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In order to capture as much information as possible large galaxies surveys have been increasing their volume and redshift depth. To face this challenge theory has responded by making cosmological simulations of huge computational volumes with equally increasing numbers of dark matter particles and supercomputing resources. Thus, it is taken for granted that the ideal situation is when a single computational box encompasses the whole volume of the observational survey, e.g. |$\sim 50\, h^{-3}\,{\rm Gpc}^3$| for the DESI and Euclid surveys. Here we study the effects of missing long waves in a finite volume using several relevant statistics: the abundance of dark matter haloes, the probability distribution function (PDF), the correlation function and power spectrum, and covariance matrices. Finite volume effects can substantially modify the results if the computational volumes are less than |$\sim (500\mbox{$\, h^{-1}$Mpc})^3$|⁠. However, the effects become extremely small and practically can be ignored when the box size exceeds ∼1 Gpc3. We find that the average power spectra of dark matter fluctuations show remarkable lack of dependence on the computational box size with less than 0.1 per cent differences between |$1$| and |$4\mbox{$\, h^{-1}\,$Gpc}$| boxes. No measurable differences are expected for the halo mass functions for these volumes. The covariance matrices are scaled trivially with volume, and small corrections due to supersample modes can be added. We conclude that there is no need to make those extremely large simulations when a box size of |$1-1.5\mbox{$\, h^{-1}$Gpc}$| is sufficient to fulfil most of the survey science requirements. [ABSTRACT FROM AUTHOR]

The one-point probability distribution function (PDF) of the matter density field in the universe is a fundamental property that plays an essential role in cosmology for estimates such as gravitational weak lensing, non-linear clustering, massive production of mock galaxy catalogues, and testing predictions of cosmological models. Here we make a comprehensive analysis of the dark matter PDF, using a suite of ∼7000 N -body simulations that covers a wide range of numerical and cosmological parameters. We find that the PDF has a simple shape: it declines with density as a power-law P ∝ ρ−2, which is exponentially suppressed on both small and large densities. The proposed double-exponential approximation provides an accurate fit to all our N -body results for small filtering scales R < 5 h−1Mpc with rms density fluctuations σ > 1. In combination with the spherical infall model that works well for small fluctuations σ < 1, the PDF is now approximated with just few per cent errors over the range of 12 orders of magnitude – a remarkable example of precision cosmology. We find that at |${\sim } 5{-}10{{\ \rm per\ cent}}$| level the PDF explicitly depends on redshift (at fixed σ) and on cosmological density parameter Ωm. We test different existing analytical approximations and find that the often-used lognormal approximation is always 3–5 times less accurate than either the double-exponential approximation or the spherical infall model. [ABSTRACT FROM AUTHOR]

Predicting structural properties of dark matter haloes is one of the fundamental goals of modern cosmology. We use the suite of MultiDark cosmological simulations to study the evolution of dark matter halo density profiles, concentrations, and velocity anisotropies. We find that in order to understand the structure of dark matter haloes and to make 1-2 per cent accurate predictions for density profiles, one needs to realize that halo concentration is more complex than the ratio of the virial radius to the core radius in the Navarro-Frenk-White (NFW) profile. For massive haloes, the average density profile is far from the NFW shape and the concentration is defined by both the core radius and the shape parameter α in the Einasto approximation. We show that haloes progress through three stages of evolution. They start as rare density peaks and experience fast and nearly radial infall that brings mass closer to the centre, producing a highly concentrated halo. Here, the halo concentration increases with increasing halo mass and the concentration is defined by the a parameter with a nearly constant core radius. Later haloes slide into the plateau regime where the accretion becomes less radial, but frequent mergers still affect even the central region. At this stage, the concentration does not depend on halo mass. Once the rate of accretion and merging slows down, haloes move into the domain of declining concentration-mass relation because new accretion piles up mass close to the virial radius while the core radius is staying constant. Accurate analytical fits are provided. [ABSTRACT FROM AUTHOR]

We present newmeasurements of the abundance of galaxies with a given circular velocity in the Local Volume: a region centred on the MilkyWay Galaxy and extending to distance ~10 Mpc. The sample of ~750 mostly dwarf galaxies provides a unique opportunity to study the abundance and properties of galaxies down to absolute magnitudes MB ≈ -10 and virial masses Mvir = 109M⊙. We find that the standard Λ cold dark matter (ΛCDM) model gives remarkably accurate estimates for the velocity function of galaxies with circular velocities V #8819; 70 kms-1 and corresponding virial masses Mvir ≳ 5 × 1010M⊙, but it badly fails by overpredicting ~5 times the abundance of large dwarfs with velocities V = 30-40 kms-1. The warm dark matter (WDM) models cannot explain the data either, regardless of mass of WDM particle. Just as in previous observational studies, we find a shallow asymptotic slope dN/dlog V ∝ Vα, α ≈ -1 of the velocity function, which is inconsistent with the standard ΛCDMmodel that predicts the slope α=-3. Though reminiscent to the known overabundance of satellite problem, the overabundance of field galaxies is a much more difficult problem. For the standard CDM model to survive, in the 10 Mpc radius of the Milky Way there should be 1000 not yet detected galaxies with virial mass Mvir ≈ 1010M⊙, extremely low surface brightness and no detectable H I gas. So far none of this type of galaxies have been discovered. [ABSTRACT FROM AUTHOR]

Dwarf spheroidal galaxies (dSphs) are extremely gas-poor, dark matter-dominated galaxies, which make them ideal to test the predictions of the cold dark matter (CDM) model. We argue that the removal of the baryonic component from gas-rich dwarf irregular galaxies, the progenitors of dSphs, can substantially reduce their central density. Thus, it may play an important role in alleviating one of the problems of the CDM model related to the structure of relatively massive satellite galaxies of the Milky Way (MW). Traditionally, collisionless cosmological N-body simulations are used when confronting theoretical predictions with observations. However, these simulations assume that the baryon fraction everywhere in the Universe is equal to the cosmic mean, an assumption which is incorrect for dSphs. We find that the combination of (i) the lower baryon fraction in dSphs compared to the cosmic mean and (ii) the concentration of baryons in the inner part of the MW halo can go a long way towards explaining the observed circular velocity profiles of dSphs. We perform controlled numerical simulations that mimic the effects of baryons. From these we find that the blowing away of baryons by ram pressure, when the dwarfs fall into larger galaxies, decreases the circular velocity profile of the satellite and reduces the density in the central ∼200–500 pc by a factor of $(1- f_{\rm b})^4\approx 0.5$, where $f_{\rm b}$ is the cosmological fraction of baryons. Additionally, the enhanced baryonic mass in the central regions of the parent galaxy generates tidal forces, which are larger than those experienced by subhaloes in traditional N-body simulations. Increased tidal forces substantially alter circular velocity profiles for satellites with pericentres less than 50 kpc. We show that these two effects are strong enough to bring the predictions of subhaloes from CDM simulations into agreement with the observed structure of MW dSphs, regardless of the details of the baryonic processes. [ABSTRACT FROM AUTHOR]

ABSTRACT We study the concentration of dark matter haloes and its evolution in N-body simulations of the standard Λ cold dark matter (ΛCDM) cosmology. The results presented in this paper are based on four large N-body simulations with ∼10 billion particles each: the Millennium-I and -II, Bolshoi and MultiDark simulations. The MultiDark (or BigBolshoi) simulation is introduced in this paper. This suite of simulations with high mass resolution over a large volume allows us to compute with unprecedented accuracy the concentration over a large range of scales (about six orders of magnitude in mass), which constitutes the state of the art of our current knowledge on this basic property of dark matter haloes in the ΛCDM cosmology. We find that there is consistency among the different simulation data sets, despite the different codes, numerical algorithms and halo/subhalo finders used in our analysis. We confirm a novel feature for halo concentrations at high redshifts: a flattening and upturn with increasing mass. The concentration c( M, z) as a function of mass and the redshift and for different cosmological parameters shows a remarkably complex pattern. However, when expressed in terms of the linear rms fluctuation of the density field σ( M, z), the halo concentration c(σ) shows a nearly universal simple U-shaped behaviour with a minimum at a well-defined scale at σ∼ 0.71. Yet, some small dependences with redshift and cosmology still remain. At the high-mass end (σ < 1), the median halo kinematic profiles show large signatures of infall and highly radial orbits. This c-σ( M, z) relation can be accurately parametrized and provides an analytical model for the dependence of concentration on halo mass. When applied to galaxy clusters, our estimates of concentrations are substantially larger - by a factor up to 1.5 - than previous results from smaller simulations, and are in much better agreement with results of observations. [ABSTRACT FROM AUTHOR]

We study dynamics of bars in models of disc galaxies embedded in realistic dark matter haloes. We find that disc thickness plays an important, if not dominant, role in the evolution and structure of the bars. We also make extensive numerical tests of different N-body codes used to study bar dynamics. Models with thick discs typically used in this type of modelling (height-to-length ratio ) produce slowly rotating, and very long, bars. In contrast, more realistic thin discs with the same parameters as in our Galaxy produce bars with normal length , which rotate quickly with the ratio of the corotation radius to the bar radius compatible with observations. Bars in these models do not show a tendency to slow down, and may lose as little as 2–3 per cent of their angular momentum due to dynamical friction with the dark matter over cosmological time. We attribute the differences between the models to a combined effect of high phase-space density and smaller Jeans mass in the thin-disc models, which result in the formation of a dense central bulge. Special attention is paid to numerical effects, such as the accuracy of orbital integration, force and mass resolution. Using three N-body codes –gadget, adaptive refinement tree (art) andpkdgrav – we find that numerical effects are very important and, if not carefully treated, may produce incorrect and misleading results. Once the simulations are performed with sufficiently small time-steps and with adequate force and mass resolution, all the codes produce nearly the same results: we do not find any systematic deviations between the results obtained withtree codes (gadget andpkdgrav) and with the adaptive mesh refinement (art) code. [ABSTRACT FROM AUTHOR]

We present 2000 mock galaxy catalogues for the analysis of baryon acoustic oscillations (BAOs) in the Emission Line Galaxy (ELG) sample of the extended Baryon Oscillation Spectroscopic Survey Data Release 16 (eBOSS DR16). Each mock catalogue has a number density of |$6.7 \times 10^{-4} h^3 \rm Mpc^{-3}$|⁠ , covering a redshift range from 0.6 to 1.1. The mocks are calibrated to small-scale eBOSS ELG clustering measurements at scales of |$\lesssim 30\, h^{-1}$| Mpc. The mock catalogues are generated using a combination of GaLAxy Mocks (GLAM) simulations and the quick particle-mesh (QPM) method. GLAM simulations are used to generate the density field, which is then assigned dark matter haloes using the QPM method. Haloes are populated with galaxies using a halo occupation distribution. The resulting mocks match the survey geometry and selection function of the data, and have slightly higher number density that allows room for systematic analysis. The large-scale clustering of mocks at the BAO scale is consistent with data and we present the correlation matrix of the mocks. [ABSTRACT FROM AUTHOR]

We present a method for computing the six-dimensional coarse-grained phase-space density f( x, v) in an N-body system, and derive its distribution function v( f). The method is based on Delaunay tessellation, where v( f) is obtained with an effective fixed smoothing window over a wide f range. The errors are estimated, and v( f) is found to be insensitive to the sampling resolution or the simulation technique. We find that in gravitationally relaxed haloes built by hierarchical clustering, v( f) is well approximated by a robust power law, , over more than four decades in f, from its virial level to the numerical resolution limit. This is tested to be valid in the Λ cold dark matter cosmology for haloes with masses , indicating insensitivity to the slope of the initial fluctuation power spectrum. By mapping the phase-space density in position space, we find that the high- f end of v( f) is dominated by the ‘cold’ subhaloes rather than the parent-halo central region and its global spherical profile. The value of f in subhaloes near the virial radius is typically >100 times higher than its value at the halo centre, and it decreases gradually from the outside in toward its value at the halo centre. This seems to reflect phase mixing due to mergers and tidal effects involving puffing up and heating. The phase-space density can thus provide a sensitive tool for studying the evolution of subhaloes during the hierarchical build-up of haloes. It remains to be understood why the evolved substructure adds up to the actual universal power law of . It seems that this behaviour results from the hierarchical clustering process and is not a general result of violent relaxation. [ABSTRACT FROM AUTHOR]

Upcoming surveys will use a variety of galaxy selections to map the large-scale structure of the Universe. It is important to make accurate predictions for the properties and clustering of such galaxies, including the errors on these statistics. Here, we describe a novel technique which uses the semi-analytical model of galaxy formation galform , embedded in the high-resolution N -body Planck-Millennium simulation, to populate a thousand halo catalogues generated using the Parallel-PM N -body glam code. Our hybrid scheme allows us to make clustering predictions on scales that cannot be modelled in the original N -body simulation. We focus on luminous red galaxies (LRGs) selected in the redshift range z = 0.6 − 1 from the galform output using similar colour-magnitude cuts in the r, z , and W 1 bands to those that will be applied in the Dark Energy Spectroscopic Instrument (DESI) survey, and call this illustrative sample 'DESI-like' LRGs. We find that the LRG-halo connection is non-trivial, leading to the prediction of a non-standard halo occupation distribution; in particular, the occupation of central galaxies does not reach unity for the most massive haloes, and drops with increasing mass. The glam catalogues reproduce the abundance and clustering of the LRGs predicted by galform. We use the glam mocks to compute the covariance matrices for the two-point correlation function and power spectrum of the LRGs and their background dark matter density field, revealing important differences. We also make predictions for the linear-growth rate and the baryon acoustic oscillations distances at z = 0.6, 0.74, and 0.93. All 'DESI-like' LRG catalogues are made publicly available. [ABSTRACT FROM AUTHOR]

Halo bias is the main link between the matter distribution and dark matter haloes. In its simplest form, halo bias is determined by halo mass, but there are known additional dependencies on other halo properties which are of consequence for accurate modelling of galaxy clustering. Here, we present the most precise measurement of these secondary-bias dependencies on halo age, concentration, and spin, for a wide range of halo masses spanning from 1010.7 to 1014.7 h −1 M⊙. At the high-mass end, we find no strong evidence of assembly bias for masses above M vir ∼1014 h −1 M⊙. Secondary bias exists, however, for halo concentration and spin, up to cluster-size haloes, in agreement with previous findings. For halo spin, we report, for the first time, two different regimes: above M vir ∼ 1011.5 h −1 M⊙, haloes with larger values of spin have larger bias, at fixed mass, with the effect reaching almost a factor 2. This trend reverses below this characteristic mass. In addition to these results, we test, for the first time, the performance of a multitracer method for the determination of the relative bias between different subsets of haloes. We show that this method increases significantly the signal to noise of the secondary-bias measurement as compared to a traditional approach. This analysis serves as the basis for follow-up applications of our multitracer method to real data. [ABSTRACT FROM AUTHOR]

We study the galaxy clustering dependence on the [O II] emission line luminosity in the SDSS DR7 Main galaxy sample at mean redshift z ∼ 0.1. We select volume-limited samples of galaxies with different [O II] luminosity thresholds and measure their projected, monopole and quadrupole two-point correlation functions. We model these observations using the 1 h-1 Gpc Multi Dark-Planck cosmological simulation and generate light cones with the SUrvey Gener A to R algorithm. To interpret our results, we adopt a modified (Sub)Halo Abundance Matching scheme, accounting for the stellar mass incompleteness of the emission line galaxies. The satellite fraction constitutes an extra parameter in this model and allows to optimize the clustering fit on both small and intermediate scales (i.e. rp ≲ 30 h-1 Mpc), with no need of any velocity bias correction. We find that, in the local Universe, the [O II] luminosity correlateswith all the clustering statistics explored and with the galaxy bias. This latter quantity correlates more strongly with the SDSS r-band magnitude than [O II] luminosity. In conclusion, we propose a straightforward method to produce reliable clustering models, entirely built on the simulation products, which provides robust predictions of the typical ELG host halo masses and satellite fraction values. The SDSS galaxy data, MultiDark mock catalogues and clustering results are made publicly available. [ABSTRACT FROM AUTHOR]

We report and provide fitting functions for the abundance of dark matter halos and subhalos as a function of mass, circular velocity, and redshift from the new Bolshoi-Planck and MultiDark-Planck ΛCDM cosmological simulations, based on the Planck cosmological parameters. We also report the halo mass accretion rates, which may be connected with galaxy star formation rates. We show that the higher cosmological matter density of the Planck parameters compared with the WMAP parameters leads to higher abundance of massive halos at high redshifts. We find that the median halo spin parameter λB=J(√2MvirRvirVvir)−1 is nearly independent of redshift, leading to predicted evolution of galaxy sizes that is consistent with observations, while the significant decrease with redshift in median λP=J|E|−1/2G−1M−5/2 predicts more decrease in galaxy sizes than is observed. Using the Tully-Fisher and Faber-Jackson relations between galaxy velocity and mass, we show that a simple model of how galaxy velocity is related to halo maximum circular velocity leads to increasing overprediction of cosmic stellar mass density as redshift increases beyond redshifts z∼1, implying that such velocity-mass relations must change at redshifts z ≳ 1. By making a realistic model of how observed galaxy velocities are related to halo circular velocity, we show that recent optical and radio observations of the abundance of galaxies are in good agreement with our ΛCDM simulations. Our halo demographics are based on updated versions of the \rockstar\ and \ctrees\ codes, and this paper includes appendices explaining all of their outputs. This paper is an introduction to a series of related papers presenting other analyses of the Bolshoi-Planck and MultiDark-Planck simulations. [ABSTRACT FROM AUTHOR]

Current and future large redshift surveys, as the Sloan Digital Sky Survey IV extended Baryon Oscillation Spectroscopic Survey (SDSS-IV/eBOSS) or the Dark Energy Spectroscopic Instrument (DESI), will use emission-line galaxies (ELGs) to probe cosmological models by mapping the large-scale structure of the Universe in the redshift range 0.6 < z < 1.7. With current data, we explore the halo-galaxy connection by measuring three clustering properties of g-selected ELGs as matter tracers in the redshift range 0.6 < z < 1: (i) the redshift-space two-point correlation function using spectroscopic redshifts from the BOSS ELG sample and VIPERS; (ii) the angular two-point correlation function on the footprint of the CFHT-LS; (iii) the galaxy-galaxy lensing signal around the ELGs using the CFHTLenS. We interpret these observations by mapping them on to the latest high-resolution MultiDark Planck N-body simulation, using a novel (Sub)Halo-Abundance Matching technique that accounts for the ELG incompleteness. ELGs at z ~ 0.8 live in haloes of (1 ± 0.5) × 1012 h-1M☉ and 22.5 ± 2.5 per cent of them are satellites belonging to a larger halo. The halo occupation distribution of ELGs indicates that we are sampling the galaxies in which stars form in the most efficient way, according to their stellar-to-halo mass ratio. [ABSTRACT FROM AUTHOR]

We present a study of the clustering and halo occupation distribution of Baryon Oscillation Spectroscopic Survey (BOSS) CMASS galaxies in the redshift range 0.43 < z < 0.7 drawn from the Final SDSS-III Data Release. We compare the BOSS results with the predictions of a halo abundance matching (HAM) clustering model that assigns galaxies to dark matter haloes selected from the large Big Multi Dark N-body simulation of a flat Λ cold dark matter Planck cosmology. We compare the observational data with the simulated ones on a light cone constructed from 20 subsequent outputs of the simulation. Observational effects such as incompleteness, geometry, veto masks and fibre collisions are included in the model, which reproduces within 1σ errors the observed monopole of the two-point correlation function at all relevant scales: from the smallest scales, 0.5 h-1 Mpc, up to scales beyond the baryon acoustic oscillation feature. This model also agrees remarkably well with the BOSS galaxy power spectrum (up to k ~ 1 h Mpc-1), and the three-point correlation function. The quadrupole of the correlation function presents some tensions with observations. We discuss possible causes that can explain this disagreement, including target selection effects. Overall, the standard HAM model describes remarkably well the clustering statistics of the CMASS sample. We compare the stellar-to-halo mass relation for the CMASS sample measured using weak lensing in the Canada--France--Hawaii Telescope Stripe 82 Survey with the prediction of our clustering model, and find a good agreement within 1σ. The BigMD-BOSS light cone including properties of BOSS galaxies and halo properties is made publicly available. [ABSTRACT FROM AUTHOR]

We model the luminosity-dependent projected and redshift-space two-point correlation functions (2PCFs) of the Sloan Digital Sky Survey (SDSS) Data Release 7 Main galaxy sample, using the halo occupation distribution (HOD) model and the subhalo abundance matching (SHAM) model and its extension. All the models are built on the same high-resolution Nbody simulations. We find that the HOD model generally provides the best performance in reproducing the clustering measurements in both projected and redshift spaces. The SHAM model with the same halo-galaxy relation for central and satellite galaxies (or distinct haloes and subhaloes), when including scatters, has a best-fitting X2 around 2-3. We therefore extend the SHAM model to the subhalo clustering and abundance matching (SCAM) by allowing the central and satellite galaxies to have different galaxy-halo relations. We infer the corresponding halo/subhalo parameters by jointly fitting the galaxy 2PCFs and abundances and consider subhaloes selected based on three properties, the mass Macc at the time of accretion, the maximum circular velocity Vacc at the time of accretion, and the peak maximum circular velocity Vpeak over the history of the subhaloes. The three subhalo models work well for luminous galaxy samples (with luminosity above L*). For low-luminosity samples, the Vacc model stands out in reproducing the data, with the Vpeak model slightly worse, while the Macc model fails to fit the data. We discuss the implications of the modelling results. [ABSTRACT FROM AUTHOR]

Future dark energy experiments will require accurate theoretical predictions for the baryon acoustic oscillations (BAOs). Here, we use large N-body simulations to study any systematic shifts and damping in BAO due to non-linear effects. The impact of cosmic variance is largely reduced by dividing the tracer power spectrum by that from a 'BAO-free' simulation starting with the same random amplitudes and phases. The accuracy of our simulations allows us to resolve well dark matter (sub)haloes, which permits us to study with high accuracy (better than 0.02 per cent for dark matter and 0.07 per cent for low-bias haloes) small BAO shifts α towards larger k, and non-linear damping Σnl of BAO in the power spectrum. For dark matter, we provide an accurate parametrization of the evolution of α as a function of the linear growth factor D(z). For halo samples, with bias from 1.2 to 2.8, we measure a typical BAO shift of ≈0.25 per cent, with no appreciable evolution with redshift. Moreover, we report a constant shift as a function of halo bias. We find a different evolution of the BAO damping in all halo samples as compared to dark matter with haloes suffering less damping, and also find some weak dependence on bias. Larger BAO shift and damping are measured in redshift-space, which can be explained by linear theory due to redshift-space distortions. A clear modulation in phase with the acoustic scale is observed in the scale-dependent halo bias due to the presence of BAOs. We compare our results with previous works. [ABSTRACT FROM AUTHOR]

We reproduce the galaxy clustering catalogue from the SDSS-III Baryon Oscillation Spectroscopic Survey Final Data Release (BOSS DR11&DR12) with high fidelity on all relevant scales in order to allow a robust analysis of baryon acoustic oscillations and redshift space distortions. We have generated (6000) 12 288 MultiDark PATCHY BOSS (DR11) DR12 light cones corresponding to an effective volume of ∼192 000 [h-1 Gpc]³ (the largest ever simulated volume), including cosmic evolution in the redshift range from 0.15 to 0.75. The mocks have been calibrated using a reference galaxy catalogue based on the halo abundance matching modelling of the BOSS DR11&DR12 galaxy clustering data and on the data themselves. The production follows three steps. First, we apply the PATCHY code to generate a dark matter field and an object distribution including non-linear stochastic galaxy bias. Secondly, we run the halo/stellar distribution reconstruction HADRON code to assign masses to the various objects. This step uses the mass distribution as a function of local density and non-local indicators (i.e. tidal field tensor eigenvalues and relative halo exclusion separation for massive objects) from the reference simulation applied to the corresponding patchy dark matter and galaxy distribution. Finally, we apply the SUGAR code to build the light cones. The resulting MultiDarkPATCHY mock light cones reproduce the number density, selection function, survey geometry, and in general within 1s, for arbitrary stellar mass bins, the power spectrum up to k = 0.3 h Mpc-1, the two-point correlation functions down to a few Mpc scales, and the three-point statistics of the BOSS DR11&DR12 galaxy samples. [ABSTRACT FROM AUTHOR]

We present the measurements and modelling of the small-to-intermediate scale (~0.1-25 h-1 Mpc) projected and three-dimensional redshift-space two-point correlation functions (2PCFs) of local galaxies in the Sloan Digital Sky Survey Data Release 7. We find a clear dependence of galaxy clustering on luminosity in both projected and redshift spaces, generally being stronger for more luminous samples. The measurements are successfully interpreted within the halo occupation distribution (HOD) framework with central and satellite velocity bias parameters to describe galaxy kinematics inside haloes and to model redshift-space distortion effects. In agreement with previous studies, we find that more luminous galaxies reside in more massive haloes. Including the redshift-space 2PCFs helps tighten the HOD constraints. Moreover, we find that luminous central galaxies are not at rest at the halo centres, with the velocity dispersion about 30 per cent that of the dark matter. Such a relative motion may reflect the consequence of galaxy and halo mergers, and we find that central galaxies in lower mass haloes tend to be more relaxed with respect to their host haloes. The motion of satellite galaxies in luminous samples is consistent with their following that of the dark matter. For faint samples, satellites tends to have slower motion, with velocity dispersion inside haloes about 85 per cent that of the dark matter. We discuss possible applications of the velocity bias constraints on studying galaxy evolution and cosmology. In the appendix, we characterize the distribution of galaxy redshift measurement errors, which is well described by a Gaussian-convolved double exponential distribution. [ABSTRACT FROM AUTHOR]