On estimating redshift and luminosity distributions in photometric redshift surveys

The luminosity functions of galaxies and quasars provide invaluable information about galaxy and quasar formation. Estimating the luminosity function from magnitude limited samples is relatively straightforward, provided that the distances to the objects in the sample are known accurately; techniques for doing this have been available for about thirty years. However, distances are usually known accurately for only a small subset of the sample. This is true of the objects in the Sloan Digital Sky Survey, and will be increasingly true of the next generation of deep multi-color photometric surveys. Estimating the luminosity function when distances are only known approximately (e.g., photometric redshifts are available, but spectroscopic redshifts are not) is more difficult. I describe two algorithms which can handle this complication: one is a generalization of the V_max algorithm, and the other is a maximum likelihood approach. Because these methods account for uncertainties in the distance estimate, they impact a broader range of studies. For example, they are useful for studying the abundances of galaxies which are sufficiently nearby that the contribution of peculiar velocity to the spectroscopic redshift is not negligible, so only a noisy estimate of the true distance is available. In this respect, peculiar velocities and photometric redshift errors have similar effects. The methods developed here are also useful for estimating the stellar luminosity function in samples where accurate parallax distances are not available.Comment: 9 pages, 6 figures, submitted to MNRA

Substructure in dark matter halos: Towards a model of the abundance and spatial distribution of subclumps

I develop a model for the abundance and spatial distribution of dark matter subclumps. The model shows that subclumps of massive parent halos formed at earlier times than subclumps of the same mass in lower mass parents; equivalently, halos in dense regions at a given time formed earlier than halos of the same mass in less dense regions. This may provide the basis for interpreting recent observations which indicate that the stellar populations of the most massive elliptical galaxies are also the oldest.Comment: 5 pages, 2 figures, submitted to MNRA

An excursion set model for the distribution of dark matter and dark matter haloes

A model of the gravitationally evolved dark matter distribution, in the Eulerian space, is developed. It is a simple extension of the excursion set model that is commonly used to estimate the mass function of collapsed dark matter haloes. In addition to describing the evolution of the dark matter itself, the model allows one to describe the evolution of the Eulerian space distribution of the haloes. It can also be used to describe density profiles, on scales larger than the virial radius, of these haloes, and to quantify the way in which matter flows in and out of Eulerian cells. When the initial Lagrangian space distribution is white noise Gaussian, the model suggests that the Inverse Gaussian distribution should provide a reasonably good approximation to the evolved Eulerian density field, in agreement with numerical simulations. Application of this model to clustering from more general Gaussian initial conditions is discussed at the end.Comment: 15 pages, 5 figures, submitted to MNRAS Sept. 199

The importance of stepping up in the excursion set approach

Recently, we provided a simple but accurate formula which closely approximates the first crossing distribution associated with random walks having correlated steps. The approximation is accurate for the wide range of barrier shapes of current interest and is based on the requirement that, in addition to having the right height, the walk must cross the barrier going upwards. Therefore, it only requires knowledge of the bivariate distribution of the walk height and slope, and is particularly useful for excursion set models of the massive end of the halo mass function. However, it diverges at lower masses. We show how to cure this divergence by using a formulation which requires knowledge of just one other variable. While our analysis is general, we use examples based on Gaussian initial conditions to illustrate our results. Our formulation, which is simple and fast, yields excellent agreement with the considerably more computationally expensive Monte-Carlo solution of the first crossing distribution, for a wide variety of moving barriers, even at very low masses.Comment: 10 pages, 5 figure

The environmental dependence of clustering in hierarchical models

In hierarchical models, density fluctuations on different scales are correlated. This induces correlations between dark halo masses, their formation histories, and their larger-scale environments. In turn, this produces a correlation between galaxy properties and environment. This correlation is entirely statistical in nature. We show how the observed clustering of galaxies can be used to quantify the importance of this statistical correlation relative to other physical effects which may also give rise to correlations between the properties of galaxies and their surroundings. We also develop a halo model description of this environmental dependence of clustering.Comment: 11 pages, 6 figures, MNRAS in pres

Small Scale Anisotropies of UHECRs from Super-Heavy Halo Dark Matter

The decay of very heavy metastable relics of the Early Universe can produce ultra-high energy cosmic rays (UHECRs) in the halo of our own Galaxy. In this model, no Greisen-Zatsepin-Kuzmin cutoff is expected because of the short propagation distances. We show here that, as a consequence of the hierarchical build up of the halo, this scenario predicts the existence of small scale anisotropies in the arrival directions of UHECRs, in addition to a large scale anisotropy, known from previous studies. We also suggest some other observable consequences of this scenario which will be testable with upcoming experiments, as Auger, EUSO and OWL.Comment: Contribution given at ICRC 2001 - August 7-15, 2001 - Hambur

How unusual are the Shapley Supercluster and the Sloan Great Wall?

We use extreme value statistics to assess the significance of two of the most dramatic structures in the local Universe: the Shapley supercluster and the Sloan Great Wall. If we assume that Shapley (volume ~ 1.2 x 10^5 (Mpc/h)^3) evolved from an overdense region in the initial Gaussian fluctuation field, with currently popular choices for the background cosmological model and the shape and amplitude sigma8 of the initial power spectrum, we estimate that the total mass of the system is within 20 percent of 1.8 x 10^16 Msun/h. Extreme value statistics show that the existence of this massive concentration is not unexpected if the initial fluctuation field was Gaussian, provided there are no other similar objects within a sphere of radius 200 Mpc/h centred on our Galaxy. However, a similar analysis of the Sloan Great Wall, a more distant (z ~ 0.08) and extended concentration of structures (volume ~ 7.2 x 10^5 (Mpc/h)^3) suggests that it is more unusual. We estimate its total mass to be within 20 percent of 1.2 x 10^17 Msun/h; even if it is the densest such object of its volume within z=0.2, its existence is difficult to reconcile with Gaussian initial conditions if sigma8 < 0.9. This tension can be alleviated if this structure is the densest within the Hubble volume. Finally, we show how extreme value statistics can be used to address the likelihood that an object like Shapley exists in the same volume which contains the Great Wall, finding, again, that Shapley is not particularly unusual. It is straightforward to incorporate other models of the initial fluctuation field into our formalism.Comment: 13 pages, 8 figure

Constrained realizations and minimum variance reconstruction of non-Gaussian random fields

With appropriate modifications, the Hoffman--Ribak algorithm that constructs constrained realizations of Gaussian random fields having the correct ensemble properties can also be used to construct constrained realizations of those non-Gaussian random fields that are obtained by transformations of an underlying Gaussian field. For example, constrained realizations of lognormal, generalized Rayleigh, and chi-squared fields having $n$ degrees of freedom constructed this way will have the correct ensemble properties. The lognormal field is considered in detail. For reconstructing Gaussian random fields, constrained realization techniques are similar to reconstructions obtained using minimum variance techniques. A comparison of this constrained realization approach with minimum variance, Wiener filter reconstruction techniques, in the context of lognormal random fields, is also included. The resulting prescriptions for constructing constrained realizations as well as minimum variance reconstructions of lognormal random fields are useful for reconstructing masked regions in galaxy catalogues on smaller scales than previously possible, for assessing the statistical significance of small-scale features in the microwave background radiation, and for generating certain non-Gaussian initial conditions for $N$-body simulations.Comment: 12 pages, gzipped postscript, MNRAS, in pres