Excursion set approach to the clustering of dark matter haloes in Lagrangian space

We present a stochastic approach to the spatial clustering of dark matter haloes in Lagrangian space. Our formalism is based on a local formulation of the 'excursion set' approach by Bond et al., which automatically accounts for the 'cloud-in-cloud' problem in the identification of bound systems. Our method allows us to calculate correlation functions of haloes in Lagrangian space using either a multidimensional Fokker–Planck equation with suitable boundary conditions, or an array of Langevin equations with spatially correlated random forces. We compare the results of our method with theoretical predictions for the halo autocorrelation function considered in the literature, and find good agreement with the results recently obtained within a treatment of halo clustering in terms of 'counting fields' by Catelan et al. Finally, the possible effect of spatial correlations on numerical simulations of halo merger trees is discussed.     

Cold dark matter models with a step-like initial power spectrum

We investigate the properties of clusters of galaxies in the ΛCDM models with a step-like initial power spectrum. We examine the mass function, the peculiar velocities and the power spectrum of clusters in models with different values of the density parameter Ω₀, the normalized Hubble constant h and the spectral parameter p that describes the shape of the initial power spectrum. The results are compared with observations. We also investigate the rms bulk velocity in the models, where the properties of clusters are consistent with the observed data. We find that the power spectrum of clusters is in good agreement with the observed power spectrum of the Abell–ACO clusters if the spectral parameter p is in the range p =0.6–0.8. The power spectrum and the rms peculiar velocity of clusters are consistent with observations only if Ω₀<0.4 . The models with Ω₀=0.3 are consistent with the observed properties of clusters if h =0.50–0.63. For h =0.65, we find that Ω₀=0.20–0.27.     

Dark matter haloes within clusters

We examine the properties of dark matter haloes within a rich galaxy cluster using a high-resolution simulation that captures the cosmological context of a cold dark matter universe. The mass and force resolution permit the resolution of 150 haloes with circular velocities larger than 80 km s⁻¹ within the cluster virial radius of 2 Mpc (with Hubble constant H ₀ = 50 km s⁻¹ Mpc⁻¹). This enables an unprecedented study of the statistical properties of a large sample of dark matter haloes evolving in a dense environment. The cumulative fraction of mass attached to these haloes varies from close to zero per cent at 200 kpc to 13 per cent at the virial radius. Even at this resolution the overmerging problem persists; haloes that pass within 100–200 kpc of the cluster centre are tidally disrupted. Additional substructure is lost at earlier epochs within the massive progenitor haloes. The median ratio of apocentric to pericentric radii is 6:1, so that the orbital distribution is close to isotropic, circular orbits are rare and radial orbits are common. The orbits of haloes are unbiased with respect to both position within the cluster and the orbits of the smooth dark matter background, and no velocity bias is detected. The tidal radii of surviving haloes are generally well-fitted using the simple analytic prediction applied to their orbital pericentres. Haloes within clusters have higher concentrations than those in the field. Within the cluster, halo density profiles can be modified by tidal forces and individual encounters with other haloes that cause significant mass loss —'galaxy harassment'. Mergers between haloes do not occur inside the cluster virial radius.     

The bias field of dark matter haloes

Accepted 1998 January 26. Received 1998 January 26; in original form 1997 August 13This paper presents a stochastic approach to the clustering evolution of dark matter haloes in the Universe. Haloes, identified by a Press–Schechter-type algorithm in Lagrangian space, are described in terms of 'counting fields', acting as non-linear operators on the underlying Gaussian density fluctuations. By ensemble-averaging these counting fields, the standard Press–Schechter mass function as well as analytic expressions for the halo correlation function and corresponding bias factors of linear theory are obtained, extending the recent results by Mo & White. The non-linear evolution of our halo population is then followed by solving the continuity equation, under the sole hypothesis that haloes move by the action of gravity. This leads to an exact and general formula for the bias field of dark matter haloes, defined as the local ratio between their number density contrast and the mass density fluctuation. Besides being a function of position and 'observation' redshift, this random field depends upon the mass and formation epoch of the objects and is both non-linear and non-local. The latter features are expected to leave a detectable imprint on the spatial clustering of galaxies, as described, for instance, by statistics like the bispectrum and the skewness. Our algorithm may have several interesting applications, among which is the possibility of generating mock halo catalogues from low-resolution N -body simulations.     

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.     

Universal profile of dark matter haloes and the spherical infall model

I propose a modification of the spherical infall model for the evolution of density fluctuations with initially Gaussian probability distribution and scale-free power spectra in the Einsteinde Sitter universe as developed by Hoffman & Shaham. I introduce a generalized form of the initial density distribution around an overdense region and cut it off at half the interpeak separation, accounting in this way for the presence of the neighbouring fluctuations. Contrary to the original predictions of Hoffman & Shaham, the resulting density profiles within virial radii no longer have a power-law shape, but their steepness increases with distance. The profiles of haloes of galactic mass are well fitted by the universal profile formula of changing slope obtained as a result of N -body simulations by Navarro, Frenk & White. The trend of steeper profiles for smaller masses and higher spectral indices is also reproduced. The agreement between the model and simulations is better for smaller masses and lower spectral indices, which suggests that galaxies form mainly by accretion, while formation of clusters involves merging.