Halo Asymmetry in the Modeling of Galaxy Clustering

Conventional studies of galaxy clustering within the framework of halo models typically assume that the density profile of all dark matter halos can be approximated by the Navarro–Frenk–White (NFW) spherically symmetric profile. However, both modern N-body simulations and observational data suggest that most halos are either oblate or prolate, and almost never spherical. In this paper we present a modified model of the galaxy correlation function. In addition to the five “classical” halo occupation distribution (HOD) parameters proposed by Zheng et al., it includes an additional free parameter ϕ in the modified NFW density profile describing the asymmetry of the host dark matter halo. Using a subhalo abundance matching model, we populate galaxies within BolshoiP N-body simulations. We compute the projected two-point correlation function w p (r p ) for six stellar-mass volume-limited galaxy samples. We fit our model to the results and then compare the best-fit asymmetry parameter ϕ (and other halo parameters) to the asymmetry of dark matter halos measured directly from the simulations and find that they agree within 1σ. We then fit our model to the w p (r p ) results from Zehavi et al. and compare halo parameters. We show that our model accurately retrieves the halo asymmetry and other halo parameters. Additionally, we find 2%–6% differences between the halo masses ( logMmin and logM 1) estimated by our model and those estimated by “classical” HOD models. The model proposed in this paper can serve as an alternative to multiparameter HOD models, since it can be used for relatively small samples of galaxies.


Introduction
In the standard cosmological model (ΛCDM), the baryonic components of galaxies, in the form of stars, gas, and dust, are thought to be embedded in dark matter (DM) halos.This DM component is dominant and, as such, drives the evolution of the large-scale structure of the Universe.In this scenario, the growth of structure is thought to be hierarchical.DM overdensities first collapsed into small halos, which then grew progressively over time both through the steady gravitational inflow of surrounding DM and through halo mergers.The baryonic component accreted at the centers of these halos indirectly followed the evolution of the DM structure, forming the complex structures observed in the local Universe (e.g., White & Rees 1978;Kaiser 1984; Bardeen et al. 1986;Mo & White 1996;Kauffmann et al. 1997).
Today, direct observations of DM are impossible, except for those provided by weak-lensing studies (Hoekstra et al. 2013;Mandelbaum 2015Mandelbaum , 2018)).Therefore, methods that use a combination of galaxy observations and theoretical models that describe the relationship between galaxies and DM have been developed to study the properties of DM halos.On large cosmic scales, one of the most widely used methods are halo occupation distribution (HOD) models (e.g., Peacock & Smith 2000;Seljak 2000;Magliocchetti & Porciani 2003;Zehavi et al. 2004;Zheng et al. 2005), which are able to constrain the properties of the DM halos by modeling the clustering properties of the galaxies that reside within them.
Within the framework of empirical halo modeling, it is common practice to assume a spherical symmetry of DM halos, with halo density profiles following the form proposed by Navarro et al. (1997; often referred to as the Navarro-Frenk-White (NFW) profile).However, both N-body numerical simulations and observations suggest that these assumptions, while a good first approximation, may not reflect the true shape and mass distribution of DM halos.As a result, most studies lack information about the true shape of DM halos and their asymmetries.The assumption of spherical symmetry could also influence the measurements of the halo masses and thus the final conclusions of such studies.Indeed, it has been shown in weak-lensing studies that halo triaxiality is the main source of uncertainty in halo mass estimates (Osato et al. 2018;McClintock et al. 2019;Zhang et al. 2022).
Moreover, spherically symmetric halos are largely ruled out by N-body numerical simulations, which show that a typical DM halo is a triaxial spheroid (often asymmetric), which tends to be prolate in shape (e.g., Frenk et al. 1988;Dubinski & Carlberg 1991;Warren et al. 1992;Cole & Lacey 1996).In addition, the specific shape of a DM halo depends strongly on both its mass and redshift, with more massive (or/and highredshift) halos being less spherical and more prolate than the less massive and/or lower-redshift ones (Jing & Suto 2002;Hopkins et al. 2005;Kasun & Evrard 2005;Allgood et al. 2006;Bett et al. 2007;Macciò et al. 2007;Muñoz-Cuartas et al. 2011;Schneider et al. 2012;Vega-Ferrero et al. 2017).Deviations from the spherical shape in the most massive halos, such as those hosting clusters of galaxies, are mostly due to their frequent mergers with smaller/less massive halos, usually from one direction (i.e., along filaments), which prevents the DM halo from maintaining a relaxed ellipsoidal shape (e.g., Łokas 2000).
The results of N-body simulations are also strongly supported by observational evidence, mostly from strong and weak gravitational lensing observations.Most of these observations suggest that either oblate or prolate spheroidal shapes of DM halos are the most common, especially for the most massive, cluster-sized DM halos (e.g., Carter & Metcalfe 1980;Sackett & Sparke 1990;Evans & Bridle 2009;Kawahara 2010;Oguri et al. 2010Oguri et al. , 2012;;Sayers et al. 2011;Despali et al. 2017;Chiu et al. 2018;Okabe et al. 2020;Hellwing et al. 2021;Lau et al. 2021;Gonzalez et al. 2022).Measurements based on the proper motions of globular clusters in our own Milky Way from the Gaia data (Gaia Collaboration et al. 2018) also favor a prolate rather than a spherical shape of the Milky Way DM halo (Posti & Helmi 2019;Watkins et al. 2019).
Simplified assumptions about the symmetry and mass profile can then lead to an under/overestimation of the size of the DM halo and an erroneous estimate of the number of galaxies that may reside in it.Hayashi & Chiba (2012) found that mass estimates of DM-dominated dwarf spheroidal galaxies are sensitively dependent on the assumed mass profiles and shapes of the DM halos.They showed that the M 300 mass (mass enclosed in a spheroid with a major-axis length of 300 pc) can be significantly overestimated when spherical symmetry is assumed for both stellar and dark halo density profiles.On larger scales, Corless & King (2007) investigated the effect of the assumption of spherical symmetry of the DM halo on measurements of massive clusters based on weak gravitational lensing and found that halo masses can be overestimated by up to 50% if halo asymmetry is not taken into account.
Independent of the problem of the shape of DM halos, numerous studies based on hydrodynamical simulations and semianalytical models have shown that galaxy occupation is strongly related to many other secondary halo properties (e.g., Artale et al. 2018;Zehavi et al. 2018;Hadzhiyska et al. 2020;Xu et al. 2021;Yuan et al. 2021).These effects are commonly referred to as galaxy assembly bias and have been shown to be a significant source of error in studies of the galaxy-halo relationship (e.g., Zentner et al. 2014).
In recent years, many studies have focused on improving and extending HOD models to account for these effects.On the empirical modeling side, Hearin et al. (2016) created so-called decorated HOD models that minimally expand the parameter space with respect to "classical" HODs to account for the assembly bias.Other groups took advantage of the precision and volume of the latest cosmological simulations and created HOD frameworks that include multiple parameters to account for additional effects such as velocity bias, environment-based bias, and concentration bias (e.g., Zheng & Guo 2016;Wibking et al. 2019;Zhai et al. 2019;Wibking et al. 2020;Yuan et al. 2022).
However, a "classical" HOD model coupled with the NFW density profile is still widely used in studies of galaxy clustering, despite its known weaknesses and in the presence of superior models (e.g., see recent work by Gao et al. 2022;Harikane et al. 2022;Lange et al. 2022;Linke et al. 2022;Qin et al. 2022;Yung et al. 2022;Herrero Alonso et al. 2023;Petter et al. 2023;Zhai et al. 2023).This shows the need for simple, easy-to-apply, and computationally cheap HOD models that describe observed galaxy clustering reasonably well.
In this paper we take such a minimalist approach.We present an empirical two-point correlation function model that consists of the combination of the "classical" HOD model and a modified NFW density profile that includes an additional free parameter.This parameter accounts for the DM halo asymmetry.Our main goal is to show that a simple modification of the NFW density profile is sufficient to make a relatively unbiased prediction of the galaxy-halo connection, especially when measurements are made on relatively small data samples.We focus mainly on the modeling of the projected correlation function w p (r p ), as this is the most commonly used statistic to measure galaxy clustering based on observations.To be clear, our model is certainly not designed to be used to populate large N-body simulations, as it does not take into account any of the secondary effects mentioned above.Instead, we focus on its applicability to clustering studies on small samples of galaxies, for which extended multiparameter HODs prove to be overly complex.
We begin by fitting the projected two-point correlation function w p (r p ) of the mock galaxy catalog with our model.Using the results of this fitting, we show the accuracy of this new model and its usability for clustering measurements.We then apply our model to correlation functions measured by Zehavi et al. (2011) and show the differences between results obtained with and without the DM halo shape assumption.
The paper is structured as follows.In Section 2 we briefly describe the properties of the cosmological simulations used in this work and the methods used to identify DM halos and to populate them with galaxies.We then examine the main properties of DM halos relevant to this work, and we finally present the selection of the galaxy samples.Next, in Section 3 we present the formulae we use to describe the DM halo shape and introduce a free parameter f, which is then included in the modified NFW density profile to account for the DM halo asymmetry.The methods for measuring and modeling the galaxy correlation function are presented in Section 4. Our results are presented in Section 5.There we also present the differences between the halo masses estimated using our model and the results based on the HOD model assuming halo spherical symmetry from Zehavi et al. (2011).The limitations and applicability of our model are discussed in Section 6.Finally, a summary and conclusions are presented in Section 7.
Throughout this paper, we refer to distances in comoving units.We adopt a flat ΛCDM cosmology, with Ω M = 0.307, Ω Λ = 0.693 (Planck Collaboration et al. 2014) for measurements based on the mock galaxy catalog and Ω M = 0.3, Ω Λ = 0.7 for measurements based on the correlation functions from Zehavi et al. (2011).In both cases the distances are given in units of h −1 Mpc (where h = H 0 /100 km s −1 Mpc −1 ).

Data
Cosmological N-body simulations are proving to be a unique tool for detailed studies of large-scale structure evolution, especially within the ΛCDM model framework.This is also true in the context of our work.The very good resolution of modern N-body simulations allows detailed studies of DM structures and their asymmetries even on small cosmic scales (<1 Mpc).
In this section we briefly describe the main features of the large N-body simulation used in our work-Bolshoi-Planck (BolshoiP).For a detailed description of this data set, we refer the reader to the dedicated paper by Klypin et al. (2016).We also present the methods used to populate the galaxies within the simulated DM halos and the selection of stellar-mass subsamples from these galaxy sets.

Cosmological N-body Simulations
In this paper we use the BolshoiP N-body DM-only cosmological simulation (Klypin et al. 2011(Klypin et al. , 2016)).The parameters describing this simulation, such as comoving volume, number of particles, and mass resolution, are listed in Table 1.Below we briefly describe only some of its aspects that are important in the context of our work.
The BolshoiP is the high-resolution simulation of 2048 3 (≈8.6 × 10 9 ) collisionless DM particles distributed within a comoving volume of (250 h −1 Mpc) 3 over the redshift range from z = 80 to the present day.In this paper we focus on the redshift z = 0. BolshoiP assumes a flat ΛCDM cosmology, with cosmological parameters obtained from the Planck mission data and published by Planck Collaboration et al. (2014;i.e., Ω M = 0.307, Ω Λ = 0.693, h = 0.7, n = 0.96, σ 8 = 0.82).The high mass resolution, relatively large volume box, and updated cosmological parameters make this simulation ideal for clustering studies such as those presented in this paper.

Creating the Galaxy Mock Catalog
We used the publicly available Python package Halotools (v0.7; see Hearin et al. 2017, for a description of the first release) to identify the DM halos in the BolshoiP simulations.This package includes the ROCKSTAR halo finder algorithm (Behroozi et al. 2013), which uses the precise approximation of the DM halos by associating nonspherically symmetric ellipsoids with their mass distribution (Allgood et al. 2006).We focus on DM halos with masses in the range of 10 10.5 -10 15 M e at z = 0.
In the next step we populated the halos and subhalos with galaxies-creating mock catalogs.We use a subhalo abundance matching technique (SHAM; Kravtsov et al. 2004).To implement this method, we again used the Halotools Python package and one of its prebuilt galaxy-halo connecting models: the stellar-to-(sub)halo (SMHM) models proposed by Behroozi et al. (2010).This model connects the stellar mass of galaxies to the mass of their host DM halos and subhalos by assuming a direct relation given in parameterized form of the SMHM function by Behroozi et al. (2010, see Table 2 therein).The level of intrinsic scatter is set to 0.2 dex.The central galaxies are placed at the center of their host DM halo and the satellite galaxies at the centers of subhalos.The most important aspect of this model is that the true distribution of subhalos is not assumed to be spherically symmetric, so the distribution of satellite galaxies reflects the asymmetries of the host DM halo.These mock catalogs have been created without any additional observational strategies.
We then further divide the galaxy mock catalog into six stellarmass volume-limited subsamples: M1 ( ( ).The general properties of these subsamples, such as number of galaxies, mean stellar mass, and mean host DM halo asymmetry parameters-the triaxiallity parameter T (see Section 3.1) and the parameter f proposed in this work (described in Section 3.2)-are presented in Table 2.

Dark Matter Halo Shape
In this section we define the asymmetry parameter f.To better explain how this parameter relates to the halo shape, we first show its relation to the triaxiality parameter T. We then show how it can be used as a free parameter to modify the NFW density profile later used in the modeling of the galaxy correlation function.

Standard Description of Halo Asymmetry-Triaxiality T
The shape of a halo is usually characterized by three ellipsoidal axes a, b, c, with a b c, usually expressed as the ratio of the second-largest axis to the largest axis (μ = b/a) and the ratio of the smallest axis to the largest axis (η = c/a).The triaxiality parameter T is a combination of these two parameters (Franx et al. 1991): Three shapes of DM halos can be defined based on the value of the parameter T: oblate when 0 < T < 1/3, triaxial when 1/3 < T < 2/3, and prolate when 2/3 < T < 1.
The value of the parameter T depends strongly on the mass of the DM halo, with more massive halos being more asymmetric and their overall halo shape often being prolate (e.g., Vega-Ferrero et al. 2017).This is also the case in our halo catalogs.The prolate halo shape dominates and accounts for more than 50% of all possible halos in our sample.
It is well established that the halo mass is strongly correlated with the stellar mass of the galaxy (see, e.g., Meneux et al. 2008;Beutler et al. 2013;Marulli et al. 2013;Dolley et al. 2014;Skibba et al. 2015;Durkalec et al. 2018).Since the most massive halos are also the most asymmetric, it is not surprising that in our mock galaxy catalogs we found a strong correlation between the host halo asymmetry parameter T and the stellar mass of the hosted galaxies.As shown in Figure 1, about 67% of the most massive galaxies, with stellar masses greater than 10 11.25 M e , reside in halos of asymmetric, prolate shape.This percentage drops to 45% for galaxies with stellar masses less than 10 10 M e .This means that the introduction of halo shape dependence in halo modeling of galaxy clustering will be particularly important for high-mass samples.

Defining a Novel Asymmetry Parameter f
Introducing the common description of halo asymmetrythe triaxiality T-into the halo model would require us to use two additional free parameters (μ and η).This could be computationally demanding for many studies.As an alternative, we propose a new asymmetry parameter f.It has been constructed to measure how much the shape of the DM halo deviates from spherical symmetry.In our definition of f we assume spherical symmetry along one axis and measure the deviation from this symmetry along the other axis.This approach greatly simplifies the implementation of halo asymmetry in halo models.The asymmetry parameter f is defined as follows: where These two parameters-the triaxiality parameter and the newly defined asymmetry parameter f-are obviously not independent.Their relationship is shown in Figure 2.There are two distinct areas in the plot-regions with f less than 1 and regions with f greater than 1.The region of f < 1 correlates with values of T between 0.5 and 1, indicating that host DM halos have mostly prolate shapes.On the other hand, f > 1 are correlated with values of T between 0 and 0.5, i.e., host DM halos are predominantly oblate in shape.
By construction, the parameter f assumes a value of f = 1 for an idealized case of spherically symmetric halos (i.e., when the ellipsoidal axes are equal to each other).The farther away from this value, the greater the deviation from spherical symmetry (i.e., the greater the asymmetry of the DM halo).f > 1 defines halos with oblate shape asymmetry, while f < 1 defines a prolate shape.
As an example of how this parameter should be interpreted, we use the mock galaxy catalog.Since the majority of mock galaxies are located within prolate halos and most massive galaxies are more likely to be in the most asymmetrical halos (see Figure 1), we expect f < 1 for all galaxy stellar-mass samples, and we expect its value to decrease with increasing stellar mass.Indeed, as shown in Figure 3, f is less than 1 for all samples, and the median, f med , systematically moves away from f = 1 (marking the spherically symmetric halo) with increasing stellar mass of the hosted galaxies, from f med = 0.75 The percentage of galaxies residing in a halo with one of the three types of asymmetry varies with stellar mass.The most massive galaxies are most likely to be in the prolate halos.This means that the halo shape might be the important factor in galaxy clustering modeling of these galaxies.for the least massive galaxies with mean stellar masses 10 10.01 M e (sample M1) to f med = 0.57 for the most massive galaxies with median stellar masses of 10 11.31 M e (sample M6).Similar results are therefore expected from the projected correlation function fit.

Defining Modified NFW Density Profile with Asymmetry
Parameter f One of the most important components of the halo model is the density profile of the DM halo.The most commonly used in the literature is the so-called NFW density profile proposed by Navarro et al. (1997).This symmetric profile is easy to adapt into halo models, but it may not properly reflect the real DM halo density profile.In particular, it does not take into account the possibility that the DM halo may have a nonspherical shape, which is a common occurrence as shown in Section 3. To account for this halo property, we propose to extend the standard NFW density profile by adding an additional parameter f (see Equation (2)), which can be interpreted as a deviation of the halo shape from spherical symmetry.
In the general case, the DM NFW profile is defined as where ρ crit is a present (z = 0) critical mass density, δ c is the overdensity of the DM halo, R s is a characteristic DM halo radius, and R is a three-dimensional vector described by the three ellipsoidal axes a, b, and c as follows: When a = b = c, the NFW density profile becomes spherically symmetric.However, if we assume that only one of two pairs of axes a and b or c and b is approximately equal, then we can rewrite this equation as where the two-dimensional vector x 2 + y 2 or y 2 + z 2 is represented by r.
If we also assume (in the first approximation) that r is proportional to z, then where c/a and a/b measure the asymmetry of the DM halo profile shape (in two dimensions) with respect to the spherical shape.Rather than using c a and a b , we can take the average length of the two axes divided by the third one.Now Equation (6) assumes the form ( ) and the variable in parentheses can be rewritten making use of the parameter f (as defined in Equation ( 2)).Finally, we can express R as where parameter f has values in the range 0 < f < ∞ , and a factor 1 2 is present owing to the fact that the f = 1 profile is required to be spherically symmetric.
The density profile modifications and the addition of the asymmetry parameter f of course affect the modeled two-point correlation function.In Figure 4 we show how the modeled correlation function changes for different halo asymmetries while the other HOD parameters are fixed at the same values.As shown, the asymmetry of the halo mostly affects the correlations on scales of <1 h −1 Mpc (one-halo term), where it has a significant influence on the shape of the correlation function.The difference between a spherically symmetric halo (marked with f = 1) and an asymmetric (prolate) halo with f = 0.4 at scales r p = 0.2 h −1 Mpc reaches Δw p (r p ) = 663.35(3.5 dex) in the case presented.The assumption of spherical symmetry of the DM halo can therefore influence the results of the w p (r p ) fitting.

Correlation Function, the Baseline HOD Model, and
w p (r p ) Fitting

Correlation Function
To measure the correlation functions presented in this paper,4 we used the Haltools v0.7 Python package (v0.7;see Hearin et al. 2017, for a description of the first version).For details we refer the reader to documentation of this code.Here we will only present the most important details.A commonly used estimator for the two-point correlation function was introduced by Landy & Szalay (1993): where N G and N R are the total number of objects in the galaxy sample and random points generated in the same volume and with the same geometric properties as the real sample, respectively.GG, GR, and RR are the number of galaxygalaxy, galaxy-random, and random-random pairs at a given separation radius, respectively.The random points catalog has been constructed to contain 100 times more objects than the galaxy catalog.The projected two-point correlation function is computed by integrating ξ over π between 0 and p = max 60 Mpc h -1 : For each volume-limited stellar-mass galaxy sample this projected correlation has been measured in 12 equally spaced (on the logarithmic scale) radial bins, except for the most massive sample M6, where we used eight bins owing to the small number of galaxies available.In each case the correlation functions have been measured over the range from The statistical errors of the correlation function measurements were estimated using a jackknife resampling method.We created N = 125 equal-volume cubic boxes, each of size 50 3 Mpc 3 h -3 covering the entire volume of the galaxy sample.Then, we created subsamples, systematically leaving out one of the boxes.The error covariance matrix, which describes a total dispersion between these samples, was computed using where wp i , and wp j , are the means of the correlation function in bins i and j, respectively.

Halo Occupation Distribution
In our work we use the baseline five-parameter HOD model described in Zheng et al. (2007): and α.M min denotes the minimum halo mass for which half of the DM halos contain a central galaxy above the adopted stellar-mass (or luminosity) threshold for this sample.M 1 denotes the satellite halo mass for which a DM halo contains on average one additional satellite galaxy, while M 0 denotes the cutoff mass scale.The scatter between the stellar mass (or luminosity) of the galaxies and the halo mass is given by s M log h , while α is the power-law slope of the galaxy mean occupation function.
The core assumption of this HOD model is that the number of galaxies residing within the host DM halo is a function of the mass of that halo 〈N g (M h )〉 and that the total number of galaxies within an average halo is a sum of the average occupation of central 〈N cen (M h )〉 and satellite 〈N sat (M h )〉 galaxies: Finally, using the best-fit HOD parameters, we are able to obtain the average DM halo mass 〈M h 〉 hosting a given galaxy population with and the large-scale galaxy bias b g as is the DM mass function for which we adopted the fitting formula proposed by Tinker et al. (2008) and ( ) n z g represents the number density of galaxies,

Model Fitting Procedure
To fit our model to the correlation functions, we use Markov Chain Monte Carlo (MCMC) methods.The MCMC sampling was done by implementing the affine-invariant ensemble sampler of Goodman & Weare (2010), which is provided by the publicly available Python library emcee (Foreman-Mackey et al. 2013).To generate the posterior parameters for each fit, we run an MCMC with n = 25 random walkers (chains), each of which explores the parameter space starting from different, randomly chosen initial parameters.At each step of the random walk, a new set of HOD parameters is generated from a Gaussian distribution with a fixed variance σ 2 and is accepted if is less than a random number generated from a uniform distribution in the range [0, 1].To compute χ 2 , we use the measured values of the projected correlation function w p (r p ) with the full error covariance matrix C and the number density of galaxies n g in each subsample as where w p is a vector containing measurements of the two-point correlation function.We assume a 1% uncertainty s n g for the observed number density.The suffixes "obs" and "mod" denote the values measured from galaxy mock (or observation) catalogs and HOD model predictions, respectively.The best-fit HOD parameters are determined by finding the 50th percentile of the marginal posterior probability distribution of all random walk realizations.Uncertainties are taken as the 16th and 84th percentiles.
For each fit, we use the following methods to ensure its convergence: 1.The total number of random walk steps for each chain realization is determined by using the integrated autocorrelation time (IAT) τ (Foreman- Mackey et al. 2013).
Following this method, we first determine the N burn steps required for convergence.We assume , 18 burn where τ is calculated using methods available in the emcee library.After reaching N burn , we then continue for an additional number of steps not less than the size of N burn for the given galaxy sample.2. We ensure that the χ 2 value for each random walk chain converges and stabilizes at the lowest possible value.
As an example, we show an implementation of these methods in Figure 5.For this figure we implement the above methods for the correlation function modeling of the results from mock galaxy sample M2.Presented results are representative of the other samples.The top panel shows the IAT as a function of the chain length N iter .As shown, the IAT or τ increases and reaches a plateau (a true autocorrelation time) after N burn iterations (shaded area), marking the number of iterations sufficient for the fit to converge.In this case, for the M2 subsample, N burn = 41,243.This number of course varies from sample to sample, as listed in Table 3.

Results
We computed the projected two-point correlation function w p (r p ) for six volume-limited stellar-mass galaxy samples selected from mock galaxy catalogs populated in the BolshoiP N-body simulation (see Section 2 for the description of the subsample selection).
For each correlation function measurement, we performed model fitting using the modified model with the DM halo asymmetry parameter included (see Section 3.3).All measured correlation functions with the best fit are shown in the upper right corner of Figure 6, while the obtained best-fit parameters are listed in Table 4.
In Figure 6 we show a corner plot with the result of the MCMC fitting of the six-parameter model to the projected correlation function of mock galaxy sample M5 (as a representative of the other results).Overall, all of the HOD parameters are slightly correlated, with the strongest correlation between s M log and satellite mass M 1 .The correlations with parameter f are not significantly different from those between other HOD parameters.Parameter f is the most strongly correlated with the satellite halo mass M 1 , which is expected since these two are related to the one-halo term.

Model Reliability
In this section we show that the model introduced in this paper provides accurate measurements of the characteristic DM halo masses and halo shape.
First, the modeled w p (r p ) can reproduce the shape of the measured correlation function very well, as shown in the upper right corner of Figure 6.The model proves to be suitable for typical correlation function measurements covering distances r p from 0.1 to 20 h −1 Mpc.
Second, the inferred HODs are similar to the "true" HODs obtained from the simulations, as is illustrated in Figure 7, where we plot 〈N g 〉(M) for all the samples selected from the mock stellar masses compared to the results obtained from the fit.The largest differences are seen for samples M5 and M6.This may be related to the fact that these samples are the least numerous, making the measurement of the correlation function and the fitting of the model less reliable.
In addition, as shown in Figure 8, the characteristic host halo masses M min and M 1 obtained from the HOD model fit are similar (within 1σ) to the true values for all stellar-massselected galaxy samples.The largest differences are seen for the low-mass galaxy samples, where the M min masses are underestimated.This could be related to the fact that the onehalo term is weakest in these subsamples, which reduces the accuracy of the fit on small scales, affecting the halo mass estimate.Similarly, the underestimation of M min for the most massive galaxy sample may be related to the relatively small number of galaxies affecting the reliability of the correlation function measurement.
Finally, the most important aspect of the introduction of our model is to provide accurate information about the shape of DM halos.Our model successfully describes the average asymmetry of the host halo.In the bottom panel of Figure 8 we compare the best-fit parameter for the DM halo asymmetry f (filled symbols) with the "real" values measured directly from  the BolshoiP mock catalogs (dashed line).The results are in very good agreement for all stellar-mass subsamplesdiscrepancies between the best-fit parameter f and a corresponding "true" value are in the 1σ range.

Application to SDSS Clustering Results from Zehavi et al. (2011)
In their paper Zehavi et al. (2011) present the luminosity and color dependence of galaxy clustering as seen in the Sloan  Digital Sky Survey (SDSS).They measure the two-point correlation function and quantify it using five-parameter HOD models including an NFW halo density profile and thereby assuming a spherical symmetry of DM halos.We use their w p (r p ) measurements along with the covariance matrices for luminosity-threshold-selected samples, kindly provided by the authors of that paper, and repeat the HOD modeling, this time using our proposed model.To ease the comparison, we adopt the same cosmology and concentration −halo mass relation as Zehavi et al. (2011).In this section we present the results of this fitting and compare our results to the original measurements presented in Zehavi et al. (2011).
The fitting methods are exactly the same as described in Section 4.3.The correlation function with the best-fitting models is shown in the upper right corner of Figure 9.For each fit, our model is able to reproduce the shape of w p (r p ).As a representative example of other fits, Figure 9 also shows a corner plot with the results of the MCMC fit to the M r < −21.0 projected correlation function from Zehavi et al. (2011).Parameter f is well constrained, and we observe a mild correlation of this parameter with logM 1 and α, which is expected, as both these parameters are related to the one-halo term.The parameters of the best fit are shown in Table 5.
The most interesting aspect of the six-parameter halo modeling proposed in this paper is the information about the halo asymmetry.In the top panel of Figure 10 we show how the best-fit host halo asymmetry parameter f changes with the luminosity of the galaxy sample.In general, we observe that the host halo asymmetry becomes stronger with increasing sample luminosity and hence with increasing host halo mass.This observation is consistent with our previous conclusions based on mock catalogs: the most massive halos tend to be the most asymmetric (more prolate).However, in the case of measurements based on observations, the change in asymmetry is not continuous.It starts with a plateau of slightly oblate host , the asymmetry of the host halo increases up to f ∼ 0.8, finally changing to the most asymmetric (prolate) halos of f from 0.5 to 0.3 for the most luminous samples.We also note the changes in the parameter measurement uncertainty, which is largest for the lowluminosity subsamples.This could be related to the weak one-halo term observed for these samples.The accuracy of the fit is therefore lower at small scales, which is reflected in a higher uncertainty of the best-fit parameter.In each plot, the solid black lines represent the average number of galaxies 〈N g 〉, the dotted line represents the average number of central galaxies 〈N c 〉, and the dashed line represents the average number of central galaxies 〈N s 〉.The series of gray lines in each plot represent 50 randomly selected HODs from the MCMC chains that are within Δχ 2 < 1 relative to the best-fitting model.The value of f measured for the sample of galaxies < -M 19.5 r max deviates significantly from the observed trend.This deviation can be explained by the cosmic variance effect on the measurement caused by the Sloan Great Wall (SGW), a supercluster observed at z ∼ 0.8 (see Gott et al. 2005).According to tests performed by Zehavi et al. (2011), for the luminosity threshold samples used in our work, the presence of the SGW mainly affects the < -M 20.0 r max and < M r max -19.5 samples.
Nevertheless, the observed behavior of the asymmetry parameter is related to the clustering dependence on luminosity observed by Zehavi et al. (2011).The correlation functions for < -M 18.5 r max and < − 19.5 samples are nearly identical (hence the plateau of nearly identical results).Then, there is an increase in clustering strength when moving to samples with < -M 20.5 Another point of comparison are the two characteristic halo masses-the minimum halo mass M min and the satellite halo mass M 1 -obtained from the best fit. 5 As shown in the second panel of Figure 10, these halo masses differ between the two models.The M min halo masses obtained by Zehavi et al. (2011) are consistently underestimated (on average by 3% in M log min ) with respect to the values obtained with our model.The differences are larger for low-luminosity galaxy samples (M r > −19.0), but these also have larger uncertainties.Similarly, the M 1 halo masses obtained by Zehavi et al. (2011) are smaller than those obtained in our work, but only for samples with M r > −20.0; the differences are also stronger, reaching on average 4.6% (in logM 1 ) for these samples.For galaxies brighter than M r = −20.0, the values of the satellite halo mass M 1 obtained by the two models are almost identical.This indicates that the satellite masses are not affected by the halo asymmetry in this luminosity range.
Although we use the same cosmology and concentration −halo mass relation as Zehavi et al. (2011), the differences in M min and M 1 described in the previous paragraph cannot be related to the halo asymmetry alone.There are other subtle differences between model components that have been shown to play a role in halo mass estimates.The most important are the halo mass function, the large-scale halo bias, and the halo exclusion method.In addition, even if we use the same model for the concentration-mass relation, our virial mass definition is modified by the parameter f (see Appendix A), while in Zehavi et al. (2011) it is not.We therefore proceed with the more direct comparison, using exactly the same model as proposed in this paper but fixing f = 1, which represents spherically symmetric halos.The results of the best-fitting parameters of this model are represented in Figure 10 by open symbols.In this case the difference between the model with a free asymmetry parameter and the model with a fixed f = 1 is on average 1% for M log min .However, toward the highest-luminosity samples (M r < −20.5) this difference increases to ∼3% and exceeds 1σ errors.
The situation is similar for satellite halo masses M 1 .On average the difference is 2% for logM 1 .However, noticeably for the two most luminous samples it reaches 6%.This indicates that the halo shape has the strongest influence on the halo mass estimates of the most massive halos that host the brightest galaxies.In the case of the average host halo masses 〈M h 〉, obtained using the best-fit parameters, the results are very similar for both models, as shown in the third panel of Figure 10.However, we tentatively observe a trend where for the brightest galaxies M r < −21.0, hosted by the most massive halos, the estimates of 〈M h 〉 both from Zehavi et al. (2011) and from the model with fixed f = 1 are underestimated with respect to the asymmetric six-parameter model, while remaining within 1σ for low-and intermediate-luminosity galaxies.This suggests that for the most massive halos, which are also the most asymmetric (prolate), the assumption of spherical symmetry may influence the average halo mass estimates.These results need to be confirmed using correlation function measurements based on more numerous samples.
Finally, when comparing the large-scale galaxy bias b g , obtained using results from both the "classical" model and the six-parameter model proposed in this paper, we see no difference between the models.The trend of b g increasing with the luminosity of the galaxy sample is preserved, and the results from all discussed models are well within 1σ errors-as shown in the bottom panel of Figure 10-indicating that the halo asymmetry does not affect the galaxy bias measurements.

Model Limitations and Realistic Usability
Our model can be described as a conventional HOD model.As such, it suffers from the well-known shortcomings of this type of model.In particular, it assumes that the halo mass is the main driver of the galaxy−halo connection.There are known violations of this assumption, commonly referred to as galaxy assembly bias (or simply assembly bias), meaning that the galaxy occupation is strongly related to several secondary halo properties other than halo mass (see, e.g., Miyatake et al. 2016;Artale et al. 2018;Zehavi et al. 2018;Hadzhiyska et al. 2020;Xu et al. 2021;Yuan et al. 2021).Zentner et al. (2014) showed that ignoring the assembly bias in halo occupation modeling leads to significant systematic errors, especially for extreme populations, such as star-forming or quenched galaxies.
In our model we address only one of the possible sources of assembly bias: halo asymmetry.We aim to answer the question, can we build a simple halo model that accounts for halo asymmetry and is able to make reasonably good predictions about galaxy−halo connections?In particular, can it be used to model w p (r p ) measured for observational data and provide information about the mass and asymmetry of the halo?As we show in Section 5, the best-fit parameters from the w p (r p ) fitting of our model are in good agreement with the "true" values from the simulations on which the correlation function measurements are based.We are also able to fit our model to observational data and obtain the average host halo asymmetry for any given galaxy sample.
It should be noted, however, that our model makes a number of assumptions that may not hold for all galaxy samples.The first assumption is in the definition of the halo asymmetry parameter f itself-it does not describe the shape of the halo, but rather the deviation from spherical symmetry.At this stage we assume that the two axes of the ellipsoidal halo are the same and measure how the other axes deviate from this symmetry.This is a good first approximation (allowing us to limit the number of free parameters), and as we show in Section 5, the model performs well and is able to accurately retrieve the "true" values from the w(r p ) fit.
The other known weakness is the use of the NFW density profile as a universal recipe for modeling DM halos, regardless of their size and mass.Many studies point out that the NFW density profile can only reproduce the real mass distribution of DM halos with a very limited accuracy.For example, at the scale of single galaxy halos, Gentile et al. (2004) showed that the NFW density profiles are inconsistent with the measured (using velocity curves) DM distributions of spiral galaxies because they do not take into account the central density core that occurs in these galaxies.It has also been shown that the deviations from the spherical NFW profile increase when we consider the most massive halos (e.g., Klypin et al. 2016).These shortcomings are mitigated in our model by the introduction of an asymmetry dependence.
Despite all these known problems, the NFW density profile, coupled with "classical" HOD models, is still widely used (see, e.g., recent studies by Gao et al. 2022;Harikane et al. 2022;Lange et al. 2022;Linke et al. 2022;Qin et al. 2022;Yung et al. 2022;Herrero Alonso et al. 2023;Petter et al. 2023;Zhai et al. 2023).The reason for this is its applicability to a wide range of data, especially those of limited size.For these samples, systematic errors related to the assembly bias are negligible compared to other uncertainties related to the sample size.The model proposed in this paper can easily be used in these types of studies, complementing the "standard" halo mass measurement with information on halo asymmetry.However, it should not be used as a method to associate galaxies with the simulated halos.

Comparison with Different Models
With all these limitations in mind, we examine how our modified six-parameter model compares with other models.We use exactly the same measurements of the correlation functions for stellar-mass-selected mock galaxy samples (see Section 2.2) and fit two additional models.The first model, henceforth called classicHOD, is virtually identical to the model proposed in this paper.All model components (e.g., concentration-mass relation, halo mass function) are the same, but we fix the parameter f = 1.With the second model, hereafter called concentrationHOD, we test the influence of the concentrationmass relation and the asymmetry on the modeled correlation function.Again, we fix f = 1 and keep the other components the same, except for the halo concentration-mass relation, which we change from the power-law relation (see Appendix A) to the one proposed by Ludlow et al. (2016).
In Figure 11 we show the comparison of the best-fit characteristic halo masses M min and M 1 obtained from these three models (for mock samples M1-M6).In both plots, the gray area represents the 1σ deviation from the mean true value obtained from the mock galaxy catalogs, and different points represent the best-fit results from three models as labeled.As shown in the left panel of Figure 11, we do not observe any significant differences between the best-fit M min values, which are consistent within the uncertainties.However, we note that the classicHOD minimum mass estimates are typically lower than our six-parameter model results.On average these differences fluctuate around 2% for M log min .Similarly, in the right panel of Figure 11, the best-fit M 1 results from classicHOD and our six-parameter model are in very good agreement and are well within 1σ of the true values.
Notably, however, for low-mass samples ( < M log 10.5 * ) the best-fit M 1 values from the concentrationHOD model are higher than the true values, but they are in agreement for higher-mass samples.This result is another proof that the concentration-mass relation plays a significant role in correlation function models (Artale et al. 2018;Zehavi et al. 2018;Bose et al. 2019;Hadzhiyska et al. 2020).What is important in the context of our work is that the concentration-mass relation has a stronger influence on the satellite halo mass than the halo asymmetry, especially for low-mass halos.
However, at the fundamental level of usability, our model does not deviate significantly from the classicHOD and at the same time provides information about the shape of the halo.

DM Halo Asymmetry and Its Stellar Mass Dependence
In Section 3 we have shown that the shape of the DM halo correlates with the stellar mass of the galaxy.The majority (∼67%) of galaxies with stellar masses ( )>  M M log 11.25 tend to occupy DM halos of prolate shape (see Figure 1).Using our modified halo model, we are able to reproduce the same results.Based on the values of f obtained by fitting the model, we observe that the asymmetry of the host halo varies with the stellar mass of the galaxy.The parameter f decreases from 0.85 ± 0.10 for the least massive galaxies to 0.59 ± 0.21 for the most massive ones.This means that the average DM halo shape changes from almost spherically symmetric (values of f close to 1) to increasingly asymmetric with increasing stellar mass, just as shown in Section 3.This observation is in broad agreement with previous studies of DM halo shape.Allgood et al. (2006) examined the dependence of the shape parameters on halo mass and radius in ΛCDM N-body simulations over the redshift range z = 0−3.They found that the majority of halos are prolate at all redshifts, with the fraction of halos that are prolate increasing for halos more massive than the characteristic mass M * at a given redshift (in the simulations used by Allgood et al. 2006, M * for z = 0 is 8.0 × 10 12 M e ).
Similarly, Hahn et al. (2007bHahn et al. ( , 2007a)), who studied the environmental dependence of the DM halo shape, found that the environment influences the halo shape.Halos of masses below M * tend to be more oblate, and those above this mass favor the prolate shape.These studies are extended by Despali et al. (2014), who found, based on three cosmological simulations, that DM halos tend to be prolate regardless of redshift (with a slight tendency to become more triaxial in earlier epochs).As for the correlation between halo mass and its asymmetry, in the same paper they found that the more massive halos are less spherical, regardless of the cosmic epoch.
From an observational point of view, massive galaxies tend to cluster more, i.e., they occupy denser environments.In terms of models, in particular the HOD model, it is usually interpreted that their DM halo mass is strongly dependent on the galaxy properties, with more luminous and massive galaxies occupying more massive halos (see, e.g., Norberg et al. 2002;Abbas & Sheth 2006;Pollo et al. 2006;Abbas & Sheth 2007;de la Torre et al. 2007;Coil et al. 2008;Meneux et al. 2008;Abbas et al. 2010;Hartley et al. 2010;Zehavi et al. 2011;Coupon et al. 2012;Beutler et al. 2013;Marulli et al. 2013;Mostek et al. 2013;Guo et al. 2015;Skibba et al. 2015;Durkalec et al. 2018;Paul et al. 2019).Considering that the most massive halos are the most asymmetric, we conclude that the DM halo shape must be taken into account when modeling the galaxy correlation function, especially for galaxies with high luminosity and stellar mass.

Summary and Conclusions
In this paper we present a six-parameter model designed to account for halo asymmetry in the modeling of the galaxy's two-point correlation function.The proposed model includes, in addition to the classical five-parameter HOD, an additional parameter f (implemented in the NFW density profile) describing the deviation of the halo shape from spherical symmetry.This parameter is largely related to the more commonly used triaxiality parameter T, as it is based on the ratios of the ellipsoid axes (see Figure 2).
In the first part of the paper, we test our model on a sample of mock galaxies populated (using the subhalo abundance matching method) in BolshoiP N-body simulations.We measure the real-space two-point correlation w p (r p ) function for six stellar-mass-selected mock galaxy samples and model these functions with a six-parameter model including a newly proposed asymmetry parameter f.We then compare our bestfit results with the "real" values provided by the simulations.
In the second part of the paper we fit our model to the twopoint correlation function measurements of Zehavi et al. (2011), which are based on SDSS observations.Zehavi et al. (2011) performed the traditional five-parameter HOD modeling, which assumes spherical symmetry for DM halos.We compare their results with those of our model.
The main results and conclusions can be summarized as follows: 1. We find that (1) the six-parameter model can reproduce the measured shape of the galaxy correlation function and the halo occupation function quite well and (2) the new asymmetry parameter f and other halo mass parameters computed from the best w p (r p ) fits are in good agreement (within 1σ error) with the analogs measured directly from the simulations.2. Using best-fit estimates of the asymmetry parameter f obtained from modeling the correlation functions of Zehavi et al. (2011) using SDSS data, we show that the halo asymmetry increases with the luminosity of the galaxy samples.The most luminous galaxies < -M 21.0 r max , are located in the most massive and asymmetric (prolate) halos.The intermediately luminous galaxies - > 19.0 > -M 20.5 r max reside in halos that are almost spherical (but still prolate), while the least luminous galaxies M r > −19.0 reside in halos that are slightly oblate.In none of the samples is the halo shape perfectly symmetrical.

Comparison of the best-fit characteristic halo masses
M min and M 1 from Zehavi et al. (2011; assuming spherical symmetry of the halo) with those from this work (including halo asymmetry as a free parameter) shows a 3% difference between the minimum halo masses M log min for all galaxy samples, with values from Zehavi et al. (2011) being consistently lower.On the other hand, satellite masses logM 1 differ by 4.6% for lowand intermediate-luminosity samples, while they are in good agreement for bright samples.In the case of the symmetrical model assuming f = 1, the characteristic masses are comparable except for the most luminous samples, where the difference reaches 6% in the case of logM 1 and is well above the 1σ level.4. Comparison of the estimates of the mean halo masses 〈M h 〉 shows agreement between the models for low-and intermediate-luminosity samples in the SDSS data.For the brightest galaxies, the 〈M h 〉 estimated using the halo sphericity assumption are lower than those obtained in our work.These differences suggest that for massive halos the estimates of their mass are sensitive to the assumption of their shape.5. Lastly, we present that the galaxy bias b g for the SDSS data is not influenced by the shape of the DM halos.
Overall, the modifications to the halo model proposed in this paper allow us to complement measurements of traditional halo parameters with information about halo asymmetry.We have shown that this modified model can reproduce both simulated results and observational measurements of the correlation function very well.Thus, this model can serve as an alternative model when the extended multiparameter HOD models prove too complex for a given data sample.At the fundamental level, our model performs similarly to "classical" HOD models, but it additionally provides information about the shape of the halo.
Future developments of this model may include (i) the introduction of a more complex two-parameter description of the halo asymmetry, which will allow for more precise discrimination between oblate, prolate, and triaxial halos; and (ii) the inclusion of different and more detailed mass functions, bias, and concentration-mass dependencies.Our model as it stands can also be used in studies of the dependence of halo asymmetry on redshift.
the asymmetry parameter f (see Equation (A3)), there is no need to additionally shape halos by the probability of them being flattened to an ellipsoidal shape, as done by Tinker et al. (2005) in their "ellipsoidal" halo exclusion model (Equations (B12) and (B13) therein).In our model R vir1 and R vir2 are already corrected to be "ellipsoidal" by the parameter f.This model has not been tested for different cosmological models or parameters other than those mentioned in this paper, nor for redshifts higher than z = 0.1.The model is easily adaptable to higher redshifts given the appropriate redshift dependencies in the various components.Furthermore, the model as proposed in this paper can be subject to improvements such as the inclusion of different and more detailed mass functions, bias, and concentration parameterizations.Bear in mind, however, one of the main aims of this paper, as mentioned in the Introduction: to keep the model as simple and computationally lightweight as possible.

Figure 1 .
Figure1.Triaxiality parameter T of the host DM halo as a function of the stellar mass of the galaxies residing in this halo.Results shown in this figure are based on the mock galaxy catalogs populated in BolshoiP simulations.The percentage of galaxies residing in a halo with one of the three types of asymmetry varies with stellar mass.The most massive galaxies are most likely to be in the prolate halos.This means that the halo shape might be the important factor in galaxy clustering modeling of these galaxies.

Figure 2 .
Figure 2. Parameter T as a function of f measured for host DM halos.Each point represents a single host DM halo.f = 1 indicates a spherically symmetric halo.For host DM halos with T > 0.5 (indicating a more prolate type of shape asymmetry), ∼95% of the sample halos take values of f < 1.For T < 0.5 (i.e., DM halos with oblate shape asymmetry), the parameter is f > 1.

Figure 3 .
Figure 3. Median value of the host halo asymmetry parameter f as a function of a median stellar mass of galaxy subsamples selected from the BolshoiP mock galaxy sample.The value of f is obtained using the ellipsoid axes a, b, and c available in the halo catalog.Error bars represent the standard deviation.As expected, the asymmetry of the host halo increases with stellar mass and is strongest for the most massive sample.

Figure 4 .
Figure 4.The modeled two-point correlation function for different halo asymmetries.Different lines mark the modeled correlation functions for different f (as labeled) and other parameters fixed to the same value: = M log 12.40 min

Figure 5 .
Figure 5. Top panel: average IAT τ as a function of the number of chain iterations for the correlation function modeling of mock galaxies from the M2 subsample.The filled circles mark the average autocorrelation time obtained for all six free parameters.The shaded area shows the N burn iterations necessary for the fit to converge, i.e., the number of iterations for which τ = N burn /50.Bottom panel: χ 2 /dof value as a function of iteration number for all 25 random walk chains.

Figure 6 .
Figure 6.Corner plot: the result of the MCMC fitting of the six-parameter model to the projected correlation function of the M5 mock galaxy sample.We show only this one for brevity, but each of the correlation function models shown in the upper right corner has a corresponding corner plot similar to this one.The off-diagonal plots show the density maps for a given set of model parameters.The contours represent regions containing 68.3% and 95.5% of the posterior density.The histograms on the diagonal show the probability distribution functions (pdf's) for the six fitting parameters.The best-fit parameters are indicated by blue solid lines, while the dashed lines show the 16th and 84th percentiles for each parameter.Upper right: projected two-point correlation function w p (r p ) (filled circles) with the best-fitting sixparameter halo model (solid lines) for volume-limited stellar-mass-selected mock galaxy subsamples populated in the BolshoiP simulations.For clarity, both the data points and the best-fitting curves have been shifted by 0.3 dex.

Figure 7 .
Figure7.Comparison of the best-fit HODs (black lines) for different mock subsamples with the "true" HODs (circles).In each plot, the solid black lines represent the average number of galaxies 〈N g 〉, the dotted line represents the average number of central galaxies 〈N c 〉, and the dashed line represents the average number of central galaxies 〈N s 〉.The series of gray lines in each plot represent 50 randomly selected HODs from the MCMC chains that are within Δχ 2 < 1 relative to the best-fitting model.

Figure 8 .
Figure 8.Comparison of the characteristic host halo massesM log min (top panel) and logM 1 (middle panel) and the asymmetry parameter f (bottom panel) obtained directly from the mock catalog (dashed line) and using our best-fit six-parameter model (filled circles for M min , filled squares for M 1 , and filled triangles for f).In all figures, the shaded area represents the standard deviation from the mean value obtained from the mock catalogs.

Figure 9 .
Figure 9. Corner plot: the result of the MCMC fitting of the six-parameter model to the projected correlation function from the Zehavi et al. (2011) sample = -M 21.0 r max.We show only this one for brevity, but each of the correlation function fits shown in the upper right corner has a corresponding corner plot similar to this one.The off-diagonal plots show the density maps for a given set of model parameters.The contours represent regions containing 68.3% and 95.5% of the posterior density.Histograms on the diagonal show the pdf's for the six fitted parameters.Best-fitting parameters are indicated with orange solid lines, while the dashed lines show the 16th and 84th percentiles for each parameter.Upper right: projected two-point correlation function w p (r p ) (filled circles) fromZehavi et al. (2011) with the best-fitting six-parameter models (solid lines) from the SDSS M r luminosity-selected subsamples.For clarity, offsets are applied to both the data points and the best-fitting curves, i.e., they have been offset by 0.3 dex each.
with the rapid change in halo asymmetry that we observe.

Figure 10 .
Figure 10.Top two panels: comparison of the best-fit host halo asymmetry parameter f, minimum halo mass M min , and satellite halo mass M 1 obtained in this work (filled symbols as labeled) and by Zehavi et al. (2011) using the model that assumes spherical symmetry of DM halos (dashed lines).In the case of results from Zehavi et al. (2011), due to the difference in notation, we use their ¢ M 1 .Bottom two panels: comparison of the average halo masses á ñ M log h and galaxy bias b g estimated using best-fit parameters from this work (filled points) and from Zehavi et al. (2011; dashed line) as a function of the absolute magnitude M r of the galaxy sample.

Figure 11 .
Figure 11.Comparison of best-fit characteristic halo masses M min and M 1 obtained via a six-parameter model fit (solid line), standard HOD model with NFW density profile (dotted line), and HOD model with halo concentration-mass relation from Ludlow et al. (2016) (dotted line).In both panels, the shaded area represents the 1σ deviation from the mean true value obtained from the mock catalogs. >

Table 1
Main Properties of BolshoiP Cosmological Simulations Used in This Work

Table 3
Number of Iterations N iter in Each Chain and Number of Steps Necessary to Reach Convergence of Fit N burn for Different Mock Subsamples

Table 4
Best-fit Parameters for Six Volume-limited Stellar-mass-selected Mock Galaxy Samples Note.M1 to M6 denote different stellar-mass subsamples as in Table2.All DM halo masses are given in M e .The number of degrees of freedom is equal to 8 for samples from M1 to M5 (13 measured w p values plus the number density n g minus six fitted model parameters); for the most massive sample M6, dof = 4, due to a smaller number of correlation function bins (nine measured w p values).

Table 5
Zehavi et al. (2011)Obtained in This Work for M r Luminosity-selected Samples fromZehavi et al. (2011) Note.All DM halo masses are given in M e .The number of degrees of freedom is equal to 8 for all samples (13 measured w p values plus the number density n g minus six fitted HOD parameters).