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.     

Simulations of deep pencil-beam redshift surveys

We create mock pencil-beam redshift surveys from very large cosmological N -body simulations of two cold dark matter (CDM) cosmogonies, an Einstein–de Sitter model ( τ CDM) and a flat model with Ω₀=0.3 and a cosmological constant (ΛCDM). We use these to assess the significance of the apparent periodicity discovered by Broadhurst et al. Simulation particles are tagged as 'galaxies' so as to reproduce observed present-day correlations. They are then identified along the past light-cones of hypothetical observers to create mock catalogues with the geometry and the distance distribution of the Broadhurst et al. data. We produce 1936 (2625) quasi-independent catalogues from our τ CDM (ΛCDM) simulation. A couple of large clumps in a catalogue can produce a high peak at low wavenumbers in the corresponding one-dimensional power spectrum, without any apparent large-scale periodicity in the original redshift histogram. Although the simulated redshift histograms frequently display regularly spaced clumps, the spacing of these clumps varies between catalogues and there is no 'preferred' period over our many realizations. We find only a 0.72 (0.49) per cent chance that the highest peak in the power spectrum of a τ CDM (ΛCDM) catalogue has a peak-to-noise ratio higher than that in the Broadhurst et al. data. None of the simulated catalogues with such high peaks shows coherently spaced clumps with a significance as high as that of the real data. We conclude that in CDM universes, the regularity on a scale of ∼130  h ⁻¹ Mpc observed by Broadhurst et al. has a priori probability well below 10⁻³.     

Constraints on σ8 from galaxy clustering in N-body simulations and semi-analytic models

We generate mock galaxy catalogues for a grid of different cosmologies, using rescaled N -body simulations in tandem with a semi-analytic model run using consistent parameters. Because we predict the galaxy bias, rather than fitting it as a nuisance parameter, we obtain an almost pure constraint on σ₈ by comparing the projected two-point correlation function we obtain to that from the Sloan Digital Sky Survey (SDSS). A systematic error arises because different semi-analytic modelling assumptions allow us to fit the r -band luminosity function equally well. Combining our estimate of the error from this source with the statistical error, we find  σ₈= 0.97 ± 0.06  . We obtain consistent results if we use galaxy samples with a different magnitude threshold, or if we select galaxies by b _J-band rather than r -band luminosity and compare to data from the 2dF Galaxy Redshift Survey (2dFGRS). Our estimate for σ₈ is higher than that obtained for other analyses of galaxy data alone, and we attempt to find the source of this difference. We note that in any case, galaxy clustering data provide a very stringent constraint on galaxy formation models.     

The real- and redshift-space density distribution functions for large-scale structure in the spherical collapse approximation

We use the spherical collapse (SC) approximation to derive expressions for the smoothed redshift-space probability distribution function (PDF), as well as the p -order hierarchical amplitudes S _p , in both real and redshift space. We compare our results with numerical simulations, focusing on the     standard CDM model, where redshift distortions are strongest. We find good agreement between the SC predictions and the numerical PDF in real space even for     , where σ _L is the linearly evolved rms fluctuation on the smoothing scale. In redshift space, reasonable agreement is possible only for     . Numerical simulations also yield a simple empirical relation between the real-space PDF and the redshift-space PDF: we find that for     , the redshift-space PDF, [ P δ _{( z )}], is, to a good approximation, a simple rescaling of the real-space PDF, P [ δ ], i.e.,     where σ and σ _{( z )} are the real-space and redshift-space rms fluctuations, respectively. This result applies well beyond the validity of linear perturbation theory, and it is a good fit for both the standard CDM model and the ΛCDM model. It breaks down for SCDM at     , but provides a good fit to the ΛCDM models for σ _L as large as 0.8.     

The abundance and radial distribution of satellite galaxies

Using detailed mock galaxy redshift surveys (MGRSs) we investigate the abundance and radial distribution of satellite galaxies. The mock surveys are constructed using large numerical simulations and the conditional luminosity function (CLF), and are compared against data from the Two Degree Field Galaxy Redshift Survey (2dFGRS). We use Monte Carlo Markov chains to explore the full posterior distribution of the CLF parameter space, and show that the average relation between light and mass is tightly constrained and in excellent agreement with our previous models and with that of Vale & Ostriker. The radial number density distribution of satellite galaxies in the 2dFGRS reveals a pronounced absence of satellites at small projected separations from their host galaxies. This is (at least partly) owing to the overlap and merging of galaxy images in the 2dFGRS parent catalogue. Owing to the resulting close-pair incompleteness we are unfortunately unable to put meaningful constraints on the radial distribution of satellite galaxies; the data are consistent with a radial number density distribution that follows that of the dark matter particles, but we cannot rule out alternatives with a constant number density core. Marginalizing over the full CLF parameter space, we show that in a ΛCDM concordance cosmology the observed abundances of host and satellite galaxies in the 2dFGRS indicate a power spectrum normalization of  σ₈≃ 0.7  . The same cosmology but with  σ₈= 0.9  is unable to match simultaneously the abundances of host and satellite galaxies. This confirms our previous conclusions based on the pairwise peculiar velocity dispersions and the group multiplicity function.     

Measuring Ω0 from the entropy evolution of clusters

In this paper we have extended the entropy-driven model of cluster evolution developed by Bower in order to be able to predict the evolution of galaxy clusters for a range of cosmological scenarios. We have applied this model to recent measurements of the evolution of the L _x− T normalization and X-ray luminosity function in order to place constraints on cosmological parameters. We find that these measurements alone do not select a particular cosmological framework. An additional constraint is required on the effective slope of the power spectrum to break the degeneracy that exists between this and the background cosmology. We therefore include a theoretical calculation of the Ω₀ dependence on the power spectrum, based on the cold dark matter paradigm, which infers Ω₀<0.55 (0.1<Ω₀<0.7 for Ω₀+Λ₀=1), at the 95 per cent confidence level. Alternatively, an independent measurement of the slope of the power spectrum from galaxy clustering requires Ω₀<0.6 (Ω₀<0.65 for Ω₀+Λ₀=1), again to 95 per cent confidence. The rate of entropy evolution is insensitive to the values of Ω₀ considered, although it is sensitive to changes in the distribution of the intracluster medium.     

Measuring Ω0 using cluster evolution

The evolution of the abundance of galaxy clusters depends sensitively on the value of the cosmological density parameter, Ω₀. Recent ASCA data are used to quantify this evolution as measured by the cluster X-ray temperature function. A χ² minimization fit to the cumulative temperature function, as well as a maximum-likelihood estimate (which requires additional assumptions about cluster luminosities), leads to the estimate Ω₀ ≈ 0.45 ± 0.25 (1σ statistical error). Various systematic uncertainties are considered, none of which significantly enhances the probability that Ω₀ = 1. These conclusions hold for models with or without a cosmological constant, i.e., with Λ₀ = 0 or Λ₀ = 1 − Ω₀. The statistical uncertainties are at least as large as any of the individual systematic errors that have been considered here, suggesting that additional temperature measurements of distant clusters will allow an improvement in this estimate. An alternative method that uses the highest redshift clusters to place an upper limit on Ω₀ is also presented and tentatively applied, with the result that Ω₀  1 can be ruled out at the 98 per cent confidence level. Whilst this method does not require a well-defined statistical sample of distant clusters, there are still modelling uncertainties that preclude a firmer conclusion at this time.     

Peculiar velocities and the mean density parameter

We study the peculiar velocity field inferred from the Mark III spirals using a new method of analysis. We estimate optimal values of Tully–Fisher scatter and zero-point offset, and we derive the three-dimensional rms peculiar velocity ( σ _v ) of the galaxies in the samples analysed. We check our statistical analysis using mock catalogues derived from numerical simulations of cold dark matter (CDM) models considering measurement uncertainties and sampling variations. Our best determination for the observations is σ _v =(660±50) km s⁻¹. We use the linear theory relation between σ _v , the density parameter Ω, and the galaxy correlation function ξ ( r ) to infer the quantity     , where b is the linear bias parameter of optical galaxies and the uncertainties correspond to bootstrap resampling and an estimated cosmic variance added in quadrature. Our findings are consistent with the results of cluster abundances and redshift-space distortion of the two-point correlation function. These statistical measurements suggest a low value of the density parameter Ω∼0.4 if optical galaxies are not strongly biased tracers of mass.     

On the errors on Ω0: Monte Carlo simulations of the EMSS cluster sample

We perform Monte Carlo simulations of synthetic EMSS cluster samples, to quantify the systematic errors and the statistical uncertainties on the estimate of Ω₀ derived from fits to the cluster number density evolution and to the X-ray temperature distribution up to z =0.83 . We identify the scatter around the relation between cluster X-ray luminosity and temperature to be a source of systematic error, of the order of Δ_systΩ₀=0.09 , if not properly taken into account in the modelling. After correcting for this bias, our best Ω₀ is 0.66. The uncertainties on the shape and normalization of the power spectrum of matter fluctuations imply relatively large uncertainties on this estimate of Ω₀, of the order of Δ_statΩ₀=0.1 at the 1 σ level. On the other hand, the statistical uncertainties due to the finite size of the high-redshift sample are twice as small. Therefore, what is needed in order to improve the accuracy of Ω₀ estimates based on cluster number density evolution is a more reliable measure of the local temperature function and a better understanding of the cluster observed properties both in the local Universe and at high redshift, that is the relation between cluster mass, temperature and luminosity. This requires detailed observations of X-ray selected cluster samples, in comparison with hydrodynamic simulations including refined physics.     

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.     

The WARPS survey – IV. The X-ray luminosity–temperature relation of high-redshift galaxy clusters

We present a measurement of the cluster X-ray luminosity–temperature ( L – T ) relation out to high redshift ( z ∼0.8). Combined ROSAT PSPC spectra of 91 galaxy clusters detected in the Wide Angle ROSAT Pointed Survey (WARPS) are simultaneously fitted in redshift and luminosity bins. The resulting temperature and luminosity measurements of these bins, which occupy a region of the high-redshift L – T relation not previously sampled, are compared with existing measurements at low redshift in order to constrain the evolution of the L – T relation. We find the best fit to low-redshift ( z <0.2) cluster data, at T >1 keV, to be L ∝ T ^3.15±0.06. Our data are consistent with no evolution in the normalization of the L – T relation up to z ∼0.8. Combining our results with ASCA measurements taken from the literature, we find η =0.19±0.38 (for Ω₀=1, with 1 σ errors) where L _Bol∝(1+ z ) ^η T ^3.15, or η =0.60±0.38 for Ω₀=0.3. This lack of evolution is considered in terms of the entropy-driven evolution of clusters. Further implications for cosmological constraints are also discussed.     

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.     

The evolution of galactic discs

We use recent observations of high-redshift galaxies to study the evolution of galactic discs over the redshift range 0 <  z ≲1. The data are inconsistent with models in which discs were already assembled at z  = 1 and have evolved only in luminosity since that time. Assuming that disc properties change with redshift as powers of 1 +   z and analysing the observations assuming an Einstein–de Sitter universe, we find that for given rotation speed, disc scalelength decreases with z as ∼ (1 +  z )⁻¹, total B -band mass-to-light ratio decreases with z as ∼ (1 +  z )⁻¹, and disc luminosity (again in B ) depends only weakly on z . These scalings are consistent with current data on the evolution of disc galaxy abundance as a function of size and luminosity. Both the scalings and the abundance evolution are close to the predictions of hierarchical models for galaxy formation. If different cosmogonies are compared, the observed evolution in disc size and disc abundance favours a flat low-Ω₀ universe over an Einstein–de Sitter universe.     

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.     

The K-band Hubble diagram for the brightest cluster galaxies: a test of hierarchical galaxy formation models

We analyse the K -band Hubble diagram for a sample of brightest cluster galaxies (BCGs) in the redshift range 0< z <1. In good agreement with earlier studies, we confirm that the scatter in the absolute magnitudes of the galaxies is small (0.3 mag). The BCGs exhibit very little luminosity evolution in this redshift range: if q ₀=0.0, we detect no luminosity evolution; for q ₀=0.5, we measure a small negative evolution (i.e., BCGs were about 0.5 mag fainter at z =1 than today). If the mass in stars of these galaxies had remained constant over this period of time, substantial positive luminosity evolution would be expected: BCGs should have been brighter in the past, since their stars were younger. A likely explanation for the observed zero or negative evolution is that the stellar mass of the BCGs has been assembled over time through merging and accretion, as expected in hierarchical models of galaxy formation. The colour evolution of the BCGs is consistent with that of an old stellar population ( z _for>2) that is evolving passively. We can thus use evolutionary population synthesis models to estimate the rate of growth in stellar mass for these systems. We find that the stellar mass in a typical BCG has grown by a factor ≃2 since z ≃1 if q ₀=0.0, or by factor ≃4 if q ₀=0.5. These results are in good agreement with the predictions of semi-analytic models of galaxy formation and evolution set in the context of a hierarchical scenario for structure formation. The models predict a scatter in the luminosities of the BCGs that is somewhat larger than the observed one, but that depends on the criterion used to select the model clusters.     

The luminosity function around isolated spiral galaxies

We determine the companion galaxy luminosity function (LF) for regions around isolated spiral galaxies. If we assume that any excess in the galaxy number counts in the vicinity of a spiral galaxy is due to galaxies at the same distance, then a system LF can be determined from the variation of excess numbers with apparent magnitude. By studying the excess over many field 'centre' galaxies, a good statistical accuracy can be obtained for the companion galaxy LF. Since redshift information is not required for the faint galaxies, it is possible to sample further down the LF as compared with redshift surveys. For 23 primary galaxies of known redshift, we find a dwarf satellite Schechter LF with a characteristic magnitude M _V *( D )≃−19 and a faint-end slope α=−1.7, down to M_V =−14 ( H ₀=50 km s⁻¹ Mpc⁻¹).     

Testing deprojection algorithms on mock angular catalogues: evidence for a break in the power spectrum

We produce mock angular catalogues from simulations with different initial power spectra to test methods that recover measures of clustering in three dimensions, such as the power spectrum, variance and higher order cumulants. We find that the statistical properties derived from the angular mock catalogues are in good agreement with the intrinsic clustering in the simulations. In particular, we concentrate on the detailed predictions for the shape of the power spectrum, P ( k ). We find that there is good evidence for a break in the galaxy P ( k ) at scales in the range 0.02< k <0.06 h Mpc⁻¹, using an inversion technique applied to the angular correlation function measured from the APM Galaxy Survey. For variants on the standard cold dark matter (CDM) model, a fit at the location of the break implies Ω h =0.45±0.10, where Ω is the ratio of the total matter density to the critical density, and Hubble's constant is parametrized as H ₀=100 h km s⁻¹ Mpc⁻¹. On slightly smaller, though still quasi-linear scales, there is a feature in the APM power spectrum where the local slope changes appreciably, with the best match to CDM models obtained for Ω h ≃0.2. Hence the location and narrowness of the break in the APM power spectrum combined with the rapid change in its slope on quasi-linear scales cannot be matched by any variant of CDM, including models that have a non-zero cosmological constant or a tilt to the slope of the primordial P ( k ). These results are independent of the overall normalization of the CDM models or any simple bias that exists betwen the galaxy and mass distributions.