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All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Subhaloes are anisotropically distributed and aligned with the smooth matter distribution of their host haloes

ABSTRACT We investigate the distributions of subhaloes about their hosts in two suites of zoom-in N-body simulations of halo growth – one suite focused on Milky Way-Mass haloes ($\sim 10^{12} \, \mathrm{M}_{\odot }$) and another focused on cluster-mass haloes ($\sim 10^{15} \, \mathrm{M}_{\odot }$) in the Symphony simulation suite. We find, in agreement with previous work on this subject, that subhaloes are distributed anisotropically about their host haloes. In particular, the positions of subhaloes lie preferentially near the major axes of their host haloes, possibly implying that satellite galaxies will exhibit a similar alignment. Furthermore, we show that in two-dimensional projection subhaloes are more likely to be observed near the halo centre (where the central galaxy presumably resides) when the host halo is projected nearly along its major axis. This projection effect is significant. Within projected radii of a few per cent of the virial radius of the host halo, the fraction of mass in subhaloes is $\sim 175~{{\ \rm per\ cent}}$ larger for Milky Way-mass haloes and as much as $\sim 195~{{\ \rm per\ cent}}$ larger for cluster haloes when projected along the major axis as compared to the average from a random projection. This result has consequences for many applications including the interpretation of gravitational lenses. Finally, we find that the orbital angular momentum vector of subhaloes is aligned with the angular momentum vector of their host halo, indicating that a significant component of a halo’s angular momentum may be carried in its subhaloes. This has consequences for galaxy formation models which use host halo angular momentum as a proxy for galaxy momentum.

Mezini, Lorena (ORCID:0000000300053973)↗
DOE OSTI2021

On the influence of halo mass accretion history on galaxy properties and assembly bias

Halo assembly bias is the secondary dependence of the clustering of dark matter haloes on their assembly histories at fixed halo mass. This established dependence is expected to manifest itself on galaxy clustering, a potential effect commonly known as galaxy assembly bias. Using the IllustrisTNG300 magnetohydrodynamical simulation, in this work we analyse the dependence of the properties and clustering of galaxies on the specific mass accretion history of their hosting haloes (sMAH). We first show that several halo and galaxy properties strongly correlate with the slope of the sMAH (β) at fixed halo mass. Haloes with increasingly steeper β increment their masses faster early on, and their hosted galaxies present larger stellar-to-halo mass ratios, lose their gas faster, reach the peak of their star formation histories at higher redshift, and become quenched earlier. We also demonstrate that β provides a more stable link to these key galaxy formation properties than other broadly employed halo proxies, such as formation time. Finally, we measure the secondary dependence of galaxy clustering on β at fixed halo mass. By tracing back the evolution of individual haloes, we show that the amplitude of the galaxy assembly bias signal for the progenitors of z = 0 galaxies increases with redshift, reaching a factor of 2 at z = 1 for haloes of Mhalo = 10 11.5 –10 12 h –1 M ⊙ . The measurement of the evolution of assembly bias along the merger tree provides a new theoretical perspective to the study of secondary bias. Our findings have also important implications for the generation of mock catalogues for upcoming cosmological surveys.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

The influence of subhaloes on host halo properties

Within the ΛCDM cosmology, dark matter haloes are composed of both a smooth component and a population of smaller gravitationally bound subhaloes. These components are often treated as a single halo when properties, such as density profiles, are extracted from simulations. Recent work has shown that density profiles change substantially when subhalo mass is excluded. In this paper, we expand on this result by analysing three specific host halo properties – concentration (cNFW), spin (λB), and shape (c/a) – when calculated only from the smooth component of the halo. This analysis is performed on both Milky Way-mass haloes and cluster-mass haloes in high-resolution zoom-in N-body simulations. We find that when subhaloes are excluded, the median value of (1) cNFW is enhanced by $\approx 30\pm 11$ and $\approx 77\pm 8.1~{{\ \rm per\ cent}}$ for Milky Way-mass ($10^{12.1}\, \text{M}_\odot$) and cluster-mass ($10^{14.8}\, \text{M}_\odot$) haloes, respectively, (2) λB is reduced for Milky Way-mass by $\approx 11\pm 4.9~{{\ \rm per\ cent}}$ and cluster-mass haloes by $\approx 27\pm 3.5~{{\ \rm per\ cent}}$. Additionally, with the removal of subhaloes, cluster-mass haloes tend to become more spherical as the ratio of minor-to-major axis, c/a, increases by $\approx 11\pm 3.6~{{\ \rm per\ cent}}$, whereas Milky Way-mass haloes remain approximately the same shape with c/a changed by $\approx 1.0\pm 5.8~{{\ \rm per\ cent}}$. Fractional changes of each of these properties depend primarily on the amount of mass in subhaloes and, to a lesser extent, mass accretion history. Our findings demonstrate that the properties of the smooth components of dark matter haloes are biased relative to the total halo mass.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

High mass and halo resolution from fast low resolution simulations

Generating mocks for future sky surveys requires large volumes and high resolutions, which is computationally expensive even for fast simulations. Here we try to develop numerical schemes to calibrate various halo and matter statistics in fast low resolution simulations compared to high resolution N-body and hydrodynamic simulations. For the halos, we improve the initial condition resolution and develop a halo finder "relaxed-FoF", where we allow different linking lengths for different halo mass and velocity dispersions. We show that our relaxed-FoF halo finder improves the common statistics, such as halo bias, halo mass function, halo auto power spectrum, cross correlation coefficient with the reference halo catalog, and halo-matter cross power spectrum. We also calibrate small-scale velocities of small halos to improve the power spectrum in redshift space. For the matter statistics, we incorporate the potential gradient descent (PGD) method into fast simulations to improve the matter distribution at nonlinear scales. By building a lightcone output, we show that the PGD method significantly improves the weak lensing convergence tomographic power spectrum. With these improvements FastPM is comparable to the high resolution full N-body simulation of the same mass resolution, with two orders of magnitude fewer time steps. These techniques can be used to improve the halo and matter statistics of FastPM simulations for mock catalogs of future surveys such as DESI and LSST.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Galaxy–halo size relation from Sloan Digital Sky Survey Data Release 7 and the ELUCID simulation

ABSTRACT Based on galaxies in the Sloan Digital Sky Survey Data Release 7 and dark matter haloes in the dark matter only, cosmological, and constrained ELUCID simulation, we investigate the relation between the observed radii of central galaxies with stellar mass $\gtrsim\!{10}^{8} \, h^{-2}\, {\rm M}_\odot$ and the virial radii of their host dark matter haloes with virial mass $\gtrsim\!{10}^{10.5} \, h^{-1}\, {\rm M}_\odot$, and the dependence of galaxy–halo size relation on the halo spin and concentration. Galaxies in observation are matched to dark matter (sub)haloes in the ELUCID simulation using a novel neighbourhood subhalo abundance matching method. For galaxy two-dimensional (2D) half-light radii R50, we find that early- and late-type galaxies have the same power-law index 0.55 with $R_{50} \propto R_{\rm vir}^{0.55}$, although early-type galaxies have smaller 2D half-light radii than late-type galaxies at fixed halo virial radii. When converting the 2D half-light radii R50 to 3D half-mass radii r1/2, both early- and late-type galaxies display similar galaxy–halo size relations with $\log r_{1/2} = 0.55 \log (R_{\rm vir}/210 \, h^{-1}\, {\rm kpc}) + 0.39$. We find that the galaxy–halo size ratio r1/2/Rvir decreases with increasing halo mass. At fixed halo mass, there is no significant dependence of galaxy–halo size ratio on the halo spin or concentration.

Astronomy & Astrophysics↗
DOE OSTI2020

Testing dark matter halo properties using self-similarity

ABSTRACT We use self-similarity in N-body simulations of scale-free models to test for resolution dependence in the mass function and two-point correlation functions of dark matter haloes. We use 10243 particle simulations performed with abacus , and compare results obtained with two halo finders: friends-of-friends (fof ), and rockstar . The fof mass functions show a systematic deviation from self-similarity which is explained by resolution dependence of the fof mass assignment previously reported in the literature. Weak evidence for convergence is observed only starting from haloes of several thousand particles, and mass functions are overestimated by at least as much as $20-25{{\ \rm per\ cent}}$ for haloes of 50 particles. The mass function of the default rockstar halo catalogue (with bound virial spherical overdensity mass), on the other hand, shows good convergence of the order of 50 to 100 particles per halo, with no detectable evidence at the few percent level of any systematic dependence for larger particle number. Tests show that the mass unbinding procedure in rockstar is the key factor in obtaining this much improved resolution. Applying the same analysis to the halo–halo two point correlation function, we find again strong evidence for convergence only for rockstar haloes, at separations sufficiently large so that haloes do not overlap. At these separations, we can exclude dependence on resolution at the $5-10{{\ \rm per\ cent}}$ level once haloes have of the order of 50 to 100 particles. At smaller separations results are not converged even at significantly larger particle number, and bigger simulations would be required to establish the resolution required for convergence.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

The beyond-halo mass effects of the cosmic web environment on galaxies

ABSTRACT Galaxy properties primarily depend on their host halo mass. Halo mass, in turn, depends on the cosmic web environment. We explore if the effect of the cosmic web on galaxy properties is entirely transitive via host halo mass, or if the cosmic web has an effect independent of mass. The secondary galaxy bias, sometimes referred to as ‘galaxy assembly bias’, is the beyond-mass component of the galaxy–halo connection. We investigate the link between the cosmic web environment and the secondary galaxy bias in simulations. We measure the secondary galaxy bias through the following summary statistics: projected two-point correlation function, $w_{\mathrm{p}}(r_{\mathrm{p}})$, and counts-in-cylinders statistics, $P(N_{\mathrm{CIC}})$. First, we examine the extent to which the secondary galaxy bias can be accounted for with a measure of the environment as a secondary halo property. We find that the total secondary galaxy bias preferentially places galaxies in more strongly clustered haloes. In particular, haloes at fixed mass tend to host more galaxies when they are more strongly associated with nodes or filaments. This tendency accounts for a significant portion, but not the entirety, of the total secondary galaxy bias effect. Secondly, we quantify how the secondary galaxy bias behaves differently depending on the host halo proximity to nodes and filaments. We find that the total secondary galaxy bias is relatively stronger in haloes more associated with nodes or filaments. We emphasize the importance of removing halo mass effects when considering the cosmic web environment as a factor in the galaxy–halo connection.

Astronomy & Astrophysics↗
DOE OSTI2026

Density profile of dynamical halos

Among the most fundamental properties of a dark matter halo is its density profile. Motivated by the recent proposal by García et al . [Mon. Not. R. Astron. Soc. 521 , 2464 (2023)] to define a dynamical halo as the collection of orbiting particles in a gravitationally bound structure, we characterize the mean and scatter of the orbiting profile of dynamical halos as a function of their orbiting mass. Here, we demonstrate that the orbiting profile of individual halos at fixed mass depends on a single dynamical variable, the halo radius 𝑟 h , which characterizes the spatial extent of the profile. The scatter in halo radius at fixed orbiting mass is ≈16%. Only a small fraction of this scatter arises due to differences in halo formation time, with late-forming halos being more compact (smaller halo radii). Accounting for this additional correlation results in an ≈11% scatter in halo radius at fixed mass and halo formation time.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

The halo light-cone catalogues of ABACUSSUMMIT

We describe a method for generating halo catalogues on the light-cone using the AbacusSummit suite of N-body simulations. The main application of these catalogues is the construction of realistic mock galaxy catalogues and weak lensing maps on the sky. Our algorithm associates the haloes from a set of coarsely spaced snapshots with their positions at the time of light-cone crossing by matching halo particles to on-the-fly light-cone particles. It then records the halo and particle information into an easily accessible product, which we call the AbacusSummit halo light-cone catalogues. Our recommended use of this product is in the halo mass regime of M halo > 2.1 x 10 11 M ⊙ h -1 for the base resolution simulations, i.e. haloes containing at least 100 particles, where the interpolated halo properties are most reliable. To test the validity of the obtained catalogues, we perform various visual inspections and consistency checks. In particular, we construct galaxy mock catalogues of emission-line galaxies (ELGs) at z ~ 1 by adopting a modified version of the AbacusHOD script, which builds on the standard halo occupation distribution (HOD) method by including various extensions. We find that the multipoles of the autocorrelation function are consistent with the predictions from the full-box snapshot, implicitly validating our algorithm. In addition, we compute and output CMB convergence maps and find that the auto- and cross-power spectrum agrees with the theoretical prediction at the sub-per-cent level.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

MultiCAM: a multivariable framework for connecting the mass accretion history of haloes with their properties

ABSTRACT Models that connect galaxy and halo properties often summarize a halo’s mass accretion history (MAH) with a single value and use this value as the basis for predictions. However, a single-value summary fails to capture the complexity of MAHs and information can be lost in the process. We present MultiCAM, a generalization of traditional abundance matching frameworks, which can simultaneously connect the full MAH of a halo with multiple halo and/or galaxy properties. As a first case study, we apply MultiCAM to the problem of connecting dark matter halo properties to their MAHs in the context of a dark matter-only simulation. While some halo properties, such as concentration, are more strongly correlated to the early-time mass growth of a halo, others, like the virial ratio, have stronger correlations with late-time mass growth. This highlights the necessity of considering the impact of the entire MAH on halo properties. For most of the halo properties we consider, we find that MultiCAM models that use the full MAH achieve higher accuracy than conditional abundance matching models that use a single epoch. We also demonstrate an extension of MultiCAM that captures the covariance between predicted halo properties. This extension provides a baseline model for applications where the covariance between predicted properties is important.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Testing the thermal Sunyaev-Zel’dovich power spectrum of a halo model using hydrodynamical simulations

Statistical properties of large-scale cosmological structures serve as powerful tools for constraining the cosmological properties of our Universe. Tracing the gas pressure, the thermal Sunyaev-Zel’dovich (tSZ) effect is a biased probe of mass distribution and, hence, can be used to test the physics of feedback or cosmological models. Therefore, it is crucial to develop robust modelling of hot gas pressure for applications to tSZ surveys. Since gas collapses into bound structures, it is expected that most of the tSZ signal is within halos produced by cosmic accretion shocks. Hence, simple empirical halo models can be used to predict the tSZ power spectra. In this study, we employed the HMx halo model to compare the tSZ power spectra with those of several hydrodynamical simulations: the Horizon suite and the Magneticum simulation. We examine various contributions to the tSZ power spectrum across different redshifts, including the one- and two-halo term decomposition, the amount of bound gas, the importance of different masses, and the electron pressure profiles. Our comparison of the tSZ power spectrum reveals discrepancies between the halo model and cosmological simulations that increase with redshift. We find a 20% to 50% difference between the measured and predicted tSZ angular power spectrum over the multipole range ℓ = 10 3 − 10 4 . Our analysis reveals that these differences are driven by the excess of power in the predicted two-halo term at low k and in the one-halo term at high k . At higher redshifts ( z ∼ 3), simulations indicate that more power comes from outside the virial radius than from inside, suggesting a limitation in the applicability of the halo model. We also observe differences in the pressure profiles, despite the fair level of agreement on the tSZ power spectrum at low redshift with the default calibration of the halo model. In conclusion, our study suggests that the properties of the halo model need to be carefully controlled against real or mock data to be proven useful for cosmological purposes.

Ayçoberry, Emma (ORCID:0000000292351195)↗
DOE OSTI2020

Concentrations of dark haloes emerge from their merger histories

ABSTRACT The concentration parameter is a key characteristic of a dark matter halo that conveniently connects the halo’s present-day structure with its assembly history. Using ‘Dark Sky’, a suite of cosmological N-body simulations, we investigate how halo concentration evolves with time and emerges from the mass assembly history. We also explore the origin of the scatter in the relation between concentration and assembly history. We show that the evolution of halo concentration has two primary modes: (1) smooth increase due to pseudo-evolution; and (2) intense responses to physical merger events. Merger events induce lasting and substantial changes in halo structures, and we observe a universal response in the concentration parameter. We argue that merger events are a major contributor to the uncertainty in halo concentration at fixed halo mass and formation time. In fact, even haloes that are typically classified as having quiescent formation histories experience multiple minor mergers. These minor mergers drive small deviations from pseudo-evolution, which cause fluctuations in the concentration parameters and result in effectively irreducible scatter in the relation between concentration and assembly history. Hence, caution should be taken when using present-day halo concentration parameter as a proxy for the halo assembly history, especially if the recent merger history is unknown.

Wang, Kuan↗
DOE OSTI2023

A differentiable model of the evolution of dark matter halo concentration

ABSTRACT We introduce a new model of the evolution of the concentration of dark matter haloes, c(t). For individual haloes, our model approximates c(t) as a power law with a time-dependent index, such that at early times, concentration has a nearly constant value of c ≈ 3–4, and as cosmic time progresses, c(t) smoothly increases. Using large samples of halo merger trees taken from the Bolshoi–Planck and MultiDark Planck 2 cosmological simulations, we demonstrate that our three-parameter model can approximate the evolution of the concentration of individual haloes with a typical accuracy of 0.1 dex for $t\gtrsim 2\, {\rm Gyr}$ for all Bolshoi–Planck and MultiDark Planck 2 haloes of present-day peak mass $M_{0}\gtrsim 10^{11.5}\, {\rm M}_{\odot }$. We additionally present a new model of the evolution of the concentration of halo populations, which we show faithfully reproduces both average concentration growth and the diversity of smooth trajectories of c(t), including capturing correlations with halo mass and halo assembly history. Our publicly available source code, diffprof, can be used to generate Monte Carlo realizations of the concentration histories of cosmologically representative halo populations. diffprof is differentiable due to its implementation in the jax autodiff library, which facilitates the incorporation of our model into existing analytical halo model frameworks.

Stevanovich, Dash↗
DOE OSTI2025

Association between optically identified galaxy clusters and the underlying dark matter halos

Clusters of galaxies trace massive dark matter halos in the Universe, but they can include multiple halos projected along lines of sight. Here, we study the halos contributing to clusters using the Cardinal simulation, which mimics the Dark Energy Survey data. We use the red-sequence-based cluster finding algorithm redMaPPer as a case study. For each cluster, we identify the halos hosting its member galaxies, and we define the main halo as the one contributing the most to the cluster's richness ($λ$, the estimated number of member galaxies). At $z=0.3$, for clusters with $λ> 60$, the main halo typically contributes to $92\%$ of the richness, and this fraction drops to $67\%$ for $λ\approx 20$. Defining "clean" clusters as those with $\geq50\%$ of the richness contributed by the main halo, we find that $100\%$ of the $λ> 60$ clusters are clean, while $73\%$ of the $λ\approx 20$ clusters are clean. Three halos can usually account for more than $80\%$ of the richness of a cluster. The main halos associated with redMaPPer clusters have a completeness ranging from $98\%$ at virial mass $10^{14.6}~h^{-1}M_{\odot}$ to $64\%$ at $10^{14}~h^{-1}M_{\odot}$. In addition, we compare the inferred cluster centers with true halo centers, finding that $30\%$ of the clusters are miscentered with a mean offset $40\%$ of the cluster radii, in agreement with recent X-ray studies. These systematics worsen as redshift increases, but we expect that upcoming surveys extending to longer wavelengths will improve the cluster finding at high redshifts. Our results affirm the robustness of the redMaPPer algorithm and provide a framework for benchmarking other cluster-finding strategies.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

Illuminating dark matter halo density profiles without subhaloes

Cold dark matter haloes consist of a relatively smooth dark matter component as well as a system of bound subhaloes. It is the prevailing practice to include all mass, including mass in subhaloes, in studies of halo density profiles in simulations. However, often in observational studies satellites are treated as having their own distinct dark matter density profiles in addition to the profile of the host. This difference can make comparisons between theoretical and observed results difficult. In this work, we investigate density profiles of the smooth components of host haloes by excluding mass contained within subhaloes. We find that the density profiles of the smooth halo component (without subhaloes) differ substantially from the conventional halo density profile, declining more rapidly at large radii. We also find that concentrations derived from smooth density profiles exhibit less scatter at fixed mass and a weaker mass dependence than standard concentrations. Both smooth and standard halo profiles can be described by a generalized Einasto profile, an Einasto profile with a modified central slope, with smaller residuals than either a Navarro–Frenk–White or Einasto profile. Furthermore, these results hold for both Milky Way-mass and cluster-mass haloes. This new characterization of smooth halo profiles can be useful for many analyses, such as lensing and dark matter annihilation, in which the smooth and clumpy components of a halo should be accounted for separately.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

The outer stellar mass of massive galaxies: a simple tracer of halo mass with scatter comparable to richness and reduced projection effects

ABSTRACT Using the weak gravitational lensing data from the Hyper Suprime-Cam Subaru Strategic Program (HSC survey), we study the potential of different stellar mass estimates in tracing halo mass. We consider galaxies with log10(M⋆/M⊙) > 11.5 at 0.2 < z < 0.5 with carefully measured light profiles, and clusters from the redMaPPer and CAMIRA richness-based algorithms. We devise a method (the ‘Top-N test’) to evaluate the scatter in the halo mass–observable relation for different tracers, and to inter-compare halo mass proxies in four number density bins using stacked galaxy–galaxy lensing profiles. This test reveals three key findings. Stellar masses based on CModel photometry and aperture luminosity within R <30 kpc are poor proxies of halo mass. In contrast, the stellar mass of the outer envelope is an excellent halo mass proxy. The stellar mass within R = [50, 100] kpc, M⋆, [50, 100], has performance comparable to the state-of-the-art richness-based cluster finders at log10Mvir ≳ 14.0 and could be a better halo mass tracer at lower halo masses. Finally, using N-body simulations, we find that the lensing profiles of massive haloes selected by M⋆, [50, 100] are consistent with the expectation for a sample without projection or mis-centring effects. Richness-selected clusters, on the other hand, display an excess at R ∼ 1 Mpc in their lensing profiles, which may suggest a more significant impact from selection biases. These results suggest that M⋆-based tracers have distinct advantages in identifying massive haloes, which could open up new avenues for cluster cosmology. The codes and data used in this work can be found here:

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

Insights into the origin of halo mass profiles from machine learning

ABSTRACT The mass distribution of dark matter haloes is the result of the hierarchical growth of initial density perturbations through mass accretion and mergers. We use an interpretable machine-learning framework to provide physical insights into the origin of the spherically-averaged mass profile of dark matter haloes. We train a gradient-boosted-trees algorithm to predict the final mass profiles of cluster-sized haloes, and measure the importance of the different inputs provided to the algorithm. We find two primary scales in the initial conditions (ICs) that impact the final mass profile: the density at approximately the scale of the haloes’ Lagrangian patch RL ($R\sim 0.7\, R_L$) and that in the large-scale environment (R ∼ 1.7 RL). The model also identifies three primary time-scales in the halo assembly history that affect the final profile: (i) the formation time of the virialized, collapsed material inside the halo, (ii) the dynamical time, which captures the dynamically unrelaxed, infalling component of the halo over its first orbit, (iii) a third, most recent time-scale, which captures the impact on the outer profile of recent massive merger events. While the inner profile retains memory of the ICs, this information alone is insufficient to yield accurate predictions for the outer profile. As we add information about the haloes’ mass accretion history, we find a significant improvement in the predicted profiles at all radii. Our machine-learning framework provides novel insights into the role of the ICs and the mass assembly history in determining the final mass profile of cluster-sized haloes.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

A better way to define dark matter haloes

ABSTRACT Dark matter haloes have long been recognized as one of the fundamental building blocks of large-scale structure formation models. Despite their importance – or perhaps because of it! – halo definitions continue to evolve towards more physically motivated criteria. Here, we propose a new definition that is physically motivated, effectively unique, and parameter-free: ‘A dark matter halo is comprised of the collection of particles orbiting in their own self-generated potential’. This definition is enabled by the fact that, even with as few as ≈300 particles per halo, nearly every particle in the vicinity of a halo can be uniquely classified as either orbiting or infalling based on its dynamical history. For brevity, we refer to haloes selected in this way as physical haloes. We demonstrate that (1) the mass function of physical haloes is Press–Schechter, provided the critical threshold for collapse is allowed to vary slowly with peak height; and (2) the peak-background split prediction of the clustering amplitude of physical haloes is statistically consistent with the simulation data, with accuracy no worse than ≈5 per cent.

Astronomy & Astrophysics↗