Excursion Set Research Articles

Morphology of 21cm brightness temperature during the Epoch of Reionization using Contour Minkowski Tensor

We use morphological descriptors, Betti numbers and Contour Minkowski Tensor (CMT) on 21cm brightness temperature excursion sets, to study the ionization and heating history of the intergalactic medium (IGM) during and before the Epoch of Reionization (EoR) . The ratio of eigenvalues of the CMT denoted by β, gives shape information while it's trace gives the contour length of holes and connected regions. We simulate the matter density, neutral hydrogen fraction, spin temperature and brightness temperature field using the publicly available code 21cmFAST in a redshift range of z=20.22 to z=6. We study the redshift evolution of three quantities—the Betti number counts Ncon,hole, the characteristic size rchcon,hole and shape anisotropy parameter βchcon,hole of connected regions and holes for these fields and investigate the different physical origins of their evolution. We make a qualitative comparison of different models of heating and ionization during the EoR . We obtain different regimes of morphological evolution of brightness temperature, depending upon how the shapes and sizes of connected regions and holes change with redshift for different astrophysical settings affecting the ionization and heating history of the IGM during and before the EoR . We find that the morphology of the brightness temperature field traces the morphology of ionized regions below a certain redshift value depending upon the model, where Δ rchhole<10 % and Δ βchhole<1 % relative to the xHI field. This difference decreases with redshift. Therefore, the ionization history of the IGM can be reconstructed using the morphological description of δ Tb in real space.

Artist: fast radiative transfer for large-scale simulations of the epoch of reionization

ABSTRACT We introduce the ‘Asymmetric Radiative Transfer In Shells Technique’ (artist), a new method for photon propagation on large scales that explicitly conserves photons, propagates photons at the speed of light, approximately accounts for photon directionality, and closely reproduces results of more detailed radiative transfer (RT) methods. Crucially, it is computationally fast enough to evolve the large cosmological volumes required to predict the 21cm power spectrum on scales that will be probed by future experiments targeting the epoch of reionization (EoR). Most seminumerical models aimed at predicting the EoR 21cm signal on these scales use an excursion set formalism (ESF) to model the gas ionization, which achieves computational viability by making a number of approximations. While artist is still roughly two orders of magnitude slower than ESF, it does allow to model the EoR without the need for such approximations. This is particularly important when considering a wide range of reionization scenarios for which artist would help limit the assumptions made. By implementing our RT method within the seminumerical code simfast21, we show that Artist predicts a significantly different evolution for the EoR ionization field compared to the code’s native ESF. In particular, artist predicts up to a factor of two difference in the power spectra, depending on the physical parameters assumed. Its application to large-scale EoR simulations will therefore allow more physically motivated constraints to be obtained for key EoR parameters. In particular, it will remove the need for the artificial rescaling of the escape fraction.

Cosmological filaments in the light of excursion set of saddle points

ABSTRACT The universe in large scales is structured as a network known as cosmic web. Filaments are one of the structural components of this web, which can be introduced as a novel probe to study the formation and evolution of structures and as a probe to study the cosmological models and to address the missing baryon problem. The aim of this work is to introduce an analytical framework to study the statistics of filaments such as number density of them and also to obtain the length-mass relation. For this objective, we model filaments as collapsed objects which have an extension in one direction, accordingly we use the ellipsoidal collapse to study the evolution of an over-dense region via gravitational instability. We find that the non-linear density of filaments in the epoch of formation is almost mass independent and is in order of ∼30. By introducing filament’s extended condition, we find a fitting function for length-mass relation. For the statistics of filaments, we propose a novel framework named excursion set of saddle points. In this approach, we count the saddle points of the density field Hessian matrix, and relate it to the count of filaments. In addition, we addressed the filament in filament problem with up-crossing approximation.

A reduction principle for the critical values of random spherical harmonics

We study here the random fluctuations in the number of critical points with values in an interval I⊂R for Gaussian spherical eigenfunctions fℓ, in the high energy regime where ℓ→∞. We show that these fluctuations are asymptotically equivalent to the centred L2-norm of fℓ times the integral of a (simple and fully explicit) function over the interval under consideration. We discuss also the relationships between these results and the asymptotic behaviour of other geometric functionals on the excursion sets of random spherical harmonics.

On macroscopic holes in some supercritical strongly dependent percolation models

We consider $Z^d$, with d bigger or equal to three. We investigate the vacant set of random interlacements in the strongly percolative regime, the vacant set of the simple random walk, and the excursion set above a given level of the Gaussian free field in the strongly percolative regime. We derive asymptotic upper and lower exponential bounds for the large deviation probability that the adequately thickened component of the boundary of a large box centered at the origin in the respective vacant sets or excursion set leaves in the box a macroscopic volume in its complement. We also derive geometric information on the shape of the left-out volume. It is plausible, but open at the moment, that certain critical levels coincide, both in the case of random interlacements and of the Gaussian free field. If this holds true, the asymptotic upper and lower bounds that we obtain are matching in principal order for all three models, and the macroscopic holes are nearly spherical. We heavily rely on the recent work arXiv:1706.07229 by Maximilian Nitzschner and the author for the coarse graining procedure, which we employ in the derivation of the upper bounds.

Unexpected topology of the temperature fluctuations in the cosmic microwave background

We study the topology generated by the temperature fluctuations of the cosmic microwave background (CMB) radiation, as quantified by the number of components and holes, formally given by the Betti numbers, in the growing excursion sets. We compare CMB maps observed by the Planck satellite with a thousand simulated maps generated according to the ΛCDM paradigm with Gaussian distributed fluctuations. The comparison is multi-scale, being performed on a sequence of degraded maps with mean pixel separation ranging from 0.05 to 7.33°. The survey of the CMB over 𝕊2 is incomplete due to obfuscation effects by bright point sources and other extended foreground objects like our own galaxy. To deal with such situations, where analysis in the presence of “masks” is of importance, we introduce the concept of relative homology. The parametric χ2-test shows differences between observations and simulations, yielding p-values at percent to less than permil levels roughly between 2 and 7°, with the difference in the number of components and holes peaking at more than 3σ sporadically at these scales. The highest observed deviation between the observations and simulations for b0 and b1 is approximately between 3σ and 4σ at scales of 3–7°. There are reports of mildly unusual behaviour of the Euler characteristic at 3.66° in the literature, computed from independent measurements of the CMB temperature fluctuations by Planck’s predecessor, the Wilkinson Microwave Anisotropy Probe (WMAP) satellite. The mildly anomalous behaviour of the Euler characteristic is phenomenologically related to the strongly anomalous behaviour of components and holes, or the zeroth and first Betti numbers, respectively. Further, since these topological descriptors show consistent anomalous behaviour over independent measurements of Planck and WMAP, instrumental and systematic errors may be an unlikely source. These are also the scales at which the observed maps exhibit low variance compared to the simulations, and approximately the range of scales at which the power spectrum exhibits a dip with respect to the theoretical model. Non-parametric tests show even stronger differences at almost all scales. Crucially, Gaussian simulations based on power-spectrum matching the characteristics of the observed dipped power spectrum are not able to resolve the anomaly. Understanding the origin of the anomalies in the CMB, whether cosmological in nature or arising due to late-time effects, is an extremely challenging task. Regardless, beyond the trivial possibility that this may still be a manifestation of an extreme Gaussian case, these observations, along with the super-horizon scales involved, may motivate the study of primordial non-Gaussianity. Alternative scenarios worth exploring may be models with non-trivial topology, including topological defect models.

A numerical investigation on the high‐frequency geometry of spherical random eigenfunctions

AbstractA lot of attention has been drawn over the last few years by the investigation of the geometry of spherical random eigenfunctions (random spherical harmonics) in the high‐frequency regime, that is, for diverging eigenvalues. In this paper, we present a review of these results and we collect for the first time a comprehensive numerical investigation, focussing on particular on the behavior of Lipschitz‐Killing curvatures/Minkowski functionals (i.e., the area, the boundary length, and the Euler‐Poincaré characteristic of excursion sets) and on critical points. We show in particular that very accurate analytic predictions exist for their expected values and variances, for the correlation among these functionals, and for the cancellation that occurs for some specific thresholds (the variances becoming an order of magnitude smaller—the so‐called Berry's cancellation phenomenon). Most of these functionals can be used for important statistical applications, for instance, in connection to the analysis of cosmic microwave background data.

Profile Extrema for Visualizing and Quantifying Uncertainties on Excursion Regions: Application to Coastal Flooding

ABSTRACTWe consider the problem of describing excursion sets of a real-valued function f, that is, the set of inputs where f is above a fixed threshold. Such regions are hard to visualize if the input space dimension, d, is higher than 2. For a given projection matrix from the input space to a lower dimensional (usually 1, 2) subspace, we introduce profile sup (inf) functions that associate to each point in the projection’s image the sup (inf) of the function constrained over the pre-image of this point by the considered projection. Plots of profile extrema functions convey a simple, although intrinsically partial, visualization of the set. We consider expensive to evaluate functions where only a very limited number of evaluations, n, is available, for example, , and we surrogate f with a posterior quantity of a Gaussian process (GP) model. We first compute profile extrema functions for the posterior mean given n evaluations of f. We quantify the uncertainty on such estimates by studying the distribution of GP profile extrema with posterior quasi-realizations obtained from an approximating process. We control such approximation with a bound inherited from the Borell-TIS inequality. The technique is applied to analytical functions (d = 2, 3) and to a five-dimensional coastal flooding test case for a site located on the Atlantic French coast. Here f is a numerical model returning the area of flooded surface in the coastal region given some offshore conditions. Profile extrema functions allowed us to better understand which offshore conditions impact large flooding events.

Analytic Formulation of 21 cm Signal from Cosmic Dawn: Lyα Fluctuations

Abstract We present an analytic formalism to compute the fluctuating component of the H i signal and extend it to take into account the effects of partial Lyα coupling during the era of cosmic dawn. We use excursion set formalism to calculate the size distribution of randomly distributed self-ionized regions. These ionization bubbles are surrounded by partially heated and Lyα coupled regions, which create spin temperature T S fluctuations. We use the ratio of number of Lyα to ionizing photons (f L ) and number of X-ray photons emitted per stellar baryon (N heat) as modeling parameters. Using our formalism, we compute the global H i signal, its autocorrelation, and its power spectrum in the redshift range 10 ≤ z ≤ 30 for the ΛCDM model. We check the validity of this formalism for various limits and simplified cases. Our results agree reasonably well with existing results from N-body simulations, in spite of following a different approach and requiring orders of magnitude less computation power and time. We further apply our formalism to study the fluctuating component corresponding to the recent observation by the Experiment to Detect the Global Epoch of reionization Signature (EDGES) that shows an unexpectedly deep absorption trough in the global H i signal in the redshift range 15 &lt; z &lt; 19. We show that, generically, the EDGES observation predicts a larger signal in this redshift range but a smaller signal at higher redshifts. We also explore the possibility of negative real-space autocorrelation of spin temperature and show that it can be achieved for partial Lyα coupling in many cases corresponding to simplified models and a complete model without density perturbations.

Mean geometry for 2D random fields: Level perimeter and level total curvature integrals

We introduce the level perimeter integral and the total curvature integral associated with a real valued function f defined on the plane R^2 as integrals allowing to compute the perimeter of the excursion set of f above level t and the total (signed) curvature of its boundary for almost every level t. Thanks to the Gauss-Bonnet theorem, the total curvature is directly related to the Euler Characteristic of the excursion set. We show that the level perimeter and the total curvature integrals can be explicitly computed in two different frameworks: piecewise constant functions (also called here elementary functions) and smooth (at least C^2) functions. Considering 2D random fields (in particular considering shot noise random fields), we compute their mean perimeter and total curvature integrals, and this provides new explicit computations of the mean perimeter and Euler Characteristic densities of excursion sets, beyond the Gaussian framework.

Halo concentrations from extended Press–Schechter merger histories

We apply the model relating halo concentration to formation history proposed by Ludlow et al. to merger trees generated using an algorithm based on excursion set theory. We find that while the model correctly predicts the median relation between halo concentration and mass, it underpredicts the scatter in concentration at fixed mass. Since the same model applied to N-body merger trees predicts the correct scatter, we postulate that the missing scatter is due to the lack of any environmental dependence in merger trees derived from excursion set theory. We show that a simple modification to the merger tree construction algorithm, which makes merger rates dependent on environment, can increase the scatter by the required amount, and simultaneously provide a qualitatively correct correlation between environment and formation epoch in the excursion set merger trees.

Studying the morphology of H iisodensity surfaces during reionization using Shapefinders and percolation analysis

Minkowski functionals and Shapefinders shed light on the connectedness of large-scale structure by determining its topology and morphology. We use a sophisticated code, SURFGEN2, to measure the Minkowski functionals and Shapefinders of individual clusters by modelling cluster surfaces using the 'Marching Cube 33' triangulation algorithm. In this paper, we study the morphology of simulated neutral hydrogen (HI) density fields using Shapefinders at various stages of reionization from the excursion set approach. Accompanying the Shapefinders, we also employ the 'largest cluster statistic' (LCS) to understand the percolation process. Percolation curves demonstrate that the non-Gaussianity in the HI field increases as reionization progresses. The large clusters in both the HI overdense and underdense excursion sets possess similar values of "thickness" ($T$), as well as "breadth" ($B$), but their third Shapefinder - "length" ($L$) - becomes almost proportional to their volume. The large clusters in both HI overdense and underdense segments are overwhelmingly filamentary. The 'cross-section' of a filamentary cluster can be estimated using the product of the first two Shapefinders, $T \times B$. Hence the cross sections of the large clusters at the onset of percolation do not vary much with volume and their sizes only differ in terms of their lengths. This feature appears more vividly in HI overdense regions than in underdense regions and is more pronounced at lower redshifts which correspond to an advanced stage of reionization.

A quantitative central limit theorem for the excursion area of random spherical harmonics over subdomains of S2

In recent years, considerable interest has been drawn by the analysis of geometric functionals for the excursion sets of random eigenfunctions on the unit sphere (spherical harmonics). In this paper, we extend those results to proper subsets of the sphere S2, i.e., spherical caps, focussing, in particular, on the excursion area. Precisely, we show that the asymptotic behaviour of the excursion area is dominated by the so-called second-order chaos component and we exploit this result to establish a quantitative central limit theorem, in the high energy limit. These results generalize analogous findings for the full sphere; their proofs, however, require more sophisticated techniques, in particular, a careful analysis (of some independent interest) for smooth approximations of the indicator function for spherical cap subsets.

Lipschitz-Killing curvatures of excursion sets for two-dimensional random fields

In the present paper we study three geometrical characteristics for the excursion sets of a two-dimensional stationary isotropic random field. First, we show that these characteristics can be estimated without bias if the considered field satisfies a kinematic formula, this is for instance the case of fields given by a function of smooth Gaussian fields or of some shot noise fields. By using the proposed estimators of these geometric characteristics, we describe some inference procedures for the estimation of the parameters of the field. An extensive simulation study illustrates the performances of each estimator. Then, we use the Euler characteristic estimator to build a test to determine whether a given field is Gaussian or not, when compared to various alternatives. The test is based on a sparse information, i.e., the excursion sets for two different levels of the field to be tested. Finally, the proposed test is adapted to an applied case, synthesized 2D digital mammograms.

Neoproterozoic glacial origin of the Great Unconformity

The Great Unconformity, a profound gap in Earth's stratigraphic record often evident below the base of the Cambrian system, has remained among the most enigmatic field observations in Earth science for over a century. While long associated directly or indirectly with the occurrence of the earliest complex animal fossils, a conclusive explanation for the formation and global extent of the Great Unconformity has remained elusive. Here we show that the Great Unconformity is associated with a set of large global oxygen and hafnium isotope excursions in magmatic zircon that suggest a late Neoproterozoic crustal erosion and sediment subduction event of unprecedented scale. These excursions, the Great Unconformity, preservational irregularities in the terrestrial bolide impact record, and the first-order pattern of Phanerozoic sedimentation can together be explained by spatially heterogeneous Neoproterozoic glacial erosion totaling a global average of 3-5 vertical kilometers, along with the subsequent thermal and isostatic consequences of this erosion for global continental freeboard.

Nonlinear evolution of initially biased tracers in modified gravity

In this work we extend the perturbation theory for modified gravity (MG) in two main aspects. First, the construction of matter overdensities from Lagrangian displacement fields is shown to hold in a general framework, allowing us to find Standard Perturbation Theory (SPT) kernels from known Lagrangian Perturbation Theory (LPT) kernels. We then develop a theory of biased tracers for generalized cosmologies, extending already existing formalisms for ΛCDM. We present the correlation function in Convolution-LPT and the power spectrum in SPT for ΛCDM, f(R) Hu-Sawicky, and DGP braneworld models. Our formalism can be applied to many generalized cosmologies and to facilitate it, we are making public a code to compute these statistics. We further study the relaxation of bias with the use of a simple model and of excursion set theory, showing that in general the bias parameters have smaller values in MG than in General Relativity.

A quantitative central limit theorem for the Euler–Poincaré characteristic of random spherical eigenfunctions

We establish here a quantitative central limit theorem (in Wasserstein distance) for the Euler–Poincaré characteristic of excursion sets of random spherical eigenfunctions in dimension 2. Our proof is based upon a decomposition of the Euler–Poincaré characteristic into different Wiener-chaos components: we prove that its asymptotic behaviour is dominated by a single term, corresponding to the chaotic component of order two. As a consequence, we show how the asymptotic dependence on the threshold level $u$ is fully degenerate, that is, the Euler–Poincaré characteristic converges to a single random variable times a deterministic function of the threshold. This deterministic function has a zero at the origin, where the variance is thus asymptotically of smaller order. We discuss also a possible unifying framework for the Lipschitz–Killing curvatures of the excursion sets for Gaussian spherical harmonics.

Photon number conservation and the large-scale 21 cm power spectrum in seminumerical models of reionization

Seminumerical models of the reionization of neutral hydrogen (HI) based on the excursion set (ES) approach are known to violate photon number conservation at the few per cent level. In this work, we highlight a more severe, previously unrecognized shortcoming of ES models: the large-scale 21 cm power spectrum (equivalently, H i bias |$b_{\rm H\, \small {I}}$|⁠) is a relatively strong function of the spatial resolution used to generate ES ionization maps. We trace this problem to the fact that photon non-conservation in these models arises from a resolution-dependent mixture of spatially resolved, photon non-conserving bubbles, and partially ionized grid cells which are perfectly photon conserving (PC) by construction. We argue that this inevitably leads to a resolution dependence of |$b_{\rm H\, \small {I}}$|⁠, with the correct, converged value only emerging at very coarse resolution. Quantitatively, we find that |$b_{\rm H\,{\small I}}$| can be non-converged by as much as ∼20–|$25{{\ \rm per\ cent}}$| in conservative ES implementations with grid sizes Δx = 5–10h−1cMpc, even when photon non-conservation is as small as ∼3–|$4{{\ \rm per\ cent}}$|⁠. Thus, although numerically efficient, ES ionization maps coarse enough to produce a converged H i bias would wash out all topological features of the ionization field at scales |$k\gtrsim 0.05\, h{^{-1}}$|cMpc. We therefore present a new, explicitlyPC seminumerical algorithm which distributes photons isotropically around sources while also accounting for anisotropic overlaps between nearby bubbles. Our PC algorithm predicts a resolution-independent value of |$b_{\rm H\,{\small I}}$| consistent with the result of low-resolution ES maps, thus serving as a useful compromise between standard ES implementations and more expensive radiative transfer simulations.

A supermartingale approach to Gaussian process based sequential design of experiments

Gaussian process (GP) models have become a well-established framework for the adaptive design of costly experiments, and notably of computer experiments. GP-based sequential designs have been found practically efficient for various objectives, such as global optimization (estimating the global maximum or maximizer(s) of a function), reliability analysis (estimating a probability of failure) or the estimation of level sets and excursion sets. In this paper, we study the consistency of an important class of sequential designs, known as stepwise uncertainty reduction (SUR) strategies. Our approach relies on the key observation that the sequence of residual uncertainty measures, in SUR strategies, is generally a supermartingale with respect to the filtration generated by the observations. This observation enables us to establish generic consistency results for a broad class of SUR strategies. The consistency of several popular sequential design strategies is then obtained by means of this general result. Notably, we establish the consistency of two SUR strategies proposed by Bect, Ginsbourger, Li, Picheny and Vazquez (Stat. Comput. 22 (2012) 773–793) – to the best of our knowledge, these are the first proofs of consistency for GP-based sequential design algorithms dedicated to the estimation of excursion sets and their measure. We also establish a new, more general proof of consistency for the expected improvement algorithm for global optimization which, unlike previous results in the literature, applies to any GP with continuous sample paths.

Identification and isotropy characterization of deformed random fields through excursion sets

Abstract A deterministic application θ:ℝ2→ℝ2 deforms bijectively and regularly the plane and allows the construction of a deformed random field X∘θ:ℝ2→ℝ from a regular, stationary, and isotropic random field X:ℝ2→ℝ. The deformed field X∘θ is, in general, not isotropic (and not even stationary), however, we provide an explicit characterization of the deformations θ that preserve the isotropy. Further assuming that X is Gaussian, we introduce a weak form of isotropy of the field X∘θ, defined by an invariance property of the mean Euler characteristic of some of its excursion sets. We prove that deformed fields satisfying this property are strictly isotropic. In addition, we are able to identify θ, assuming that the mean Euler characteristic of excursion sets of X∘θ over some basic domain is known.