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

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

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

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

Mass growth and density profiles of dark matter halos in hierarchical clustering

We develop a model for the growth of dark matter halos and use it to study their evolved density profiles. In this model, halos are spherical and form by quiescent accretion of matter in clumps, called satellites. The halo mass as a function of redshift is given by the mass of the most massive progenitor, and is determined from Monte-Carlo realizations of the merger-history tree. Inside the halo, satellites move under the action of the gravitational force of the halo and a dynamical friction drag force. The associated equation of motion is solved numerically. The energy lost to dynamical friction is transferred to the halo in the form of kinetic energy. As they sink into the halo, satellites continually lose matter as a result of tidal stripping. The stripped matter moves inside the halo free of dynamical friction. The evolved density profiles are steeper than those obtained by assuming that, once they have been accreted onto the parent halo, satellites remain at a fixed distance from the halo center. We find that the final density profile depends mainly on the rate of infall of matter onto the halo. This, in turn, depends on the initial fluctuation field as well as on cosmology. For mass scales where the effective spectral index of the initial density field is less than -1, the model predicts a profile which can only approximately be matched by the one parameter family of curves suggested by Navarro, Frenk and White (1997). For scale-free power-spectra with initial slope $n$, the density profile within about 1% of the virial radius is $\rho\propto r^{-\beta}$, with $3(3+n)/(5+n)\le\beta\le 3(3+n)/(4+n)$.Comment: submitted to MNRAS, 19pages, 8 figure

Stochasticity in halo formation and the excursion set approach

The simplest stochastic halo formation models assume that the traceless part of the shear field acts to increase the initial overdensity (or decrease the underdensity) that a protohalo (or protovoid) must have if it is to form by the present time. Equivalently, it is the difference between the overdensity and (the square root of the) shear that must be larger than a threshold value. To estimate the effect this has on halo abundances using the excursion set approach, we must solve for the first crossing distribution of a barrier of constant height by the random walks associated with the difference, which is now (even for Gaussian initial conditions) a non-Gaussian variate. The correlation properties of such non-Gaussian walks are inherited from those of the density and the shear, and, since they are independent processes, the solution is in fact remarkably simple. We show that this provides an easy way to understand why earlier heuristic arguments about the nature of the solution worked so well. In addition to modelling halos and voids, this potentially simplifies models of the abundance and spatial distribution of filaments and sheets in the cosmic web.Comment: 5 pages, 1 figure. Matches published versio

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