Abstract:
This paper reviews the measurements of galaxy correlations at high redshifts, and discusses how these may be understood in models of hierarchical gravitational collapse. The clustering of galaxies at redshift one is much weaker than at present, and this is consistent with the rate of growth of structure expected in an open universe. If $\Omega=1$, this observation would imply that bias increases at high redshift, in conflict with observed $M/L$ values for known high-$z$ clusters. At redshift 3, the population of Lyman-limit galaxies displays clustering which is of similar amplitude to that seen today. This is most naturally understood if the Lyman-limit population is a set of rare recently-formed objects. Knowing both the clustering and the abundance of these objects, it is possible to deduce empirically the fluctuation spectrum required on scales which cannot be measured today owing to gravitational nonlinearities. Of existing physical models for the fluctuation spectrum, the results are most closely matched by a low-density spatially flat universe. This conclusion is reinforced by an empirical analysis of CMB anisotropies, in which the present-day fluctuation spectrum is forced to have the observed form. Open models are strongly disfavoured, leaving $\Lambda$CDM as the most successful simple model for structure formation.

Abstract:
We investigate the effects on cosmological clustering statistics of empirical biasing, where the galaxy distribution is a local transformation of the present-day Eulerian density field. The effects of the suppression of galaxy numbers in voids, and their enhancement in regions of high density, are considered, independently and in combination. We compare results from numerical simulations with the predictions of simple analytic models. We find that the bias is generally scale-dependent, so that the shape of the galaxy power spectrum differs from that of the underlying mass distribution. The degree of bias is always a monotonic function of scale, tending to an asymptotic value on scales where the density fluctuations are linear. The scale dependence is often rather weak, with many reasonable prescriptions giving a bias which is nearly independent of scale. We have investigated whether such an Eulerian bias can reconcile a range of theoretical power spectra with the twin requirements of fitting the galaxy power spectrum and reproducing the observed mass-to-light ratios in clusters. It is not possible to satisfy these constraints for any member of the family of CDM-like power spectra in an Einstein - de Sitter universe when normalised to match COBE on large scales and galaxy cluster abundances on intermediate scales. We discuss what modifications of the mass power spectrum might produce agreement with the observational data.

Abstract:
We outline a simple approach to understanding the physical origin of bias in the distribution of galaxies relative to that of dark matter. The first step is to specify how collapsed, virialized halos of dark matter trace the overall matter distribution. The next step is to make a connection between halos and the luminous galaxies we observe. We appeal to the results of semi-analytic models of galaxy formation that are tuned to fit the observed luminosity functions of local groups and clusters. We have also used a high-resolution N-body simulation of a cold dark matter (CDM) universe to study the bias relation in more detail. The differences between the galaxy and dark matter distributions are quantified using a number of different clustering statistics. We arrive at the following general conclusions: 1) A comparison of the galaxy and dark matter density fields shows that linear biasing is a good description on large scales. 2) The bias factor b depends on the shape and normalization of the power spectrum. The lower the normalization, the larger the bias. More bias is obtained for spectra with more power on large scales. For "realistic" models, b ranges from 1 to 2.5. 3) Galaxies of different luminosity or morphology have different bias factors. 4) The scale dependence of the bias factor is weak.

Abstract:
Local non-Gaussianity, parametrized by $f_{\rm NL}$, introduces a scale-dependent bias that is strongest at large scales, precisely where General Relativistic (GR) effects also become significant. With future data, it should be possible to constrain $f_{\rm NL} = {\cal O}(1)$ with high redshift surveys. GR corrections to the power spectrum and ambiguities in the gauge used to define bias introduce effects similar to $f_{\rm NL}= {\cal O}(1)$, so it is essential to disentangle these effects. For the first time in studies of primordial non-Gaussianity, we include the consistent GR calculation of galaxy power spectra, highlighting the importance of a proper definition of bias. We present observable power spectra with and without GR corrections, showing that an incorrect definition of bias can mimic non-Gaussianity. However, these effects can be distinguished by their different redshift and scale dependence, so as to extract the true primordial non-Gaussianity.

Abstract:
We propose a heuristic model that displays the main features of realistic theories for galaxy bias. We show that the low-order clustering statistics of the dark-matter distribution depend almost entirely on the locations and density profiles of dark-matter haloes. A hypothetical galaxy catalogue depends on (i) the efficiency of galaxy formation, as manifested by the halo occupation number -- the number of galaxies brighter than some sample limit contained in a halo of a given mass; (ii) the location of these galaxies within their halo. The first factor is constrained by the empirical luminosity function of groups. For the second factor, we assume that one galaxy marks the halo centre, with any remaining galaxies acting as satellites that trace the halo mass. These simple assumptions amount to a recipe for non-local bias, in which the probability of finding a galaxy is not a simple function of its local mass density. We have applied this prescription to some CDM models of current interest, and find that the predictions are close to the observed galaxy correlations for a flat $\Omega=0.3$ model ($\Lambda$CDM), but not for an $\Omega=1$ model with the same power spectrum ($\tau$CDM). This is an inevitable consequence of cluster normalization for the power spectra: cluster-scale haloes of given mass have smaller core radii for high $\Omega$, and hence display enhanced small-scale clustering. Finally, the pairwise velocity dispersion of galaxies in the $\Lambda$CDM model is lower than that of the mass, allowing cluster-normalized models to yield a realistic Mach number for the peculiar velocity field. This is largely due to the strong variation of galaxy-formation efficiency with halo mass that is required in this model.

Abstract:
When dealing with observables, one needs to generalize the bias relation between the observed galaxy fluctuation field to the underlying matter distribution in a gauge-invariant way. We provide such relation at second-order in perturbation theory adopting the local Eulerian bias model and starting from the observationally motivated uniform-redshift gauge. Our computation includes the presence of primordial non-Gaussianity. We show that large scale-dependent relativistic effects in the Eulerian bias arise independently from the presence of some primordial non-Gaussianity. Furthermore, the Eulerian bias inherits from the primordial non-Gaussianity not only a scale-dependence, but also a modulation with the angle of observation when sources with different biases are correlated.

Abstract:
On large scales, the higher order moments of the mass distribution, $S_J=\xibar_J/\xibar_2^{J-1}$, e.g., the skewness $S_3$ and kurtosis $S_4$, can be predicted using non-linear perturbation theory. Comparison of these predictions with moments of the observed galaxy distribution probes the bias between galaxies and mass. Applying this method to models with initially Gaussian fluctuations and power spectra $P(k)$ similar to that of galaxies in the APM survey, we find that the predicted higher order moments $S_J(R)$ are in good agreement with those directly inferred from the APM survey {\it in the absence of bias}. We use this result to place limits on the linear and non-linear bias parameters. Models in which the extra power observed on large scales (with respect to standard CDM) is produced by scale-dependent bias match the APM higher order amplitudes only if non-linear bias (rather than non-linear gravity) generates the observed higher order moments. When normalized to COBE DMR, these models are significantly ruled out by the $S_3$ observations. The cold plus hot dark matter model normalized to COBE can reproduce the APM higher order correlations if one introduces non-linear bias terms, while the low-density CDM model with a cosmological constant does not require any bias to fit the large-scale amplitudes.

Abstract:
When clusters of galaxies are viewed in projection, one cannot avoid picking up foreground/background interlopers (FBIs), that lie within the virial cone (VC), but outside the virial sphere. Structural & kinematic deprojection equations are not known for an expanding Universe, where the Hubble flow (HF) stretches the line-of-sight (LOS) distribution of velocities. We analyze 93 mock relaxed clusters, built from a cosmological simulation. The stacked mock cluster is well fit by an m=5 Einasto DM density profile (but only out to 1.5 virial radii [r_v]), with velocity anisotropy (VA) close to the Mamon-Lokas model with VA radius equal to that of density slope -2. The surface density of FBIs is nearly flat out to r_v, while their LOS velocity distribution shows a dominant gaussian cluster-outskirts component and a flat field component. This distribution of FBIs in projected phase space is nearly universal in mass. A local k=2.7 sigma velocity cut returns the LOS velocity dispersion profile (LOSVDP) expected from the NFW density and VA profiles measured in 3D. The HF causes a shallower outer LOSVDP that cannot be well matched by the Einasto model for any k. After this velocity cut, FBIs still account for 23% of DM particles within the VC (close to the observed fraction of cluster galaxies lying off the Red Sequence). The best-fit projected NFW/Einasto models underestimate the 3D concentration by 6+/-6% (16+/-7%) after (before) the velocity cut, unless a constant background is included in the fit. Assuming the correct mass profile, the VA profile is well recovered from the measured LOSVDP, with a slight bias towards more radial orbits in the outer regions. These small biases are overshadowed by large cluster-cluster variations caused by cosmic variance. An appendix provides an analytical approximation to the surface density, projected mass and tangential shear profiles of the Einasto model.

Abstract:
We have used a combination of high resolution cosmological N-body simulations and semi-analytic modelling of galaxy formation to investigate the processes that determine the spatial distribution of galaxies in cold dark matter (CDM) models. The galaxy distribution depends sensitively on the efficiency with which galaxies form in halos of different mass. In small mass halos, galaxy formation is inhibited by the reheating of cooled gas by feedback processes, whereas in large mass halos, it is inhibited by the long cooling time of the gas. As a result, the mass-to-light ratio of halos has a deep minimum at the halo mass associated with L* galaxies. This leads to a scale dependent bias in the distribution of galaxies relative to the distribution of mass. On large scales, the bias in the galaxy distribution is related in a simple way to the bias in the distribution of massive halos. On small scales, the correlation function is determined by the interplay between various effects including the spatial exclusion of dark matter halos, the distribution function of the number of galaxies occupying a single dark matter halo and, to a lesser extent, dynamical friction. Remarkably, these processes conspire to produce a correlation function in a flat, Omega_0=0.3, CDM model that is close to a power-law over nearly four orders of magnitude in amplitude. This model agrees well with the correlation function of galaxies measured in the APM survey. On small scales, the model galaxies are less strongly clustered than the dark matter whereas on large scales, they trace the occupied halos. Our clustering predictions are robust to changes in the parameters of the galaxy formation model, provided only those models that match the bright end of the galaxy luminosity function are considered. (abridged)

Abstract:
Assembly bias describes the finding that the clustering of dark matter haloes depends on halo formation time at fixed halo mass. In this paper, we analyse the influence of assembly bias on galaxy clustering using both semi-analytical models (SAMs) and observational data. At fixed stellar mass, SAMs predict that the clustering of {\it central} galaxies depends on the specific star formation rate (sSFR), with more passive galaxies having a higher clustering amplitude. We find similar trends using SDSS group catalogues, and verify that these are not affected by possible biases due to the group finding algorithm. Low mass central galaxies reside in narrow bins of halo mass, so the observed trends of higher clustering amplitude for galaxies with lower sSFR is not driven by variations of the parent halo mass. We argue that the clustering dependence on sSFR represent a direct detection of assembly bias. In addition, contrary to what expected based on clustering of dark matter haloes, we find that low-mass central galaxies in SAMs with larger host halo mass have a {\it lower} clustering amplitude than their counter-parts residing in lower mass haloes. This results from the fact that, at fixed stellar mass, assembly bias has a stronger influence on clustering than the dependence on the parent halo mass.