_3 = {34/7}$ for the exact solution. Our results show that, of all the approximations considered, the Zeldovich approximation is the most accurate in describing the weakly nonlinear effects of gravity. Moreover, the Zeldovich approximation is much closer to the exact results for matter and velocity distributions than the other approximations if the slope of the power spectrum of density perturbations is $-3< n \le -1$. Publication: The Astrophysical Journal Pub Date: June 1994 DOI: 10.1086/174255 arXiv: arXiv:astro-ph/9311056 Bibcode: 1994ApJ...428..433M Keywords: Approximation; Computational Astrophysics; Dynamic Stability; Galactic Structure; Gravitational Fields; Nonlinear Equations; Accuracy; Density Distribution; Perturbation Theory; Velocity Distribution; Astronomy; GRAVITATION; INSTABILITIES; COSMOLOGY: LARGE-SCALE STRUCTURE OF UNIVERSE; Astrophysics E-Print: Latex file, pages 20 , Two figures (available at 144.16.31.6 through anonymous ftp), IUCAA 31/93 full text sources arXiv | ADS |

The richness dependence of cluster correlations

We report on a study of the richness dependence of the spatial correlations of clusters of galaxies. We employ the method devised by Bond & Couchman, which combines the theory of the statistics of peaks in Gaussian random fields with the evolution of the cosmological density field by the Zeldovich Approximation, to calculate analytically both the statistical and dynamical contributions to the clustering. We compute the cluster correlation function for a variety of popular cosmological models and compare our results with data from four recent cluster samples. We find no model able to account for all of the observations, although, conversely, the observational data are of insufficient quality to rule out firmly any of the models. The model that fares best is one which has been advocated as accounting for the angular correlation function of APM galaxies: that the same power spectrum gives the best fit to both the galaxy and cluster correlation data may be taken as support for the standard picture, due to Kaiser, in which objects form at the sites of peaks in an initial cosmological density field which obeyed Gaussian statistics. No model is able to reproduce the correlation length of a sample of Abell R ≥ 2 clusters. This result may just indicate the inadequacy of the correlation length when taken alone as a diagnostic statistic, or that this cluster sample is seriously corrupted by projection effects. We consider, however, alternative explanations, including the possibility that non-Gaussian initial conditions are required and that the identification of peaks in the linear density field as sites of nascent clusters may break down for the highest peaks, corresponding to the richest clusters.

Non-linear evolution of density perturbations using the approximate constancy of the gravitational potential

During the evolution of density inhomogeneties in an $\Omega=1$, matter dominated universe, the typical density contrast changes from $\delta\simeq 10^{-4}$ to $\delta\simeq 10^2$. However, during the same time, the typical value of the gravitational potential generated by the perturbations changes only by a factor of order unity. This significant fact can be exploited to provide a new, powerful, approximation scheme for studying the formation of nonlinear structures in the universe. This scheme, discussed in this paper, evolves the initial perturbation using a Newtonian gravitational potential frozen in time. We carry out this procedure for different intial spectra and compare the results with the Zeldovich approximation and the frozen flow approximation (proposed by Mattarrese et al. recently). Our results are in far better agreement with the N-body simulations than the Zeldovich approximation. It also provides a dynamical explanation for the fact that pancakes remain thin during the evolution. While there is some superficial similarity between the frozen flow results and ours, they differ considerably in the velocity information. Actual shell crossing does occur in our approximation; also there is motion of particles along the pancakes leading to further clumping. These features are quite different from those in frozen flow model. We also discuss the evolution of the two-point correlation function in various approximations.

Very large scale structure in an open cosmology of cold dark matter and baryons

The standard cosmology of cold dark matter (CDM) in a flat universe with Gaussian, scale-invariant fluctuations, which is so successful in explaining the major properties of galaxies and their clustering on moderate scales, fails to account for the observed clustering of clusters and streaming velocities on scales of 30-100/h Mpc. Assuming that these observations are being interpreted correctly, an open Friedmann model where CDM and baryons contribute comparably to the mean mass density is considered. In this model the CDM is responsible for galaxy formation, while the presence of the baryons induces a large-scale feature in the spectrum of fluctuations at the precombination Jeans mass, giving rise to structure on very large scales. Using linear analysis and simulations based on the Zeldovich approximation, the resultant cluster-cluster correlation function and mean streaming velocity are found to be compatible with the observations. 71 references.

Rotation Profile of a String-Seeded Halo

We have used an N-body code to simulate the collapse of a wake left by a loop of cosmic string in a cold dark matter universe (Ω = 1, h = 0.5). Initial perturbations were introduced via the Zeldovich approximation in a high-resolution sphere of radius 1.5 Mpc (comoving) filled with a lattice of ∼ 104 points masses of 1.38 × 108M⊙. The top plot shows, at z = 24, a wake of a loop of Ms = 3 × 108M⊙ which has moved up and left through the center of the sphere with zeq velocity v = 0.01c directed at an 18° angle out of the plane of the figure. The collapsed object (center) is quite prolate. The outer shell of more massive particles is now also visible. The spherically averaged rotation profile of the “halo” is shown in the bottom diagram. The almost flat portion of the rotation curve corresponds to the size of the wake. There is also evidence for a soft core of ∼ 10 kpc in radius (our dynamical softening was 3 kpc). While the adopted initial velocity and mass are appropriate for loops with galaxy - galaxy separations, a one loop - one object correspondence is not a good approximation on the galactic scale. Moreover, baryons settling in the core of the halo would have steepened the profile and made it incompatible with observations. Halos seeded by several loops may have rotation profiles more compatible with those observed. The results of our simulations may be 10 applied - after an appropriate rescaling - to Abell clusters. There one loop typically seeds a single cluster. String-seeded Abell clusters should be therefore significantly prolate.

Richness distribution of string-seeded Abell clusters - Accretion by a loop versus hierarchical clustering

view Abstract Citations (7) References (36) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Richness Distribution of String-seeded Abell Clusters: Accretion by a Loop versus Hierarchical Clustering Zurek, W. H. Abstract We discuss formation of objects around loops of cosmic string in an Einstein-de Sitter universe dominated by either cold or hot dark matter. In particular, we solve, in the Zeldovich approximation, the problem of accretion of cold dark matter onto a stationary point mass introduced into the universe either in radiation or matter-dominated epoch. This process can be used to model formation of Abell clusters around loops of cosmic string. Predicted distribution of masses of Abell clusters is consistent with the observed richness distribution only when the loops seeding clusters of richness 1 enter the horizon at about the time when accretion can begin in earnest, that is, just before t_eq_. The formation epoch is also consistent with Abell cluster separations and with the estimates of their masses, provided that the dimensionless string tension is of the order of 10^-6^. If the loops were laid down much earlier, or if-as could be the case in the universe with the sufficiently hot dark matter-accretion could not start until some time after they were deposited, supermassive Abell clusters would be more common than observed. Loops laid down too late (t>=3t_eq_) yield, on the other hand, richness distribution far too steep to explain observations. We also estimate mass distribution of Abell clusters in hierarchical clustering models: It reproduces observations especially well when the power spectrum on the Abell cluster scales is close to that of white noise. Publication: The Astrophysical Journal Pub Date: January 1988 DOI: 10.1086/165877 Bibcode: 1988ApJ...324...19Z Keywords: Astronomical Models; Dark Matter; Galactic Clusters; Galactic Evolution; Mass Distribution; String Theory; Big Bang Cosmology; Hierarchies; Loops; Universe; White Noise; Astrophysics; COSMOLOGY; DARK MATTER; GALAXIES: CLUSTERING; GALAXIES: FORMATION full text sources ADS |

Premixed Combustion in Boundary Layers for Moderate Values of the Zeldovich Numbers

Abstract The classical problem of the ignition and flame generation of combustible premixed gases by a hot flat plate is analyzed by asymptotic techniques for moderate values of the Zeldovich numbers. For temperatures of the plate smaller than the adiabatic flame temperature it was found that the corrections to the zeroth-order solution (Zeldovich approximation) are very important for finite Zeldovich numbers. The asymptotic analysis was carried out with the inclusion of reactant consumption showing different responses depending if the reactant consumption parameter is larger or smaller than a critical one. For temperatures of the plate larger than the adiabatic flame temperatures (that is, the reactant consumption parameter larger than the critical one), the ignition is not characterized by a thermal runaway and a smooth transition from frozen to equilibrium flow is obtained. The evolution towards the premixed flame formation is analyzed and the flame position and flame temperature are obtained.