Topology and geometry of the CfA2 redshift survey

We analyse the redshift space topology and geometry of the nearby Universe by computing the Minkowski functionals of the Updated Zwicky Catalogue (UZC). The UZC contains the redshifts of almost 20 000 galaxies, is 96 per cent complete to the limiting magnitude m _Zw=15.5, and includes the Center for Astrophysics (CfA) Redshift Survey (CfA2). From the UZC we can extract volume-limited samples reaching a depth of 70  h ⁻¹ Mpc before sparse sampling dominates. We quantify the shape of the large-scale galaxy distribution by deriving measures of planarity and filamentarity from the Minkowski functionals. The nearby Universe shows a large degree of planarity and a small degree of filamentarity. This quantifies the sheet-like structure of the Great Wall, which dominates the northern region (CfA2N) of the UZC. We compare these results with redshift space mock catalogues constructed from high-resolution N -body simulations of two cold dark matter (CDM) models with either a decaying massive neutrino ( τ CDM) or a non-zero cosmological constant (ΛCDM). We use semi-analytic modelling to form and evolve galaxies in these dark matter‐only simulations. We are thus able, for the first time, to compile redshift space mock catalogues which contain galaxies, along with their observable properties, rather than dark matter particles alone. In both models the large-scale galaxy distribution is less coherent than the observed distribution, especially with regard to the large degree of planarity of the real survey. However, given the small volume of the region studied, this disagreement can still be a result of cosmic variance, as shown by the agreement between the ΛCDM model and the southern region of CfA2.     

Clustering of galaxy clusters in cold dark matter universes

We use very large cosmological N -body simulations to obtain accurate predictions for the two-point correlations and power spectra of mass-limited samples of galaxy clusters. We consider two currently popular cold dark matter (CDM) cosmogonies, a critical density model ( τ CDM) and a flat low density model with a cosmological constant (ΛCDM). Our simulations each use 10⁹ particles to follow the mass distribution within cubes of side 2  h ⁻¹ Gpc ( τ CDM) and 3  h ⁻¹ Gpc (ΛCDM) with a force resolution better than 10⁻⁴ of the cube side. We investigate how the predicted cluster correlations increase for samples of increasing mass and decreasing abundance. Very similar behaviour is found in the two cases. The correlation length increases from     for samples with mean separation     to     for samples with     The lower value here corresponds to τ CDM and the upper to ΛCDM. The power spectra of these cluster samples are accurately parallel to those of the mass over more than a decade in scale. Both correlation lengths and power spectrum biases can be predicted to better than 10 per cent using the simple model of Sheth, Mo & Tormen. This prediction requires only the linear mass power spectrum and has no adjustable parameters. We compare our predictions with published results for the automated plate measurement (APM) cluster sample. The observed variation of correlation length with richness agrees well with the models, particularly for ΛCDM. The observed power spectrum (for a cluster sample of mean separation     ) lies significantly above the predictions of both models.     

The alignment of clusters using large-scale simulations

The alignment of clusters of galaxies with their nearest neighbours and between clusters within a supercluster is investigated using simulations of 512³ dark matter particles for ΛCDM and τ CDM cosmological models. Strongly significant alignments are found for separations of up to 15  h ⁻¹ Mpc in both cosmologies, but for the ΛCDM model the alignments extend up to separations of 30  h ⁻¹ Mpc. The effect is strongest for nearest neighbours, but is not significant enough to be useful as an observational discriminant between cosmologies. As a check of whether this difference in alignments is present in other cosmologies, smaller simulations with 256³ particles are investigated for four different cosmological models. Because of poor number statistics, only the standard CDM model shows indications of having different alignments from the other models.     

Clustering of galaxies in a hierarchical universe – III. Mock redshift surveys

This is the third paper in a series which combines N -body simulations and semi-analytic modelling to provide a fully spatially resolved simulation of the galaxy formation and clustering processes. Here we extract mock redshift surveys from our simulations: a cold dark matter model with either Ω₀=1 ( τ CDM) or Ω₀=0.3 and Λ=0.7 (ΛCDM). We compare the mock catalogues with the northern region (CfA2N) of the Center for Astrophysics (CfA) Redshift Surveys. We study the properties of galaxy groups and clusters identified using standard observational techniques, and also the relation of these groups to real virialized systems. Most features of CfA2N groups are reproduced quite well by both models with no obvious dependence on Ω₀. Redshift‐space correlations and pairwise velocities are also similar in the two cosmologies. The luminosity functions predicted by our galaxy formation models depend sensitively on the treatment of star formation and feedback. For the particular choices of Paper I they agree poorly with the CfA survey. To isolate the effect of this discrepancy on our mock redshift surveys, we modify galaxy luminosities in our simulations to reproduce the CfA luminosity function exactly. This adjustment improves agreement with the observed abundance of groups, which depends primarily on the galaxy luminosity density, but other statistics, connected more closely with the underlying mass distribution, remain unaffected. Regardless of the luminosity function adopted, modest differences with observation remain. These can be attributed to the presence of the 'Great Wall' in the CfA2N. It is unclear whether the greater coherence of the real structure is a result of cosmic variance, given the relatively small region studied, or reflects a physical deficiency of the models.     

Evolution of the mass function of dark matter haloes

We use a high-resolution ΛCDM numerical simulation to calculate the mass function of dark matter haloes down to the scale of dwarf galaxies, back to a redshift of 15, in a  50 h ⁻¹ Mpc  volume containing 80 million particles. Our low-redshift results allow us to probe low-σ density fluctuations significantly beyond the range of previous cosmological simulations. The Sheth & Tormen mass function provides an excellent match to all of our data except for redshifts of 10 and higher, where it overpredicts halo numbers increasingly with redshift, reaching roughly 50 per cent for the  10¹⁰–10¹¹ M_⊙  haloes sampled at redshift 15. Our results confirm previous findings that the simulated halo mass function can be described solely by the variance of the mass distribution, and thus has no explicit redshift dependence. We provide an empirical fit to our data that corrects for the overprediction of extremely rare objects by the Sheth & Tormen mass function. This overprediction has implications for studies that use the number densities of similarly rare objects as cosmological probes. For example, the number density of high-redshift  ( z ≃ 6) QSOs  , which are thought to be hosted by haloes at 5σ peaks in the fluctuation field, are likely to be overpredicted by at least a factor of 50 per cent. We test the sensitivity of our results to force accuracy, starting redshift and halo-finding algorithm.     

Constraints on cosmological and biasing models using active galactic nucleus clustering

We attempt to put constraints on different cosmological and biasing models by combining the recent clustering results of X-ray sources in the local ( z ≤0.1) and distant Universe ( z ∼1) . To this end we compare the measured angular correlation function for bright (Akylas et al.) and faint (Vikhlinin & Forman) ROSAT X-ray sources respectively with those expected in three spatially flat cosmological models. Taking into account the different functional forms of the bias evolution, we find that there are two cosmological models which match the data well. In particular, low-Ω_○ cosmological models (Ω_Λ=1−Ω_○=0.7) that contain either (i) high σ ₈^mass=1.13 value with galaxy merging bias, b ( z )∝(1+ z )^1.8 or (ii) low σ ₈^mass=0.9 with non-bias, b ( z ) ≡ 1 best reproduce the AGN clustering results, while τ CDM models with different bias behaviour are ruled out at a high significance level.     

Future evolution of bound superclusters in an accelerating Universe

The evolution of marginally bound supercluster-like objects in an accelerating Λ cold dark matter (ΛCDM) Universe is followed, by means of cosmological simulations, from the present time to an expansion factor   a = 100  . The objects are identified on the basis of the binding density criterion introduced by Dünner et al. Superclusters are identified with the ones whose mass   M > 10¹⁵  h ⁻¹ M_⊙  , the most massive one with   M ∼ 8 × 10¹⁵  h ⁻¹ M_⊙  , comparable to the Shapley supercluster. The spatial distribution of the superclusters remains essentially the same after the present epoch, reflecting the halting growth of the cosmic web as Λ gets to dominate the expansion of the Universe. The same trend can be seen in the stagnation of the development of the mass function of virialized haloes and bound objects. The situation is considerably different when looking at the internal evolution, quantified in terms of their shape, compactness and density profile, and substructure in terms of their multiplicity function. We find a continuing evolution from a wide range of triaxial shapes at   a = 1  to almost perfect spherical shapes at   a = 100  . We also find a systematic trend towards a higher concentration. Meanwhile, we see their substructure gradually disappearing, as the surrounding subclumps fall in and merge to form one coherent, virialized system.     

Approaching a homogeneous galaxy distribution: results from the Stromlo–APM redshift survey

Recent results from a number of redshift surveys suggest that the Universe is well described by an inhomogeneous, fractal distribution on the largest scales probed. This distribution has been found to have fractal dimension, D , approximately equal to 2.1, in contrast to a homogeneous distribution in which the dimension should approach the value 3 as the scale is increased. In this paper we demonstrate that estimates of D , based on the conditional density of galaxies, are prone to bias from several sources. These biases generally result in a smaller measured fractal dimension than the true dimension of the sample. We illustrate this behaviour in application to the Stromlo–APM redshift survey, showing that this data set in fact provides evidence for fractal dimension increasing with survey depth. On the largest scale probed, r ≈60  h ⁻¹ Mpc, we find evidence for a distribution with dimension D =2.76±0.10. A comparison between this sample and mock Stromlo–APM catalogues taken from N -body simulations (which assume a CDM cosmology) reveals a striking similarity in the behaviour of the fractal dimension. Thus we find no evidence for inhomogeneity in excess of that expected from conventional cosmological theory. We consider biases affecting future large surveys and demonstrate, using mock SDSS catalogues, that this survey will be able to measure the fractal dimension on scales at which we expect to see full turn-over to homogeneity, in an accurate and unbiased way.     

Peculiar velocity reconstruction with the fast action method: tests on mock redshift surveys

We present extensive tests of the fast action method (FAM) for recovering the past orbits of mass tracers in an expanding universe from their redshift-space coordinates at the present epoch. The tests focus on the reconstruction of present-day peculiar velocities using mock catalogues extracted from high-resolution N -body simulations. The method allows for a self-consistent treatment of redshift-space distortions by direct minimization of a modified action for a cosmological gravitating system. When applied to ideal, volume-limited catalogues, FAM recovers unbiased peculiar velocities with a one-dimensional, 1σ error of ∼220 km  s⁻¹  , if velocities are smoothed on a scale of  5 h ⁻¹  Mpc. Alternatively, when no smoothing is applied, FAM predicts nearly unbiased velocities for objects residing outside the highest density regions. In this second case the 1σ error decreases to a level of ∼150 km  s⁻¹  . The correlation properties of the peculiar velocity fields are also correctly recovered on scales larger than  5 h ⁻¹  Mpc. Similar results are obtained when FAM is applied to flux-limited catalogues mimicking the IRAS PSC z survey. In this case FAM reconstructs peculiar velocities with similar intrinsic random errors, while velocity–velocity correlation properties are well reproduced beyond scales of  ∼8 h ⁻¹  Mpc. We also show that FAM provides better velocity predictions than other, competing methods based on linear theory or the Zel'dovich approximation. These results indicate that FAM can be successfully applied to presently available galaxy redshift surveys such as IRAS PSC z .     

The cluster distribution as a test of dark matter models — IV. Topology and geometry

We study the geometry and topology of the large-scale structure traced by galaxy clusters in numerical simulations of a box of side 320 h ⁻¹ Mpc, and compare them with available data on real clusters. The simulations we use are generated by the Zel'dovich approximation, using the same methods as we have used in the first three papers in this series. We consider the following models to see if there are measurable differences in the topology and geometry of the superclustering they produce: (i) the standard cold dark matter model (SCDM); (ii) a CDM model with Ω₀ = 0.2 (OCDM); (iii) a CDM model with a 'tilted' power spectrum having n  = 0.7 (TCDM); (iv) a CDM model with a very low Hubble constant, h  = 0.3 (LOWH); (v) a model with mixed CDM and HDM (CHDM); (vi) a flat low-density CDM model with Ω₀ = 0.2 and a non-zero cosmological Λ term (ΛCDM). We analyse these models using a variety of statistical tests based on the analysis of: (i) the Euler–Poincaré characteristic; (ii) percolation properties; (iii) the minimal spanning tree construction. Taking all these tests together we find that the best-fitting model is ΛCDM and, indeed, the others do not appear to be consistent with the data. Our results demonstrate that despite their biased and extremely sparse sampling of the cosmological density field, it is possible to use clusters to probe subtle statistical diagnostics of models, which go far beyond the low-order correlation functions usually applied to study superclustering.     

Observational biases in Lagrangian reconstructions of cosmic velocity fields

Lagrangian reconstruction of large-scale peculiar velocity fields can be strongly affected by observational biases. We develop a thorough analysis of these systematic effects by relying on specially selected mock catalogues. For the purpose of this paper, we use the Monge–Ampère–Kantorovitch (MAK) reconstruction method, although any other Lagrangian reconstruction method should be sensitive to the same problems. We extensively study the uncertainty in the mass-to-light assignment due to incompleteness (missing luminous mass tracers), and the poorly determined relation between mass and luminosity. The impact of redshift distortion corrections is analysed in the context of MAK and we check the importance of edge and finite-volume effects on the reconstructed velocities. Using three mock catalogues with different average densities, we also study the effect of cosmic variance. In particular, one of them presents the same global features as found in observational catalogues that extend to 80 h ⁻¹ Mpc scales. We give recipes, checked using the aforementioned mock catalogues, to handle these particular observational effects, after having introduced them into the mock catalogues so as to quantitatively mimic the most densely sampled currently available galaxy catalogue of the nearby Universe. Once biases have been taken care of, the typical resulting error in reconstructed velocities is typically about a quarter of the overall velocity dispersion, and without significant bias. We finally model our reconstruction errors to propose an improved Bayesian approach to measure Ω_m in an unbiased way by comparing the reconstructed velocities to the measured ones in distance space, even though they may be plagued by large errors. We show that, in the context of observational data, it is possible to build a nearly unbiased estimator of Ω_m using MAK reconstruction.     

Voids in the Las Campanas Redshift Survey versus cold dark matter models

We analysed the distribution of void sizes in the two-dimensional slices of the Las Campanas Redshift Survey (LCRS). Fourteen volume-limited subsamples were extracted from the six slices to cover a large part of the survey and to test the robustness of the results against cosmic variance. Thirteen samples were randomly culled to produce homogeneously selected samples. We then studied the relationship between the cumulative area covered by voids and the void size as a property of the void hierarchy. We found that the distribution of void sizes scaled with the mean galaxy separation, λ . In particular, we found that the size of voids covering half of the area is given by D _med≈ λ +(12±3)  h ⁻¹ Mpc. Next, by employing an environmental density threshold criterion to identify mock galaxies, we were able to extend this analysis to mock samples from dynamical N -body simulations of cold dark matter (CDM) models. To reproduce the observed void statistics, overdensity thresholds of δ _th≈0,…,1 are necessary. We compared standard (SCDM), open (OCDM), vacuum energy dominated (ΛCDM) and broken scale invariant CDM models (BCDM): we found that both the void size distribution and the two-point correlation function provided important and complementary information on the large-scale matter distribution. The dependence of the void statistics on the threshold criterion for the mock galaxy identification showed that the galaxy biasing was more crucial for the void size distribution than were differences between the cosmological models.     

Constraints on cosmological anisotropy out to z= 1 from Type Ia supernovae

A combined sample of 79 high- and low-redshift Type Ia supernovae (SNe) is used to set constraints on the degree of anisotropy in the Universe out to z ≃1. First, we derive the global most probable values of matter density Ω_M, the cosmological constant Ω_Λ and the Hubble constant H ₀, and find them to be consistent with the published results from the two data sets of Riess et al. and Perlmutter et al. We then examine the Hubble diagram (HD, i.e., the luminosity–redshift relation) in different directions on the sky by utilizing spherical harmonic expansion. In particular, via the analysis of the dipole anisotropy, we divide the sky into the two hemispheres that yield the most discrepant of the three cosmological parameters, and the scatter χ _HD² in each case. The most discrepant values roughly move along the locus −4Ω_M+3Ω_Λ=1 (cf. Perlmutter et al.), but by no more than Δ≈2.5 along this line. For a perfect Friedmann–Robertson–Walker universe, Monte Carlo realizations that mimic the current set of SNe yield values higher than the measured Δ in ∼1/5 of the cases (for Ω_M). We discuss implications for the validity of the Cosmological Principle, and possible calibration problems in the SNe data sets.     

Resolving cosmic structure formation with the Millennium-II Simulation

We present the Millennium-II Simulation (MS-II), a very large N -body simulation of dark matter evolution in the concordance Λ cold dark matter (ΛCDM) cosmology. The MS-II assumes the same cosmological parameters and uses the same particle number and output data structure as the original Millennium Simulation (MS), but was carried out in a periodic cube one-fifth the size  (100  h ⁻¹ Mpc)  with five times better spatial resolution (a Plummer equivalent softening of  1.0  h ⁻¹ kpc  ) and with 125 times better mass resolution (a particle mass of  6.9 × 10⁶  h ⁻¹ M_⊙  ). By comparing results at MS and MS-II resolution, we demonstrate excellent convergence in dark matter statistics such as the halo mass function, the subhalo abundance distribution, the mass dependence of halo formation times, the linear and non-linear autocorrelations and power spectra, and halo assembly bias. Together, the two simulations provide precise results for such statistics over an unprecedented range of scales, from haloes similar to those hosting Local Group dwarf spheroidal galaxies to haloes corresponding to the richest galaxy clusters. The 'Milky Way' haloes of the Aquarius Project were selected from a lower resolution version of the MS-II and were then resimulated at much higher resolution. As a result, they are present in the MS-II along with thousands of other similar mass haloes. A comparison of their assembly histories in the MS-II and in resimulations of 1000 times better resolution shows detailed agreement over a factor of 100 in mass growth. We publicly release halo catalogues and assembly trees for the MS-II in the same format within the same archive as those already released for the MS.     

PSCz superclusters: detection, shapes and cosmological implications

We study the possibility of correctly identifying, from the smooth galaxy density field of the PSC z flux-limited catalogue, high-density regions (superclusters) and recovering their true shapes in the presence of a bias introduced by the coupling between the selection function and the constant radius smoothing. We quantify such systematic biases in the smoothed PSC z density field and after applying the necessary corrections we study supercluster multiplicity and morphologies using a differential geometry definition of shape. Our results strongly suggest that filamentary morphology is the dominant feature of PSC z superclusters. Finally, we compare our results with those expected in three different cosmological models and find that the Λ cold dark matter (CDM) model (Ω_Λ=1−Ω_m=0.7) performs better than Ω_m=1 CDM models.     

Massive dark matter haloes around bright isolated galaxies in the 2dFGRS

We identify a large sample of isolated bright galaxies and their fainter satellites in the 2dF Galaxy Redshift Survey (2dFGRS). We analyse the dynamics of ensembles of these galaxies selected according to luminosity and morphological type by stacking the positions of their satellites and estimating the velocity dispersion of the combined set. We test our methodology using realistic mock catalogues constructed from cosmological simulations. The method returns an unbiased estimate of the velocity dispersion provided that the isolation criterion is strict enough to avoid contamination and that the scatter in halo mass at fixed primary luminosity is small. Using a maximum likelihood estimator that accounts for interlopers, we determine the satellite velocity dispersion within a projected radius of 175  h ⁻¹ kpc. The dispersion increases with the luminosity of the primary and is larger for elliptical galaxies than for spiral galaxies of similar b _J luminosity. Calibrating the mass–velocity dispersion relation using our mock catalogues, we find a dynamical mass within 175  h ⁻¹ kpc of     for elliptical galaxies and     for spiral galaxies. Finally, we compare our results with recent studies and investigate their limitations using our mock catalogues.     

The redshift evolution of clustering in the Hubble Deep Field

We present a correlation function analysis for the catalogue of photometric redshifts obtained from the Hubble Deep Field image by Fernandez-Soto, Lanzetta & Yahil. By dividing the catalogue into redshift bins of width Δ z =0.4 we measured the angular correlation function w ( θ ) as a function of redshift up to z ∼4.8. From these measurements we derive the trend of the correlation length r ₀. We find that r ₀( z ) is roughly constant with look-back time up to z ≃2, and then increases to higher values at z ≳2.4. We estimate the values of r ₀, assuming ξ ( r , z )=[ r r ₀( z )]^{− γ} , γ =1.8 and various geometries. For Ω₀=1 we find r ₀( z =3)≃7.00±4.87  h ⁻¹ Mpc, in good agreement with the values obtained from analysis of the Lyman break galaxies.     

Spatial clustering of Ultra Steep Spectrum sources and galaxies

We present measurements of the clustering properties of galaxies in the field of redshift range 0.5 ≲ z ≲ 1.5 Ultra Steep Spectrum radio sources selected from the Sydney University Molonglo Sky Survey and the National Radio Astronomy Observatories Very Large Array Sky Survey. Galaxies in these USS fields were identified in deep near-infrared observations, complete down to   K _s= 20  , using the IRIS2 instrument at the Anglo-Australian Telescope. We used the redshift distribution of   K _s < 20  galaxies taken from Cimatti et al. (2002) to constrain the correlation length r ₀. We find a strong correlation signal of galaxies with   K _s < 20  around our USS sample. A comoving correlation length   r ₀= 14.0 ± 2.8  h ⁻¹ Mpc  and γ= 1.98 ± 0.15 are derived in a flat cosmological model universe.
We compare our findings with those obtained in a cosmological N -body simulation populated with galform semi-analytic galaxies. We find that clusters of galaxies with masses in the range   M = 10^13.4–14.2  h ⁻¹ M_⊙  have a cluster–galaxy cross-correlation amplitude comparable to those found between the USS hosts and galaxies. These results suggest that distant radio galaxies are excellent tracers of galaxy overdensities and pinpoint the progenitors of present day rich clusters of galaxies.