Topology Of Large-scale Structure Research Articles

Wasserstein distance as a new tool for discriminating cosmologies through the topology of large-scale structure

ABSTRACT In this work, we test Wasserstein distance in conjunction with persistent homology as a tool for discriminating large-scale structures of simulated universes with different values of σ8 cosmological parameter (present root-mean-square matter fluctuation averaged over a sphere of radius 8 Mpc comoving). The Wasserstein distance (a.k.a. the pair-matching distance) was proposed to measure the difference between two networks in terms of persistent homology. The advantage of this approach consists in its non-parametric way of probing the topology of the cosmic web, in contrast to graph-theoretical approach depending on linking length. By treating the haloes of the cosmic web as points in a point cloud, we calculate persistent homologies, build persistence (birth–death) diagrams, and evaluate Wasserstein distance between them. The latter showed itself as a convenient tool to compare simulated cosmic webs. We show that one can discern two cosmic webs (simulated or real) with different σ8 parameter. It turns out that Wasserstein distance’s discrimination ability depends on redshift z, as well as on the dimensionality of considered homology features. We find that the highest discriminating power this tool obtains is at z = 2 snapshots, among the considered z = 2, 1, and 0.1 ones.

Fisher forecasts for primordial non-Gaussianity from persistent homology

We study the information content of summary statistics built from the multi-scale topology of large-scale structures on primordial non-Gaussianity of the local and equilateral type. We use halo catalogs generated from numerical N-body simulations of the Universe on large scales as a proxy for observed galaxies. Besides calculating the Fisher matrix for halos in real space, we also check more realistic scenarios in redshift space. Without needing to take a distant observer approximation, we place the observer on a corner of the box. We also add redshift errors mimicking spectroscopic and photometric samples. We perform several tests to assess the reliability of our Fisher matrix, including the Gaussianity of our summary statistics and convergence. We find that the marginalized 1-σ uncertainties in redshift space are Δf NL loc ∼ 16 and Δf NL equi ∼ 41 on a survey volume of 1 (Gpc/h)3. These constraints are weakly affected by redshift errors. We close by speculating as to how this approach can be made robust against small-scale uncertainties by exploiting (non)locality.

The persistence of large scale structures. Part I. Primordial non-Gaussianity

We develop an analysis pipeline for characterizing the topology of large scale structure and extracting cosmological constraints based on persistent homology. Persistent homology is a technique from topological data analysis that quantifies the multiscale topology of a data set, in our context unifying the contributions of clusters, filament loops, and cosmic voids to cosmological constraints. We describe how this method captures the imprint of primordial local non-Gaussianity on the late-time distribution of dark matter halos, using a set of N-body simulations as a proxy for real data analysis. For our best single statistic, running the pipeline on several cubic volumes of size 40 (Gpc/h)3, we detect fNL loc=10 at 97.5% confidence on ∼ 85% of the volumes. Additionally wetest our ability to resolve degeneracies betweenthe topological signature of fNL loc and variation of σ8 and argue that correctly identifying nonzero fNL loc in this case is possible via an optimal template method. Our method relies on information living at \U0001d4aa(10) Mpc/h, a complementary scale with respect to commonly used methods such as the scale-dependent bias in the halo/galaxy power spectrum. Therefore, while still requiring a large volume, our method does not require sampling long-wavelength modes to constrain primordial non-Gaussianity. Moreover, our statistics are interpretable: we are able to reproduce previous results in certain limits and we make new predictions for unexplored observables, such as filament loops formed by dark matter halos in a simulation box.

Large-scale structure topology in non-standard cosmologies: impact of dark sector physics

Even as our measurements of cosmological parameters improve, the physical nature of the dark sector of the universe largely remains a mystery. Many effects of dark sector models are most prominent at very large scales and will rely on future galaxy surveys to elucidate. In this paper we compare the topological properties of the large scale dark matter distribution in a number of cosmological models using hydrodynamical simulations and the cosmological genus statistic. Genus curves are computed from z = 11 to z = 0 for {\Lambda}CDM, Quintessence and Warm Dark Matter models, over a scale range of 1 to 20 Mpc/h. The curves are analysed in terms of their Hermite spectra to describe the power contained in non-Gaussian deformations to the cosmological density field. We find that the {\Lambda}CDM and {\Lambda}WDM models produce nearly identical genus curves indicating no topological differences in structure formation. The Quintessence model, which differs solely in its expansion history, produces significant differences in the strength and redshift evolution of non-Gaussian modes associated with higher cluster abundances and lower void abundances. These effects are robust to cosmic variance and are characteristically different from those produced by tweaking the parameters of a {\Lambda}CDM model. Given the simplicity and similarity of the models, detecting these discrepancies represents a promising avenue for understanding the effect of non-standard cosmologies on large-scale structure.

Topology of Large-scale Structures of Galaxies in Two Dimensions—Systematic Effects

Abstract We study the two-dimensional topology of galactic distribution when projected onto two-dimensional spherical shells. Using the latest Horizon Run 4 simulation data, we construct the genus of the two-dimensional field and consider how this statistic is affected by late-time nonlinear effects—principally gravitational collapse and redshift space distortion (RSD). We also consider systematic and numerical artifacts, such as shot noise, galaxy bias, and finite pixel effects. We model the systematics using a Hermite polynomial expansion and perform a comprehensive analysis of known effects on the two-dimensional genus, with a view toward using the statistic for cosmological parameter estimation. We find that the finite pixel effect is dominated by an amplitude drop and can be made less than 1% by adopting pixels smaller than 1/3 of the angular smoothing length. Nonlinear gravitational evolution introduces time-dependent coefficients of the zeroth, first, and second Hermite polynomials, but the genus amplitude changes by less than 1% between z = 1 and z = 0 for smoothing scales R G > 9 Mpc / h . Non-zero terms are measured up to third order in the Hermite polynomial expansion when studying RSD. Differences in the shapes of the genus curves in real and redshift space are small when we adopt thick redshift shells, but the amplitude change remains a significant ∼  ( 10 % ) effect. The combined effects of galaxy biasing and shot noise produce systematic effects up to the second Hermite polynomial. It is shown that, when sampling, the use of galaxy mass cuts significantly reduces the effect of shot noise relative to random sampling.

A TOPOLOGICAL ANALYSIS OF LARGE-SCALE STRUCTURE, STUDIED USING THE CMASS SAMPLE OF SDSS-III

We study the three-dimensional genus topology of large-scale structure using the northern region of the CMASS Data Release 10 (DR10) sample of the SDSS-III Baryon Oscillation Spectroscopic Survey. We select galaxies with redshift 0.452 1011.56 M ☉. We study the topology at two smoothing lengths: R G = 21 h –1 Mpc and R G = 34 h –1 Mpc. The genus topology studied at the R G = 21 h –1 Mpc scale results in the highest genus amplitude observed to date. The CMASS sample yields a genus curve that is characteristic of one produced by Gaussian random phase initial conditions. The data thus support the standard model of inflation where random quantum fluctuations in the early universe produced Gaussian random phase initial conditions. Modest deviations in the observed genus from random phase are as expected from shot noise effects and the nonlinear evolution of structure. We suggest the use of a fitting formula motivated by perturbation theory to characterize the shift and asymmetries in the observed genus curve with a single parameter. We construct 54 mock SDSS CMASS surveys along the past light cone from the Horizon Run 3 (HR3) N-body simulations, where gravitationally bound dark matter subhalos are identified as the sites of galaxy formation. We study the genus topology of the HR3 mock surveys with the same geometry and sampling density as the observational sample and find the observed genus topology to be consistent with ΛCDM as simulated by the HR3 mock samples. We conclude that the topology of the large-scale structure in the SDSS CMASS sample is consistent with cosmological models having primordial Gaussian density fluctuations growing in accordance with general relativity to form galaxies in massive dark matter halos.

OBSERVATIONAL REQUIREMENTS FOR Lyα FOREST TOMOGRAPHIC MAPPING OF LARGE-SCALE STRUCTURE ATz∼ 2

The z > 2 Lyman-alpha (Lya) forest traces the underlying dark-matter distribution on large scales and, given sufficient sightlines, can be used to create 3D maps of large-scale structure. We examine the observational requirements to construct such maps and estimate the signal-to-noise as a function of exposure time and sightline density. Sightline densities at z = 2.25 are n_los = [360, 1200,3300] deg^{-2} at limiting magnitudes of g =[24.0, 24.5,25.0], resulting in transverse sightline separations of d_perp = [3.6, 1.9, 1.2] h^{-1} Mpc, which roughly sets the reconstruction scale. We simulate these reconstructions using mock spectra with realistic noise properties, and find that spectra with S/N = 4 per angstrom can be used to generate maps that clearly trace the underlying dark-matter at overdensities of rho/<rho> ~ 1. For the VLT/VIMOS spectrograph, exposure times t_exp = [4, 6, 10] hrs are sufficient for maps with spatial resolution epsilon_3d = [5.0, 3.2, 2.3] h^{-1} Mpc. Assuming ~ 250 h^{-1} Mpc is probed along the line-of-sight, 1 deg^2 of survey area would cover a comoving volume of ~ 10^6 h^{-3} Mpc^3 at <z>=2.3, enabling efficient mapping of large volumes with 8-10m telescopes. These maps could be used to study galaxy environments, detect proto-clusters, and study the topology of large-scale structure at high-z.

Using the topology of large-scale structure in the WiggleZ Dark Energy Survey as a cosmological standard ruler

We present new and accurate measurements of the cosmic distance–redshift relation, spanning 0.2 < z < 1, using the topology of large-scale structure as a cosmological standard ruler. Our results derive from an analysis of the Minkowski functionals of the density field traced by the WiggleZ Dark Energy Survey. The Minkowski functionals are a set of statistics which completely describe the topological nature of each isodensity surface within the field, as a function of the density value. Given the shape of the underlying matter power spectrum, measured by fluctuations in the cosmic microwave background radiation, the expected amplitudes of the Minkowski functionals are specified as an excursion set of a Gaussian random field, with minimal non-Gaussian corrections for the smoothing scales ≥10 h−1 Mpc considered in this analysis. The measured amplitudes then determine the cosmic distance DV(z), which we obtain with 3–7 per cent accuracies in six independent redshift slices, with the standard ruler originating in the known curvature of the model power spectrum at the smoothing scale. We introduce a new method for correcting the topological statistics for the sparse-sampling of the density field by the galaxy tracers, and validate our overall methodology using mock catalogues from N-body simulations. Our distance measurements are consistent with standard models which describe the cosmic expansion history, and with previous analyses of baryon acoustic oscillations (BAOs) detected by the WiggleZ Survey, with the topological results yielding a higher distance precision by a factor of 2. However, the full redshift-space power-spectrum shape is required to recover the topological distances, in contrast to the preferred length scale imprinted by BAOs, which is determined by simpler physics.

BETTI NUMBERS OF GAUSSIAN FIELDS

We present the relation between the genus in cosmology and the Betti numbers for excursion sets of three- and two-dimensional smooth Gaussian random fields, and numerically investigate the Betti numbers as a function of threshold level. Betti numbers are topological invariants of figures that can be used to distinguish topological spaces. In the case of the excursion sets of a three-dimensional field there are three possibly non-zero Betti numbers; $\beta_0$ is the number of connected regions, $\beta_1$ is the number of circular holes, and $\beta_2$ is the number of three-dimensional voids. Their sum with alternating signs is the genus of the surface of excursion regions. It is found that each Betti number has a dominant contribution to the genus in a specific threshold range. $\beta_0$ dominates the high-threshold part of the genus curve measuring the abundance of high density regions (clusters). $\beta_1$ dominates the genus near the median thresholds which measures the topology of negatively curved iso-density surfaces, and $\beta_2$ corresponds to the low-threshold part measuring the void abundance. We average the Betti number curves (the Betti numbers as a function of the threshold level) over many realizations of Gaussian fields and find that both the amplitude and shape of the Betti number curves depend on the slope of the power spectrum $n$ in such a way that their shape becomes broader and their amplitude drops less steeply than the genus as $n$ decreases. This behaviour contrasts with the fact that the shape of the genus curve is fixed for all Gaussian fields regardless of the power spectrum. Even though the Gaussian Betti number curves should be calculated for each given power spectrum, we propose to use the Betti numbers for better specification of the topology of large scale structures in the universe.

PHYSICS OF NON-GAUSSIAN FIELDS AND THE COSMOLOGICAL GENUS STATISTIC

We report a technique to calculate the impact of distinct physical processes inducing non-Gaussianity on the cosmological density field. A natural decomposition of the cosmic genus statistic into an orthogonal polynomial sequence allows complete expression of the scale-dependent evolution of the topology of large-scale structure, in which effects including galaxy bias, nonlinear gravitational evolution, and primordial non-Gaussianity may be delineated. The relationship of this decomposition to previous methods for analyzing the genus statistic is briefly considered and the following applications are made: (1) the expression of certain systematics affecting topological measurements, (2) the quantification of broad deformations from Gaussianity that appear in the genus statistic as measured in the Horizon Run simulation, and (3) the study of the evolution of the genus curve for simulations with primordial non-Gaussianity. These advances improve the treatment of flux-limited galaxy catalogs for use with this measurement and further the use of the genus statistic as a tool for exploring non-Gaussianity.

TOPOLOGY OF A LARGE-SCALE STRUCTURE AS A TEST OF MODIFIED GRAVITY

The genus of the iso-density contours is a robust measure of the topology of large scale structure, and it is relatively insensitive to nonlinear gravitational evolution, galaxy bias and redshift-space distortion. We show that the growth of density fluctuations is scale-dependent even in the linear regime in some modified gravity theories, which opens a new possibility of testing the theories observationally. We propose to use the genus of the iso-density contours, an intrinsic measure of the topology of large scale structure, as a statistic to be used in such tests. In Einstein's general theory of relativity, density fluctuations are growing at the same rate on all scales in the linear regime, and the genus per comoving volume is almost conserved as structures are growing homologously, so we expect that the genus-smoothing scale relation is basically time-independent. However, in some modified gravity models where structures grow with different rates on different scales, the genus-smoothing scale relation should change over time. This can be used to test the gravity models with large scale structure observations. We studied the case of the f(R) theory, DGP braneworld theory as well as the parameterized post-Friedmann (PPF) models. We also forecast how the modified gravity models can be constrained with optical/IR or redshifted 21cm radio surveys in the near future.

THE NEW HORIZON RUN COSMOLOGICAL N-BODY SIMULATIONS

We present two large cosmological N-body simulations, called Horizon Run 2 (HR2) and Horizon Run 3 (HR3), made using <TEX>$6000^3$</TEX> = 216 billions and <TEX>$7210^3$</TEX> = 374 billion particles, spanning a volume of <TEX>$(7.200\;h^{-1}Gpc)^3$</TEX> and <TEX>$(10.815\;h^{-1}Gpc)^3$</TEX>, respectively. These simulations improve on our previous Horizon Run 1 (HR1) up to a factor of 4.4 in volume, and range from 2600 to over 8800 times the volume of the Millennium Run. In addition, they achieve a considerably finer mass resolution, down to <TEX>$1.25{\times}10^{11}h^{-1}M_{\odot}$</TEX>, allowing to resolve galaxy-size halos with mean particle separations of <TEX>$1.2h^{-1}$</TEX>Mpc and <TEX>$1.5h^{-1}$</TEX>Mpc, respectively. We have measured the power spectrum, correlation function, mass function and basic halo properties with percent level accuracy, and verified that they correctly reproduce the CDM theoretical expectations, in excellent agreement with linear perturbation theory. Our unprecedentedly large-volume N-body simulations can be used for a variety of studies in cosmology and astrophysics, ranging from large-scale structure topology, baryon acoustic oscillations, dark energy and the characterization of the expansion history of the Universe, till galaxy formation science - in connection with the new SDSS-III. To this end, we made a total of 35 all-sky mock surveys along the past light cone out to z = 0.7 (8 from the HR2 and 27 from the HR3), to simulate the BOSS geometry. The simulations and mock surveys are already publicly available at http://astro.kias.re.kr/Horizon-Run23/.

Using the topology of large-scale structure to constrain dark energy

The use of standard rulers, such as the scale of the Baryonic Acoustic oscillations (BAO), has become one of the more powerful techniques employed in cosmology to probe the entity driving the accelerating expansion of the Universe. In this paper, the topology of large scale structure (LSS) is used as one such standard ruler to study this mysterious `dark energy'. By following the redshift evolution of the clustering of luminous red galaxies (LRGs) as measured by their 3D topology (counting structures in the cosmic web), we can chart the expansion rate and extract information about the equation of state of dark energy. Using the technique first introduced in (Park & Kim, 2009), we evaluate the constraints that can be achieved using 3D topology measurements from next-generation LSS surveys such as the Baryonic Oscillation Spectroscopic Survey (BOSS). In conjunction with the information that will be available from the Planck satellite, we find a single topology measurement on 3 different scales is capable of constraining a single dark energy parameter to within 5% and 10% when dynamics are permitted. This offers an alternative use of the data available from redshift surveys and serves as a cross-check for BAO studies.

LARGE-SCALE STRUCTURE OF THE UNIVERSE AS A COSMIC STANDARD RULER

We propose to use the large-scale structure of the universe as a cosmic standard ruler, based on the fact that the pattern of galaxy distribution should be maintained in the course of time on large scales. By examining the scale-dependence of the pattern in different redshift intervals it is possible to reconstruct the expansion history of the universe, and thus to measure the cosmological parameters governing the expansion of the universe. The features in the galaxy distribution that can be used as standard rulers include the topology of large-scale structure and the overall shapes of galaxy power spectrum and correlation function. The genus, being an intrinsic topology measure, is resistant against the non-linear gravitational evolution, galaxy biasing, and redshift-space distortion effects, and thus is ideal for quantifying the primordial topology of the large-scale structure. The expansion history of the universe can be constrained by comparing among the genus measured at different redshifts. In the case of initially Gaussian fluctuations the genus accurately recovers the slope of the primordial power spectrum near the smoothing scale, and the expansion history can be constrained by comparing between the predicted and measured genus.

THREE-DIMENSIONAL GENUS TOPOLOGY OF LUMINOUS RED GALAXIES

We measure the 3D genus topology of large scale structure using Luminous Red Galaxies (LRGs) in the Sloan Digital Sky Survey and find it consistent with the Gaussian random phase initial conditions expected from the simplest scenarios of inflation. This studies 3D topology on the largest scales ever obtained. The topology is sponge-like. We measure topology in two volume-limited samples: a dense shallow sample studied with smoothing length of 21h^{-1}Mpc, and a sparse deep sample studied with a smoothing length of 34h^{-1}Mpc. The amplitude of the genus curve is measured with 4% uncertainty. Small distortions in the genus curve expected from non-linear biasing and gravitational effects are well explained (to about 1-\sigma accuracy) by N-body simulations using a subhalo-finding technique to locate LRGs. This suggests the formation of LRGs is a clean problem that can be modeled well without any free fitting parameters. This bodes well for using LRGs to measure the characteristic scales such as the baryon oscillation scale in future deep redshift surveys.

Genus Topology of Structure in the Sloan Digital Sky Survey: Model Testing

We measure the three-dimensional topology of large-scale structure in the Sloan Digital Sky Survey (SDSS). This allows the genus statistic to be measured with unprecedented statistical accuracy. The sample size is now sufficiently large to allow the topology to be an important tool for testing galaxy formation models. For comparison, we make mock SDSS samples using several state-of-the-art N-body simulations: the Millennium run of Springel et al. (10 billion particles), the Kim & Park CDM models (1.1 billion particles), and the Cen & Ostriker hydrodynamic code models (8.6 billion cell hydro mesh). Each of these simulations uses a different method for modeling galaxy formation. The SDSS data show a genus curve that is broadly characteristic of that produced by Gaussian random-phase initial conditions. Thus, the data strongly support the standard model of inflation where Gaussian random-phase initial conditions are produced by random quantum fluctuations in the early universe. But on top of this general shape there are measurable differences produced by nonlinear gravitational effects and biasing connected with galaxy formation. The N-body simulations have been tuned to reproduce the power spectrum and multiplicity function but not topology, so topology is an acid test for these models. The data show a meatball shift (only partly due to the Sloan Great Wall of galaxies) that differs at the 2.5 ? level from the results of the Millenium run and the Kim & Park dark halo models, even including the effects of cosmic variance.

The effect of primordial non-Gaussianity on the topology of large-scale structure

We study the effect of primordial non-Gaussianity on the development of large-scale cosmic structure using high-resolution N-body simulations. In particular, we focus on the topological properties of the "cosmic web", quantitatively characterized by the Minkowski Functionals, for models with quadratic non-linearities with different values of the usual non-Gaussianity parameter fNL. In the weakly non-linear regime, we find that analytic formulae derived from perturbation theory agree with the numerical results within a few percent of the amplitude of each MF when |fNL|<1000. In the non-linear regime, the detailed behavior of the MFs as functions of threshold density deviates more strongly from the analytical curves, while the overall amplitude of the primordial non-Gaussian effect remains comparable to the perturbative prediction. When smaller-scale information is included, the influence of primordial non-Gaussianity becomes increasingly significant statistically due to decreasing sample variance. We find that the effect of the primordial non-Gaussianity with |fNL|=50 is comparable to the sample variance of mass density fields with a volume of 0.125(Gpc/h)^3 when they are smoothed by Gaussian filter at a scale of 5Mpc/h. The detectability of this effect in actual galaxy surveys will strongly depend upon residual uncertainties in cosmological parameters and galaxy biasing.

Topology of large-scale structure in the 2dF Galaxy Redshift Survey

We investigate the topology of the completed 2dF Galaxy Redshift Survey (2dFGRS), drawing two flux-limited samples of the local Universe from the 2dFGRS catalogue, which contains over 220000 galaxies at a median redshift of z = 0.11. The samples are cut at z = 0.2 and corrected for selection effects. We then use the three-dimensional genus statistic to probe the connectedness of the galaxy distribution on scales ranging from 8 to 20 Mpc, and compare these measurements with the analytical result for a Gaussian random density field, a generic prediction of inflationary models. We demonstrate consistency with inflation on the range of scales considered. We then introduce a parametrization of the analytic genus curve formula that is sensitive to asymmetries in genus number as a function of density and use it to demonstrate that such phenomena are ruled out with 95 per cent confidence between 8 and 16 Mpc.

Effects of Gravitational Evolution, Biasing, and Redshift Space Distortion on Topology

We have studied the dependence of topology of large-scale structure on tracer, gravitational evolution, redshift space distortion, and cosmology. A series of large N-body simulations of the ΛCDM and SCDM models that have evolved 1.1 or 8.6 billion particles are used in the study. Evolution of the genus statistic, used as a topology measure, from redshift 8 to 0 is accurately calculated over a wide range of smoothing scales using the simulations. The tracers of large-scale structure considered are the cold dark matter (CDM), biased peaks in the initial density field, dark halos, and populating the dark halos in accordance with a halo occupation distribution (HOD) model. We have found that the effects of biasing, gravitational evolution, and initial conditions on topology of large-scale structure are all comparable. The redshift space distortion effects are relatively small down to about 5 h-1 Mpc for all tracers except for the high-threshold part of the genus curve. The gravitational effects are found to be well modeled by analytic perturbation theory when the CDM distribution is considered. But the direction of gravitational evolution of topology can be even reversed for different tracers. For example, the shift parameter of the genus curve evolves in opposite directions for matter and HOD galaxies at large scales. At small scales, there are interesting deviations of the genus curve of dark halos and galaxies from that of matter in our initially Gaussian simulations. The deviations should be understood as due to combined effects of gravitational evolution and biasing. This fact gives us an important opportunity: topology of large-scale structure can be used as a strong constraint on galaxy formation mechanisms. At scales larger than 20 h-1 Mpc all the above effects gradually decrease. With good knowledge of the effects of nonlinear gravitational evolution and galaxy biasing on topology, one can also constrain the Gaussian random phase initial conditions hypothesis to high accuracy.

Spatial characterization of the numerically simulated vorticity fields of a flow in a flume

The topology of large scale structures in a turbulent boundary layer is investigated numerically. Spatial characteristics of the large scale structure are presented through an original method, proper orthogonal decomposition (POD) of the three-dimensional vorticity fields. The DNS results, obtained by Tiselj et al. [23] for a fully developed turbulent flow in a flume, are used in the present work to analyze coherent structures with the proposed methodology. In contrast to the reconstruction methods that use instantaneous flow quantities, this approach utilizes the whole dataset of the numerical simulation. The analysis uses one thousand 3D vorticity fields from 50000 time steps of the simulation for the Reynolds number of 2600 (the turbulent Reynolds number Re*=171). The computational domain is 2146×171×537 wall units and the grid resolution is 128×65×72 points (in streamwise, wall-normal and spanwise directions, respectively). Experimental results obtained by using particle image velocimetry (PIV) in a fully developed turbulent boundary layer in a flume, which were analyzed with the same statistical characterization method, are in agreement with the DNS analysis: the dominant vortical structure appears to have a longitudinal streamwise orientation, an inclination angle of about 8°, streamwise length of several hundred wall units, and a distance between the neighboring structures of about 100 wall units in the spanwise direction.