Excursion Set Research Articles

Origin and Full Characterization of the Secondary (Assembly) Halo Bias

The clustering of dark matter halos depends not only on their mass, the so-called primary bias, but also on their internal properties, the so-called secondary bias. While the former effect is well understood within the Press–Schechter and excursion set models of structure formation, the latter is not. In those models, protohalos are fully characterized by their height and scale, which determine the halo mass and collapse time, so there is no room for any other halo property. This is why the secondary bias was believed not to be innate but due to the distinct merger rate of halos lying in different backgrounds, and dubbed assembly bias. However, it has now been determined that mergers leave no imprint in the inner halo properties. In fact, the innate origin of the secondary bias cannot be discarded because, in the more realistic peak model of structure formation, halo seeds are characterized by one additional property: the peak curvature. Here, we use the confluent system of peak trajectory formalism to show that peaks lying in different backgrounds have different mean curvatures, which in turn cause them to evolve into halos with different typical inner properties. The dependence we find of the properties on halo background (or halo clustering) reproduces the results of simulations.

Halo Bias in the Peak Model: A First-principles Nonparametric Approach

The Press–Schechter (PS) and excursion set (ES) models of structure formation fail in reproducing the halo bias found in simulations, while the ES-peaks' formalism built in the peak model reproduces it only at high masses and does not address in a fully satisfactory manner peak nesting, and the mass and time of ellipsoidal collapse of triaxial peaks in the Gaussian-smoothed density field. Here, we apply the confluent system of peak trajectories formalism fixing all these issues from first principles and with no free parameters to infer the Lagrangian local peak bias parameters, which adopt very simple analytic expressions similar to those found in the PS and ES models. The predicted Eulerian linear halo bias recovers the results of simulations. More specifically, we show that the only small departure observed at intermediate and low masses can be due to the spurious halo splitting and grouping caused by the spherical overdensity halo-finding algorithm used in simulations.

Mass scaling relations for dark halos from an analytic universal outer density profile

Context. The average matter density within the turnaround scale, which demarcates where galaxies shift from clustering around a structure to joining the expansion of the Universe, is an important cosmological probe. However, a measurement of the mass enclosed by the turnaround radius is difficult. Analyses of the turnaround scale in simulated galaxy clusters place the turnaround radius at about three times the virial radius in a ΛCDM universe and at a (present-day) density contrast with the background matter density of the Universe of δ ~ 11. Assessing the mass at such extended distances from a cluster’s center is a challenge for current mass measurement techniques. Consequently, there is a need to develop and validate new mass-scaling relations, to connect observable masses at cluster interiors with masses at greater distances. Aims. Our research aims to establish an analytical framework for the most probable mass profile of galaxy clusters, leading to novel mass scaling relations, allowing us to estimate masses at larger scales. We derive such analytical mass profiles and compare them with those from cosmological simulations. Methods. We used excursion set theory, which provides a statistical framework for the density and local environment of dark matter halos, and complement it with the spherical collapse model to follow the non-linear growth of these halos. Results. The profile we developed analytically showed good agreement (better than 30%, and dependent on halo mass) with the mass profiles of simulated galaxy clusters. Mass scaling relations were obtained from the analytical profile with offset better than 15% from the simulated ones. This level of precision highlights the potential of our model for probing structure formation dynamics at the outskirts of galaxy clusters.

Planet Formation Regulated by Galactic-scale Interstellar Turbulence

Planet formation occurs over a few Myr within protoplanetary disks of dust and gas, which are often assumed to evolve in isolation. However, extended gaseous structures have been uncovered around many protoplanetary disks, suggestive of late-stage infall from the interstellar medium (ISM). To quantify the prevalence of late-stage infall, we apply an excursion set formalism to track the local density and relative velocity of the ISM over the disk lifetime. We then combine the theoretical Bondi–Hoyle–Lyttleton (BHL) accretion rate with a simple disk evolution model, anchoring stellar accretion timescales to observational constraints. Disk lifetimes, masses, stellar accretion rates, and gaseous outer radii as a function of stellar mass and age are remarkably well reproduced by our simple model that includes only ISM accretion. We estimate that 20%−70% of disks may be mostly composed of material accreted in the most recent half of their lifetime, suggesting that disk properties are not a direct test of isolated evolution models. Our calculations indicate that BHL accretion can also supply sufficient energy to drive turbulence in the outer regions of protoplanetary disks with viscous α SS ∼ 10−5 to 10−1, although we emphasize that angular momentum transport and particularly accretion onto the star may still be driven by internal processes. Our simple approach can be easily applied to semianalytic models. Our results represent a compelling case for regulation of planet formation by large-scale turbulence, with broad consequences for planet formation theory. This possibility urgently motivates deep observational surveys to confirm or refute our findings.

Surface area and volume of excursion sets observed on point cloud based polytopic tessellations

Surface area and volume of excursion sets observed on point cloud based polytopic tessellations

Kernel-based Sensitivity Analysis for (Excursion) Sets

In this article, we aim to perform sensitivity analysis of set-valued models and, in particular, to quantify the impact of uncertain inputs on feasible sets, which are key elements in solving a robust optimization problem under constraints. While most sensitivity analysis methods deal with scalar outputs, this article introduces a novel approach to perform sensitivity analysis with set-valued outputs. Our innovative methodology is designed for excursion sets, but is versatile enough to be applied to set-valued simulators, including those found in viability fields, or when working with maps like pollutant concentration maps or flood zone maps. We propose to use the Hilbert-Schmidt Independence Criterion (HSIC) with a kernel designed for set-valued outputs. After setting a probabilistic framework for random sets, a first contribution is the proof that this kernel is characteristic, an essential property in a kernel-based sensitivity analysis context. To measure the contribution of each input, we then propose to use HSIC-ANOVA indices. With these indices, we can identify which inputs should be neglected (screening) and we can rank the others according to their influence (ranking). The estimation of these indices is also adapted to the set-valued outputs. Finally, we test the proposed method on three test cases of excursion sets.

A central limit theorem for the number of excursion set components of Gaussian fields

A central limit theorem for the number of excursion set components of Gaussian fields

Morphology of nanoporous glass: Stochastic 3D modeling, stereology and the influence of pore width

Excursion sets of Gaussian random fields are used to model the three-dimensional (3D) morphology of differently manufactured porous glasses (PGs), which vary with respect to their mean pore widths measured by mercury intrusion porosimetry. The stochastic 3D model is calibrated by means of volume fractions and two-point coverage probability functions estimated from tomographic image data. Model validation is performed by comparing model realizations and image data in terms of morphological descriptors which are not used for model fitting. For this purpose, we consider mean geodesic tortuosity and constrictivity of the pore space, quantifying the length of the shortest transportation paths and the strength of bottleneck effects, respectively. Additionally, a stereological approach for parameter estimation is presented, i.e., the 3D model is calibrated using merely two-dimensional (2D) cross-sections of the 3D image data. Doing so, on average, a comparable goodness of fit is achieved as well. The variance of the calibrated model parameters is discussed, which is estimated on the basis of randomly chosen, individual 2D cross-sections. Moreover, interpolating between the model parameters calibrated to differently manufactured glasses enables the predictive simulation of virtual but realistic PGs with mean pore widths that have not yet been manufactured. The predictive power is demonstrated by means of cross-validation. Using the presented approach, relationships between parameters of the manufacturing process and descriptors of the resulting morphology of PGs are quantified, which opens possibilities for an efficient optimization of the underlying manufacturing process. Published by the American Physical Society 2024

Smooth Matérn Gaussian random fields: Euler characteristic, expected number and height distribution of critical points

This paper studies Gaussian random fields with Matérn covariance functions with smooth parameter ν>2. Two cases of parameter spaces, the Euclidean space and N-dimensional sphere, are considered. For such smooth Gaussian fields, we have derived the explicit formulae for the expected Euler characteristic of the excursion set, the expected number and height distribution of critical points. The results are valuable for approximating the excursion probability in family-wise error control and for computing p-values in peak inference.

Probing the Anisotropy and Non-Gaussianity in the Redshift Space through the Conditional Moments of the First Derivative

Focusing on the redshift space observations with plane-parallel approximation and relying on the rotational dependency of the general definition of excursion sets, we introduce the so-called conditional moments of the first derivative (cmd) measures for the smoothed matter density field in three dimensions. We derive the perturbative expansion of cmd for the real space and redshift space where peculiar velocity disturbs the galaxies’ observed locations. Our criteria can successfully recognize the contribution of linear Kaiser and Finger-of-God effects. Our results demonstrate that the cmd measure has significant sensitivity for pristine constraining the redshift space distortion parameter β = f/b and interestingly, the associated normalized quantity in the Gaussian linear Kaiser limit has only β dependency. Implementation of the synthetic anisotropic Gaussian field approves the consistency between the theoretical and numerical results. Including the first-order contribution of non-Gaussianity perturbatively in the cmd criterion implies that the N-body simulations for the Quijote suite in the redshift space have been mildly skewed with a higher value for the threshold greater than zero. The non-Gaussianity for the perpendicular direction to the line of sight in the redshift space for smoothing scales R ≳ 20 Mpc h −1 is almost the same as in the real space. In contrast, the non-Gaussianity along the line-of-sight direction in the redshift space is magnified. The Fisher forecasts indicate a significant enhancement in constraining the cosmological parameters Ω m , σ 8, and n s when using cmd + cr jointly.

Efficient 3D real-time adaptive AUV sampling of a river plume front

The coastal environment faces multiple challenges due to climate change and human activities. Sustainable marine resource management necessitates knowledge, and development of efficient ocean sampling approaches is increasingly important for understanding the ocean processes. Currents, winds, and freshwater runoff make ocean variables such as salinity very heterogeneous, and standard statistical models can be unreasonable for describing such complex environments. We employ a class of Gaussian Markov random fields that learns complex spatial dependencies and variability from numerical ocean model data. The suggested model further benefits from fast computations using sparse matrices, and this facilitates real-time model updating and adaptive sampling routines on an autonomous underwater vehicle. To justify our approach, we compare its performance in a simulation experiment with a similar approach using a more standard statistical model. We show that our suggested modeling framework outperforms the current state of the art for modeling such spatial fields. Then, the approach is tested in a field experiment using two autonomous underwater vehicles for characterizing the three-dimensional fresh-/saltwater front in the sea outside Trondheim, Norway. One vehicle is running an adaptive path planning algorithm while the other runs a preprogrammed path. The objective of adaptive sampling is to reduce the variance of the excursion set to classify freshwater and more saline fjord water masses. Results show that the adaptive strategy conducts effective sampling of the frontal region of the river plume.

Accurate halo mass functions from the simplest excursion set theory

ABSTRACT Excursion set theory is a powerful and widely used tool for describing the distribution of dark matter haloes, but it is normally applied with simplifying approximations. We use numerical sampling methods to study the mass functions predicted by the theory without approximations. With a spherical top-hat window and a constant δ = 1.5 threshold, the theory accurately predicts mass functions with the M200 mass definition, both unconditional and conditional, in simulations of a range of matter-dominated cosmologies. For Λ cold dark matter at the present epoch, predictions lie between the M200m and M200c mass functions. In contrast, with the same window function, a non-constant threshold based on ellipsoidal collapse predicts uniformly too few haloes. This work indicates a new way to simply and accurately evaluate halo mass functions, clustering bias, and assembly histories for a range of cosmologies. We provide a fitting function that accurately represents the predictions of the theory for a wide range of parameters.

Uniqueness of Unbounded Component for Level Sets of Smooth Gaussian Fields

Abstract For a large family of stationary continuous Gaussian fields $f$ on ${\mathbb {R}}^{d}$, including the Bargmann–Fock and Cauchy fields, we prove that there exists at most one unbounded connected component in the level set $\{f=\ell \}$ (as well as in the excursion set $\{f\geq \ell \}$) almost surely for every level $\ell \in {\mathbb {R}}$, thus proving a conjecture proposed by Duminil-Copin, Rivera, Rodriguez, and Vanneuville. As the fields considered are typically very rigid (e.g., analytic almost surely), there is no sort of finite energy property available and the classical approaches to prove uniqueness become difficult to implement. We bypass this difficulty using a soft shift argument based on the Cameron–Martin theorem.

Detection of long range dependence in the time domain for (in)finite-variance time series

Empirical detection of long range dependence (LRD) of a time series often consists of deciding whether an estimate of the memory parameter d corresponds to LRD. Surprisingly, the literature offers only a few estimators for d in the time domain. And those that exist are criticized for relying on visual inspection to determine an observation window [ n 1 , n 2 ] for a linear regression. Here, we provide rigorous asymptotic conditions on [ n 1 , n 2 ] to ensure consistency of the well-known variance plot estimator under LRD. We do this for a large class of square-integrable time series models so that the estimator can be used to detect LRD for infinite-variance time series in the sense of indicators of excursion sets. Thus, detection of LRD for infinite-variance time series is another novelty of our paper. Finally, a simulation study compares the variance plot estimator with a popular spectral domain estimator.

Spatial confidence regions for combinations of excursion sets in image analysis.

The analysis of excursion sets in imaging data is essential to a wide range of scientific disciplines such as neuroimaging, climatology, and cosmology. Despite growing literature, there is little published concerning the comparison of processes that have been sampled across the same spatial region but which reflect different study conditions. Given a set of asymptotically Gaussian random fields, each corresponding to a sample acquired for a different study condition, this work aims to provide confidence statements about the intersection, or union, of the excursion sets across all fields. Such spatial regions are of natural interest as they directly correspond to the questions 'Where do all random fields exceed a predetermined threshold?', or 'Where does at least one random field exceed a predetermined threshold?'. To assess the degree of spatial variability present, our method provides, with a desired confidence, subsets and supersets of spatial regions defined by logical conjunctions (i.e. set intersections) or disjunctions (i.e. set unions), without any assumption on the dependence between the different fields. The method is verified by extensive simulations and demonstrated using task-fMRI data to identify brain regions with activation common to four variants of a working memory task.

Voids and haloes-in-voids statistics as a probe of the expansion history of the Universe

ABSTRACT Structures in the Universe are arranged into the cosmic web. Distributions, statistics, and evolutions of the structures can be used as probes for cosmological models. We investigate the number density of voids and dark matter haloes-in-voids in the excursion set theory (EST). We study the Markov and non-Markov frameworks of EST in both spherical and ellipsoidal collapse models. Afterward, we compare the number density of voids and haloes-in-voids in the standard lambda cold dark matter and the reconstructed model. The reconstructed model is a model-independent reconstruction based on background observations. This work explores the effects of the collapse model barrier in the different EST frameworks on the statistics of voids and the statistics of haloes-in-voids. Finally, we find the hint that cosmological models can be distinguished by the number density of haloes-in-voids in the 1.0−2.5 redshift range. The maximum difference is observed in z ∼ 1.9.

On the perimeter estimation of pixelated excursion sets of two‐dimensional anisotropic random fields

AbstractWe are interested in creating statistical methods to provide informative summaries of random fields through the geometry of their excursion sets. To this end, we introduce an estimator for the length of the perimeter of excursion sets of random fields on observed over regular square tilings. The proposed estimator acts on the empirically accessible binary digital images of the excursion regions and computes the length of a piecewise linear approximation of the excursion boundary. The estimator is shown to be consistent as the pixel size decreases, without the need of any normalization constant, and with neither assumption of Gaussianity nor isotropy imposed on the underlying random field. In this general framework, even when the domain grows to cover , the estimation error is shown to be of smaller order than the side length of the domain. For affine, strongly mixing random fields, this translates to a multivariate Central Limit Theorem for our estimator when multiple levels are considered simultaneously. Finally, we conduct several numerical studies to investigate statistical properties of the proposed estimator in the finite‐sample data setting.

Asymptotic expansion of the expected Minkowski functional for isotropic central limit random fields

AbstractThe Minkowski functionals, including the Euler characteristic statistics, are standard tools for morphological analysis in cosmology. Motivated by cosmic research, we examine the Minkowski functional of the excursion set for an isotropic central limit random field, whose k-point correlation functions (kth-order cumulants) have the same structure as that assumed in cosmic research. Using 3- and 4-point correlation functions, we derive the asymptotic expansions of the Euler characteristic density, which is the building block of the Minkowski functional. The resulting formula reveals the types of non-Gaussianity that cannot be captured by the Minkowski functionals. As an example, we consider an isotropic chi-squared random field and confirm that the asymptotic expansion accurately approximates the true Euler characteristic density.

3-D Adaptive AUV Sampling for Classification of Water Masses

Autonomous underwater vehicles with onboard computing units foster innovative approaches for sampling oceanographic phenomena. Feedback of observations via the onboard model for planning algorithms enable adaptive sampling for such robotic units. In this work, we develop, implement, and test an adaptive sampling algorithm for efficient sampling of water masses in a 3-D frontal system. Focusing on a river plume, salinity variations are used to characterize the water masses. A threshold in salinity is assumed to distinguish the ocean and river waters, so that excursions below the threshold define river waters. The onboard model builds on a Gaussian random field representation of the salinity variations in (north, east, depth) coordinates. This model is initially trained from numerical ocean model data, and then updated with data gathered by the vehicle sensor. The Gaussian random field model further allows closed-form expressions of the expected spatially integrated Bernoulli variance of the salinity excursion set, which is used to reward sampling efforts. Combining these results with forward-looking planning algorithms, we suggest a workflow for 3-D adaptive sampling to map river plume systems. Simulation studies are used to compare the suggested approach with others. Results of field trials in the Nidelva river plume in Norway are presented and discussed.

A data-driven modeling approach to quantify morphology effects on transport properties in nanostructured NMC particles

We present a data-driven modeling approach to quantify morphology effects on transport properties in nanostructured materials. Our approach is based on the combination of stochastic modeling of the 3D nanostructure and numerical modeling of effective transport properties, which is used to investigate process-structure–property relationships of hierarchically structured cathode materials for lithium-ion batteries. We focus on nanostructured LiNi1/3Mn1/3Co1/3O2 (NMC) particles, the nanoporous morphology of which has a crucial impact on their effective transport properties (i.e, effective ionic and electric conductivity) and thus on the performance of the cell. First, we develop a parametric stochastic model for the 3D morphology of the nanostructured NMC particles based on excursion sets of so-called χ2-fields. This model, which has only two parameters, is then fitted to FIB-SEM image data of the NMC particles manufactured with different calcination temperatures and different particle sizes. This way it is possible to generate digital twins of the NMC particles. In a second step, measured 3D image data and corresponding digital twins are used as input for the numerical simulation of effective transport properties. Based on the results obtained by these simulations, we can quantify process-structure–property relationships. Overall, we present a methodological framework that allows for an efficient optimization of the fabrication process of nanostructured NMC particles.