Maximum Likelihood Random Galaxy Catalogues and Luminosity Function Estimation

We present a new algorithm to generate a random (unclustered) version of an magnitude limited observational galaxy redshift catalogue. It takes into account both galaxy evolution and the perturbing effects of large scale structure. The key to the algorithm is a maximum likelihood (ML) method for jointly estimating both the luminosity function (LF) and the overdensity as a function of redshift. The random catalogue algorithm then works by cloning each galaxy in the original catalogue, with the number of clones determined by the ML solution. Each of these cloned galaxies is then assigned a random redshift uniformly distributed over the accessible survey volume, taking account of the survey magnitude limit(s) and, optionally, both luminosity and number density evolution. The resulting random catalogues, which can be employed in traditional estimates of galaxy clustering, make fuller use of the information available in the original catalogue and hence are superior to simply fitting a functional form to the observed redshift distribution. They are particularly well suited to studies of the dependence of galaxy clustering on galaxy properties as each galaxy in the random catalogue has the same list of attributes as measured for the galaxies in the genuine catalogue. The derivation of the joint overdensity and LF estimator reveals the limit in which the ML estimate reduces to the standard 1/Vmax LF estimate, namely when one makes the prior assumption that the are no fluctuations in the radial overdensity. The new ML estimator can be viewed as a generalization of the 1/Vmax estimate in which Vmax is replaced by a density corrected Vdc,max.

Generating Dark Matter Halo Merger Trees

We present a new Monte-Carlo algorithm to generate merger trees describing the formation history of dark matter halos. The algorithm is a modification of the algorithm of Cole et al (2000) used in the GALFORM semi-analytic galaxy formation model. As such, it is based on the Extended Press-Schechter theory and so should be applicable to hierarchical models with a wide range of power spectra and cosmological models. It is tuned to be in accurate agreement with the conditional mass functions found in the analysis of merger trees extracted from the LCDM Millennium N-body simulation. We present a comparison of its predictions not only with these conditional mass functions, but also with additional statistics of the Millennium Simulation halo merger histories. In all cases we find it to be in good agreement with the Millennium Simulation and thus it should prove to be a very useful tool for semi-analytic models of galaxy formation and for modelling hierarchical structure formation in general. We have made our merger tree generation code and code to navigate the trees available at http://star-www.dur.ac.uk/~cole/merger_trees .

The statistical properties of LCDM halo formation

We present a comparison of the statistical properties of dark matter halo merger trees extracted from the Millennium Simulation with Extended Press-Schechter (EPS) formalism and the related GALFORM Monte-Carlo method for generating ensembles of merger trees. The volume, mass resolution and output frequency make the Millennium Simulation a unique resource for the study of the hierarchical growth of structure. We construct the merger trees of present day friends-of-friends groups and calculate a variety of statistics that quantify the masses of their progenitors as a function of redshift; accretion rates; and the redshift distribution of their most recent major merger. We also look in the forward direction and quantify the present day mass distribution of halos into which high redshift progenitors of a specific mass become incorporated. We find that EPS formalism and its Monte-Carlo extension capture the qualitative behaviour of all these statistics but, as redshift increases they systematically underestimate the masses of the most massive progenitors. This shortcoming is worst for the Monte-Carlo algorithm. We present a fitting function to a scaled version of the progenitor mass distribution and show how it can be used to make more accurate predictions of both progenitor and final halo mass distributions.