Non-linear redshift distortions: the two-point correlation function

We consider a situation where the density and peculiar velocities in real space are linear, and we calculate ξ _s , the two-point correlation function in redshift space, incorporating all non-linear effects which arise as a consequence of the map from real to redshift space. Our result is non-perturbative and it includes the effects of possible multi-streaming in redshift space. We find that the deviations from the predictions of the linear redshift distortion analysis increase for the higher spherical harmonics of ξ _s . While the deviations are insignificant for the monopole ξ ₀, the hexadecapole ξ ₄ exhibits large deviations from the linear predictions. For a COBE normalized     ,     cold dark matter (CDM) power spectrum, our results for ξ ₄ deviate from the linear predictions by a factor of two on the scale of ∼10  h ⁻¹ Mpc. The deviations from the linear predictions depend separately on f (Ω) and b . This holds the possibility of removing the degeneracy that exists between these two parameters in the linear analysis of redshift surveys which yields only     .
We also show that the commonly used phenomenological model, where the non-linear redshift two-point correlation function is calculated by convolving the linear redshift correlation function with an isotropic pair velocity distribution function, is a limiting case of our result.     

The bias field of dark matter haloes

Accepted 1998 January 26. Received 1998 January 26; in original form 1997 August 13This paper presents a stochastic approach to the clustering evolution of dark matter haloes in the Universe. Haloes, identified by a Press–Schechter-type algorithm in Lagrangian space, are described in terms of 'counting fields', acting as non-linear operators on the underlying Gaussian density fluctuations. By ensemble-averaging these counting fields, the standard Press–Schechter mass function as well as analytic expressions for the halo correlation function and corresponding bias factors of linear theory are obtained, extending the recent results by Mo & White. The non-linear evolution of our halo population is then followed by solving the continuity equation, under the sole hypothesis that haloes move by the action of gravity. This leads to an exact and general formula for the bias field of dark matter haloes, defined as the local ratio between their number density contrast and the mass density fluctuation. Besides being a function of position and 'observation' redshift, this random field depends upon the mass and formation epoch of the objects and is both non-linear and non-local. The latter features are expected to leave a detectable imprint on the spatial clustering of galaxies, as described, for instance, by statistics like the bispectrum and the skewness. Our algorithm may have several interesting applications, among which is the possibility of generating mock halo catalogues from low-resolution N -body simulations.     

Variance and skewness in the FIRST survey

We investigate the large-scale clustering of radio sources in the FIRST 1.4-GHz survey by analysing the distribution function ( counts in cells ). We select a reliable sample from the the FIRST catalogue, paying particular attention to the problem of how to define single radio sources from the multiple components listed. We also consider the incompleteness of the catalogue. We estimate the angular two-point correlation function w (θ), the variance Ψ₂ and skewness Ψ₃ of the distribution for the various subsamples chosen on different criteria. Both w (θ) and Ψ₂ show power-law behaviour with an amplitude corresponding to a spatial correlation length of r ₀ ∼ 10  h ⁻¹Mpc. We detect significant skewness in the distribution, the first such detection in radio surveys. This skewness is found to be related to the variance through Ψ₃ =  S ₃(Ψ₂)^α, with α = 1.9 ± 0.1, consistent with the non-linear gravitational growth of perturbations from primordial Gaussian initial conditions. We show that the amplitude of variance and the skewness are consistent with realistic models of galaxy clustering.     

The Durham/UKST Galaxy Redshift Survey – IV. Redshift-space distortions in the two-point correlation function

We have investigated the redshift-space distortions in the optically selected Durham/UKST Galaxy Redshift Survey using the two-point galaxy correlation function perpendicular and parallel to the observer's line of sight, ξ(σ, π). On small, non-linear scales we observe an elongation of the constant ξ(σ, π) contours in the line-of-sight direction. This is a result of the galaxy velocity dispersion and is the common 'Finger of God' effect seen in redshift surveys. Our result for the one-dimensional pairwise rms velocity dispersion is 〈 w ²〉^1/2=416±36 km s⁻¹, which is consistent with those from recent redshift surveys and canonical values, but inconsistent with SCDM or LCDM models. On larger, linear scales we observe a compression of the ξ(σ, π) contours in the line-of-sight direction. This is caused by the infall of galaxies into overdense regions, and the Durham/UKST data favours a value of (Ω^0.6/ b )∼0.5, where Ω is the mean mass density of the Universe and b is the linear bias factor that relates the galaxy and mass distributions. Comparison with other optical estimates yields consistent results, with the conclusion that the data do not favour an unbiased critical-density universe.     

Predictions for the clustering properties of the Lyman-alpha forest – I. One-point statistics

We present predictions for the one-point probability distribution and cumulants of the transmitted QSO flux in the high redshift Lyman- α forest. We make use of the correlation between the Lyman- α optical depth and the underlying matter density predicted by gravitational instability theory and seen in numerical hydrodynamic simulations. We have modelled the growth of matter fluctuations using the non-linear shear‐free dynamics, an approximation which reproduces well the results of perturbation theory for the cumulants in the linear and weakly non-linear clustering regime. As high matter overdensities tend to saturate in spectra, the statistics of the flux distribution are dominated by weakly non-linear overdensities. As a result, our analytic approach can produce accurate predictions, when tested against N -body simulation results, even when the underlying matter field has root-mean-square fluctuations larger than unity. Our treatment can be applied to either Gaussian or non-Gaussian initial conditions. Here we concentrate on the former case, but also include a study of a specific non-Gaussian model. We discuss how the methods and predictions we present can be used as a tool to study the generic clustering properties of the Lyman- α forest at high redshift. With such an approach, rather than concentrating on simulating specific cosmological models, we may be in a position to directly test our assumptions for the Gaussian nature of the initial conditions, and the gravitational instability origin of structure itself. In a separate paper we present results for two-point statistics.     

The star formation histories of galaxies in the Sloan Digital Sky Survey

We present the results of a moped analysis of  ∼3 × 10⁵  galaxy spectra from the Sloan Digital Sky Survey Data Release 3 (SDSS DR3), with a number of improvements in data, modelling and analysis compared with our previous analysis of DR1. The improvements include: modelling the galaxies with theoretical models at a higher spectral resolution of 3 Å, better calibrated data, an extended list of excluded emission lines and a wider range of dust models. We present new estimates of the cosmic star formation rate (SFR), the evolution of stellar mass density and the stellar mass function from the fossil record. In contrast to our earlier work the results show no conclusive peak in the SFR out to a redshift around 2 but continue to show conclusive evidence for 'downsizing' in the SDSS fossil record. The star formation history is now in good agreement with more traditional instantaneous measures. The galaxy stellar mass function is determined over five decades of mass, and an updated estimate of the current stellar mass density is presented. We also investigate the systematic effects of changes in the stellar population modelling, the spectral resolution, dust modelling, sky lines, spectral resolution and the change of data set. We find that the main changes in the results are due to the improvements in the calibration of the SDSS data, changes in the initial mass function and the theoretical models used.     

Mass selection bias in galaxy cluster peculiar velocities from the kinetic Sunyaev–Zel'dovich effect

Upcoming surveys for galaxy clusters using the Sunyaev–Zel'dovich effect are potentially sensitive enough to create a peculiar velocity catalogue. The statistics of these peculiar velocities are sensitive to cosmological parameters. We develop a method to explore parameter space using N -body simulations in order to quantify dark matter halo velocity statistics which will be useful for cluster peculiar velocity observations. We show that mass selection bias from a kinetic Sunyaev–Zel'dovich velocity catalogue forecasts rms peculiar velocities with a much more complicated  Ω_m  dependency than suggested by linear perturbation theory. In addition, we show that both two-point functions for velocities disagree with linear theory predictions out to  ∼40  h ⁻¹ Mpc  separations. A pedagogical appendix is included developing linear theory notation with respect to the two-point peculiar velocities functions.     

Effects of correlation between merging steps on the global halo formation

The excursion set theory of halo formation is modified by adopting the fractional Brownian motion, to account for possible correlation between merging steps. We worked out analytically the conditional mass function, halo merging rate and formation time distribution in the spherical collapse model. We also developed an approximation for the ellipsoidal collapse model and applied it to the calculation of the conditional mass function and the halo formation time distribution. For models in which the steps are positively correlated, the halo merger rate is enhanced when the accreted mass is less than  ∼25 M *  , while for the negatively correlated case this rate is reduced. Compared with the standard model in which the steps are uncorrelated, the models with positively correlated steps produce more aged population in small mass haloes and more younger population in large mass haloes, while for the models with negatively correlated steps the opposite is true. An examination of simulation results shows that a weakly positive correlation between successive merging steps appears to fit best. We have also found a systematic effect in the measured mass function due to the finite volume of simulations. In future work, this will be included in the halo model to accurately predict the three-point correlation function estimated from simulations.     

Towards breaking the Ω-bias degeneracy in density–velocity comparisons

I derive a second-order local relation between the redshift-space mass density field and the real-space velocity field. This relation can be useful for comparisons between the cosmic density and peculiar velocity fields, for a number of reasons. First, relating the real-space velocity directly to the redshift-space density enables one to avoid the Ω-dependent reconstruction of the density field in real space. Secondly, the reconstruction of the three-dimensional velocity field in redshift space, questionable because of its vorticity, is also unnecessary. Finally, a similar relation between the galaxy density field and the velocity field offers a way to break the Ω-bias degeneracy in density–velocity comparisons, when combined with an additional measurement of the redshift-space galaxy skewness. I derive the latter relation under the assumption of non-linear but local bias; accounting for stochasticity of bias is left for further study.     

Peculiar velocities and the mean density parameter

We study the peculiar velocity field inferred from the Mark III spirals using a new method of analysis. We estimate optimal values of Tully–Fisher scatter and zero-point offset, and we derive the three-dimensional rms peculiar velocity ( σ _v ) of the galaxies in the samples analysed. We check our statistical analysis using mock catalogues derived from numerical simulations of cold dark matter (CDM) models considering measurement uncertainties and sampling variations. Our best determination for the observations is σ _v =(660±50) km s⁻¹. We use the linear theory relation between σ _v , the density parameter Ω, and the galaxy correlation function ξ ( r ) to infer the quantity     , where b is the linear bias parameter of optical galaxies and the uncertainties correspond to bootstrap resampling and an estimated cosmic variance added in quadrature. Our findings are consistent with the results of cluster abundances and redshift-space distortion of the two-point correlation function. These statistical measurements suggest a low value of the density parameter Ω∼0.4 if optical galaxies are not strongly biased tracers of mass.     

Companions around the nearest luminous galaxies: Segregation and selection effects

Using the “Updated Nearby Galaxy Catalog”, we consider different properties of companion galaxies around luminous hosts in the Local Volume. The data on stellar masses, linear diameters, surface brightnesses, HI‐richness, specific star formation rate (sSFR), and morphological types are discussed for members of the nearest groups, including the Milky Way and M 31 groups, as a function of their separation from the hosts. Companion galaxies in groups tend to have lower stellar masses, smaller linear diameters, and fainter mean surface brightnesses as the distance to their host decreases. The hydrogen‐to‐stellar mass ratio of the companions increases with their linear projected separation from the dominant luminous galaxy. This tendency is more expressed around the bulge‐dominated hosts. While linear separation of the companions decreases, their mean sSFR becomes lower, accompanied with the increasing sSFR scatter. the typical linear projected separation of dSphs around the bulge‐dominated hosts, 350 kpc, is substantially larger than that around the disk‐dominated ones, 130 kpc. This difference probably indicates the presence of larger hot/warm gas haloes around the early‐type host galaxies. The mean fraction of dSph (quenched) companions in the 11 nearest groups as a function of their projected separation Rp can be expressed as ƒ(E) = (0.55–0.69)×R_p. The fraction of dSphs around the Milky Way and M 31 looks much higher than in other nearby groups because the quenching efficiency dramatically increases towards the ultra‐low mass companions. We emphasize that the observed properties of the Local Group are not typical for other groups in the Local Volume due to the role of selection effects caused by our location inside the Local Group. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Predicting the clustering of X-ray selected galaxy clusters in flux-limited surveys

We present a model to predict the clustering properties of X-ray selected clusters in flux-limited surveys. Our technique correctly accounts for past light-cone effects on the observed clustering and follows the non-linear evolution in redshift of the underlying dark matter correlation function and cluster bias factor. The conversion of the limiting flux of a survey into the corresponding minimum mass of the hosting dark matter haloes is obtained by using theoretical and empirical relations between mass, temperature and X-ray luminosity of galaxy clusters. Finally, our model is calibrated to reproduce the observed cluster counts adopting a temperature–luminosity relation moderately evolving with redshift. We apply our technique to three existing catalogues: the ROSAT Brightest Cluster Sample (BCS); the X-ray Brightest Abell-type Cluster sample (XBACs); and the ROSAT –ESO Flux-Limited X-ray sample (REFLEX). Moreover, we consider an example of possible future space missions with fainter limiting flux. In general, we find that the amplitude of the spatial correlation function is a decreasing function of the limiting flux and that the Einstein–de Sitter models always give smaller correlation amplitudes than open or flat models with low matter density parameter Ω_0m. In the case of the XBACs catalogue, the comparison with previous estimates of the observational spatial correlation shows that only the predictions of models with Ω_0m=0.3 are in good agreement with the data, while the Einstein–de Sitter models have too low a correlation strength. Finally, we use our technique to discuss the best strategy for future surveys. Our results show that, to study the clustering properties of X-ray selected clusters, the choice of a wide area catalogue, even with a brighter limiting flux, is preferable to a deeper, but smaller area, survey.     

Peculiar velocities of galaxies and clusters

We present a simple model for the shape of the distribution function of galaxy peculiar velocities. We show how both non-linear and linear theory terms combine to produce a distribution which has an approximately Gaussian core with exponential wings. The model is easily extended to study how the statistic depends on the type of particle used to trace the velocity field (dark matter particles, dark matter haloes, galaxies), and on the density of the environment in which the test particles are located. Comparisons with simulations suggest that our model is accurate. We also show that the evolution of the peculiar velocities depends on the local, rather than the global, density. Since clusters populate denser regions on average, using cluster velocities with the linear theory scaling may lead to an overestimate of the global value of Ω₀. Conversely, using linear theory with the global value of Ω₀ to scale cluster velocities from the initial to the present time results in an underestimate of their true velocities. In general, however, the directions of motions of haloes are rather well described by linear theory. Our results help to simplify models of redshift-space distortions considerably.     

Clustering of galaxy clusters in cold dark matter universes

We use very large cosmological N -body simulations to obtain accurate predictions for the two-point correlations and power spectra of mass-limited samples of galaxy clusters. We consider two currently popular cold dark matter (CDM) cosmogonies, a critical density model ( τ CDM) and a flat low density model with a cosmological constant (ΛCDM). Our simulations each use 10⁹ particles to follow the mass distribution within cubes of side 2  h ⁻¹ Gpc ( τ CDM) and 3  h ⁻¹ Gpc (ΛCDM) with a force resolution better than 10⁻⁴ of the cube side. We investigate how the predicted cluster correlations increase for samples of increasing mass and decreasing abundance. Very similar behaviour is found in the two cases. The correlation length increases from     for samples with mean separation     to     for samples with     The lower value here corresponds to τ CDM and the upper to ΛCDM. The power spectra of these cluster samples are accurately parallel to those of the mass over more than a decade in scale. Both correlation lengths and power spectrum biases can be predicted to better than 10 per cent using the simple model of Sheth, Mo & Tormen. This prediction requires only the linear mass power spectrum and has no adjustable parameters. We compare our predictions with published results for the automated plate measurement (APM) cluster sample. The observed variation of correlation length with richness agrees well with the models, particularly for ΛCDM. The observed power spectrum (for a cluster sample of mean separation     ) lies significantly above the predictions of both models.     

Thermal instability in a weakly ionized plasma

We revisit the problem of clump formation due to thermal instabilities in a weakly ionized plasma with the help of a linear perturbation analysis, as discussed by Nejad-Asghar & Ghanbari. In the absence of a magnetic field and ambipolar diffusion the characteristic equation reduces to the thermal instability described by Field. We derive the critical wavelengths, which separate the spatial ranges of stability and instability. Contrary to the original analysis of Nejad-Asghar & Ghanbari, perturbations with a wavelength larger than the critical wavelength destabilize the cloud. Moreover, the instability regime of isentropic perturbations is drastically reduced. Isobaric modes with real values of the critical wavelength appear only if the density dependence of the cooling rate is more pronounced than the temperature dependence. Isentropic modes arise only if the power of the density in the cooling rate is smaller than 1/2, which is not fulfilled for CO cooling. We find that ambipolar diffusion is not a dominating heating process in molecular gas.     

Dark energy and the evolution of spherical overdensities

We use the non-linear spherical model in cold dark matter (CDM) cosmologies with dark energy to investigate the effects of dark energy on the growth of structure and the formation of virialized structures. We consider dark energy models with a constant equation-of-state parameter w . For  −1 < w < −1/3  , clusters form earlier and are more concentrated in quintessence than in ΛCDM models, but they form later and are less concentrated than in the corresponding open model with the same matter density and no dark energy. We point out some confusion in the literature around the expression of the collapse factor (ratio of the radius of the sphere at virialization to that at turnaround) derived from the virial theorem. We use the Sheth & Tormen extension of the Press–Schechter framework to calculate the evolution of the cluster abundance in different models and show the sensitivity of the cluster abundance to both the amplitude of the mass fluctuations, σ₈, and the σ₈– w normalization, selected to match either the cosmic microwave background observations or the abundance of X-ray clusters.     

Correlations of cosmic tidal fields

We study correlations amongst tidal fields originated by the large-scale distribution of matter in the Universe. The two-point tidal correlation is described as a rank-4 tensor, the elements of which can be written in terms of four fundamental scalar functions ranging, with respect to the spatial separation, from purely transversal to purely longitudinal correlations. Tidal fields, on both galaxy and cluster scales, are revealed to be correlated over distances larger than the mass–density correlation length, although traceless tidal fields show anti-correlation between diagonal terms along orthogonal directions. The cross-correlation between mass and tidal field is also analysed. These results are relevant for galaxy formation and the interpretation of large-scale weak lensing phenomena.     

Dry mergers: a crucial test for galaxy formation

We investigate the role that dry mergers play in the build-up of massive galaxies within the cold dark matter paradigm. Implementing an empirical shut-off mass scale for star formation, we find a nearly constant dry merger rate of  ∼6 × 10⁻⁵ Mpc⁻³ Gyr⁻¹  at   z ≤ 1  and a steep decline at larger z . Less than half of these mergers are between two galaxies that are morphologically classified as early-types, and the other half is mostly between an early- and late-type galaxy. Latter are prime candidates for the origin of tidal features around red elliptical galaxies. The introduction of a transition mass scale for star formation has a strong impact on the evolution of galaxies, allowing them to grow above a characteristic mass scale of   M _{*, c} ∼ 6.3 × 10¹⁰ M_⊙  by mergers only. As a consequence of this transition, we find that around   M _{*, c}   , the fraction of 1:1 mergers is enhanced with respect to unequal mass major mergers. This suggests that it is possible to detect the existence of a transition mass scale by measuring the relative contribution of equal mass mergers to unequal mass mergers as a function of galaxy mass. The evolution of the high-mass end of the luminosity function is mainly driven by dry mergers at low z . We however find that only 10–20 per cent of galaxies more massive than   M _{*, c}   experience dry major mergers within their last Gyr at any given redshift   z ≤ 1  .