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.     

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.     

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.     

The effect of formation redshifts on the cluster mass–temperature relation

I employ an ensemble of hydrodynamical simulations and the xspec mekal emission model to reproduce observable spectral and flux-weighted temperatures for 24 clusters. Each cluster is imaged at 16 points in its history, which allows the investigation of evolutionary effects on the mass–temperature relation. In the zero-redshift scaling relations, I find no evidence for a relationship between cluster temperature and formation epoch for those clusters that acquired 75 per cent of their final mass since a redshift of 0.6. This result holds for both observable and intrinsic intracluster medium temperatures, and implies that halo formation epochs are not an important variable in analysis of observable cluster temperature functions.     

The Gaussian cell two-point 'energy-like' equation: application to large-scale galaxy redshift and peculiar motion surveys

We introduce a simple linear equation relating the line-of-sight peculiar-velocity and density contrast correlation functions. The relation, which we call the Gaussian cell two-point 'energy-like' equation , is valid at the distant-observer limit and requires Gaussian smoothed fields. In the variance case, i.e. at zero lag, the equation is similar in its mathematical form to the Irvine–Layzer cosmic energy equation. β estimation with this equation from the Point Source Catalogue Redshift (PSC) survey and the SEcat catalogue of peculiar velocities is carried out, returning a value of  β= 0.44 ± 0.08  . The applicability of the method for the 6dF galaxy redshift and peculiar motions survey is demonstrated with mock data where it is shown that β could be determined with ≈10 per cent accuracy. The prospects for constraining the dark energy equation of state with this method from the kinematic and thermal Sunyaev–Zel'dovich cluster surveys are discussed. The equation is also used to construct a non-parametric mass-density power-spectrum estimator from peculiar-velocity data.     

The cluster substructure–alignment connection

Using a sample of 903 APM clusters we investigate whether their dynamical status, as evidenced by the presence of significant substructures, is related to the large-scale structure of the Universe. We find that the cluster dynamical activity is strongly correlated with the tendency of clusters to be aligned with their nearest neighbour and in general with the nearby clusters that belong to the same supercluster. Furthermore, dynamically active clusters are more clustered than the overall cluster population. These are strong indications that clusters develop in a hierarchical fashion by anisotropic merging along the large-scale filaments within which they are embedded.     

The XMM–Newton/2dF survey – VIII. The extended X-ray sources

We present a sample of eight extended X-ray sources detected in the wide-field (∼2.3 deg²), bright (2–10 ks) XMM–Newton /2dF survey, reaching a flux limit of  ∼2 × 10⁻¹⁴ erg s⁻¹ cm⁻²  . Of these, seven are identified as secure X-ray clusters in the soft 0.3–2 keV band using a standard wavelet algorithm on either the PN or the MOS images. Spectroscopic or photometric redshifts are available for five clusters, spanning a range between 0.12 and 0.68. The X-ray spectral fittings show temperatures between 1 and 4.6 keV, characteristic of poor clusters and groups of galaxies. We derive for the first time the XMM–Newton cluster number count  log  N –log  S   distribution albeit with poor statistics. Both the  log  N –log  S   and the luminosity–temperature relation are in good agreement with previous ROSAT results.     

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.     

Constraints on the clustering, biasing and redshift distribution of radio sources

We discuss how different theoretical predictions for the variance σ ² of the counts-in-cells distribution of radio sources can be matched to measurements from the FIRST survey at different flux limits. The predictions are given by the integration of models for the angular correlation function w ( θ ) for three different functional forms of the redshift distribution N ( z ), different spatial correlation functions that match the observed present-day shape and by different evolutions of the bias b ( z ) with redshift. We also consider the two cases of open and flat universes. Although the predicted w ( θ ) show substantial differences because of differences in the values of N ( z ), these differences are not significant compared to the uncertainties in the current observations. It turns out that, independent of the geometry of the universe and the flux limit, the best fit is provided by models with constant biasing at all times, although the difference between models with epoch-independent bias and models with bias that evolves linearly with redshift is not very large. All models with strong evolution of bias with epoch are ruled out, as they grossly overestimate the amplitude of the variance over the whole range of angular scales sampled by the counts-in-cells analysis. As a further step we directly calculated w _obs( θ ) at 3 mJy from the catalogue and matched it with our models for the angular correlation function, in the hypothesis that the clustering signal comes from two different populations, namely AGN-powered sources and starbursting galaxies. The results are consistent with a scenario for hierarchical clustering where the fainter starbursting galaxies trace the mass at all epochs, while the brighter AGNs are strongly biased, with b ( z ) evolving linearly with redshift, as suggested by some theories of galaxy formation and evolution.     

The apparent and intrinsic shape of the APM galaxy clusters

We estimate the distribution of intrinsic shapes of APM galaxy clusters from the distribution of their apparent shapes. We measure the projected cluster ellipticities using two alternative methods. The first method is based on moments of the discrete galaxy distribution while the second is based on moments of the smoothed galaxy distribution. We study the performance of both methods using Monte Carlo cluster simulations covering the range of APM cluster distances and including a random distribution of background galaxies. We find that the first method suffers from severe systematic biases, whereas the second is more reliable. After excluding clusters dominated by substructure and quantifying the systematic biases in our estimated shape parameters, we recover a corrected distribution of projected ellipticities. We use the non-parametric kernel method to estimate the smooth apparent ellipticity distribution, and numerically invert a set of integral equations to recover the corresponding distribution of intrinsic ellipticities under the assumption that the clusters are either oblate or prolate spheroids. The prolate spheroidal model fits the APM cluster data best.     

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.     

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.     

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.