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.     

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.     

Determining bias with cumulant correlators

correlator of the galaxy density field Q ₂₁ is examined from the point of view of biasing. It is shown that, to leading order, it depends on two biasing parameters b b ₂, and on q ₂₁, the underlying cumulant correlator of the mass. As the skewness Q ₃ has analogous properties, the slope of the correlation function −γ, Q ₃ and Q ₂₁ uniquely determine the bias parameter on a particular scale to be b  = γ/6( Q ₂₁ −  Q ₃), when working in the context of gravitational instability with Gaussian initial conditions. Thus on large scales, easily accessible with the future Sloan Digital Sky Survey and the 2 Degree Field Survey, it will be possible to extract b b ₂ from simple counts-in-cells measurements. Moreover, the higher order cumulants, Q _N , successively determine the higher order biasing parameters. From these it is possible to predict higher order cumulant correlators as well. Comparison of the predictions with the measurements will provide internal consistency checks on the validity of the assumptions in the theory, most notably perturbation theory of the growth of fluctuations by gravity and Gaussian initial conditions. Since the method is insensitive Ω, it can be successfully combined with results from velocity fields, which determine Ω^0.6/b, to measure the total density parameter in the Universe.     

Sources of contamination to weak lensing tomography: redshift-dependent shear measurement bias

The current methods available to estimate gravitational shear from astronomical images of galaxies introduce systematic errors which can affect the accuracy of weak lensing cosmological constraints. We study the impact of KSB shape measurement bias on the cosmological interpretation of tomographic two-point weak lensing shear statistics.
We use a set of realistic image simulations produced by the Shear Testing Programme (STEP) collaboration to derive shape measurement bias as a function of redshift. We define biased two-point weak lensing statistics and perform a likelihood analysis for two fiducial surveys. We present a derivation of the covariance matrix for tomography in real space and a fitting formula to calibrate it for non-Gaussianity.
We find the biased aperture mass dispersion is reduced by  ∼20 per cent  at redshift ∼1, and has a shallower scaling with redshift. This effect, if ignored in data analyses, biases σ₈ and w ₀ estimates by a few per cent. The power of tomography is significantly reduced when marginalizing over a range of realistic shape measurement biases. For a Canada-France-Hawaii Telescope Legacy Survey (CFHTLS)-Wide-like survey,  [Ω_m, σ₈]  confidence regions are degraded by a factor of 2, whereas for a Kilo-Degree Survey (KIDS)-like survey the factor is 3.5. Our results are strictly valid only for KSB methods, but they demonstrate the need to marginalize over a redshift-dependent shape measurement bias in all future cosmological analyses.     

The redshift evolution of clustering in the Hubble Deep Field

We present a correlation function analysis for the catalogue of photometric redshifts obtained from the Hubble Deep Field image by Fernandez-Soto, Lanzetta & Yahil. By dividing the catalogue into redshift bins of width Δ z =0.4 we measured the angular correlation function w ( θ ) as a function of redshift up to z ∼4.8. From these measurements we derive the trend of the correlation length r ₀. We find that r ₀( z ) is roughly constant with look-back time up to z ≃2, and then increases to higher values at z ≳2.4. We estimate the values of r ₀, assuming ξ ( r , z )=[ r r ₀( z )]^{− γ} , γ =1.8 and various geometries. For Ω₀=1 we find r ₀( z =3)≃7.00±4.87  h ⁻¹ Mpc, in good agreement with the values obtained from analysis of the Lyman break galaxies.     

Power spectrum shape from peculiar velocity data

We constrain the velocity power spectrum shape parameter Γ in linear theory using the nine bulk flow and shear moments estimated from four recent peculiar velocity surveys. For each survey, a likelihood function for Γ was found after marginalizing over the power spectrum amplitude  σ₈Ω^0.6_m  using constraints obtained from comparisons between redshift surveys and peculiar velocity data. In order to maximize the accuracy of our analyses, the velocity noise σ_* was estimated directly for each survey. A statistical analysis of the differences between the values of the moments estimated from different surveys showed consistency with theoretical predictions, suggesting that all the surveys investigated reflect the same large-scale flows. The peculiar velocity surveys were combined into a composite survey yielding the constraint  Γ= 0.13^+0.09_−0.05  . This value is lower than, but consistent with, values obtained using redshift surveys and cosmic microwave background data.     

Measuring Ω0 using cluster evolution

The evolution of the abundance of galaxy clusters depends sensitively on the value of the cosmological density parameter, Ω₀. Recent ASCA data are used to quantify this evolution as measured by the cluster X-ray temperature function. A χ² minimization fit to the cumulative temperature function, as well as a maximum-likelihood estimate (which requires additional assumptions about cluster luminosities), leads to the estimate Ω₀ ≈ 0.45 ± 0.25 (1σ statistical error). Various systematic uncertainties are considered, none of which significantly enhances the probability that Ω₀ = 1. These conclusions hold for models with or without a cosmological constant, i.e., with Λ₀ = 0 or Λ₀ = 1 − Ω₀. The statistical uncertainties are at least as large as any of the individual systematic errors that have been considered here, suggesting that additional temperature measurements of distant clusters will allow an improvement in this estimate. An alternative method that uses the highest redshift clusters to place an upper limit on Ω₀ is also presented and tentatively applied, with the result that Ω₀  1 can be ruled out at the 98 per cent confidence level. Whilst this method does not require a well-defined statistical sample of distant clusters, there are still modelling uncertainties that preclude a firmer conclusion at this time.     

On the errors on Ω0: Monte Carlo simulations of the EMSS cluster sample

We perform Monte Carlo simulations of synthetic EMSS cluster samples, to quantify the systematic errors and the statistical uncertainties on the estimate of Ω₀ derived from fits to the cluster number density evolution and to the X-ray temperature distribution up to z =0.83 . We identify the scatter around the relation between cluster X-ray luminosity and temperature to be a source of systematic error, of the order of Δ_systΩ₀=0.09 , if not properly taken into account in the modelling. After correcting for this bias, our best Ω₀ is 0.66. The uncertainties on the shape and normalization of the power spectrum of matter fluctuations imply relatively large uncertainties on this estimate of Ω₀, of the order of Δ_statΩ₀=0.1 at the 1 σ level. On the other hand, the statistical uncertainties due to the finite size of the high-redshift sample are twice as small. Therefore, what is needed in order to improve the accuracy of Ω₀ estimates based on cluster number density evolution is a more reliable measure of the local temperature function and a better understanding of the cluster observed properties both in the local Universe and at high redshift, that is the relation between cluster mass, temperature and luminosity. This requires detailed observations of X-ray selected cluster samples, in comparison with hydrodynamic simulations including refined physics.     

Constraints on perfect fluid and scalar field dark energy models from future redshift surveys

We discuss the constraints that future photometric and spectroscopic redshift surveys can put on dark energy through the baryon oscillations of the power spectrum. We model the dark energy either with a perfect fluid or a scalar field and take into account the information contained in the linear growth function. We show that the growth function helps to break the degeneracy in the dark energy parameters and reduce the errors on   w ₀, w ₁  roughly by 30 per cent, making more appealing multicolour surveys based on photometric redshifts. We find that a 200-deg² spectroscopic survey reaching   z ≈ 3  can constrain   w ₀, w ₁  to within  Δ w ₀= 0.21, Δ w ₁= 0.26  , to  Δ w ₀= 0.39, Δ w ₁= 0.54  using photometric redshifts with an absolute uncertainty of 0.02, and to  Δ w ₀= 0.43, Δ w ₁= 0.66  with an uncertainty of 0.04. In the scalar field case, we show that the slope n of the inverse power-law potential for dark energy can be constrained to  Δ n = 0.26  (spectroscopic redshifts) or  Δ n = 0.40  (photometric redshifts), i.e. better than with future ground-based supernovae surveys or cosmic microwave background data.     

The luminosity function around isolated spiral galaxies

We determine the companion galaxy luminosity function (LF) for regions around isolated spiral galaxies. If we assume that any excess in the galaxy number counts in the vicinity of a spiral galaxy is due to galaxies at the same distance, then a system LF can be determined from the variation of excess numbers with apparent magnitude. By studying the excess over many field 'centre' galaxies, a good statistical accuracy can be obtained for the companion galaxy LF. Since redshift information is not required for the faint galaxies, it is possible to sample further down the LF as compared with redshift surveys. For 23 primary galaxies of known redshift, we find a dwarf satellite Schechter LF with a characteristic magnitude M _V *( D )≃−19 and a faint-end slope α=−1.7, down to M_V =−14 ( H ₀=50 km s⁻¹ Mpc⁻¹).     

Bulk flow and shear moments of the SFI++ survey

We find the nine bulk flow and shear moments from the SFI++ survey, as well as for subsamples of group and field galaxies. We constrain the velocity power spectrum shape parameter Γ in linear theory using these moments. A likelihood function for Γ was found after marginalizing over the power spectrum amplitude  σ₈Ω^0.6_m  using constraints obtained from comparisons between redshift surveys and peculiar velocity data. We have estimated the velocity noise  σ_*  from the data since without it our results may be biased. We also performed a statistical analysis of the difference between the field and group catalogues and found that the results from each reflect the same underlying large-scale flows. We found that we can constrain the power spectrum shape parameter to be  Γ= 0.15^+0.18_−0.08  for the groups catalogue and  Γ= 0.09^+0.04_−0.04  for the field galaxy catalogue in fair agreement with the value from Wilkinson Microwave Anisotropy Probe .     

Spatial clustering of Ultra Steep Spectrum sources and galaxies

We present measurements of the clustering properties of galaxies in the field of redshift range 0.5 ≲ z ≲ 1.5 Ultra Steep Spectrum radio sources selected from the Sydney University Molonglo Sky Survey and the National Radio Astronomy Observatories Very Large Array Sky Survey. Galaxies in these USS fields were identified in deep near-infrared observations, complete down to   K _s= 20  , using the IRIS2 instrument at the Anglo-Australian Telescope. We used the redshift distribution of   K _s < 20  galaxies taken from Cimatti et al. (2002) to constrain the correlation length r ₀. We find a strong correlation signal of galaxies with   K _s < 20  around our USS sample. A comoving correlation length   r ₀= 14.0 ± 2.8  h ⁻¹ Mpc  and γ= 1.98 ± 0.15 are derived in a flat cosmological model universe.
We compare our findings with those obtained in a cosmological N -body simulation populated with galform semi-analytic galaxies. We find that clusters of galaxies with masses in the range   M = 10^13.4–14.2  h ⁻¹ M_⊙  have a cluster–galaxy cross-correlation amplitude comparable to those found between the USS hosts and galaxies. These results suggest that distant radio galaxies are excellent tracers of galaxy overdensities and pinpoint the progenitors of present day rich clusters of galaxies.     

Measuring Ω0 from the entropy evolution of clusters

In this paper we have extended the entropy-driven model of cluster evolution developed by Bower in order to be able to predict the evolution of galaxy clusters for a range of cosmological scenarios. We have applied this model to recent measurements of the evolution of the L _x− T normalization and X-ray luminosity function in order to place constraints on cosmological parameters. We find that these measurements alone do not select a particular cosmological framework. An additional constraint is required on the effective slope of the power spectrum to break the degeneracy that exists between this and the background cosmology. We therefore include a theoretical calculation of the Ω₀ dependence on the power spectrum, based on the cold dark matter paradigm, which infers Ω₀<0.55 (0.1<Ω₀<0.7 for Ω₀+Λ₀=1), at the 95 per cent confidence level. Alternatively, an independent measurement of the slope of the power spectrum from galaxy clustering requires Ω₀<0.6 (Ω₀<0.65 for Ω₀+Λ₀=1), again to 95 per cent confidence. The rate of entropy evolution is insensitive to the values of Ω₀ considered, although it is sensitive to changes in the distribution of the intracluster medium.     

Cold dark matter models with a step-like initial power spectrum

We investigate the properties of clusters of galaxies in the ΛCDM models with a step-like initial power spectrum. We examine the mass function, the peculiar velocities and the power spectrum of clusters in models with different values of the density parameter Ω₀, the normalized Hubble constant h and the spectral parameter p that describes the shape of the initial power spectrum. The results are compared with observations. We also investigate the rms bulk velocity in the models, where the properties of clusters are consistent with the observed data. We find that the power spectrum of clusters is in good agreement with the observed power spectrum of the Abell–ACO clusters if the spectral parameter p is in the range p =0.6–0.8. The power spectrum and the rms peculiar velocity of clusters are consistent with observations only if Ω₀<0.4 . The models with Ω₀=0.3 are consistent with the observed properties of clusters if h =0.50–0.63. For h =0.65, we find that Ω₀=0.20–0.27.     

Investigating cosmological weak lensing with the ray-bundle method

Using the ray-bundle method for calculating gravitational lens magnifications, we outline a method by which the magnification probability may be determined specifically in the weak lensing limit for cosmological models obtained from N -body simulations.
16 different models are investigated, which are variations on three broad classes of cold dark matter model: the standard model with  (Ω₀, λ ₀)=(1.0,0.0)  , the open model with  (Ω₀, λ ₀)=(0.3,0.0)  and the lambda model, which is a flat model with a cosmological constant  (Ω₀, λ ₀)=(0.3,0.7)  .
The effects of varying the Hubble parameter, H ₀, the power spectrum shape parameter, Γ, and the cluster mass normalization, σ ₈, are studied. It is shown that there is no signature of these parameters in the weak lensing magnification distributions. The magnification probability distributions are also shown to be independent of the numerical parameters such as the lens mass and simulation box size in the N -body simulations.     

On the H i content, dust-to-gas ratio and nature of Mg ii absorbers

We estimate the mean dust-to-gas ratio of Mg  ii absorbers as a function of rest equivalent width W ₀ and redshift over the range  0.5 < z < 1.4  . Using the expanded Sloan Digital Sky Survey/ Hubble Space Telescope sample of low-redshift Lyman-α absorbers, we first show the existence of a 8σ correlation between the mean hydrogen column density  〈 N _{H  i} 〉  and W ₀, an indicator of gas velocity dispersion. By combining these results with recent dust-reddening measurements, we show that the mean dust-to-gas ratio of Mg  ii absorbers does not appreciably depend on rest equivalent width. Assuming that, on average, dust-to-gas ratio is proportional to metallicity, we find its redshift evolution to be consistent with that of   L ^★  galaxies from   z = 0.5  to 1.4, and we show that our constraints disfavour dwarf galaxies as the origin of such absorbers. We discuss other scenarios and favour galactic outflows from  ∼ L ^★  galaxies as the origin of the majority of strong Mg  ii absorbers. Finally, we show that, once evolutionary effects are taken into account, the Bohlin et al. relation between A _V and N _H is also satisfied by strong Mg  ii systems down to lower column densities than those probed in our Galaxy.     

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.     

Clustering of luminous red galaxies – II. Small-scale redshift-space distortions

This is the second paper of a series where we study the clustering of luminous red galaxies (LRG) in the recent spectroscopic Sloan Digital Sky Survey (SDSS) data release, DR6, which has 75 000 LRG covering over  1 Gpc³  h ⁻³  for  0.15 < z < 0.47  . Here, we focus on modelling redshift-space distortions in  ξ(σ, π)  , the two-point correlation in separate line-of-sight and perpendicular directions, at small scales and in the line-of-sight. We show that a simple Kaiser model for the anisotropic two-point correlation function in redshift space, convolved with a distribution of random peculiar velocities with an exponential form, can describe well the correlation of LRG on all scales. We show that to describe with accuracy the so-called 'fingers-of-God' (FOG) elongations in the radial direction, it is necessary to model the scale dependence of both bias b and the pairwise rms peculiar velocity σ₁₂ with the distance. We show how both quantities can be inferred from the  ξ(σ, π)  data. From   r ≃ 10 Mpc  h ⁻¹  to   r ≃ 1 Mpc  h ⁻¹  , both the bias and σ₁₂ are shown to increase by a factor of 2: from   b = 2  to 4 and from  σ₁₂= 400  to  800 km s⁻¹  . The latter is in good agreement, within a 5 per cent accuracy in the recovered velocities, with direct velocity measurements in dark matter simulations with  Ω_m= 0.25  and  σ₈= 0.85  .