Excursion Set Research Articles

Cosmology with galaxy clusters: Systematic effects in the halo mass function

We investigate potential systematic effects in constraining the amplitude of primordial fluctuations \sigma_8 arising from the choice of halo mass function in the likelihood analysis of current and upcoming galaxy cluster surveys. We study the widely used N-body simulation fit of Tinker et al. (T08) and, as an alternative, the recently proposed analytical model of Excursion Set Peaks (ESP). We first assess the relative bias between these prescriptions when constraining \sigma_8 by sampling the ESP mass function to generate mock catalogs and using the T08 fit to analyse them, for various choices of survey selection threshold, mass definition and statistical priors. To assess the level of absolute bias in each prescription, we then repeat the analysis on dark matter halo catalogs in N-body simulations designed to mimic the mass distribution in the current data release of Planck SZ clusters. This N-body analysis shows that using the T08 fit without accounting for the scatter introduced when converting between mass definitions (alternatively, the scatter induced by errors on the parameters of the fit) can systematically over-estimate the value of \sigma_8 by as much as 2\sigma\ for current data, while analyses that account for this scatter should be close to unbiased in \sigma_8. With an increased number of objects as expected in upcoming data releases, regardless of accounting for scatter, the T08 fit could over-estimate the value of \sigma_8 by ~1.5\sigma. The ESP mass function leads to systematically more biased but comparable results. A strength of the ESP model is its natural prediction of a weak non-universality in the mass function which closely tracks the one measured in simulations and described by the T08 fit. We suggest that it might now be prudent to build new unbiased ESP-based fitting functions for use with the larger datasets of the near future.

Direct Minkowski Functional analysis of large redshift surveys: a new high-speed code tested on the luminous red galaxy Sloan Digital Sky Survey-DR7 catalogue

As deeper galaxy catalogues are soon to come, it becomes even more importantto measure large-scale fluctuations in the catalogues with robust statistics that cover all moments of the galaxy distribution.In this paper we reinforce a direct analysis of galaxy data by employing the Germ-Grain method to calculate thefamily of Minkowski Functionals. We introduce a new code, suitable for the analysis of large data sets without smoothingand without the construction of excursion sets. We provide new tools to measure correlation properties, putting emphasis onexplicitly isolating non-Gaussian correlations with the help of integral-geometric relations. As a first application we present the analysis of large-scale fluctuations in the luminous red galaxy sample of Sloan Digital Sky Survey data release 7 data. We findsignificant deviations from the $\Lambda$ cold dark matter mock catalogues on samples as large as $500h^{-1}$Mpc (more than $3\sigma$)and slight deviations of around $2\sigma$ on $700h^{-1}$Mpc, and we investigate possible sources of these deviations.

Statistics of dark matter halos in the excursion set peak framework

We derive approximated, yet very accurate analytical expressions for the abundance and clustering properties of dark matter halos in the excursion set peak framework; the latter relies on the standard excursion set approach, but also includes the effects of a realistic filtering of the density field, a mass-dependent threshold for collapse, and the prescription from peak theory that halos tend to form around density maxima. We find that our approximations work excellently for diverse power spectra, collapse thresholds and density filters. Moreover, when adopting a cold dark matter power spectra, a tophat filtering and a mass-dependent collapse threshold (supplemented with conceivable scatter), our approximated halo mass function and halo bias represent very well the outcomes of cosmological N-body simulations.

An improved model of H ii bubbles during the epoch of reionization

The size distribution of ionized regions during the epoch of reionization -- a key ingredient in understanding the HI power spectrum observable by 21cm experiments -- can be modelled analytically using the excursion set formalism of random walks in the smoothed initial density field. To date, such calculations have been based on simplifying assumptions carried forward from the earliest excursion set models of two decades ago. In particular, these models assume that the random walks have uncorrelated steps and that haloes can form at arbitrary locations in the initial density field. We extend these calculations by incorporating recent technical developments that allow us to (a) include the effect of correlations in the steps of the walks induced by a realistic smoothing filter and (b) more importantly, account for the fact that dark matter haloes preferentially form near peaks in the initial density. A comparison with previous calculations shows that including these features, particularly the peaks constraint on halo locations, has large effects on the size distribution of the HII bubbles surrounding these haloes. For example, when comparing models at the same value of the globally averaged ionized volume fraction, the typical bubble sizes predicted by our model are more than a factor 2 larger than earlier calculations. Our results can potentially have a significant impact on estimates of the observable HI power spectrum.

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.

Early Pliensbachian (Early Jurassic) C-isotope perturbation and the diffusion of the Lithiotis Fauna: Insights from the western Tethys

High-resolution carbonate carbon-isotope stratigraphy of Lower Jurassic marine shallow-water limestones are compared with the distribution of the Lithiotis Fauna on the Trento Platform (Southern Alps, Italy). The Lithiotis Fauna is the first example of globally distributed mound-building bivalves in the geological record and experienced global diffusion in the Early Jurassic. A set of carbonate carbon-isotope excursions of 2–3‰, illustrating three distinct negative shifts followed by positive rebounds, are recorded in the isotope stratigraphy and can be correlated with the global negative δ13C shift of the Sinemurian–Pliensbachian boundary Event (S–P Event) and to the subsequent phase of C-isotope perturbation that characterized the lower Pliensbachian. In the studied stratigraphic sections, the S–P Event likely triggered eutrophic conditions illustrated by the presence of organic-rich facies and by fossil associations characteristic of poorly oxygenated waters. After the eutrophic phase, the amelioration of environmental conditions was marked by a positive ~3‰ rebound of the δ13Ccarb values, and by the occurrence of marine stenotypic faunas. On the Trento Platform, the stabilization of the δ13Ccarb values coincided with the appearance of the Lithiotis Fauna that subsequently became widely distributed in the entire range of platform environments and thrived in the late Pliensbachian when metric-scale bivalve mounds were generated. During the same time, the maximum proliferation of the Lithiotis Fauna is recorded both in the Tethyan and Panthalassa regions. Hence, the reported relationships between the δ13Ccarb data and the distribution and ecological characteristics of the genera contained in the Lithiotis Fauna suggest that the S–P Event and its aftermath, possibly coupled with the undergoing coeval continent reorganization that led to the opening of the Hispanic Corridor, could have set the stage for the rapid diffusion of these unusual bivalves across many parts of the globe.

The Peak/Dip Picture of the Cosmic Web

AbstractThe initial shear field plays a central role in the formation of large-scale structures, and in shaping the geometry, morphology, and topology of the cosmic web. We discuss a recent theoretical framework for the shear tensor, termed the ‘peak/dip picture’, which accounts for the fact that halos/voids may form from local extrema of the density field – rather than from random spatial positions; the standard Doroshkevich's formalism is generalized, to include correlations between the density Hessian and shear field at special points in space around which halos/voids may form. We then present the ‘peak/dip excursion-set-based’ algorithm, along with its most recent applications – merging peaks theory with the standard excursion set approach.

Multiobjective optimization using Gaussian process emulators via stepwise uncertainty reduction

Optimization of expensive computer models with the help of Gaussian process emulators is now commonplace. However, when several (competing) objectives are considered, choosing an appropriate sampling strategy remains an open question. We present here a new algorithm based on stepwise uncertainty reduction principles. Optimization is seen as a sequential reduction of the volume of the excursion sets below the current best solutions (Pareto set), and our sampling strategy chooses the points that give the highest expected reduction. The method is tested on several numerical examples and on an agronomy problem, showing that it provides an efficient trade-off between exploration and intensification.

Double extreme on joint sets for Gaussian random fields

For a centered Gaussian random field X={X(t),t∈RN}, let T1 and T2 be two compact sets in RN such that I=T1∩T2≠0̸ and denote by χ(Au(I)) the Euler characteristic of the excursion set Au(I)={t∈I:X(t)≥u}. We show that under certain smoothness and regularity conditions, as u→∞, the joint excursion probability P{supt∈T1X(t)≥u,sups∈T2X(s)≥u} can be approximated by the expected Euler characteristic E{χ(Au(I))} such that the error is super-exponentially small.

Measuring non-local Lagrangian peak bias

We investigate nonlocal Lagrangian bias contributions involving gradients of the linear density field, for which we have predictions from the excursion set peak formalism. We begin by writing down a bias expansion which includes all the bias terms, including the nonlocal ones. Having checked that the model furnishes a reasonable fit to the halo mass function, we develop a 1-point cross-correlation technique to measure bias factors associated with 2-distributed quantities. We validate the method with numerical realizations of peaks of Gaussian random fields before we apply it to N-body simulations. We focus on the lowest (quadratic) order nonlocal contributions. We can reproduce our measurement of \chi_{10} if we allow for an offset between the Lagrangian halo center-of-mass and the peak position. The sign and magnitude of \chi_{10} is consistent with Lagrangian haloes sitting near linear density maxima. The resulting contribution to the halo bias can safely be ignored for M = 10^13 Msun/h, but could become relevant at larger halo masses. For the second nonlocal bias \chi_{01} however, we measure a much larger magnitude than predicted by our model. We speculate that some of this discrepancy might originate from nonlocal Lagrangian contributions induced by nonspherical collapse.

On the existence of paths between points in high level excursion sets of Gaussian random fields

The structure of Gaussian random fields over high levels is a well researched and well understood area, particularly if the field is smooth. However, the question as to whether or not two or more points which lie in an excursion set belong to the same connected component has constantly eluded analysis. We study this problem from the point of view of large deviations, finding the asymptotic probabilities that two such points are connected by a path laying within the excursion set, and so belong to the same component. In addition, we obtain a characterization and descriptions of the most likely paths, given that one exists.

Excursion and Contour Uncertainty Regions for Latent Gaussian Models

SummaryIn several areas of application ranging from brain imaging to astrophysics and geostatistics, an important statistical problem is to find regions where the process studied exceeds a certain level. Estimating such regions so that the probability for exceeding the level in the entire set is equal to some predefined value is a difficult problem connected to the problem of multiple significance testing. In this work, a method for solving this problem, as well as the related problem of finding credible regions for contour curves, for latent Gaussian models is proposed. The method is based on using a parametric family for the excursion sets in combination with a sequential importance sampling method for estimating joint probabilities. The accuracy of the method is investigated by using simulated data and an environmental application is presented.

Halo shapes, initial shear field, and cosmic web

The ellipsoidal collapse model, combined with the excursion set theory, allows one to estimate the shapes of dark matter halos as seen in high-resolution numerical simulations. The same theoretical framework predicts a quasi-universal behaviour for the conditional axis ratio distributions at later times, set by initial conditions and unaltered by non-linear evolution. The formalism for halo shapes is also useful in making the connection with the initial shear field of the cosmic web, which plays a crucial role in the formation of large-scale structures. The author has briefly discussed the basic aspects of the modelling, as well as the implications of a new formula for the constrained eigenvalues of the initial shear field, given the fact that positions are peaks or dips in the corresponding density field – and not random locations. This formula leads to a new generalized excursion set algorithm for peaks in Gaussian random fields. The results highlighted, here, are relevant for a number of applications, especially for weak lensing studies and for devising algorithms to find and classify structures in the cosmic web.

The excursion set approach in non-Gaussian random fields

Insight into a number of interesting questions in cosmology can be obtained from the first crossing distributions of physically motivated barriers by random walks with correlated steps. We write the first crossing distribution as a formal series, ordered by the number of times a walk upcrosses the barrier. Since the fraction of walks with many upcrossings is negligible if the walk has not taken many steps, the leading order term in this series is the most relevant for understanding the massive objects of most interest in cosmology. This first term only requires knowledge of the bivariate distribution of the walk height and slope, and provides an excellent approximation to the first crossing distribution for all barriers and smoothing filters of current interest. We show that this simplicity survives when extending the approach to the case of non-Gaussian random fields. For non-Gaussian fields which are obtained by deterministic transformations of a Gaussian, the first crossing distribution is simply related to that for Gaussian walks crossing a suitably rescaled barrier. Our analysis shows that this is a useful way to think of the generic case as well. Although our study is motivated by the possibility that the primordial fluctuation field was non-Gaussian, our results are general. In particular, they do not assume the non-Gaussianity is small, so they may be viewed as the solution to an excursion set analysis of the late-time, nonlinear fluctuation field rather than the initial one. They are also useful for models in which the barrier height is determined by quantities other than the initial density, since most other physically motivated variables (such as the shear) are usually stochastic and non-Gaussian. We use the Lognormal transformation to illustrate some of our arguments.

Halo mass definition and multiplicity function

Comparing the excursion set and CUSP formalisms for the derivation of the halo mass function, we investigate the role of the mass definition in the properties of the multiplicity function of cold dark matter (CDM) haloes. We show that the density profile for haloes formed from triaxial peaks that undergo ellipsoidal collapse and virialisation is such that the ratio between the mean inner density and the outer local density is essentially independent of mass. This causes that, for suited values of the spherical overdensity $\Delta$ and the linking length $b$, SO and FoF masses are essentially equivalent to each other and the respective multiplicity functions are essentially the same. The overdensity for haloes having undergone ellipsoidal collapse is the same as if they had formed according to the spherical top-hat model, which leads to a value of $b$ corresponding to the usual virial overdensity, $\Delta_{vir}$, equal to $\sim$ 0.2. The multiplicity function resulting from such mass definitions, expressed as a function of the top-hat height for spherical collapse, is very approximately universal in all CDM cosmologies. The reason for this is that, for such mass definitions, the top-hat density contrast for ellipsoidal collapse and virialisation is close to a universal value, equal to $\sim$ 0.9 times the usual top-hat density contrast for spherical collapse.

DARK-MATTER HALO ASSEMBLY BIAS: ENVIRONMENTAL DEPENDENCE IN THE NON-MARKOVIAN EXCURSION-SET THEORY

In the standard excursion set model for the growth of structure, the statistical properties of halos are governed by the halo mass and are independent of the larger scale environment in which the halos reside. Numerical simulations, however, have found the spatial distributions of halos to depend not only on their mass but also on the details of their assembly history and environment. Here we present a theoretical framework for incorporating this "assembly bias" into the excursion set model. Our derivations are based on modifications of the path integral approach of Maggiore & Riotto (2010) that models halo formation as a non-Markovian random walk process. The perturbed density field is assumed to evolve stochastically with the smoothing scale and exhibits correlated walks in the presence of a density barrier. We write down conditional probabilities for multiple barrier crossings, and derive from them analytic expressions for descendant and progenitor halo mass functions and halo merger rates as a function of both halo mass and the linear overdensity of the larger-scale environment of the halo. Our results predict a higher halo merger rate and higher progenitor halo mass function in regions of higher overdensity, consistent with the behavior seen in N-body simulations.

AN ANALYTICAL MODEL OF THE LARGE NEUTRAL REGIONS DURING THE LATE STAGE OF REIONIZATION

In this paper we investigate the nature and distribution of large neutral regions during the late epoch of reionization. In the "bubble model" of reionization, the mass distribution of large ionized regions ("bubbles") during the early stage of reionization is obtained by using the excursion set model, where the ionization of a region corresponds to the first up-crossing of a barrier by random trajectories. We generalize this idea, and develop a method to predict the distribution of large scale neutral regions during the late stage of reionization, taking into account the ionizing background after the percolation of HII regions. The large scale neutral regions which we call "neutral islands" are not individual galaxies or minihalos, but larger regions where fewer galaxies formed and hence ionized later, and they are identified in the excursion set model with the first down-crossings of the island barrier. Assuming that the consumption rate of ionizing background photons is proportional to the surface area of the neutral islands, we obtained the size distribution of the neutral islands. We also take the "bubbles-in-island" effect into account by considering the conditional probability of up-crossing a bubble barrier after down-crossing the island barrier. We find that this effect is very important. An additional barrier is set to avoid islands being percolated through. We find that there is a characteristic scale for the neutral islands, while the small islands are rapidly swallowed up by the ionizing background, this characteristic scale does not change much as the reionization proceeds.

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.

Extended void merging tree algorithm for self-similar models

In hierarchical evolution, voids exhibit two different behaviors related with their surroundings and environments, they can merge or collapse. These two different types of void processes can be described by the two-barrier excursion set formalism based on Brownian random walks. In this study, the analytical approximate description of the growing void merging algorithm is extended by taking into account the contributions of voids that are embedded into overdense region(s) which are destined to vanish due to gravitational collapse. Following this, to construct a realistic void merging model that consists of both collapse and merging processes, the two-barrier excursion set formalism of the void population is used. Assuming spherical voids in the Einstein de Sitter Universe, the void merging algorithm which allows us to consider the two main processes of void hierarchy in one formalism is constructed. In addition to this, the merger rates, void survival probabilities, void size distributions in terms of the collapse barrier and finally, the void merging tree algorithm in the self-similar models are defined and derived.

Detecting sparse cone alternatives for Gaussian random fields, with an application to fMRI

Our problem is to find a good approximation to the P-value of the maximum of a random field of test statistics for a cone alternative at each point in a sample of Gaussian random fields. These test statistics have been proposed in the neuroscience literature for the analysis of fMRI data allowing for unknown delay in the hemodynamic response. However the null distribution of the maximum of this 3D random field of test statistics, and hence the threshold used to detect brain activation, was unsolved. To find a solution, we approximate the P-value by the expected Euler characteristic (EC) of the excursion set of the test statistic random field. Our main result is the required EC density, derived using Taylor’s Gaussian kinematic formula. We apply this to a set of fMRI data on pain perception.