Using a mapping from dark matter halos to galaxy properties based on hydrodynamical simulations, we explore the impact of galaxy properties on the void size function and the void-galaxy correlation function. We replicate the properties of galaxies from Illustris on MassiveNus halos, to perform both luminosity and star formation rate cuts on MassiveNus halos. We compare the impact of such cuts on voids properties with respect to cuts on halo mass (as usually performed on halo catalogs driven from N-body simulations). We find that void catalogs built from luminosity-selected galaxies and halos are consistent within errors, while void catalogs built from star formation rate selected galaxies differ from void catalogs built on halos. We investigate the reason for this difference. Our work suggests that voids built on galaxy catalogs (selected through luminosity cut) can be reliably studied by using halos in dark matter simulations.
Understanding the internal structure and spatial distribution of cosmic voids is crucial when considering them as probes of cosmology. We present recent advances in modeling void density- and velocity-profiles in real space, as well as void two-point statistics in redshift space, by examining voids identified via the watershed transform in state-of-the-art $Λ$CDM n-body simulations and mock galaxy catalogs. The simple and universal characteristics that emerge from these statistics indicate the self-similarity of large-scale structure and suggest cosmic voids to be among the most pristine objec…
Cosmic voids are a key component of the large-scale structure that contain a plethora of cosmological information. Typically, voids are identified from the underlying galaxy distribution, which is a biased tracer of the total matter field. Previous works have shown that 2D voids identified in weak lensing maps -- weak lensing voids -- correspond better to true underdense regions along the line of sight. In this work, we study how the properties of weak lensing voids depend on the choice of void finder, by adapting several popular void finders. We present and discuss the differences between ide…
Cosmic voids are promising tools for cosmological tests due to their sensitivity to dark energy, modified gravity and alternative cosmological scenarios. Most previous studies in the literature of void properties use cosmological N-body simulations of dark matter (DM) particles that ignore the potential effect of baryonic physics. Using a spherical underdensity finder, we analyse voids using the mass field and subhalo tracers in the EAGLE simulations, which follow the evolution of galaxies in a $\rmΛ$ cold dark matter Universe with state-of-the-art subgrid models for baryonic processes in a $(…
We study scaling behaviour of statistics of voids in the context of the halo model of nonlinear large-scale structure. The halo model allows us to understand why the observed galaxy void probability obeys hierarchical scaling, even though the premise from which the scaling is derived is not satisfied. We argue that the commonly observed negative binomial scaling is not fundamental, but merely the result of the specific values of bias and number density for typical galaxies. The model implies quantitative relations between void statistics measured for two populations of galaxies, such as SDSS r…
We introduce new statistical methods for the study of cosmic voids, focusing on the statistics of largest size voids. We distinguish three different types of distributions of voids, namely, Poisson-like, lognormal-like and Pareto-like distributions. The last two distributions are connected with two types of fractal geometry of the matter distribution. Scaling voids with Pareto distribution appear in fractal distributions with box-counting dimension smaller than three (its maximum value), whereas the lognormal void distribution corresponds to multifractals with box-counting dimension equal to t…
The voids between galaxies are identified with the volumes of the Poisson Voronoi tessellation. Two new survival functions for the apparent radii of voids are derived. The sectional normalized area of the Poisson Voronoi tessellation is modelled by the Kiang function and by the exponential function. Two new survival functions with equivalent sectional radius are therefore derived; they represent an alternative to the survival function of voids between galaxies as given by the self-similar distribution. The spatial appearance of slices of the 2dF Galaxy Redshift Survey is simulated.
We present new numerical tools to analyse cosmic void catalogues, implemented inside the CosmoBolognaLib, a large set of Open Source C++/Python numerical libraries. The CosmoBolognaLib provides a common numerical environment for cosmological calculations. This work extends these libraries by adding new algorithms for cosmological analyses of cosmic voids, covering the existing gap between theory and observations. We implemented new methods to model the size function of cosmic voids, in both observed and simulated samples of dark matter and biased tracers. Moreover, we provide new numerical too…
This is the second part of a thorough investigation of the redshift-space effects that affect void properties and the impact they have on cosmological tests. Here, we focus on the void-galaxy cross-correlation function, specifically, on the projected versions that we developed in a previous work. The pillar of the analysis is the one-to-one relationship between real and redshift-space voids above the shot-noise level identified with a spherical void finder. Under this mapping, void properties are affected by three effects: (i) a systematic expansion as a consequence of the distortions induced …
Do void statistics contain information beyond the tracer 2-point correlation function? Yes! As we vary the sum of the neutrino masses, we find void statistics contain information absent when using just tracer 2-point statistics. Massive neutrinos uniquely affect cosmic voids. We explore their impact on void clustering using both the DEMNUni and MassiveNuS simulations. For voids, neutrino effects depend on the observed void tracers. As the neutrino mass increases, the number of small voids traced by cold dark matter particles increases and the number of large voids decreases. Surprisingly, when…
We present a general analytical formalism to calculate accurately several statistics related to underdense regions in the Universe. The statistics are computed for dark matter halo and galaxy distributions both in real space and redshift space at any redshift. Using this formalism, we found that void statistics for galaxy distributions can be obtained, to a very good approximation, assuming galaxies to have the same clustering properties as halos above a certain mass. We deducted a relationship between this mass and that of halos with the same accumulated number density as the galaxies. We als…
We present a detailed description of our void finding algorithm which is an extension of the prescription by Hoyle and Vogeley (2002). We include a discussion of the reproducibility and robustness of the algorithm as well as the statistical significance of the detected voids. We apply our void finder to the Data Release 5 (DR5) of the Sloan Digital Sky Survey (SDSS) and identify 232 cosmological voids. A void catalog which contains the most salient properties of the detected voids is created. We present a statistical analysis of the distribution of the size, shape and orientation of our identi…
Cosmic voids are the major volume component in the matter distribution of the Universe. They posses great potential for constraining dark energy as well as for testing theories of gravity. Nevertheless, in spite of their growing popularity as cosmological probes, a gap of knowledge between cosmic void observations and theory still persists. In particular, the void size function models proposed in literature have been proven unsuccessful in reproducing the results obtained from cosmological simulations in which cosmic voids are detected from biased tracers of the density field, undermining the …
Università di Trieste（的里雅斯特大学）；INAF – Osservatorio Astronomico di Trieste（意大利国家天体物理研究所的里雅斯特天文台）；ICSC – Centro Nazionale di Ricerca in High Performance Computing, Big Data e Quantum Computing（高性能计算、大数据与量子计算国家研究中心）；INFN – Sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；IFPU – Institute for Fundamental Physics of the Universe（宇宙基础物理学研究所）；University Observatory, LMU Faculty of Physics（慕尼黑大学物理学院天文台）；Aix-Marseille Université, CNRS/IN2P3, CPPM（艾克斯-马赛大学，法国国家科学研究中心/欧洲粒子物理研究所，普罗旺斯粒子物理中心）
Cosmic voids offer an extraordinary opportunity to study the effects of massive neutrinos on cosmological scales. Because they are freely streaming, neutrinos can penetrate the interior of voids more easily than cold dark matter or baryons, which makes their relative contribution to the mass budget in voids much higher than elsewhere in the Universe. In simulations it has recently been shown how various characteristics of voids in the matter distribution are affected by neutrinos, such as their abundance, density profiles, dynamics, and clustering properties. However, the tracers used to ident…
Stereology allows shifting from the 3D distribution of the volumes of Poissonian Voronoi Diagrams to their 2D cross-sections. The basic assumption is that the 3D statistics of the volumes of the voids in the local Universe has a distribution function of the gamma-type. The standard rule of conversion from 3D volumes to 2D circles, adopting the standard rules of stereology, produces a new probability density function of the radii which contains the Meijer $G$-function. A non-Poissonian distribution of volumes is also considered. The distribution of the 3D radii of the Sloan Digital Sky Survey D…
We propose a novel technique to probe the expansion history of the Universe based on the clustering statistics of cosmic voids. In particular, we compute their two-point statistics in redshift space on the basis of realistic mock galaxy catalogs and apply the Alcock-Paczynski test. In contrast to galaxies, we find void auto-correlations to be marginally affected by peculiar motions, providing a model-independent measure of cosmological parameters without systematics from redshift-space distortions. Because only galaxy-galaxy and void-galaxy correlations have been considered in these types of s…
To study the impact of sparsity and galaxy bias on void statistics, we use a single large-volume, high-resolution N-body simulation to compare voids in multiple levels of subsampled dark matter, halo populations, and mock galaxies from a Halo Occupation Distribution model tuned to different galaxy survey densities. We focus our comparison on three key observational statistics: number functions, ellipticity distributions, and radial density profiles. We use the hierarchical tree structure of voids to interpret the impacts of sampling density and galaxy bias, and theoretical and empirical functi…
Upcoming surveys such as \LSST{} and \Euclid{} will significantly improve the power of weak lensing as a cosmological probe. To maximise the information that can be extracted from these surveys, it is important to explore novel statistics that complement standard weak lensing statistics such as the shear-shear correlation function and peak counts. In this work, we use a recently proposed weak lensing observable -- weak lensing voids -- to make parameter constraint forecasts for an LSST-like survey. We use the \cosmoslics{} $w$CDM simulation suite to measure void statistics as a function of cos…
Radial velocity statistics of cosmic voids as a probe of interacting dark energy
The Chinese University of Hong Kong（香港中文大学）；Sun Yat-sen University（中山大学）；Institut d’Astrophysique de Paris, UMR 7095, CNRS, Sorbonne Université（巴黎天体物理学研究所，CNRS，索邦大学）
We perform a statistical study of the global motion of cosmic voids using both a numerical simulation and observational data. We analyse their relation to large-scale mass flows and the physical effects that drive those motions. We analyse the bulk motions of voids, defined by the mean velocity of haloes in the surrounding shells in the numerical simulation, and by galaxies in the Sloan Digital Sky Survey Data Release 7. We find void mean bulk velocities close to 400 km/s, comparable to those of haloes (~ 500-600 km/s), depending on void size and the large-scale environment. Statistically, sma…
We study how primordial non-Gaussianities affect the clustering of voids at large scales. We derive a formula of the bias of voids induced from the non-Gaussianities by making use of the functional integral method. In a similar way as of haloes, we find that primordial non-Gaussianities can generate scale-dependence in the bias of voids at large scales. In addition, we show that by observing the cross power spectrum of voids and haloes we could check the consistency relation between the non-linearity parameters f_NL and tau_NL. Large voids (high peak objects) would be good targets since the ef…
We present a comparison of void properties between the standard model of cosmology, $Λ$ Cold Dark Matter ($Λ$CDM), and two alternative cosmological models with evolving and interacting dark sectors: a quintessence model ($ϕ$CDM) and a Coupled Dark Matter-Dark Energy (CDE) model. Using $N$-body simulations of these models, we derive several measures of void statistics and properties, including distributions of void volume, ellipticity, prolateness, and average density. We find that the volume distribution derived from the CDE simulation deviates from the volume distribution derived from the $Λ$…
Multiprobe cosmology forecasts from halo-occupation-distribution-based forward modeling of galaxy and void statistics
Steward Observatory, University of Arizona（亚利桑那大学斯图尔德天文台）；Department of Physics, University of Arizona（亚利桑那大学物理系）；Aix-Marseille Université, CNRS/IN2P3, CPPM（艾克斯-马赛大学，法国国家科学研究中心/法国国家核子和粒子物理研究所，粒子与宇宙学中心）；Department of Astrophysical Sciences, Peyton Hall, Princeton University（普林斯顿大学天体物理学系）；Universitäts-Sternwarte München, Fakultät für Physik, Ludwig-Maximilians-Universität（慕尼黑大学天文台，慕尼黑大学物理学院）
Optimization of Tessellation-based Statistics: Void Statistics
École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Tsinghua University（清华大学）；Shanghai Jiao Tong University（上海交通大学）；Nanjing Institute of Astronomical Optics and Technology, Chinese Academy of Sciences（中国科学院南京天文光学技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）
Cosmic voids are a promising environment to characterize neutrino-induced effects on the large-scale distribution of matter in the universe. We perform a comprehensive numerical study of the statistical properties of voids, identified both in the matter and galaxy distributions, in massive and massless neutrino cosmologies. The matter density field is obtained by running several independent $N$-body simulations with cold dark matter and neutrino particles, while the galaxy catalogs are modeled by populating the dark matter halos in simulations via a halo occupation distribution (HOD) model to …
We study the formation of voids in a modified gravity model in which gravity is generically stronger or weaker on large scales. We show that void abundances provide complementary information to halo abundances: if normalized to the CMB, models with weaker large-scale gravity have smaller large scale power, fewer massive halos and fewer large voids, although the scalings are not completely degenerate with $σ_8$. Our results suggest that, in addition to their abundances, halo and void density profiles may also provide interesting constraints on such models: stronger large scale gravity produces …
We study the statistics of various large-scale structure tracers in gravity-only cosmological simulations including baryons and cold dark matter (CDM) initialized with two different transfer functions, and simulated as two distinct fluids. This allows us to study the impact of baryon-CDM relative perturbations on these statistics. In particular, we focus on the statistics of cosmic voids, as well as on the matter and halo real-space 2-point correlation function and baryon acoustic oscillations (BAO) peak. We find that the void size function is affected at the 1-2% level at maximum, and that th…
Modelling cosmic voids as spheres in Euclidean space, the notion of a de-Sitter configuration space is introduced. It is shown that a uniform distribution over this configuration space yields a power-law approximating the void size distribution in an intermediate range of volumes, as well as an estimate for the fractal dimension of the large scale structure.
Starting from the geometrical concept of a 4-dimensional de-Sitter configuration of spheres in Euclidean 3-space and modelling voids in the Universe as spheres, we show that a uniform distribution over this configuration space implies a power-law for the void number density which is consistent with results from the excursion set formalism and with data, for an intermediate range of void volumes. The scaling dimension of the large scale structure can be estimated as well. We also discuss the effect of restricting the survey geometry on the void statistics. This work is a new application of de-S…
We study the prospects for measuring the dark matter distribution of voids with stacked weak lensing. We select voids from a large set of $N$-body simulations, and explore their lensing signals with the full ray-tracing simulations including the effect of the large-scale structure along the line-of-sight. The lensing signals are compared with simple void model predictions to infer the three-dimensional mass distribution of voids. We show that the stacked weak lensing signals are detected at significant level (S/N$\geq5$) for a 5000 degree$^2$ survey area, for a wide range of void radii up to $…
Cosmic voids are promising cosmological laboratories for studying the dark energy phenomenon and alternative gravity theories. They are receiving special attention nowadays in view of the new generation of galaxy spectroscopic surveys, which are covering an unprecedented volume and redshift range. There are two primary statistics in void studies: (i) the void size function, which characterises the abundance of voids, and (ii) the void-galaxy cross-correlation function, which contains information about the density and velocity fields in these regions. However, it is necessary a complete descrip…
We report on the first application of the Alcock-Paczynski test to stacked voids in spectroscopic galaxy redshift surveys.We use voids from the Sutter et al. (2012) void catalog, which was derived from the Sloan Digital Sky Survey Data Release 7 main sample and luminous red galaxy catalogs. The construction of that void catalog removes potential shape measurement bias by using a modified version of the ZOBOV algorithm and by removing voids near survey boundaries and masks. We apply the shape-fitting procedure presented in Lavaux & Wandelt (2012) to ten void stacks out to redshift z=0.36. Combi…
The bias of dark matter halos and galaxies is a crucial quantity in many cosmological analyses. In this work, using large cosmological simulations, we explore the halo mass function and halo bias within cosmic voids. For the first time to date, we show that they are scale-dependent along the void profile, and provide a predictive theoretical model of both the halo mass function and halo bias inside voids, recovering for the latter and 1% accuracy against simulated data. These findings may help shed light on the dynamics of halo formation within voids and improve the analysis of several void st…
We present catalogues of voids for the SDSS DR7 redshift survey and for Millennium I simulation mock data. We aim to compare the observations with simulations based on a $Λ$CDM model and a semi-analytic galaxy formation model. We use the void statistics as a test for these models. We assembled a mock catalogue that closely resembles the SDSS DR7 catalogue and carried out a parallel statistical analysis of the observed and simulated catalogue. We find that in the observation and the simulation, voids tend to be equally spherical. The total volume occupied by the voids and their total number are…
We apply the Alcock-Paczynski (AP) test to the stacked voids identified using the large-scale structure galaxy catalog from the Baryon Oscillation Spectroscopic Survey (BOSS). This galaxy catalog is part of the Sloan Digital Sky Survey (SDSS) Data Release 12 and is the final catalog of SDSS-III. We also use 1000 mock galaxy catalogs that match the geometry, density, and clustering properties of the BOSS sample in order to characterize the statistical uncertainties of our measurements and take into account systematic errors such as redshift space distortions. For both BOSS data and mock catalog…
Majority of all galaxies reside in groups of less than 50 member galaxies. These groups are distributed in various large-scale environments from voids to superclusters. Evolution of galaxies is affected by the environment in which they reside. Our aim is to study the effects that the local group scale and the supercluster scale environment have on galaxies. We use a luminosity-density field to determine density of the large-scale environment of galaxies in groups of various richness. We calculate fractions of different types of galaxies in groups with richnesses up to 50 member galaxies and in…
We discuss various applications of VIDE, the Void IDentification and Examination toolkit, an open-source Python/C++ code for finding cosmic voids in galaxy redshift surveys and N-body simulations. Based on a substantially enhanced version of ZOBOV, VIDE not only finds voids, but also summarizes their properties, extracts statistical information, and provides a Python-based platform for more detailed analysis, such as manipulating void catalogs and particle members, filtering, plotting, computing clustering statistics, stacking, comparing catalogs, and fitting density profiles. VIDE also provid…
In this paper, we analysed the halos and voids properties of a GR-based N-body simulation carried out at redshifts z= 0.0 and z= 0.8 as differences between dynamical dark energy models (namely PL and CPL) with respect to LCDM. Analysing the halos demonstrates that both models, PL and CPL, behave like LCDM, despite the velocity dispersion of halos was more sensitive to the dynamical dark energy model. In addition, a void finder was developed to extract the properties of voids from simulated data. Further statistical model on voids confirms that the PL model produces larger voids. In summary, ou…
Robustness of cosmic void statistics: insights from SDSS DR7 and the ELUCID simulation
Shanghai Astronomical Observatory（上海天文台）；State Key Laboratory of Dark Matter Physics, Tsung-Dao Lee Institute & School of Physics and Astronomy, Shanghai Jiao Tong University（暗物质物理全国重点实验室、李政道理论与计算科学中心、物理与天文学院，上海交通大学）；Shanghai Key Laboratory for Particle Physics and Cosmology, and Key Laboratory for Particle Physics, Astrophysics and Cosmology Ministry of Education, Shanghai Jiao Tong University（粒子物理与宇宙学上海市重点实验室、教育部粒子物理、天体物理与宇宙学重点实验室，上海交通大学）；School of Aerospace Science and Technology, Xidian University（空间科学与技术学院，西安电子科技大学）
A modification of the Press-Schechter theory allowing for presence of a background large-scale structure (LSS) - a supercluster or a void, is proposed. The LSS is accounted as the statistical constraints in form of linear functionals of the random overdensity field. The deviation of the background density within the LSS is interpreted in a pseudo-cosmological sense. Using the constraints formalism may help us to probe non-trivial spatial statistics of haloes, e.g. edge and shape effects on boundaries of the superclusters and voids. Parameters of the constraints are connected to features of the…
We use properties of void populations identified in N-body simulations to forecast the ability of upcoming galaxy surveys to differentiate models of f(R) gravity from ΛCDM cosmology. We analyze simulations designed to mimic the densities, volumes, and clustering statistics of upcoming surveys, using the public VIDE toolkit. We examine void abundances as a basic probe at redshifts 1.0 and 0.4. We find that stronger f(R) coupling strengths produce voids up to ~20% larger in radius, leading to a significant shift in the void number function. As an initial estimate of the constraining power of voi…
The upcoming new generation of spectroscopic galaxy redshift surveys will provide large samples of cosmic voids, the distinct, large underdense structures in the universe. Combining these with future galaxy imaging surveys, we study the prospects of probing the underlying matter distribution in and around cosmic voids via the weak gravitational lensing effects of stacked voids, utilizing both shear and magnification information. The statistical precision is greatly improved by stacking together a large number of voids along different lines of sight, even when taking into account the impact of …
We perform a statistical study focused on void environments. We examine galaxy density profiles around voids in the SDSS, finding a correlation between void--centric distance to the shell of maximum density and void radius when a maximum in overdensity exists. We analyze voids with and without a surrounding over-dense shell in the SDSS. We find that small voids are more frequently surrounded by over-dense shells whereas the radial galaxy density profile of large voids tends to rise smoothly towards the mean galaxy density. We analyse the fraction of voids surrounded by overdense shells finding…
We investigate the basic properties of voids from high resolution, cosmological N-body simulations of Λ-dominated cold dark matter (ΛCDM) models, in order to compare with the analytical model of Sheth and van de Weygaert (SvdW) for void statistics. For the subsample of five dark matter simulations in the ΛCDM cosmology with box sizes ranging from 1000 Mpc/h to 8 Mpc/h, we find that the standard void-in-cloud effect is too simplified to explain several properties of identified small voids in simulations. (i) The number density of voids is found to be larger than the prediction of the analytical…
We develop a methodology to optimise the measurement of Baryon Acoustic Oscillation (BAO) from a given galaxy sample. In our previous work, we demonstrated that one can measure BAO from tracers in under-dense regions (voids). In this study, we combine the over-dense and under-dense tracers (galaxies & voids) to obtain better constraints on the BAO scale. To this end, we modify the de-wiggled BAO model with an additional parameter to describe both the BAO peak and the underlying exclusion pattern of void 2PCFs. We show that after applying BAO reconstruction to galaxies, the BAO peak scale of bo…
We study the relations between the multimodality of galaxy clusters drawn from the SDSS DR8 and the environment where they reside. We find that multimodal clusters reside in higher density environment than unimodal clusters. We determine morphological types of superclusters and show that clusters in superclusters of spider morphology have higher probabilities to have substructure and larger peculiar velocities of their main galaxies than clusters in superclusters of filament morphology. Our study shows the importance of the role of superclusters as high density environment which affects the pr…
Taking N-body simulations with volumes and particle densities tuned to match the SDSS DR7 spectroscopic main sample, we assess the ability of current void catalogs (e.g., Sutter et al. 2012b) to distinguish a model of coupled dark matter-dark energy from ΛCDM cosmology using properties of cosmic voids. Identifying voids with the VIDE toolkit, we find no statistically significant differences in the ellipticities, but find that coupling produces a population of significantly larger voids, possibly explaining the recent result of Tavasoli et al. (2013). In addition, we use the universal density p…
Towards precision cosmology with Voids x CMB correlations (I): Roman-Agora mock catalogs and pipeline validation
Instituto de Astrofísica de Canarias（加那利天体物理学研究所）；Universidad de La Laguna（拉古纳大学）；MTA–CSFK Lendület “Momentum” Large-Scale Structure (LSS) Research Group（马塔-匈牙利科学院技术研究中心“动量”大尺度结构研究组）；Konkoly Observatory, HUN-REN Research Centre for Astronomy and Earth Sciences（孔科利天文台，匈牙利研究与创新地球与天文学研究中心）；CPPM, Aix-Marseille Université, CNRS/IN2P3（马赛物理中心，艾克斯-马赛大学，法国国家科学研究中心/法国国家核与粒子物理研究所）；Princeton University（普林斯顿大学）
CSST large-scale structure analysis pipeline: IV. Cosmic Voids Identified from Galaxy Group Samples as Probes of the Large-scale Structure
Shanghai Jiao Tong University（上海交通大学）；National Astronomical Observatories, Chinese Academy of Sciences（中国科学院国家天文台）；University of Chinese Academy of Sciences(UCAS)（中国科学院大学）
Aims: We assess the sensitivity of void shapes to the nature of dark energy that was pointed out in recent studies. We investigate whether or not void shapes are useable as an observational probe in galaxy redshift surveys. We focus on the evolution of the mean void ellipticity and its underlying physical cause. Methods: We analyse the morphological properties of voids in five sets of cosmological N-body simulations, each with a different nature of dark energy. Comparing voids in the dark matter distribution to those in the halo population, we address the question of whether galaxy redshift su…
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…
