Cosmic statistics through weak lenses

Many recent studies have demonstrated that scaling arguments, such as the so-called hierarchical ansatz, are extremely useful in understanding the statistical properties of weak gravitational lensing. This is especially true on small angular scales (i.e. at high resolution), where the usual perturbative calculations of matter clustering no longer apply. We build on these studies in order to develop a complete picture of weak lensing at small smoothing angles. In particular, we study the full probability distribution function, bias and other multipoint statistics for the 'hot spots' of the convergence field induced by weak lensing, and relate these to the statistics of overdense regions in the underlying mass distribution. It is already known that weak lensing can constrain the background geometry of the Universe, but we further show that it can also provide valuable information about the statistics of collapsed objects and the physics of collisionless clustering. Our results are particularly important for future observations which will, at least initially, focus on small smoothing angles.     

Tracing the nature of dark energy with galaxy distribution

Dynamical dark energy (DE) is a viable alternative to the cosmological constant. Constructing tests to discriminate between Λ and dynamical DE models is difficult, however, because the differences are not large. In this paper we explore tests based on the galaxy mass function, the void probability function (VPF), and the number of galaxy clusters. At high z , the number density of clusters shows large differences between DE models, but geometrical factors reduce the differences substantially. We find that detecting a model dependence in the cluster redshift distribution is a significant challenge. We show that the galaxy redshift distribution is potentially a more sensitive characteristic. We do this by populating dark matter haloes in N -body simulations with galaxies using well-tested halo occupation distributions. We also estimate the VPF and find that samples with the same angular surface density of galaxies, in different models, exhibition almost model-independent VPF which therefore cannot be used as a test for DE. Once again, geometry and cosmic evolution compensate each other. By comparing VPFs for samples with fixed galaxy mass limits, we find measurable differences.     

On an analytical framework for voids: their abundances, density profiles and local mass functions

We present a general analytical procedure for computing the number density of voids with radius above a given value within the context of gravitational formation of the large-scale structure of the Universe out of Gaussian initial conditions. To this end, we develop an accurate (under generally satisfied conditions) extension of the unconditional mass function to constrained environments, which allows us both to obtain the number density of collapsed objects of certain mass at any distance from the centre of the void, and to derive the number density of voids defined by collapsed objects. We have made detailed calculations for the spherically averaged mass density and halo number density profiles for particular voids. We also present a formal expression for the number density of voids defined by galaxies of a given type and luminosity. This expression contains the probability for a collapsed object of certain mass to host a galaxy of that type and luminosity (i.e. the conditional luminosity function) as a function of the environmental density. We propose a procedure to infer this function, which may provide useful clues as to the galaxy formation process, from the observed void densities.     

Non-local scaling in two-dimensional gravitational clustering

Using an ensemble of high-resolution 2D numerical simulations, we explore the scaling properties of cosmological density fluctuations in the non-linear regime. We study the scaling behaviour of the usual N -point volume-averaged correlations, but also examine the scaling of the entire probability density function (PDF) of the fluctuations. We focus on two important issues: (i) whether the scaling behaviour of 2D clustering is consistent with what one infers from radial collapse arguments; and (ii) whether there is any evidence from these high-resolution simulations that a regime of stable clustering is ever entered. We find that the answers are (i) yes and (ii) no. We further find that the behaviour of the highly non-linear regime in these simulations suggests the existence of a regime in which the correlation function is independent of the initial power spectrum.     

Probing the gravity-induced bias with weak lensing: test of analytical results against simulations

Future weak lensing surveys will directly probe the density fluctuation in the Universe. Recent studies have shown how the statistics of the weak lensing convergence field is related to the statistics of collapsed objects. Extending earlier analytical results on the probability distribution function of the convergence field, we show that the bias associated with the convergence field can be directly related to the bias associated with the statistics of underlying overdense objects. This will provide us with a direct method to study the gravity-induced bias in galaxy clustering. Based on our analytical results, which use the hierarchical Ansatz for non-linear clustering, we study how such a bias depends on the smoothing angle and the source redshift. We compare our analytical results with ray-tracing experiments through N -body simulations of four different realistic cosmological scenarios, and find a very good match. Our study shows that the bias in the convergence map strongly depends on the background geometry and hence can help us in distinguishing different cosmological models in addition to improving our understanding of the gravity-induced bias in galaxy clustering.     

The statistics of weak lensing at small angular scales: probability distribution function

Weak gravitational lensing surveys have the potential to probe mass density fluctuation in the Universe directly. Recent studies have shown that it is possible to model the statistics of the convergence field at small angular scales by modelling the statistics of the underlying density field in the highly non-linear regime. We propose a new method to model the complete probability distribution function of the convergence field as a function of smoothing angle and source redshift. The model relies on a hierarchical ansatz for the behaviour of higher order correlations of the density field. We compare our results with ray-tracing simulations and find very good agreement over a range of smoothing angles. Whereas the density probability distribution function is not sensitive to the cosmological model, the probability distribution function for the convergence can be used to constrain both the power spectrum and cosmological parameters.     

Generalized cumulant correlators and hierarchical clustering

The cumulant correlators, C _pq , are statistical quantities that generalize the better-known S _p parameters; the former are obtained from the two-point probability distribution function of the density fluctuations while the latter describe only the one-point distribution. If galaxy clustering develops from Gaussian initial fluctuations and a small-angle approximation is adopted, standard perturbative methods suggest a particular hierarchical relationship of the C _pq for projected clustering data, such as that obtained from the Automatic Plate Measuring (APM) survey. We establish the usefulness of the two-point cumulants for describing hierarchical clustering by comparing such calculations against available measurements from projected catalogues, finding very good agreement. We extend the idea of cumulant correlators to multipoint generalized cumulant correlators (related to the higher-order correlation functions). We extend previous studies in the highly non-linear regime to express the generalized cumulant correlators in terms of the underlying 'tree amplitudes' of hierarchical scaling models. Such considerations lead to a technique for determining these hierarchical amplitudes, to arbitrary order, from galaxy catalogues and numerical simulations. Knowledge of these amplitudes yields important clues about the phenomenology of gravitational clustering. For instance, we show that a three-point cumulant correlator can be used to separate the tree amplitudes up to sixth order. We also combine the particular hierarchical Ansatz of Bernardeau & Schaeffer with extended and hyper-extended perturbation theory to infer values of the tree amplitudes in the highly non-linear regime.     

Statistics of weak lensing at small angular scales: analytical predictions for lower order moments

Weak lensing surveys are expected to provide direct measurements of the statistics of the projected dark matter distribution. Most analytical studies of weak lensing statistics have been limited to quasi-linear scales as they relied on perturbative calculations. On the other hand, observational surveys are likely to probe angular scales less than 10 arcmin, for which the relevant physical length-scales are in the non-linear regime of gravitational clustering. We use the hierarchical ansatz to compute the multipoint statistics of the weak lensing convergence for these small smoothing angles. We predict the multipoint cumulants and cumulant correlators up to fourth order and compare our results with high-resolution ray-tracing simulations. Averaging over a large number of simulation realizations for four different cosmological models, we find close agreement with the analytical calculations. In combination with our work on the probability distribution function, these results provide accurate analytical models for the full range of weak lensing statistics. The models allow for a detailed exploration of cosmological parameter space and of the dependence on angular scale and the redshift distribution of source galaxies. We compute the dependence of the higher moments of the convergence on the parameters Ω and Λ.     

The probability distribution of cluster formation times and implied Einstein radii

We provide a quantitative assessment of the probability distribution function of the concentration parameter of galaxy clusters. We do so by using the probability distribution function of halo formation times, calculated by means of the excursion set formalism, and a formation redshift-concentration scaling derived from results of N -body simulations. Our results suggest that the observed high concentrations of several clusters are quite unlikely in the standard Λ cold dark matter (ΛCDM) cosmological model, but that due to various inherent uncertainties, the statistical range of the predicted distribution may be significantly wider than commonly acknowledged. In addition, the probability distribution function of the Einstein radius of A1689 is evaluated, confirming that the observed value of  ∼45 ± 5 arcsec  is very improbable in the currently favoured cosmological model. If, however, a variance of ∼20 per cent in the theoretically predicted value of the virial radius is assumed, then the discrepancy is much weaker. The measurement of similarly large Einstein radii in several other clusters would pose a difficulty to the standard model. If so, earlier formation of the large-scale structure would be required, in accord with predictions of some quintessence models. We have indeed verified that in a viable early dark energy model large Einstein radii are predicted in as many as a few tens of high-mass clusters.     

On the importance of high-redshift intergalactic voids

We investigate the properties of 1D flux 'voids' (connected regions in the flux distribution above the mean-flux level) by comparing hydrodynamical simulations of large cosmological volumes with a set of observed high-resolution spectra at z ∼ 2. After addressing the effects of box size and resolution, we study how the void distribution changes when the most significant cosmological and astrophysical parameters are varied. We find that the void distribution in the flux is in excellent agreement with predictions of the standard Λcold dark matter (ΛCDM) cosmology, which also fits other flux statistics remarkably well. We then model the relation between flux voids and the corresponding 1D gas-density field along the line of sight and make a preliminary attempt to connect the 1D properties of the gas-density field to the 3D dark matter distribution at the same redshift. This provides a framework that allows statistical interpretations of the void population at high redshift using observed quasar spectra, and eventually it will enable linking the void properties of the high-redshift universe with those at lower redshifts, which are better known.     

Statistical algorithms for identification of astronomical X‐ray sources

Observations of present and future X‐ray telescopes include a large number of ipitous sources of unknown types. They are a rich source of knowledge about X‐ray dominated astronomical objects, their distribution, and their evolution. The large number of these sources does not permit their individual spectroscopical follow‐up and classification. Here we use Chandra Multi‐Wavelength public data to investigate a number of statistical algorithms for classification of X‐ray sources with optical imaging follow‐up. We show that up to statistical uncertainties, each class of X‐ray sources has specific photometric characteristics that can be used for its classification. We assess the relative and absolute performance of classification methods and measured features by comparing the behaviour of physical quantities for statistically classified objects with what is obtained from spectroscopy. We find that among methods we have studied, multi‐dimensional probability distribution is the best for both classifying source type and redshift, but it needs a sufficiently large input (learning) data set. In absence of such data, a mixture of various methods can give a better final result.We discuss some of potential applications of the statistical classification and the enhancement of information obtained in this way. We also assess the effect of classification methods and input data set on the astronomical conclusions such as distribution and properties of X‐ray selected sources. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Scale-dependent bias induced by local non-Gaussianity: a comparison to N-body simulations

We investigate the effect of primordial non-Gaussianity of the local f _NL type on the auto- and cross-power spectra of dark matter haloes using simulations of the Λ cold dark matter cosmology. We perform a series of large N -body simulations of both positive and negative f _NL, spanning the range between 10 and 100. Theoretical models predict a scale-dependent bias correction  Δ b ( k , f _NL)  that depends on the linear halo bias   b ( M )  . We measure the power spectra for a range of halo mass and redshifts covering the relevant range of existing galaxy and quasar populations. We show that auto- and cross-correlation analyses of bias are consistent with each other. We find that for low wavenumbers with   k < 0.03  h  Mpc⁻¹  the theory and the simulations agree well with each other for biased haloes with   b ( M ) > 1.5  . We show that a scale-independent bias correction improves the comparison between theory and simulations on smaller scales, where the scale-dependent effect rapidly becomes negligible. The current limits on f _NL from Slosar et al. come mostly from very large scales   k < 0.01  h  Mpc⁻¹  and, therefore, remain valid. For the halo samples with   b ( M ) < 1.5 − 2  , we find that the scale-dependent bias from non-Gaussianity actually exceeds the theoretical predictions. Our results are consistent with the bias correction scaling linearly with f _NL.     

Scalar statistics on the sphere: application to the cosmic microwave background

A method to compute several scalar quantities of cosmic microwave background (CMB) maps on the sphere is presented. We consider here four type of scalars: the Hessian matrix scalars, the distortion scalars, the gradient-related scalars and the curvature scalars. Such quantities are obtained directly from the spherical harmonic coefficients   a _{ℓ m}   of the map. We also study the probability density function of these quantities for the case of a homogeneous and isotropic Gaussian field, which are functions of the power spectrum of the initial field. From these scalars it is possible to construct a new set of scalars which are independent of the power spectrum of the field. We test our results using simulations and find good agreement between the theoretical probability density functions and those obtained from simulations. Therefore, these quantities are proposed to investigate the presence of non-Gaussian features in CMB maps. Finally, we show how to compute the scalars in the presence of anisotropic noise and realistic masks.     

Chemical evolution of galaxies – I. A composition-dependent SPH model for chemical evolution and cooling

We describe an smooth particle hydrodynamics (SPH) model for chemical enrichment and radiative cooling in cosmological simulations of structure formation. This model includes: (i) the delayed gas restitution from stars by means of a probabilistic approach designed to reduce the statistical noise and, hence, to allow for the study of the inner chemical structure of objects with moderately high numbers of particles; (ii) the full dependence of metal production on the detailed chemical composition of stellar particles by using, for the first time in SPH codes, the   Q _ij   matrix formalism that relates each nucleosynthetic product to its sources and (iii) the full dependence of radiative cooling on the detailed chemical composition of gas particles, achieved through a fast algorithm using a new metallicity parameter ζ( T ) that gives the weight of each element on the total cooling function. The resolution effects and the results obtained from this SPH chemical model have been tested by comparing its predictions in different problems with known theoretical solutions. We also present some preliminary results on the chemical properties of elliptical galaxies found in self-consistent cosmological simulations. Such simulations show that the above ζ-cooling method is important to prevent an overestimation of the metallicity-dependent cooling rate, whereas the   Q _ij   formalism is important to prevent a significant underestimation of the [α/Fe] ratio in simulated galaxy-like objects.     

On the distribution of haloes, galaxies and mass

The stochasticity in the distribution of dark haloes in the cosmic density field is reflected in the distribution function   P _V ( N _h| δ _m)  , which gives the probability of finding N _h haloes in a volume V with mass density contrast δ _m. We study the properties of this function using high-resolution N -body simulations, and find that   P _V ( N _h| δ _m)  is significantly non-Poisson. The ratio between the variance and the mean goes from ∼1 (Poisson) at  1+ δ _m≪1  to <1 (sub-Poisson) at  1+ δ _m∼1  to >1 (super-Poisson) at  1+ δ _m≫1  . The mean bias relation is found to be well described by halo bias models based on the Press–Schechter formalism. The sub-Poisson variance can be explained as a result of halo exclusion, while the super-Poisson variance at high δ _m may be explained as a result of halo clustering. A simple phenomenological model is proposed to describe the behaviour of the variance as a function of δ _m. Galaxy distribution in the cosmic density field predicted by semi-analytic models of galaxy formation shows similar stochastic behaviour. We discuss the implications of the stochasticity in halo bias to the modelling of higher order moments of dark haloes and of galaxies.     

The universally growing mode in the solar atmosphere: coronal heating by drift waves

N -body simulations are an important tool in the study of formation of large-scale structures. Much of the progress in understanding the physics of galaxy clustering and comparison with observations would not have been possible without N -body simulations. Given the importance of this tool, it is essential to understand its limitations as ignoring these can easily lead to interesting but unreliable results. In this paper, we study the limitations due to the finite size of the simulation volume. In an earlier work, we proposed a formalism for estimating the effects of a finite box size on physical quantities and applied it to estimate the effect on the amplitude of clustering, mass function. Here, we extend the same analysis and estimate the effect on skewness and kurtosis in the perturbative regime. We also test the analytical predictions from the earlier work as well as those presented in this paper. We find good agreement between the analytical models and simulations for the two-point correlation function and skewness. We also discuss the effect of a finite box size on relative velocity statistics and find the effects for these quantities scale in a manner that retains the dependence on the averaged correlation function     .     

Power spectrum for the small-scale Universe

The first objects to arise in a cold dark matter (CDM) universe present a daunting challenge for models of structure formation. In the ultra small-scale limit, CDM structures form nearly simultaneously across a wide range of scales. Hierarchical clustering no longer provides a guiding principle for theoretical analyses and the computation time required to carry out credible simulations becomes prohibitively high. To gain insight into this problem, we perform high-resolution  ( N = 720³–1584³)  simulations of an Einstein–de Sitter cosmology where the initial power spectrum is   P ( k ) ∝ k ⁿ ,  with  −2.5 ≤ n ≤− 1  . Self-similar scaling is established for   n =−1  and −2 more convincingly than in previous, lower resolution simulations and for the first time, self-similar scaling is established for an   n =−2.25  simulation. However, finite box-size effects induce departures from self-similar scaling in our   n =−2.5  simulation. We compare our results with the predictions for the power spectrum from (one-loop) perturbation theory and demonstrate that the renormalization group approach suggested by McDonald improves perturbation theory's ability to predict the power spectrum in the quasi-linear regime. In the non-linear regime, our power spectra differ significantly from the widely used fitting formulae of Peacock & Dodds and Smith et al. and a new fitting formula is presented. Implications of our results for the stable clustering hypothesis versus halo model debate are discussed. Our power spectra are inconsistent with predictions of the stable clustering hypothesis in the high- k limit and lend credence to the halo model. Nevertheless, the fitting formula advocated in this paper is purely empirical and not derived from a specific formulation of the halo model.