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At least 73 records · Page 4
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Constraining the Milky Way Mass Profile with Phase-space Distribution of Satellite Galaxies

We estimate the Milky Way (MW) halo properties using satellite kinematic data including the latest measurements from Gaia DR2. With a simulation-based 6D phase-space distribution function (DF) of satellite kinematics, we can infer halo properties efficiently and without bias, and handle the selection function and measurement errors rigorously in the Bayesian framework. Applying our DF from the EAGLE simulation to 28 satellites, we obtain an MW halo mass of $M={1.23}_{-0.18}^{+0.21}\times {10}^{12}{M}_{\odot }$ and a concentration of $c={9.4}_{-2.1}^{+2.8}$ with the prior based on the M–c relation. The inferred mass profile is consistent with previous measurements but with better precision and reliability due to the improved methodology and data. Potential improvement is illustrated by combining satellite data and stellar rotation curves. Using our EAGLE DF and best-fit MW potential, we provide much more precise estimates of the kinematics for those satellites with uncertain measurements. Compared to the EAGLE DF, which matches the observed satellite kinematics very well, the DF from the semi-analytical model based on the dark-matter-only simulation Millennium II (SAM-MII) over-represents satellites with small radii and velocities. We attribute this difference to less disruption of satellites with small pericenter distances in the SAM-MII simulation. Finally, by varying the disruption rate of such satellites in this simulation, we estimate a ~5% scatter in the inferred MW halo mass among hydrodynamics-based simulations.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Accurately simulating core-collapse self-interacting dark matter halos

The properties of satellite halos provide a promising probe for dark matter (DM) physics. Observations have motivated current efforts to explain surprisingly compact DM halos. If DM is not collisionless, but has strong self-interactions, halos can undergo gravothermal collapse, leading to higher densities in the central region of the halo. However, it is challenging to model this collapse phase from first principles. To improve on this, we sought to better understand the numerical challenges and convergence properties of self-interacting dark matter (SIDM) N-body simulations in the collapse phase. Especially, our aim was to better understand the evolution of satellite halos. To do so, we ran SIDM N-body simulations of a low-mass halo in isolation and within an external gravitational potential. The simulation set-up was motivated by the perturber of the stellar stream GD-1. We find that the halo evolution is very sensitive to energy conservation errors, and a SIDM kernel size that is too large can artificially speed up the collapse. Moreover, we demonstrate that the King model can describe the density profile at small radii for the late stages that we have simulated. Furthermore, for our most highly resolved simulation (N = 5 × 10 7 ) we have made the data public. It can serve as a benchmark. Overall, we find that the current numerical methods do not suffer from convergence problems in the late collapse phase and provide guidance on how to choose numerical parameters, for example that the energy conservation error is better kept well below 1%. This allows simulations to be run of halos that become concentrated enough to explain observations of GD-1-like stellar streams or strong gravitational lensing systems.

dark matter↗
DOE OSTI2020

Damped Ly α absorbers and atomic hydrogen in galaxies: the view of the GAEA model

ABSTRACT Using the GAEA semi-analytic model, we analyse the connection between Damped Ly α systems (DLAs) and H i in galaxies. Our state-of-the-art semi-analytic model is tuned to reproduce the local galaxy H i mass function, and that also reproduces other important galaxy properties, including the galaxy mass–gas metallicity relation. To produce catalogues of simulated DLAs we throw 105 random lines of sight in a composite simulated volume: dark matter haloes with log$(\frac{M_{200}}{ {\rm M}_{\odot }}) \ge 11.5$ are extracted from the Millennium Simulation, while for $9.2 \le \log (\frac{M_{200}}{ \mathrm{M}_{\odot }})\lt 11.5$ we use the Millennium II, and for $8 \le \log (\frac{M_{200}}{\mathrm{M}_{\odot }}) \lt 9.2$ a halo occupation distribution model. At 2 < z < 3, where observational data are more accurate, our fiducial model predicts the correct shape of the column density distribution function, but its normalization falls short of the observations, with the discrepancy increasing at higher redshift. The agreement with observations is significantly improved increasing both the H i masses and the disc radii of model galaxies by a factor of 2, as implemented ‘a posteriori’ in our 2M−2R model. In the redshift range of interest, haloes with $M_{200} \ge {10}^{11} \, \mathrm{M}_{\odot }$ give the major contribution to ΩDLA, and the typical DLA host halo mass is $\sim \!{10}^{11} \, \mathrm{M}_{\odot }$. The simulated DLA metallicity distribution is in relatively good agreement with observations, but our model predicts an excess of DLAs at low metallicities. Our results suggest possible improvements for the adopted modelling of the filtering mass and metal ejection in low-mass haloes.

Di Gioia, Serafina↗
DOE OSTI2023

New constraints on warm dark matter from the Lyman- α forest power spectrum

The forest of Lyman-α absorption lines detected in the spectra of distant quasars encodes information on the nature and properties of dark matter and the thermodynamics of diffuse baryonic material. Its main observable—the 1D flux power spectrum (FPS)—should exhibit a suppression on small scales and an enhancement on large scales in warm dark matter (WDM) cosmologies compared to standard Λ⁢CDM. Here, we present an unprecedented suite of 1080 high-resolution cosmological hydrodynamical simulations run with the graphics processing unit-accelerated code cholla to study the evolution of the Lyman-α forest under a wide range of physically motivated gas thermal histories along with different free-streaming lengths of WDM thermal relics in the early Universe. A statistical comparison of synthetic data with the forest FPS measured down to the smallest velocity scales ever probed at redshifts 4.0≲z≲5.2 [E. Boera et al., Revealing reionization with the thermal history of the intergalactic medium: New constraints from the Ly⁢α flux power spectrum, Astrophys. J. 872, 101 (2019)] yields a lower-limit m WDM >3.1 keV (95% C.L.) for the WDM particle mass and constrains the amplitude and spectrum of the photoheating and photoionizing background produced by star-forming galaxies and active galactic nuclei at these redshifts. Interestingly, our Bayesian inference analysis appears to weakly favor WDM models with a peak likelihood value at the thermal relic mass of m WDM =4.5 keV. In conclusion, we find that the suppression of the FPS from free-streaming saturates at k≳0.1 s km -1 because of peculiar velocity smearing, and this saturated suppression combined with a slightly lower gas temperature provides a moderately better fit to the observed small-scale FPS for WDM cosmologies.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2026

Joint X-Ray, Kinetic Sunyaev–Zeldovich, and Weak Lensing Measurements: Toward a Consensus Picture of Efficient Gas Expulsion from Groups and Clusters

There is no consensus on how baryon feedback shapes the underlying matter distribution from either simulations or observations. We confront the uncertain landscape by jointly analyzing new measurements of the gas distribution around groups and clusters—DESI+ACT kinetic Sunyaev–Zel’dovich (kSZ) effect profiles and eROSITA X-ray gas masses—with mean halo masses characterized by galaxy–galaxy lensing. Across a wide range of halo masses ( M 500 = 10 13−14 M ⊙ ) and redshifts (0 < z < 1), we find evidence of more efficient gas expulsion beyond several R 500 than predicted by most state-of-the-art simulations. A like-with-like comparison reveals all kSZ and X-ray observations are inconsistent with the fiducial 1 Gpc 3 hydrodynamical FLAMINGO simulation, which was calibrated to reproduce pre-eROSITA X-ray gas fractions: eROSITA X-ray gas fractions are 2 × lower than the simulation, and the kSZ measurements are combined >8σ discrepant. The FLAMINGO simulation variant with the most gas expulsion, and therefore the most suppression of the matter power spectrum relative to a dark-matter-only simulation, provides a good description of how much gas is expelled and how far it extends; the enhanced gas depletion is achieved by more powerful but less frequent AGN outbursts. Joint kSZ, X-ray, and lensing measurements form a consistent picture of gas expulsion beyond several R 500 , implying a more suppressed matter power spectrum than predicted by most recent simulations. Complementary observables (e.g., thermal Sunyaev–Zel’dovich effect and fast radio bursts) and next-generation simulations are critical to understanding the physical mechanism behind this extreme gas expulsion and mapping its impact on the large-scale matter distribution.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2022

The DESI N -body Simulation Project – II. Suppressing sample variance with fast simulations

ABSTRACT Dark Energy Spectroscopic Instrument (DESI) will construct a large and precise three-dimensional map of our Universe. The survey effective volume reaches $\sim 20\, h^{-3}\, \mathrm{Gpc}^{3}$. It is a great challenge to prepare high-resolution simulations with a much larger volume for validating the DESI analysis pipelines. AbacusSummit is a suite of high-resolution dark-matter-only simulations designed for this purpose, with $200\, h^{-3}\, \mathrm{Gpc}^{3}$ (10 times DESI volume) for the base cosmology. However, further efforts need to be done to provide a more precise analysis of the data and to cover also other cosmologies. Recently, the CARPool method was proposed to use paired accurate and approximate simulations to achieve high statistical precision with a limited number of high-resolution simulations. Relying on this technique, we propose to use fast quasi-N-body solvers combined with accurate simulations to produce accurate summary statistics. This enables us to obtain 100 times smaller variance than the expected DESI statistical variance at the scales we are interested in, e.g. $k \lt 0.3\, h\, \mathrm{Mpc}^{-1}$ for the halo power spectrum. In addition, it can significantly suppress the sample variance of the halo bispectrum. We further generalize the method for other cosmologies with only one realization in AbacusSummit suite to extend the effective volume ∼20 times. In summary, our proposed strategy of combining high-fidelity simulations with fast approximate gravity solvers and a series of variance suppression techniques sets the path for a robust cosmological analysis of galaxy survey data.

cosmology: theory↗
NASA NTRS1991

The three-point correlation function for Abell clusters - Observations and simulations

Results are reported of a 3D study tracing the three-point correlation function of Abell clusters and of a comparison of this with the biased cold dark matter (CDM) simulations of Park (1990). For the observations, the maximum likelihood fits are Z11 = 0.85 and Q111 = 0.1, in excellent agreement with the result of the 2D study of Toth et al. (1989). For Park's simulation, maximum likelihood fits of Q11 = 0.9 and Q111 = 0.0 are found, which is in excellent agreement with the observational result. It is concluded that the standard biased CDM gravitational instability model reproduces the observed form of the three-point correlation functions for Abell clusters remarkably well. The number of triples as a function of the sides of triangles for Abell clusters, and the number of triples as a function of the sides of triangles for clusters in Park's simulation are illustrated on graphs.

Gott, J. R., III↗
DOE OSTI2022

Analytic Gaussian covariance matrices for galaxy N-point correlation functions

Here, we derive analytic covariance matrices for the N-point correlation functions (NPCFs) of galaxies in the Gaussian limit. Our results are given for arbitrary N and projected onto the isotropic basis functions given by spherical harmonics and Wigner 3j symbols. A numerical implementation of the 4PCF covariance is compared to the sample covariance obtained from a set of lognormal simulations, Quijote dark matter halo catalogues, and MultiDark-Patchy galaxy mocks, with the latter including realistic survey geometry. The analytic formalism gives reasonable predictions for the covariances estimated from mock simulations with a periodic-box geometry. Furthermore, fitting for an effective volume and number density by maximizing a likelihood based on Kullback-Leibler divergence is shown to partially compensate for the effects of a nonuniform window function. Our result is recently shown to facilitate NPCF analysis on a realistic survey data.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

An Empirical Model For Intrinsic Alignments: Insights From Cosmological Simulations

We extend current models of the halo occupation distribution (HOD) to include a flexible, empirical framework for the forward modeling of the intrinsic alignment (IA) of galaxies. A primary goal of this work is to produce mock galaxy catalogs for the purpose of validating existing models and methods for the mitigation of IA in weak lensing measurements. This technique can also be used to produce new, simulation-based predictions for IA and galaxy clustering. Our model is probabilistically formulated, and rests upon the assumption that the orientations of galaxies exhibit a correlation with their host dark matter (sub)halo orientation or with their position within the halo. We examine the necessary components and phenomenology of such a model by considering the alignments between (sub)halos in a cosmological dark matter only simulation. We then validate this model for a realistic galaxy population in a set of simulations in the Illustris-TNG suite. We create an HOD mock with Illustris-like correlations using our method, constraining the associated IA model parameters, with the $\mathcal{X}$$^{2}_{dof}$ between our model’s correlations and those of Illustris matching as closely as 1.4 and 1.1 for orientation–position and orientation–orientation correlation functions, respectively. By modeling the misalignment between galaxies and their host halo, we show that the 3-dimensional two-point position and orientation correlation functions of simulated (sub)halos and galaxies can be accurately reproduced from quasi-linear scales down to 0.1 $\mathcal{h}$ –1 Mpc. We also find evidence for environmental influence on IA within a halo. Our publicly-available software provides a key component enabling efficient determination of Bayesian posteriors on IA model parameters using observational measurements of galaxy-orientation correlation functions in the highly nonlinear regime.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Dark branches of immortal stars at the Galactic Center

We show that stars in the inner parsec of the Milky Way can be significantly affected by dark matter annihilation, producing population-level effects that are visible in a Hertzsprung-Russell (HR) diagram. We establish the dark HR diagram, where stars lie on a new stable dark main sequence with similar luminosities, but lower temperatures, than the standard main sequence. The dark matter density in these stars continuously replenishes, granting these stars immortality and solving multiple stellar anomalies. Upcoming telescopes could detect the dark main sequence, offering a new dark matter discovery avenue.

astrophysical simulations↗
DOE OSTI2021

Good and proper: self-similarity of N -body simulations with proper force softening

ABSTRACT Analysis of self-similarity in scale-free N-body simulations reveals the spatial and temporal scales for which statistics measured in cosmological simulations are converged to the physical continuum limit. We examine how the range of scales in which the two-point correlation function is converged depends on the force softening length and whether it is held constant in comoving or proper coordinates. We find that a proper softening that reaches roughly 1/30th of the inter-particle spacing by the end of the simulation resolves the same spatial and temporal scales as a comoving softening of the same length while using a third fewer time-steps, for a range of scale factors typical to Lambda cold dark matter (ΛCDM) simulations. We additionally infer an inherent resolution limit, set by the particle mass and scaling as a−1/2, beyond which reducing the softening does not improve the resolution. We postulate a mapping of these results with spectral index n = −2 to ΛCDM simulations.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Harnessing the Population Statistics of Subhalos to Search for Annihilating Dark Matter

The Milky Way's dark matter halo is expected to host numerous low-mass subhalos with no detectable associated stellar component. Such subhalos are invisible unless their dark matter annihilates to visible states such as photons. One of the established methods for identifying candidate subhalos is to search for individual unassociated gamma-ray sources with properties consistent with the dark matter expectation. However, robustly ruling out an astrophysical origin for any such candidate is challenging. In this work, we present a complementary approach that harnesses information about the entire population of subhalos—such as their spatial and mass distribution in the Galaxy—to search for a signal of annihilating dark matter. Using simulated data, we show that the collective emission from subhalos can imprint itself in a unique way on the statistics of observed photons, even when individual subhalos may be too dim to be resolved on their own. Additionally, we demonstrate that, for the models we consider, the signal can be identified even in the face of unresolved astrophysical point-source emission of extragalactic and Galactic origin. Furthermore, this establishes a new search technique for subhalos that is complementary to established methods, and that could have important ramifications for gamma-ray dark matter searches using observatories such as the Fermi Large Area Telescope and the Cerenkov Telescope Array.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

symfind : Addressing the Fragility of Subhalo Finders and Revealing the Durability of Subhalos

Abstract A major question in ΛCDM is what this theory actually predicts for the properties of subhalo populations. Subhalos are difficult to accurately simulate and to find within simulations, and this propagates into uncertainty in theoretical predictions for satellite galaxies. We present Symfind , a new particle-tracking-based subhalo finder, and demonstrate that it can track subhalos to orders-of-magnitude lower masses than commonly used halo-finding tools, with a focus on Rockstar and consistent-trees . These longer survival times mean that at a fixed peak subhalo mass, we find ≈ 15%–40% more subhalos within the virial radius, R vir , and ≈35%–120% more subhalos within R vir /4 in the Symphony dark-matter-only simulation suite. More subhalos are found as the resolution is increased, in contrast to the Rockstar halo finder, which appears to be converged at smaller subhalo counts. We perform extensive numerical testing. In agreement with idealized simulations, we show that the v max , the maximum circular velocity, is systematically biased low until high resolutions ( n peak ≳ 3 × 10 4 ) are achieved, but that mass loss itself can be resolved at much more modest resolutions ( n peak ≳ 4 × 10 3 ). We show that Rockstar converges to false solutions for the mass function, radial distribution, and disruption masses of subhalos. We argue that our new method can trace resolved subhalos until the point of typical galaxy disruption without invoking post hoc orphan modeling. We outline a concrete set of steps for determining whether other subhalo finders meet the same criteria. We publicly release Symfind catalogs and particle data for the Symphony simulation suite at http://web.stanford.edu/group/gfc/symphony .

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

On the statistical theory of self-gravitating collisionless dark matter flow: Scale and redshift variation of velocity and density distributions

The statistics of velocity and density fields are crucial for cosmic structure formation and evolution. Here, this paper extends our previous work on the two-point second-order statistics for the velocity field [Phys. Fluids 35, 077105 (2023)] to one-point probability distributions for both density and velocity fields. The scale and redshift variation of density and velocity distributions are studied by a halo-based non-projection approach. First, all particles are divided into halo and out-of-halo particles so that the redshift variation can be studied via generalized kurtosis of distributions for halo and out-of-halo particles, respectively. Second, without projecting particle fields onto a structured grid, the scale variation is analyzed by identifying all particle pairs on different scales $r$. We demonstrate that: (i) Delaunay tessellation can be used to reconstruct the density field. The density correlation, spectrum, and dispersion functions were obtained, modeled, and compared with the N-body simulation; (ii) the velocity distributions are symmetric on both small and large scales and are non-symmetric with a negative skewness on intermediate scales due to the inverse energy cascade on small scales with a constant rate $\varepsilon_u$; (iii) On small scales, the even order moments of pairwise velocity $\Delta u_L$ follow a two-thirds law $\propto{(-\varepsilon_ur)}^{2/3}$, while the odd order moments follow a linear scaling $\langle(\Delta u_L)^{2n+1}\rangle=(2n+1)\langle(\Delta u_L)^{2n}\rangle\langle\Delta u_L\rangle\propto{r}$; (iv) The scale variation of the velocity distributions was studied for longitudinal velocities $u_L$ or $u_L^{'}$, pairwise velocity (velocity difference) $\Delta u_L$=$u_L^{'}$-$u_L$ and velocity sum $\Sigma u_L$=$u^{'}_L$+$u_L$. Fully developed velocity fields are never Gaussian on any scale, despite that they can initially be Gaussian; (v) On small scales, $u_L$ and $\Sigma u_L$ can be modeled by a $X$ distribution to maximize the entropy of the system. The distribution of $\Delta u_L$ can be different; (vi) On large scales, $\Delta u_L$ and $\Sigma u_L$ can be modeled by a logistic or a $X$ distribution, while $u_L$ has a different distribution; (vii) the redshift variation of the velocity distributions follows the evolution of the $X$ distribution involving a shape parameter $\alpha(z)$ decreasing with time.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Beyond the two-point correlation: Constraining primordial non-Gaussianity with density-perturbation moments

Constraining primordial non-Gaussianity (PNG) on the large-scale cosmic structure (LSS) is an important step in understanding properties of the early Universe, specifically in distinguishing between different inflationary models. Measuring PNG relies on evaluating the scale-dependent correlations in the density field. New summary statistics beyond the two- and three-point correlation functions in configuration space and their Fourier-space counterparts, the power- and bispectrum may provide increased sensitivity. We introduce a new method for extracting the PNG signal imprinted on the LSS by using the first three Gaussian moments of the normalized correlation in density perturbations, evaluated on varying distance scales. We aim to assess this method’s sensitivity to local PNG, parameterized by f NL . We performed spherical convolutions on a range of scales on dark-matter-halo simulations to measure the scale-dependent correlations in the density field. From these, we computed the first three moments and compared them to a model expectation vector, parameterized to the second power in f NL . Our method provides about 21% improvement in sensitivity to f NL with respect to using the two-point correlation function alone. Notably, we find that the second moment alone carries nearly as much constraining power as the mean, highlighting the potential of higher order statistics. Given its simplicity and efficiency, this framework is well suited for application to current and upcoming large-scale surveys such as the Dark Energy Spectroscopic Instrument (DESI).

early universe↗
DOE OSTI2020

Extensions to models of the galaxy–halo connection

ABSTRACT We explore two widely used empirical models for the galaxy–halo connection, subhalo abundance matching (SHAM) and the halo occupation distribution (HOD), and compare them with the hydrodynamical simulation IllustrisTNG (TNG) for multiple statistics quantifying the galaxy distribution at $n_{\rm gal}\approx 1.3\times 10^{-3}\, ({\rm Mpc}\,h^{-1})^{-3}$. We observe that in their most straightforward implementations, both models fail to reproduce the two-point clustering measured in TNG. We find that SHAM models that use the relaxation velocity, Vrelax, and the peak velocity, Vpeak, perform best, and match the clustering reasonably well, although neither captures adequately the one-halo clustering. Splitting the total sample into sub-populations, we discover that SHAM overpredicts the clustering of high-mass, blue, star-forming, and late-forming galaxies and underpredicts that of low-mass, red, quiescent, and early-forming galaxies. We also study various baryonic effects, finding that subhaloes in the dark-matter-only simulation have consistently higher values of their SHAM-proxy properties than their full-physics counterparts. We then consider a 2D implementation of the HOD model augmented with a secondary parameter (environment, velocity anisotropy, σ2Rhalf-mass, and total potential) tuned so as to match the two-point clustering of the IllustrisTNG galaxies on large scales. We analyse these galaxy populations adopting alternative statistical tools such as galaxy–galaxy lensing, void–galaxy cross-correlations, and cumulants of the density field, finding that the hydrodynamical galaxy distribution disfavours σ2Rhalf-mass and the total potential as secondary parameters, while the environment and velocity anisotropy samples are consistent with full physics across all statistical probes examined. Our results demonstrate the power of examining multiple statistics for determining the secondary parameters that are vital for understanding the galaxy–halo connection.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

How Many Bursts Does It Take to Form a Core at the Center of a Galaxy?

We present a novel method for systematically assessing the impact of the central potential fluctuations associated with bursty outflows on the structures of dark matter halos for classical and ultrafaint dwarf (UFD) galaxies. Specifically, we use dark-matter-only simulations augmented with a manually added massive particle that modifies the central potential and approximately accounts for a centrally concentrated baryonic component. This approach enables precise control over the magnitude, frequency, and timing of rapid outflow events. We demonstrate that this method can reproduce the established result of core formation for systems that undergo multiple episodes of bursty outflows. In contrast, we also find that equivalent models involving only single (or a small number of) burst episodes do not form cores with the same efficacy. This is important because many UFDs in the local Universe are observed to have tightly constrained star formation histories that are best described by a single early burst of star formation. Using a suite of cosmological zoom-in simulations, we identify the regimes in which single bursts can and cannot form a cored density profile. Our results suggest that it may be difficult to form cores in UFD-mass systems with a single early burst, regardless of its magnitude.

cold dark matter↗
NASA NTRS1985

The evolution of large-scale structure in a universe dominated by cold dark matter

The results of numerical simulations of nonlinear gravitational clustering in universes dominated by weakly interacting, 'cold' dark matter are presented. The numerical methods used and the way in which initial conditions were generated are described, and the simulations performed are catalogued. The evolution of the fundamental statistical properties of the models is described and their comparability with observation is discussed. Graphical comparisons of these open models with the observed galaxy distribution in a large redshift survey are made. It is concluded that a model with a cosmological density parameter omega equal to one is quite unacceptable if galaxies trace the mass distribution, and that models with omega of roughly two, while better, still do not provide a fully acceptable match with observation. Finally, a situation in which galaxy formation is suppressed except in sufficiently dense regions is modelled which leads to models which can agree with observation quite well even for omega equal to one.

Davis, M.↗