Halo abundances and counts-in-cells: The excursion set approach with correlated steps
Aseem Paranjape, Tsz Yan Lam, Ravi K. Sheth
备注：
14 pages, 9 figures; v2 -- revised version with explicit demonstration that the original conclusions hold for LCDM, expanded discussion on stochasticity of barrier. Accepted in MNRAS
The Excursion Set approach has been used to make predictions for a number of interesting quantities in studies of nonlinear hierarchical clustering. These include the halo mass function, halo merger rates, halo formation times and masses, halo clustering, analogous quantities for voids, and the distribution of dark matter counts in randomly placed cells. The approach assumes that all these quantities can be mapped to problems involving the first crossing distribution of a suitably chosen barrier by random walks. Most analytic expressions for these distributions ignore the fact that, although different k-modes in the initial Gaussian field are uncorrelated, this is not true in real space: the values of the density field at a given spatial position, when smoothed on different real-space scales, are correlated in a nontrivial way. As a result, the problem is to estimate first crossing distribution by random walks having correlated rather than uncorrelated steps. In 1990, Peacock & Heavens presented a simple approximation for the first crossing distribution of a single barrier of constant height by walks with correlated steps. We show that their approximation can be thought of as a correction to the distribution associated with what we call smooth completely correlated walks. We then use this insight to extend their approach to treat moving barriers, as well as walks that are constrained to pass through a certain point before crossing the barrier. For the latter, we show that a simple rescaling, inspired by bivariate Gaussian statistics, of the unconditional first crossing distribution, accurately describes the conditional distribution, independently of the choice of analytical prescription for the former. In all cases, comparison with Monte-Carlo solutions of the problem shows reasonably good agreement. (Abridged)
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 spectr…
We show how correlated steps introduces significant contributions to the modification of the halo mass function in modified gravity models, taking the chameleon models as an example, in the framework of the excursion set approach. This correction applies to both Lagrangian and Eulerian environments discussed in previous studies. Correlated steps also enhances the modifications due to the fifth force in the conditional mass function as well as the halo bias. We found that abundance and clustering measurements from different environments can provide strong constraints on the chameleon models.
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) th…
The excursion set theory based on spherical or ellipsoidal gravitational collapse provides an elegant analytic framework for calculating the mass function and the large-scale bias of dark matter haloes. This theory assumes that the perturbed density field evolves stochastically with the smoothing scale and exhibits Markovian random walks in the presence of a density barrier. Here we derive an analytic expression for the halo bias in a new theoretical model that incorporates non-Markovian extension of the excursion set theory with a stochastic barrier. This model allows us to handle non-Markovi…
We exploit the excursion set approach in integral formulation to derive novel, accurate analytic approximations of the unconditional and conditional first crossing distributions, for random walks with uncorrelated steps and general shapes of the moving barrier; we find the corresponding approximations of the unconditional and conditional halo mass functions for Cold Dark Matter power spectra to represent very well the outcomes of state-of-the-art cosmological N-body simulations. In addition, we apply these results to derive, and confront with simulations, other quantities of interest in halo s…
In the Excursion Set approach, halo abundances and clustering are closely related. This relation is exploited in many modern methods which seek to constrain cosmological parameters on the basis of the observed spatial distribution of clusters. However, to obtain analytic expressions for these quantities, most Excursion Set based predictions ignore the fact that, although different k-modes in the initial Gaussian field are uncorrelated, this is not true in real space: the values of the density field at a given spatial position, when smoothed on different real-space scales, are correlated in a n…
The excursion set approach provides a framework for predicting how the abundance of dark matter halos depends on the initial conditions. A key ingredient of this formalism comes from the physics of halo formation: the specification of a critical overdensity threshold (barrier) which protohalos must exceed if they are to form bound virialized halos at a later time. Another ingredient is statistical, as it requires the specification of the appropriate statistical ensemble over which to average when making predictions. The excursion set approach explicitly averages over all initial positions, thu…
If one accounts for correlations between scales, then nonlocal, k-dependent halo bias is part and parcel of the excursion set approach, and hence of halo model predictions for galaxy bias. We present an analysis that distinguishes between a number of different effects, each one of which contributes to scale-dependent bias in real space. We show how to isolate these effects and remove the scale dependence, order by order, by cross-correlating the halo field with suitably transformed versions of the mass field. These transformations may be thought of as simple one-point, two-scale measurements t…
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 differe…
Recent analytical work on the modelling of dark halo abundances and clustering has demonstrated the advantages of combining the excursion set approach with peaks theory. We extend these ideas and introduce a model of excursion set peaks that incorporates the role of initial tidal effects or shear in determining the gravitational collapse of dark haloes. The model -- in which the critical density threshold for collapse depends on the tidal influences acting on protohaloes -- is well motivated from ellipsoidal collapse arguments and is also simple enough to be analytically tractable. We show tha…
We describe how to extend the excursion set peaks framework so that its predictions of dark halo abundances and clustering can be compared directly with simulations. These extensions include: a halo mass definition which uses the TopHat filter in real space; the mean dependence of the critical density for collapse delta_c on halo mass m; and the scatter around this mean value. All three of these are motivated by the physics of triaxial rather than spherical collapse. A comparison of the resulting mass function with N-body results shows that, if one uses delta_c(m) and its scatter as determined…
We study the impact of primordial non-Gaussianity generated during inflation on the bias of halos using excursion set theory. We recapture the familiar result that the bias scales as $k^{-2}$ on large scales for local type non-Gaussianity but explicitly identify the approximations that go into this conclusion and the corrections to it. We solve the more complicated problem of non-spherical halos, for which the collapse threshold is scale dependent.
Characterizing the level of primordial non-Gaussianity (PNG) in the initial conditions for structure formation is one of the most promising ways to test inflation and differentiate among different scenarios. The scale-dependent imprint of PNG on the large-scale clustering of galaxies and quasars has already been used to place significant constraints on the level of PNG in our observed Universe. Such measurements depend upon an accurate and robust theory of how PNG affects the bias of galactic halos relative to the underlying matter density field. We improve upon previous work by employing a mo…
In terms of the excursion set model, we used Monte Carlo methods in order to study the non-Markovian stochastic evolution of the smoothed overdensity $δ$ at scale $S$. For a Gaussian density field, smoothed by the top-hat filter, in real space, we used random walks, with the correct correlation between scales, in order to calculate constrained probabilities and to connect the number of structures with the overdensity of their environment. A constant barrier is used, lower than the usual one but which improves the consistency of our results with those of N-body simulations.We present comparison…
A sizeable level of non-Gaussianity in the primordial cosmological perturbations may be induced by a large trispectrum, i.e. by a large connected four-point correlation function. We compute the effect of a primordial non-Gaussian trispectrum on the halo mass function, within excursion set theory. We use the formalism that we have developed in a previous series of papers and which allows us to take into account the fact that, in the presence of non-Gaussianity, the stochastic evolution of the smoothed density field, as a function of the smoothing scale, is non-markovian. In the large mass limit…
We develop a new, simple, fast and parameter-free method to construct dark matter halo catalogues. This method requires as inputs only the linear matter power spectrum and the threshold density for halo formation in linear theory. It directly uses excursion set ideas and Lagrangian perturbation theory to produce halo catalogues with the correct abundance, large scale power spectrum, bispectrum and velocity field. These halo catalogues can be used for the fast construction of mock galaxy catalogues, allowing for the evaluation of covariance matrices for multiple observables. Because of its robu…
The high-mass end of the halo mass function is a sensitive probe of primordial non-Gaussianity (NG). In a recent study [9] we have computed the NG halo mass function in the context of the Excursion Set theory and shown that the primordial NG imprint is coupled to that induced by the non-linear collapse of dark matter halos. We also found an excellent agreement with N-body simulation results. Here, we perform a more accurate computation which accounts for the interval validity of the bispectrum expansion to next-to-leading order and extend the calculation to the case of a non-vanishing primordi…
We describe a model of dark matter halo abundances and clustering which combines the two most widely used approaches to this problem: that based on peaks and the other based on excursion sets. Our approach can be thought of as addressing the cloud-in-cloud problem for peaks and/or modifying the excursion set approach so that it averages over a special subset, rather than all possible walks. In this respect, it seeks to account for correlations between steps in the walk as well as correlations between walks. We first show how the excursion set and peaks models can be written in the same formali…
We use the Excursion Set formalism to compute the properties of the halo mass distribution for a stochastic barrier model which encapsulates the main features of the ellipsoidal collapse of dark matter halos. Non-markovian corrections due to the sharp filtering of the linear density field in real space are computed with the path-integral technique introduced by Maggiore & Riotto (2010). Here, we provide a detailed derivation of the results presented in Corasaniti & Achitouv (2011) and extend the mass function analysis to higher redshift. We also derive an analytical expression for the linear h…
We study the effects of primordial non-Gaussianity on the large scale structure in the excursion set approach, accounting for correlations between steps of the random walks in the smoothed initial density field. These correlations are induced by realistic smoothing filters (as opposed to a filter that is sharp in k-space), but have been ignored by many analyses to date. We present analytical arguments -- building on existing results for Gaussian initial conditions -- which suggest that the effect of the filter at large smoothing scales is remarkably simple, and is in fact identical to what hap…
Excursion set theory, where density perturbations evolve stochastically with the smoothing scale, provides a method for computing the mass function of cosmological structures like dark matter halos, sheets and filaments. The computation of these mass functions is mapped into the so-called first-passage time problem in the presence of a moving barrier. In this paper we use the path integral formulation of the excursion set theory developed recently to analytically solve the first-passage time problem in the presence of a generic moving barrier, in particular the barrier corresponding to ellipso…
Excursion set theory (EST) is an analytical framework to study the large-scale structure of the Universe. EST introduces a procedure to calculate the number density of structures by relating the cosmological linear perturbation theory to the nonlinear structures in late time. In this work, we introduce a novel approach to reformulate the EST in matrix formalism. We propose that the matrix representation of EST will facilitate the calculations in this framework. The method is to discretize the two-dimensional plane of variance and density contrast of EST, where the trajectories for each point i…
We describe a simple fully analytic model of the excursion set approach associated with two Gaussian random walks: the first walk represents the initial overdensity around a protohalo, and the second is a crude way of allowing for other factors which might influence halo formation. This model is richer than that based on a single walk, because it yields a distribution of heights at first crossing. We provide explicit expressions for the unconditional first crossing distribution which is usually used to model the halo mass function, the progenitor distributions, and the conditional distribution…
We present a new algorithm to sample the constrained eigenvalues of the initial shear field associated with Gaussian statistics, called the `peak/dip excursion-set-based' algorithm, at positions which correspond to peaks or dips of the correlated density field. The computational procedure is based on a new formula which extends Doroshkevich's unconditional distribution for the eigenvalues of the linear tidal field, to account for the fact that halos and voids may correspond to maxima or minima of the density field. The ability to differentiate between random positions and special points in spa…
Constrained realisations of Gaussian random fields are used in cosmology to design special initial conditions for numerical simulations. We review this approach and its application to density peaks providing several worked-out examples. We then critically discuss the recent proposal to use constrained realisations to modify the linear density field within and around the Lagrangian patches that form dark-matter haloes. The ambitious concept is to forge `genetically modified' haloes with some desired properties after the non-linear evolution. We demonstrate that the original implementation of th…
A classic method for computing the mass function of dark matter halos is provided by excursion set theory, where density perturbations evolve stochastically with the smoothing scale, and the problem of computing the probability of halo formation is mapped into the so-called first-passage time problem in the presence of a barrier. While the full dynamical complexity of halo formation can only be revealed through N-body simulations, excursion set theory provides a simple analytic framework for understanding various aspects of this complex process. In this series of paper we propose improvements …
The simplest models of dark matter halo formation are based on the heuristic assumption, motivated by spherical collapse, that virialized haloes originate from initial regions that are maxima of the smoothed density field. Here, we replace this notion with the dynamical requirement that protohalo patches be regions where the local gravitational flow converges to a point. For this purpose, we look for spheres whose gravitational acceleration at the boundary -- relative to their center of mass -- points towards their geometric center: that is, spheres with null dipole moment. We show that these …
The excursion set theory, where density perturbations evolve stochastically with the smoothing scale, provides a method for computing the dark matter halo mass function. The computation of the mass function is mapped into the so-called first-passage time problem in the presence of a moving barrier. The excursion set theory is also a powerful formalism to study other properties of dark matter halos such as halo bias, accretion rate, formation time, merging rate and the formation history of halos. This is achieved by computing conditional probabilities with non-trivial initial conditions, and th…
Cosmological $N$-body simulations play a vital role in studying models for the evolution of the Universe. To compare to observations and make a scientific inference, statistic analysis on large simulation datasets, e.g., finding halos, obtaining multi-point correlation functions, is crucial. However, traditional in-memory methods for these tasks do not scale to the datasets that are forbiddingly large in modern simulations. Our prior paper proposes memory-efficient streaming algorithms that can find the largest halos in a simulation with up to $10^9$ particles on a small server or desktop. How…
We investigate the behavior of the fifth force in voids in chameleon models using the spherical collapse method. Contrary to Newtonian gravity, we find the fifth force is repulsive in voids. The strength of the fifth force depends on the density inside and outside the void region as well as its radius. It can be many times larger than the Newtonian force and their ratio is in principle unbound. This is very different from the case in halos, where the fifth force is no more than 1/3 of gravity. The evolution of voids is governed by the Newtonian gravity, the effective dark energy force and the …
In excursion set theory the computation of the halo mass function is mapped into a first-passage time process in the presence of a barrier, which in the spherical collapse model is a constant and in the ellipsoidal collapse model is a fixed function of the variance of the smoothed density field. However, N-body simulations show that dark matter halos grow through a mixture of smooth accretion, violent encounters and fragmentations, and modeling halo collapse as spherical, or even as ellipsoidal, is a significant oversimplification. We propose that some of the physical complications inherent to…
Primordial non-Gaussianity (NG) affects the large scale structure (LSS) of the universe by leaving an imprint on the distribution of matter at late times. Much attention has been focused on using the distribution of collapsed objects (i.e. dark matter halos and the galaxies and galaxy clusters that reside in them) to probe primordial NG. An equally interesting and complementary probe however is the abundance of extended underdense regions or voids in the LSS. The calculation of the abundance of voids using the excursion set formalism in the presence of primordial NG is subject to the same tech…
We estimate the halo mass function (HMF) by applying the excursion set approach to the non-linear cosmic density field. Thereby, we account for the non-Gaussianity of today's density distribution and constrain the HMF independent of the linear collapse threshold $δ_{\textrm{crit}}$. We consider a spherical region as a halo, if its density today exceeds the virial overdensity threshold $Δ$. We model the probability distribution of the non-linear density field by a superposition of a Gaussian and a lognormal distribution, which we constrain with the bispectrum of density fluctuations, predicted …
Aims. We aim to study the stochastic evolution of the smoothed overdensity $δ$ at scale $S$ of the form $δ(S) = \int_{0}^S K(S,u)\mathrm{d}W(u)$, where $K$ is a kernel and $\mathrm{d}W$ is the usual Wiener process. Methods. For a Gaussian density field, smoothed by the top-hat filter, in real space, we used a simple kernel that gives the correct correlation between scales. A Monte Carlo procedure was used to construct random walks and to calculate first crossing distributions and consequently mass functions for a constant barrier. Results. We show that the evolution considered here improves th…
We have revisited the extended excursion set theory in modified gravity models, taking the chameleon model as an example. Instead of specifying their Lagrangian size, here we define the environments by the Eulerian size, chosen to be of the same order of the Compton length of the scalar field by physical arguments. We find that the Eulerian and Lagrangian environments have very different environmental density contrast probability distributions, the former more likely to have high matter density, which in turn suppressing the effect of the fifth force in matter clustering and halo formation. Th…
We compute the effect of primordial non-Gaussianity on the halo mass function, using excursion set theory. In the presence of non-Gaussianity the stochastic evolution of the smoothed density field, as a function of the smoothing scale, is non-markovian and beside "local" terms that generalize Press-Schechter (PS) theory, there are also "memory" terms, whose effect on the mass function can be computed using the formalism developed in the first paper of this series. We find that, when computing the effect of the three-point correlator on the mass function, a PS-like approach which consists in ne…
The excursion set reionization framework is widely used, due to its speed and accuracy in reproducing the 3D topology of reionization. However, it is known that it does not conserve photon number. Here, we introduce an efficient, on-the-fly recipe to approximately account for photon conservation. Using a flexible galaxy model shown to reproduce current high-$z$ observables, we quantify the bias in the inferred reionization history and galaxy properties resulting from the non-conservation of ionizing photons. Using a mock 21-cm observation, we perform inference with and without correcting for i…
We show how the excursion set moving barrier model for halo abundances may be generalized to the local non-Gaussian f_{nl} model. Our estimate assumes that the distribution of step sizes depends on f_{nl}, but that they are otherwise uncorrelated. Our analysis is consistent with previous results for the case of a constant barrier, and highlights some implicit assumptions. It also clarifies the basis of an approximate analytic solution to the moving barrier problem in the Gaussian case, and shows how it might be improved.
A Fuzzy Dark Matter (FDM) halo consists of a soliton core close to the center and an NFW-like density profile in the outer region. Previous investigations found that the soliton core exhibits temporal oscillations and random walk excursions around the halo center. Analyzing a set of numerical simulations, we show that both phenomena can be understood as the results of wave interference -- a suitable superposition of the ground (solitonic) state and excited states in a fixed potential suffices to account for the main features of these phenomena. Such an eigenmode analysis can shed light on the …
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 low…
We compute the dark matter halo mass function using the excursion set formalism for a diffusive barrier with linearly drifting average which captures the main features of the ellipsoidal collapse model. We evaluate the non-Markovian corrections due to the sharp filtering of the linear density field in real space with a path-integral method. We find an unprecedented agreement with N-body simulation data with deviations within ~5% level over the range of masses probed by the simulations. This indicates that the Excursion Set in combination with a realistic modelling of the collapse threshold can…
We provide a simple formula that accurately approximates the first crossing distribution of barriers having a wide variety of shapes, by random walks with a wide range of correlations between steps. Special cases of it are useful for estimating halo abundances, evolution, and bias, as well as the nonlinear counts in cells distribution. We discuss how it can be extended to allow for the dependence of the barrier on quantities other than overdensity, to construct an excursion set model for peaks, and to show why assembly and scale dependent bias are generic even at the linear level.
In this article we compare the halo mass function predicted by the excursion set theory with a drifting diffusive barrier against the results of N-body simulations for several cosmological models. This includes the standard LCDM case for a large range of halo masses, models with different types of primordial non-Gaussianity, and the Ratra-Peebles quintessence model of Dark Energy. We show that in all those cosmological scenarios, the abundance of dark matter halos can be described by a drifting diffusive barrier, where the two parameters describing the barrier have physical content. In the cas…
We present measurements of the number density of voids in the dark matter distribution from a series of N-body simulations of a ΛCDM cosmology. We define voids as spherical regions of ρ_v = 0.2ρ_m around density minima in order to relate our results to the predicted abundances using the excursion set formalism. Using a linear underdensity of δ_v = -2.7, from a spherical evolution model, we find that a volume conserving model, which does not conserve number density in the mapping from the linear to nonlinear regime, matches the measured abundance to within 16% for a range of void radii 1< r(Mpc…
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 t…
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, acco…
The abundance of cosmic voids can be described by an analogue of halo mass functions for galaxy clusters. In this work, we explore a number of void mass functions: from those based on excursion-set theory to new mass functions obtained by modifying halo mass functions. We show how different void mass functions vary in their predictions for the largest void expected in an observational volume, and compare those predictions to observational data. Our extreme-value formalism is shown to be a new practical tool for testing void theories against simulation and observation.
$N$-body cosmological simulations are an essential tool to understand the observed distribution of galaxies. We use the MultiDark simulation suite, run with the Planck cosmological parameters, to revisit the mass and velocity functions. At redshift $z=0$, the simulations cover four orders of magnitude in halo mass from $\sim10^{11}M_\odot$ with 8,783,874 distinct halos and 532,533 subhalos. The total volume used is $\sim$515 Gpc$^3$, more than 8 times larger than in previous studies. We measure and model the halo mass function, its covariance matrix w.r.t halo mass and the large scale halo bia…
The density field reconstruction technique has been widely used for recovering the Baryon Acoustic Oscillation (BAO) feature in galaxy surveys that has been degraded due to nonlinearities. Recent studies advocated adopting iterative steps to improve the recovery much beyond that of the standard technique. In this paper, we investigate the performance of a few selected iterative reconstruction techniques focusing on the BAO and the broadband-shape of the two-point clustering. We include redshift-space distortions, halo bias, and shot noise and inspect the components of the reconstructed field i…
In order to extract full cosmological information from next-generation large and high-precision weak lensing (WL) surveys (e.g. Euclid, Roman, LSST), higher-order statistics that probe the small-scale, non-linear regime of large scale structure (LSS) need to be utilized. WL peak counts, which trace overdensities in the cosmic web, are one promising and simple statistic for constraining cosmological parameters. The physical origin of WL peaks have previously been linked to dark matter halos along the line of sight and this peak-halo connection has been used to develop various semi-analytic halo…
We study cell count moments up to fifth order of the distributions of haloes, of halo substructures as a proxy for galaxies, and of mass in the context of the halo model and compare theoretical predictions to the results of numerical simulations. On scales larger than the size of the largest cluster, we present a simple point cluster model in which results depend only on cluster-cluster correlations and on the distribution of the number of objects within a cluster, or cluster occupancy. The point cluster model leads to expressions for moments of galaxy counts in which the volume-averaged momen…
