Abstract:
We use a new method, the cross power spectrum between the linear density field and the halo number density field, to measure the Lagrangian bias for dark matter halos. The method has several important advantages over the conventional correlation function analysis. By applying this method to a set of high-resolution simulations of 256^3 particles, we have accurately determined the Lagrangian bias, over 4 magnitudes in halo mass, for four scale-free models with the index n=-0.5, -1.0, -1.5 and -2.0 and three typical CDM models. Our result for massive halos with $M \ge M_*$ ($M_*$ is a characteristic non-linear mass) is in very good agreement with the analytical formula of Mo & White for the Lagrangian bias, but the analytical formula significantly underestimates the Lagrangian clustering for the less massive halos $M < M_*. Our simulation result however can be satisfactorily described, with an accuracy better than 15%, by the fitting formula of Jing for Eulerian bias under the assumption that the Lagrangian clustering and the Eulerian clustering are related with a linear mapping. It implies that it is the failure of the Press-Schechter theories for describing the formation of small halos that leads to the inaccuracy of the Mo & White formula for the Eulerian bias. The non-linear mapping between the Lagrangian clustering and the Eulerian clustering, which was speculated as another possible cause for the inaccuracy of the Mo & White formula, must at most have a second-order effect. Our result indicates that the halo formation model adopted by the Press-Schechter theories must be improved.

Abstract:
We study the diversity of the density profiles of dark matter halos based on a large set of high-resolution cosmological simulations of 256^3 particles. The cosmological models include four scale-free models and three representative cold dark matter models. The simulations have good force resolution, and there are about 400 massive halos with more than 10^4 particles within the virial radius in each cosmological model. Our unbiased selection of all massive halos enables to quantify how well the bulk of dark matter halos can be described by the Navarro, Frenk & White (NFW) profile which was established for equilibrium halos. We find that about seventy percent of the halos can be fitted by the NFW profile with a fitting residual dvi_{max} less than 30% in Omega_0=1 universes. This percentage is higher in lower density cosmological models. The rest of the halos exhibits larger deviations from the NFW profile for more significant internal substructures. There is a considerable amount of variation in the density profile even for the halos which can be fitted by the NFW profile (i.e. dvi_{max}<0.30). The distribution of the profile parameter, the concentration $c$, can be well described by a lognormal function with the mean value \bar c slightly smaller (15%) than the NFW result and the dispersion \sigma_c in \ln c about 0.25. The more virialized halos with dvi_{max}<0.15 have the mean value \bar c in good agreement with the NFW result and a slightly smaller dispersion \sigma_c (about 0.2). Our results can alleviate some of the conflicts found recently between the theoretical NFW profile and observational results. Implications for theoretical and observational studies of galaxy formation are discussed.

Abstract:
Cold Dark Matter (CDM) models of galaxy formation had been remarkably successful to explain a number of observations in the past decade. However, with both the theoretical modeling and the observations being improved, CDM models have been very recently shown to have excessive clustering on the sub-galactic scale. Here I discuss a solution, based on our high-resolution numerical simulations, to this outstanding problem by considering Warm Dark Matter (WDM). Our results show that the over-clustering problem on sub-galactic scales can be overcome by WDM models, and all the advantages of CDM models are preserved by WDM models. Therefore, the WDM model will become an interesting alternative to the well-studied CDM models.

Abstract:
We use a large set of state-of-the-art cosmological N-body simulations [512^3 particles] to study the intrinsic ellipticity correlation functions of halos. With the simulations of different resolutions, we find that the ellipticity correlations converge once the halos have more than 160 members. For halos with fewer members, the correlations are underestimated, and the underestimation amounts to a factor of 2 when the halos have only 20 particles. After correcting for the resolution effects, we show that the ellipticity correlations of halos in the bigger box (L=300 mpc) agree very well with those obtained in the smaller box (L=100 mpc). Combining these results from the different simulation boxes, we present accurate fitting formulae for the ellipticity correlation function c_{11}(r) and for the projected correlation functions Sigma_{11}(r_p) and Sigma_{22}(r_p) over three orders of magnitude in halo mass. The latter two functions are useful for predicting the contribution of the intrinsic correlations to deep lensing surveys. With reasonable assumptions for the redshift distribution of galaxies and for the mass of galaxies, we find that the intrinsic ellipticity correlation can contribute significantly not only to shallow surveys but also to deep surveys. Our results indicate that previous similar studies significantly underestimated this contribution for their limited simulation resolutions.

Abstract:
In this talk, I will show how to determine the biasing factor $b$ from the high-order moments of galaxies. The determination is based on the analytical modeling of primordial peaks and virialized halos and is independent of the currently unknown density parameter $\Omega_0$ and other cosmological parameters. The observed high-oder moments of the APM galaxies require that the biasing factor $b$ be very close to 1, i.e. the optical galaxies are an unbiased tracer of the underlying mass distribution (on quasilinear scale). The theoretical argument can be easily generalized to the three-point correlation function and the bispectrum both of which can used as further observational tests to the important conclusion of $b\approx 1$ drawn from the high-order moments. Finally I present our preliminary results of the three-point correlation functions for the Las Campanas Redshift Survey.

Abstract:
The density profile and the clustering of dark matter (DM) halos are two very important ingredients for understanding many observations of galaxies. In this talk, I present our new results for these two quantities from a large set of high quality N-body simulations. We show that in an Einstein de Sitter Universe about 35 percent of the halos could not be fitted by the Navarro, Frenk & White (NFW) profile because of too significant substructures, another 50 percent with less substructures can be reasonably described by the NFW profile, and the rest with the least substructures can be fitted by the NFW profile very nicely. For the halos which can be reasonably or better fitted by the NFW profile, the probability distribution function of the concentration parameter $c$ could be well described by a lognormal function with a dispersion $\sigma=0.27$. The mean value of $c$ is quite close to that predicted by NFW, though it decreases with the increase of the substructures. Our clustering analysis shows a significantly stronger clustering for low mass halos than Mo & White predict. A formula is therefore presented for the two-point correlation function of halos which can fit our simulation results (both CDM models and scale-free clustering models) very accurately.

Abstract:
The stable clustering hypothesis is a fundamental assumption about the nonlinear clustering of matter in cosmology. It states that the mean physical separation of particles is a constant on sufficiently small scales. While many authors have attempted to test the hypothesis with cosmological N-body simulations, no consensus has been reached on whether and where the hypothesis is valid, because of the limited dynamical range this type of simulations can achieve. In this Letter, we propose to test the hypothesis with high resolution halo simulations, since the individual halo simulations can resolve much better the fine structures of the halos and since almost all pairs of particles with small separations are presumed to be inside virialized halos. We calculated the mean pair velocity for 14 high resolution halos of $\sim 1$ million particles in a low-density flat cold dark matter model. The result agrees very well with the stable clustering prediction within the measurement uncertainty $\sim 30%$ over a large range of scales where the overdensity is $10^3$ to $10^6$. The accuracy of the test can be improved to $\sim 10%$ if some 100 halos with a similar resolution are analyzed.

Abstract:
An accurate fitting formula is reported for the two-point correlation function of the dark matter halos in hierarchical clustering models, It is valid for the the linearly clustering regime, and its accuracy is about 10% in the correlation amplitude for the halos with mass M greater than 1/100 -- 1/1000 of the characteristic non-linear mass M_c. The result is found on the basis of a careful analysis for a large set of scale-free simulations with 17 million particles. The fitting formula has a weak explicit dependence on the index n of the initial power spectrum, but can be equally well applied to the cold dark matter (CDM) cosmological models if the effective index n_{eff} of the CDM power spectrum at the scale of the halo mass replaces the index n. The formula agrees with the analytical formula of Mo & White (MW96) for massive halos with M>M_c, but the MW96 formula significantly underpredicts the correlation for the less massive halos. The difference between the fitting and the analytical formulae amounts to a factor => 2 in the correlation amplitude for M=0.01 M_c. One of the most interesting applications of this fitting formula would be the clustering of galaxies since the majority of halos hosting galaxies satisfies M<< M_c.

Abstract:
We apply the empirical method built for z=0 in the previous work of Wang et al. to a higher redshift, to link galaxy stellar mass directly with its hosting dark matter halo mass at z~0.8. The relation of the galaxy stellar mass and the host halo mass M_infall is constrained by fitting both the stellar mass function and the correlation functions at different stellar mass intervals of the VVDS observation, where M_infall is the mass of the hosting halo at the time when the galaxy was last the central galaxy. We find that for low mass haloes, their residing central galaxies are less massive at high redshift than those at low redshift. For high mass haloes, central galaxies in these haloes at high redshift are a bit more massive than the galaxies at low redshift. Satellite galaxies are less massive at earlier times, for any given mass of hosting haloes. Fitting both the SDSS and VVDS observations simultaneously, we also propose a unified model of the M_stars-M_infall relation, which describes the evolution of central galaxy mass as a function of time. The stellar mass of a satellite galaxy is determined by the same M_stars-M_infall relation of central galaxies at the time when the galaxy is accreted. With these models, we study the amount of galaxy stellar mass increased from z~0.8 to the present day through galaxy mergers and star formation. Low mass galaxies gain their stellar masses from z~0.8 to z=0 mainly through star formation. For galaxies of higher mass, the increase of stellar mass solely through mergers from z=0.8 can make the massive galaxies a factor ~2 larger than observed at z=0. We can also predict stellar mass functions of redshifts up to z~3, and the results are consistent with the latest observations.

Abstract:
In this paper we investigate the velocity dispersion profiles of clusters of galaxies for seven cosmological models. One model is the SCDM model, and the others are six low-density models with the density parameter $\Omega=0.1$, 0.2 or 0.3 and with or without a cosmological constant $\Lambda=1-\Omega$. We find that the velocity dispersion profiles depend both on $\Omega$ and on $\Lambda$. For $\Lambda=0$, the profiles are steeper in a lower-$\Omega$ model than in a higher-$\Omega$ one. The cosmological constant significantly weakens the dependence on $\Omega$: the difference in the profile distributions between two flat models is much smaller than that between the two corresponding open models with the same $\Omega$. These results in principle can be used to constrain the cosmological parameters when a large sample of the velocity dispersion profiles is available. Motivated by the practical situation that a sample of $\sim 100$ clusters with $\sim 100$ measured redshifts per cluster is still the best sample available in the foreseen future, we examine carefully to what degree the cosmological parameters can be constrained with the velocity dispersion profiles of such a sample of clusters. The limited sampling around clusters and the limited number of clusters seriously degrade the discriminative power of the velocity dispersion profiles among cosmological models. We find that the five models of $\Omega\ge 0.2$ cannot be distinguished by this type of observation. Due to the limited sampling, one should be very cautious in extracting information about the density profile and/or the dynamics around single clusters from the diluted velocity dispersion profiles