Galaxy–dark matter correlations applied to galaxy–galaxy lensing: predictions from the semi-analytic galaxy formation models

We use semi-analytic models of galaxy formation combined with high-resolution N -body simulations to make predictions for galaxy–dark matter correlations and apply them to galaxy–galaxy lensing. We analyse cross-power spectra between the dark matter and different galaxy samples selected by luminosity, colour or star formation rate. We compare the predictions with the recent detection by the Sloan Digital Sky Survey (SDSS). We show that the correlation amplitude and the mean tangential shear depend strongly on the luminosity of the sample on scales below 1  h ⁻¹ Mpc, reflecting the correlation between the galaxy luminosity and the halo mass. The cross-correlation cannot, however, be used to infer the halo profile directly because different halo masses dominate on different scales and because not all galaxies are at the centres of the corresponding haloes. We compute the redshift evolution of the cross-correlation amplitude and compare it with those of galaxies and dark matter. We also compute the galaxy–dark matter correlation coefficient and show that it is close to unity on scales above 1  h ⁻¹ Mpc for all considered galaxy types. This would allow one to extract the bias and the dark matter power spectrum on large scales from the galaxy and galaxy–dark matter correlations.     

The three-dimensional power spectrum of dark and luminous matter from the VIRMOS-DESCART cosmic shear survey

We present the first optimal power spectrum estimation and three-dimensional deprojections for the dark and luminous matter and their cross-correlations. The results are obtained using a new optimal fast estimator, deprojected using minimum variance and Singular Value Decomposition (SVD) techniques. We show the resulting 3D power spectra for dark matter and galaxies, and their covariance for the VIRMOS-DESCART weak lensing shear and galaxy data. The survey is most sensitive to non-linear scales   k _NL∼ 1 h Mpc⁻¹  . On these scales, our 3D power spectrum of dark matter is in good agreement with the RCS 3D power spectrum found by Tegmark & Zaldarriaga. Our galaxy power is similar to that found by the 2MASS survey, and larger than that of SDSS, APM and RCS, consistent with the expected difference in galaxy population.
We find an average bias   b = 1.24 ± 0.18  for the I -selected galaxies, and a cross-correlation coefficient   r = 0.75 ± 0.23  . Together with the power spectra, these results optimally encode the entire two point information about dark matter and galaxies, including galaxy–galaxy lensing. We address some of the implications regarding galaxy haloes and mass-to-light ratios. The best-fitting 'halo' parameter   h ≡ r / b = 0.57 ± 0.16  , suggesting that dynamical masses estimated using galaxies systematically underestimate total mass.
Ongoing surveys, such as the Canada–France–Hawaii Telescope Legacy Survey, will significantly improve on the dynamic range, and future photometric redshift catalogues will allow tomography along the same principles.     

Measuring the distribution of galaxies between haloes

We develop a method to measure the probability, P ( N;   M ), of finding N galaxies in a dark matter halo of mass M from the theoretically determined clustering properties of dark matter haloes and the observationally measured clustering properties of galaxies. Knowledge of this function and the distribution of the dark matter completely specifies all clustering properties of galaxies on scales larger than the size of dark matter haloes. Furthermore, P ( N;   M ) provides strong constraints on models of galaxy formation, since it depends upon the merger history of dark matter haloes and the galaxy–galaxy merger rate within haloes. We show that measurements from a combination of the Two Micron All Sky Survey and Sloan Digital Sky Survey or Two-degree Field Galaxy Redshift Survey data sets will allow P ( N;   M ) averaged over haloes occupied by bright galaxies to be accurately measured for N =0–2 .     

Scale-dependent galaxy bias in the Sloan Digital Sky Survey as a function of luminosity and colour

It has been known for a long time that the clustering of galaxies changes as a function of galaxy type. This galaxy bias acts as a hindrance to the extraction of cosmological information from the galaxy power spectrum or correlation function. Theoretical arguments show that a change in the amplitude of the clustering between galaxies and mass on large scales is unavoidable, but cosmological information can be easily extracted from the shape of the power spectrum or correlation function if this bias is independent of scale. Scale-dependent bias is generally small on large scales,   k < 0.1  h  Mpc⁻¹  , but on smaller scales can affect the recovery of  Ω_m h   from the measured shape of the clustering signal, and have a small effect on the Baryon Acoustic Oscillations. In this paper, we investigate the transition from scale-independent to scale-dependent galaxy bias as a function of galaxy population. We use the Sloan Digital Sky Survey Data Release 5 sample to fit various models, which attempt to parametrize the turn-off from scale-independent behaviour. For blue galaxies, we find that the strength of the turn-off is strongly dependent on galaxy luminosity, with stronger scale-dependent bias on larger scales for more luminous galaxies. For red galaxies, the scale dependence is a weaker function of luminosity. Such trends need to be modelled in order to optimally extract the information available in future surveys, and can help with the design of such surveys.     

The imprint of cosmic reionization on galaxy clustering

We consider the effect of reionization on the clustering properties of galaxy samples at intermediate redshifts ( z ∼ 0.3–5.5). Current models for the reionization of intergalactic hydrogen predict that overdense regions will be reionized early, thus delaying the build-up of stellar mass in the progenitors of massive lower redshift galaxies. As a result, the stellar populations observed in intermediate-redshift galaxies are somewhat younger and hence brighter in overdense regions of the Universe. Galaxy surveys would therefore be sensitive to galaxies with a somewhat lower dark matter mass in overdense regions. The corresponding increase in the observed number density of galaxies can be parametrized as a galaxy bias due to reionization. We model this process using merger trees combined with a stellar synthesis code. Our model demonstrates that reionization has a significant effect on the clustering properties of galaxy samples that are selected based on their star formation properties. The bias correction in Lyman-break galaxies (including those in proposed baryonic oscillation surveys at z < 1) is at the level of 10–20 per cent for a halo mass of  10¹² M_⊙  , leading to corrections factors of 1.5–2 in the halo mass inferred from measurements of clustering length. The reionization of helium could also lead to a sharp increase in the amplitude of the galaxy correlation function at z ∼ 3. We find that the reionization bias is approximately independent of scale and halo mass. However, since the traditional galaxy bias is mass dependent, the reionization bias becomes relatively more important for lower mass systems. The correction to the bias due to reionization is very small in surveys of luminous red galaxies at z < 1.     

Modelling galaxy–galaxy weak lensing with Sloan Digital Sky Survey groups

We use galaxy groups selected from the Sloan Digital Sky Survey (SDSS) together with mass models for individual groups to study the galaxy–galaxy lensing signals expected from galaxies of different luminosities and morphological types. We compare our model predictions with the observational results obtained from the SDSS by Mandelbaum et al. for the same samples of galaxies. The observational results are well reproduced in a Λ cold dark matter (ΛCDM) model based on the Wilkinson Microwave Anisotropy Probe ( WMAP ) 3-yr data, but a ΛCDM model with higher σ₈, such as the one based on the WMAP 1-yr data, significantly overpredicts the galaxy–galaxy lensing signal. We model, separately, the contributions to the galaxy–galaxy lensing signals from different galaxies: central versus satellite, early type versus late type and galaxies in haloes of different masses. We also examine how the predicted galaxy–galaxy lensing signal depends on the shape, density profile and the location of the central galaxy with respect to its host halo.     

Halo masses for optically selected and for radio-loud AGN from clustering and galaxy–galaxy lensing

We compute two-point correlation functions and measure the shear signal due to galaxy–galaxy lensing for 80 000 optically identified and 5700 radio-loud active galactic nuclei (AGN) from Data Release 4 of the Sloan Digital Sky Survey. Halo occupation models are used to estimate halo masses and satellite fractions for these two types of AGN. The large sample size allows us to separate AGN according to the stellar mass of their host galaxies. We study how the halo masses of optical and radio AGN differ from those of the parent population at fixed   M _*  . Halo masses deduced from clustering and from lensing agree satisfactorily. Radio AGN are found in more massive haloes than optical AGN: in our samples, their mean halo masses are  1.6 × 10¹³  and  8 × 10¹¹  h ⁻¹ M_⊙  , respectively. Optical AGN follow the same relation between stellar mass and halo mass as galaxies selected without regard to nuclear properties, but radio-loud AGN deviate significantly from this relation. The dark matter haloes of radio-loud AGN are about twice as massive as those of control galaxies of the same stellar mass. This boost is independent of radio luminosity, and persists even when our analysis is restricted to field galaxies. The large-scale gaseous environment of the galaxy clearly plays a crucial role in producing observable radio emission. The dark matter halo masses that we derive for the AGN in our two samples are in good agreement with recent models in which feedback from radio AGN becomes dominant in haloes where gas cools quasi-statically.     

Quasar–galaxy and galaxy–galaxy cross-correlations: model predictions with realistic galaxies

Several measurements of quasi-stellar object (QSO)–galaxy correlations have reported signals much larger than predictions of magnification by large-scale structure. We find that the expected signal depends strongly on the properties of the foreground galaxy population. On arcmin scales, it can be either larger or smaller by a factor of 2 for different galaxy types in comparison with a linearly biased version of the mass distribution. Thus the resolution of some of the excess measurements may lie in examining the halo occupation properties of the galaxy population sampled by a given survey; this is also the primary information such measurements will provide.
We use the halo model of clustering and simulations to predict the magnification-induced cross-correlations and errors for forthcoming surveys. With the full Sloan Digital Sky Survey, the statistical errors will be below 1 per cent for the galaxy–galaxy correlations and significantly larger for QSO–galaxy correlations. Thus accurate constraints on parameters of the galaxy halo occupation distribution can be obtained from small-scale measurements and on the bias parameter from large scales. Since the lensing-induced cross-correlation measures the first moment of the halo occupation number of galaxies, these measurements can provide the basis for interpreting galaxy clustering measurements that measure the second- and higher-order moments.     

SDSS galaxy clustering: luminosity and colour dependence and stochasticity

Differences in clustering properties between galaxy subpopulations complicate the cosmological interpretation of the galaxy power spectrum, but can also provide insights about the physics underlying galaxy formation. To study the nature of this relative clustering, we perform a counts-in-cells analysis of galaxies in the Sloan Digital Sky Survey in which we measure the relative bias between pairs of galaxy subsamples of different luminosities and colours. We use a generalized  χ²  test to determine if the relative bias between each pair of subsamples is consistent with the simplest deterministic linear bias model, and we also use a maximum likelihood technique to further understand the nature of the relative bias between each pair. We find that the simple, deterministic model is a good fit for the luminosity-dependent bias on scales above  ∼2  h ⁻¹ Mpc  , which is good news for using magnitude-limited surveys for cosmology. However, the colour-dependent bias shows evidence for stochasticity and/or non-linearity which increases in strength towards smaller scales, in agreement with previous studies of stochastic bias. Also, confirming hints seen in earlier work, the luminosity-dependent bias for red galaxies is significantly different from that of blue galaxies: both luminous and dim red galaxies have higher bias than moderately bright red galaxies, whereas the biasing of blue galaxies is not strongly luminosity dependent. These results can be used to constrain galaxy formation models and also to quantify how the colour and luminosity selection of a galaxy survey can impact measurements of the cosmological matter power spectrum.     

Weak gravitational flexion

Flexion is the significant third-order weak gravitational lensing effect responsible for the weakly skewed and arc-like appearance of lensed galaxies. Here we demonstrate how flexion measurements can be used to measure galaxy halo density profiles and large-scale structure on non-linear scales, via galaxy–galaxy lensing, dark matter mapping and cosmic flexion correlation functions. We describe the origin of gravitational flexion, and discuss its four components, two of which are first described here. We also introduce an efficient complex formalism for all orders of lensing distortion. We proceed to examine the flexion predictions for galaxy–galaxy lensing, examining isothermal sphere and Navarro–Frenk–White (NFW) profiles and both circularly symmetric and elliptical cases. We show that in combination with shear we can precisely measure galaxy masses and NFW halo concentrations. We also show how flexion measurements can be used to reconstruct mass maps in two-dimensional projection on the sky, and in three dimensions in combination with redshift data. Finally, we examine the predictions for cosmic flexion, including convergence–flexion cross-correlations, and we find that the signal is an effective probe of structure on non-linear scales.     

A constant dark matter halo surface density in galaxies

We confirm and extend the recent finding that the central surface density  μ_0D≡ r ₀ρ₀  of galaxy dark matter haloes, where r ₀ and  ρ₀  are the halo core radius and central density, is nearly constant and independent of galaxy luminosity. Based on the co-added rotation curves (RCs) of ∼1000 spiral galaxies, the mass models of individual dwarf irregular and spiral galaxies of late and early types with high-quality RCs, and the galaxy–galaxy weak-lensing signals from a sample of spiral and elliptical galaxies, we find that  log μ_0D= 2.15 ± 0.2  in units of  log(M_⊙ pc⁻²)  . We also show that the observed kinematics of Local Group dwarf spheroidal galaxies are consistent with this value. Our results are obtained for galactic systems spanning over 14 mag, belonging to different Hubble types and whose mass profiles have been determined by several independent methods. In the same objects, the approximate constancy of  μ_0D  is in sharp contrast to the systematical variations, by several orders of magnitude, of galaxy properties, including  ρ₀  and central stellar surface density.     

Satellite kinematics – II. The halo mass–luminosity relation of central galaxies in SDSS

The kinematics of satellite galaxies reflect the masses of the extended dark matter haloes in which they orbit, and thus shed light on the mass–luminosity relation (MLR) of their corresponding central galaxies. In this paper, we select a large sample of centrals and satellites from the Sloan Digital Sky Survey and measure the kinematics (velocity dispersions) of the satellite galaxies as a function of the r -band luminosity of the central galaxies. Using the analytical framework presented in More, van den Bosch & Cacciato, we use these data to infer both the mean and the scatter of the MLR of central galaxies, carefully taking account of selection effects and biases introduced by the stacking procedure. As expected, brighter centrals on average reside in more massive haloes. In addition, we find that the scatter in halo masses for centrals of a given luminosity,  σ_{log  M}   , also increases with increasing luminosity. As we demonstrate, this is consistent with  σ_{log  L}   , which reflects the scatter in the conditional probability function   P ( L _c| M )  , being independent of halo mass. Our analysis of the satellite kinematics yields  σ_{log  L} = 0.16  ±  0.04  , in excellent agreement with constraints from clustering and group catalogues, and with predictions from a semi-analytical model of galaxy formation. We thus conclude that the amount of stochasticity in galaxy formation, which is characterized by  σ_{log  L}   , is well constrained, independent of halo mass and in a good agreement with current models of galaxy formation.     

The properties of Lyα emitting galaxies in hierarchical galaxy formation models

We present detailed predictions for the properties of Lyα-emitting galaxies in the framework of the Λ cold dark matter cosmology, calculated using the semi-analytical galaxy formation model galform . We explore a model that assumes a top-heavy initial mass function in starbursts and that has previously been shown to explain the sub-millimetre number counts and the luminosity function of Lyman-break galaxies at high redshift. We show that this model, with the simple assumption that a fixed fraction of Lyα photons escape from each galaxy, is remarkably successful at explaining the observed luminosity function of Lyα emitters (LAEs) over the redshift range  3 < z < 6.6  . We also examine the distribution of Lyα equivalent widths and the broad-band continuum magnitudes of emitters, which are in good agreement with the available observations. We look more deeply into the nature of LAEs, presenting predictions for fundamental properties such as the stellar mass and radius of the emitting galaxy and the mass of the host dark matter halo. The model predicts that the clustering of LAEs at high redshifts should be strongly biased relative to the dark matter, in agreement with observational estimates. We also present predictions for the luminosity function of LAEs at   z > 7  , a redshift range that is starting to be be probed by near-infrared surveys and using new instruments such as the Dark Ages Z Lyman Explorer (DAzLE).     

The nature of high-redshift galaxies

Using semi-analytic models of galaxy formation set within the cold dark matter (CDM) merging hierarchy, we investigate several scenarios for the nature of the high-redshift     ) Lyman-break galaxies (LBGs). We consider a 'collisional starburst' model in which bursts of star formation are triggered by galaxy–galaxy mergers, and find that a significant fraction of LBGs are predicted to be starbursts. This model reproduces the observed comoving number density of bright LBGs as a function of redshift and the observed luminosity function at     and     with a reasonable amount of dust extinction. Model galaxies at     have star formation rates, half-light radii,     colours and internal velocity dispersions that are in good agreement with the data. Global quantities such as the star formation rate density and cold gas and metal content of the Universe as a function of redshift also agree well. Two 'quiescent' models without starbursts are also investigated. In one, the star formation efficiency in galaxies remains constant with redshift, while in the other, it scales inversely with disc dynamical time, and thus increases rapidly with redshift. The first quiescent model is strongly ruled out, as it does not produce enough high-redshift galaxies once realistic dust extinction is accounted for. The second quiescent model fits marginally, but underproduces cold gas and very bright galaxies at high redshift. A general conclusion is that star formation at high redshift must be more efficient than locally. The collisional starburst model appears to accomplish this naturally without violating other observational constraints.     

Understanding the halo-mass and galaxy-mass cross-correlation functions

We use the Millennium Simulation (MS) to measure the cross-correlation between halo centres and mass (or equivalently the average density profiles of dark haloes) in a Lambda cold dark matter (ΛCDM) cosmology. We present results for radii in the range  10  h ⁻¹ kpc < r < 30  h ⁻¹ Mpc  and for halo masses in the range  4 × 10¹⁰ < M ₂₀₀ < 4 × 10¹⁴  h ⁻¹ M_⊙  . Both at   z = 0  and at   z = 0.76  these cross-correlations are surprisingly well fitted if the inner region is approximated by a density profile of NFW or Einasto form, the outer region by a biased version of the linear mass autocorrelation function, and the maximum of the two is adopted where they are comparable. We use a simulation of galaxy formation within the MS to explore how these results are reflected in cross-correlations between galaxies and mass. These are directly observable through galaxy–galaxy lensing. Here also we find that simple models can represent the simulation results remarkably well, typically to ≲10 per cent. Such models can be used to extend our results to other redshifts, to cosmologies with other parameters, and to other assumptions about how galaxies populate dark haloes. Our galaxy formation simulation already reproduces current galaxy–galaxy lensing data quite well. The characteristic features predicted in the galaxy–galaxy lensing signal should provide a strong test of the ΛCDM cosmology as well as a route to understanding how galaxies form within it.     

The two-point correlation of galaxy groups: probing the clustering of dark matter haloes

We analyse the two-point correlation function (2PCF) of galaxy groups identified from the 2-degree Field Galaxy Redshift Survey with the halo-based group finder recently developed by Yang et al. With this group catalogue we are able to estimate the 2PCFs for systems ranging from isolated galaxies to rich clusters of galaxies. The real-space correlation length obtained for these systems ranges from ∼4 to ∼15  h ⁻¹ Mpc, respectively. The observed correlation amplitude (and the corresponding bias factor) as a function of group abundance is well reproduced by associating galaxy groups with dark matter haloes in the standard Λ-cold dark matter model. Redshift distortions are clearly detected in the redshift-space correlation function, the degree of which is consistent with the assumption of gravitational clustering and halo bias in the cosmic density field. In agreement with previous studies we find a strong increase of the correlation length with the mean intergroup separation. Although well-determined observationally, we show that current theoretical predictions are not yet accurate enough to allow for stringent constraints on cosmological parameters. Finally, we use our results to explore the power-law nature of the 2PCF of galaxies. We split the 2PCF into one- and two-group terms, equivalent to the one- and two-halo terms in halo occupation models, and show that the power-law form of the 2PCF is broken, when only including galaxies in the more massive systems.