Incorporating photometric redshift probability density information into real-space clustering measurements

The use of photometric redshifts in cosmology is increasing. Often, however these photo- z are treated like spectroscopic observations, in that the peak of the photometric redshift, rather than the full probability density function (PDF), is used. This overlooks useful information inherent in the full PDF. We introduce a new real-space estimator for one of the most used cosmological statistics, the two-point correlation function, that weights by the PDF of individual photometric objects in a manner that is optimal when Poisson statistics dominate. As our estimator does not bin based on the PDF peak, it substantially enhances the clustering signal by usefully incorporating information from all photometric objects that overlap the redshift bin of interest. As a real-world application, we measure quasi-stellar object (QSO) clustering in the Sloan Digital Sky Survey (SDSS). We find that our simplest binned estimator improves the clustering signal by a factor equivalent to increasing the survey size by a factor of 2–3. We also introduce a new implementation that fully weights between pairs of objects in constructing the cross-correlation and find that this pair-weighted estimator improves clustering signal in a manner equivalent to increasing the survey size by a factor of 4–5. Our technique uses spectroscopic data to anchor the distance scale and it will be particularly useful where spectroscopic data (e.g. from BOSS) overlap deeper photometry (e.g. from Pan-STARRS, DES or the LSST). We additionally provide simple, informative expressions to determine when our estimator will be competitive with the autocorrelation of spectroscopic objects. Although we use QSOs as an example population, our estimator can and should be applied to any clustering estimate that uses photometric objects.     

The 2dF Galaxy Redshift Survey: luminosity dependence of galaxy clustering

We investigate the dependence of the strength of galaxy clustering on intrinsic luminosity using the Anglo-Australian two degree field galaxy redshift survey (2dFGRS). The 2dFGRS is over an order of magnitude larger than previous redshift surveys used to address this issue. We measure the projected two-point correlation function of galaxies in a series of volume-limited samples. The projected correlation function is free from any distortion of the clustering pattern induced by peculiar motions and is well described by a power law in pair separation over the range     . The clustering of     galaxies in real space is well-fitted by a correlation length     and power-law slope     . The clustering amplitude increases slowly with absolute magnitude for galaxies fainter than M *, but rises more strongly at higher luminosities. At low luminosities, our results agree with measurements from the Southern Sky Redshift Survey 2 by Benoist et al. However, we find a weaker dependence of clustering strength on luminosity at the highest luminosities. The correlation function amplitude increases by a factor of 4.0 between     and −22.5, and the most luminous galaxies are 3.0 times more strongly clustered than L * galaxies. The power-law slope of the correlation function shows remarkably little variation for samples spanning a factor of 20 in luminosity. Our measurements are in very good agreement with the predictions of the hierarchical galaxy formation models of Benson et al.     

Clustering of galaxies at 3.6 μm in the Spitzer Wide-area Infrared Extragalactic legacy survey

We investigate the clustering of galaxies selected in the 3.6 μm band of the Spitzer Wide-area Infrared Extragalactic (SWIRE) legacy survey. The angular two-point correlation function is calculated for 11 samples with flux limits of S _3.6≥ 4–400 μJy, over an 8 deg² field. The angular clustering strength is measured at >5σ significance at all flux limits, with amplitudes of A = (0.49–29) × 10⁻³ at 1°, for a power-law model, A θ^−0.8. We estimate the redshift distributions of the samples using phenomological models, simulations and photometric redshifts, and so derive the spatial correlation lengths. We compare our results with the Galaxies In Cosmological Simulations (GalICS) models of galaxy evolution and with parametrized models of clustering evolution. The GalICS simulations are consistent with our angular correlation functions, but fail to match the spatial clustering inferred from the phenomological models or the photometric redshifts. We find that the uncertainties in the redshift distributions of our samples dominate the statistical errors in our estimates of the spatial clustering. At low redshifts (median z ≤ 0.5), the comoving correlation length is approximately constant,   r ₀= 6.1 ± 0.5  h ⁻¹  Mpc, and then decreases with increasing redshift to a value of 2.9 ± 0.3  h ⁻¹ Mpc for the faintest sample, for which the median redshift is z ∼ 1. We suggest that this trend can be attributed to a decrease in the average galaxy and halo mass in the fainter flux-limited samples, corresponding to changes in the relative numbers of early- and late-type galaxies. However, we cannot rule out strong evolution of the correlation length over  0.5 < z < 1  .     

Covariance of weak lensing observables

Analytical expressions for covariances of weak lensing statistics related to the aperture mass,   M _ap  , are derived for realistic survey geometries such as the Supernova Acceleration Probe (SNAP) ¹ for a range of smoothing angles and redshift bins. We incorporate the contributions to the noise due to the intrinsic ellipticity distribution and the effects of the finite catalogue size. Extending previous results to the most general case where the overlap of source populations is included in a complete analysis of error estimates, we study how various angular scales in various redshifts are correlated and how the estimation scatter changes with the survey parameters. Dependences on cosmological parameters and source redshift distributions are studied in detail. Numerical simulations are used to test the validity of various ingredients to our calculations. Correlation coefficients are defined in a way that makes them practically independent of cosmology. They can provide important tools to cross-correlate one or more different surveys, as well as various redshift bins within the same survey or various angular scales from the same or different surveys. The dependence of these coefficients on various models of underlying mass correlation hierarchy is also studied. Generalizations of these coefficients at the level of three-point statistics have the potential of probing the complete shape dependence of the underlying bi-spectrum of the matter distribution. A complete error analysis incorporating all sources of errors suggests encouraging results for studies using future space-based weak lensing surveys such as SNAP.     

Searching for the scale of homogeneity

We introduce a statistical quantity, known as the K function, related to the integral of the two-point correlation function. It gives us straightforward information about the scale where clustering dominates and the scale at which homogeneity is reached. We evaluate the correlation dimension, D ₂, as the local slope of the log–log plot of the K function. We apply this statistic to several stochastic point fields, to three numerical simulations describing the distribution of clusters and finally to real galaxy redshift surveys. Four different galaxy catalogues have been analysed using this technique: the Center for Astrophysics I, the Perseus–Pisces redshift surveys (these two lying in our local neighbourhood), the Stromlo–APM and the 1.2-Jy IRAS redshift surveys (these two encompassing a larger volume). In all cases, this cumulant quantity shows the fingerprint of the transition to homogeneity. The reliability of the estimates is clearly demonstrated by the results from controllable point sets, such as the segment Cox processes. In the cluster distribution models, as well as in the real galaxy catalogues, we never see long plateaus when plotting D ₂ as a function of the scale, leaving no hope for unbounded fractal distributions.     

Error analysis of the photometric redshift technique

We present a calculation of the systematic component of the error budget in the photometric redshift technique. We make use of it to describe a simple technique that allows the assignment of confidence limits to redshift measurements obtained through photometric methods. We show that our technique, through the calculation of a redshift probability function, gives complete information on the probable redshift of an object and its associated confidence intervals. This information can and must be used in the calculation of any observable quantity that makes use of the redshift.     

The Durham/UKST Galaxy Redshift Survey – III. Large-scale structure via the two-point correlation function

We have investigated the statistical clustering properties of galaxies by calculating the two-point galaxy correlation function from the optically selected Durham/UKST Galaxy Redshift Survey. This survey is magnitude-limited to b _J∼17, contains ∼2500 galaxies sampled at a rate of one-in-three and surveys a ∼4×10⁶ ( h ⁻¹ Mpc)³ volume of space. We have empirically determined the optimal method of estimating the two-point correlation function from just such a magnitude-limited survey. Applying our methods to this survey, we find that our redshift-space results agree well with those from previous optical surveys. In particular, we confirm the previously claimed detections of large-scale power out to ∼40 h ⁻¹ Mpc scales. We compare with two common models of cosmological structure formation and find that our two-point correlation function has power significantly in excess of the standard cold dark matter model in the 10–30 h ⁻¹ Mpc region. We therefore support the observational results of the APM galaxy survey. Given that only the redshift-space clustering can be measured directly, we use standard modelling methods and indirectly estimate the real-space two-point correlation function from the projected two-point correlation function. We then invert this projected correlation function to obtain an estimate of the spatial two-point correlation function in real space. This correlation function in real space has a lower amplitude than that in redshift space, but a steeper slope.     

Bayesian photometric redshifts for weak-lensing applications

The next generation of weak gravitational lensing surveys are capable of generating good measurements of cosmological parameters, provided that, amongst other requirements, adequate redshift information is available for the background galaxies that are measured. It is frequently assumed that photometric redshift techniques provide the means to achieve this. Here, we compare Bayesian and frequentist approaches to photometric redshift estimation, particularly at faint magnitudes. We identify and discuss the biases that are inherent in the various methods, and describe an optimum Bayesian method for extracting redshift distributions from photometric data.     

Reconstructing galaxy fundamental distributions and scaling relations from photometric redshift surveys. Applications to the SDSS early-type sample

Noisy distance estimates associated with photometric rather than spectroscopic redshifts lead to a biased estimate of the luminosity distribution, and produce a correlated misestimate of the sizes. We consider a sample of early-type galaxies from the Sloan Digital Sky Survey Data Release 6 for which both spectroscopic and photometric information is available, and apply the generalization of the V _max method to correct for these biases. We show that our technique recovers the true redshift, magnitude and size distributions, as well as the true size–luminosity relation. We find that using only 10 per cent of the spectroscopic information randomly spaced in our catalogue is sufficient for the reconstructions to be accurate within  ∼3 per cent  , when the photometric redshift error is  δ z ≃ 0.038  . We then address the problem of extending our method to deep redshift catalogues, where only photometric information is available. In addition to the specific applications outlined here, our technique impacts a broader range of studies, when at least one distance-dependent quantity is involved. It is particularly relevant for the next generation of surveys, some of which will only have photometric information.     

Morphological redshift estimates for galaxy clusters in a Sunyaev–Zel'dovich effect survey

We develop a new method to estimate the redshift of galaxy clusters through resolved images of the Sunyaev–Zel'dovich effect (SZE). Our method is based on morphological observables which can be measured by actual and future SZE experiments. We test the method with a set of high-resolution hydrodynamical simulations of galaxy clusters at different redshifts. Our method combines the observables in a principal component analysis. After calibrating the method with an independent redshift estimation for some of the clusters, we show – using a Bayesian approach – how the method can give an estimate of the redshift of the galaxy clusters. Although the error bars given by the morphological redshift estimation are large, it should be useful for future SZE surveys where thousands of clusters are expected to be detected; a first preselection of the high-redshift candidates could be done using our proposed morphological redshift estimator. Although not considered in this work, our method should also be useful to give an estimate of the redshift of clusters in X-ray and optical surveys.     

Photometric redshifts with surface brightness priors

We use galaxy surface brightness as prior information to improve photometric redshift (photo- z ) estimation. We apply our template-based photo- z method to imaging data from the ground-based VVDS survey and the space-based GOODS field from HST , and use spectroscopic redshifts to test our photometric redshifts for different galaxy types and redshifts. We find that the surface brightness prior eliminates a large fraction of outliers by lifting the degeneracy between the Lyman and 4000-Å breaks. Bias and scatter are improved by about a factor of 2 with the prior in each redshift bin in the range  0.4 < z < 1.3  , for both the ground and space data. Ongoing and planned surveys from the ground and space will benefit, provided that care is taken in measurements of galaxy sizes and in the application of the prior. We discuss the image quality and signal-to-noise ratio requirements that enable the surface brightness prior to be successfully applied.     

Probing the gravity-induced bias with weak lensing: test of analytical results against simulations

Future weak lensing surveys will directly probe the density fluctuation in the Universe. Recent studies have shown how the statistics of the weak lensing convergence field is related to the statistics of collapsed objects. Extending earlier analytical results on the probability distribution function of the convergence field, we show that the bias associated with the convergence field can be directly related to the bias associated with the statistics of underlying overdense objects. This will provide us with a direct method to study the gravity-induced bias in galaxy clustering. Based on our analytical results, which use the hierarchical Ansatz for non-linear clustering, we study how such a bias depends on the smoothing angle and the source redshift. We compare our analytical results with ray-tracing experiments through N -body simulations of four different realistic cosmological scenarios, and find a very good match. Our study shows that the bias in the convergence map strongly depends on the background geometry and hence can help us in distinguishing different cosmological models in addition to improving our understanding of the gravity-induced bias in galaxy clustering.     

MoMaF: the Mock Map Facility

We present the Mock Map Facility, a powerful tool for converting theoretical outputs of hierarchical galaxy formation models into catalogues of virtual observations. The general principle is straightforward: mock observing cones can be generated using semi-analytically post-processed snapshots of cosmological N -body simulations. These cones can then be projected to synthesize mock sky images. To this end, the paper describes in detail an efficient technique for creating such mock cones and images from the galaxies in cosmological simulations ( galics ) semi-analytic model, providing the reader with an accurate quantification of the artefacts it introduces at every step. We show that replication effects introduce a negative bias on the clustering signal – typically peaking at less than 10 per cent around the correlation length. We also thoroughly discuss how the clustering signal is affected by finite-volume effects, and show that it vanishes at scales larger than approximately one-tenth of the simulation box size. For the purpose of analysing our method, we show that number counts and redshift distributions obtained with galics / momaf compare well with K -band observations and the two-degree field galaxy redshift survey. Given finite-volume effects, we also show that the model can reproduce the automatic plate measuring machine angular correlation function. The momaf results discussed here are made publicly available to the astronomical community through a public data base. Moreover, a user-friendly Web interface ( http://galics.iap.fr ) allows any user to recover her/his own favourite galaxy samples through simple SQL queries. The flexibility of this tool should permit a variety of uses ranging from extensive comparisons between real observations and those predicted by hierarchical models of galaxy formation, to the preparation of observing strategies for deep surveys and tests of data processing pipelines.     

Photometric redshifts for weak lensing tomography from space: the role of optical and near infrared photometry

We study in detail the photometric redshift requirements needed for tomographic weak gravitational lensing in order to measure accurately the dark energy equation of state. In particular, we examine how ground-based photometry  ( u , g , r , i , z , y )  can be complemented by space-based near-infrared (near-IR) photometry  ( J , H )  , e.g. onboard the planned DUNE satellite. Using realistic photometric redshift simulations and an artificial neural network photo- z method we evaluate the figure of merit for the dark energy parameters  ( w ₀, w _a )  . We consider a DUNE -like broad optical filter supplemented with ground-based multiband optical data from surveys like the Dark Energy Survey, Pan-STARRS and LSST. We show that the dark energy figure of merit would be improved by a factor of 1.3–1.7 if IR filters are added onboard DUNE . Furthermore we show that with IR data catastrophic photo- z outliers can be removed effectively. There is an interplay between the choice of filters, the magnitude limits and the removal of outliers. We draw attention to the dependence of the results on the galaxy formation scenarios encoded into the mock galaxies, e.g. the galaxy reddening. For example, very deep u -band data could be as effective as the IR. We also find that about  10⁵–10⁶  spectroscopic redshifts are needed for calibration of the full survey.     

Gravitational lens magnification by Abell 1689: distortion of the background galaxy luminosity function

Gravitational lensing magnifies the observed flux of galaxies behind the lens. We use this effect to constrain the total mass in the cluster Abell 1689 by comparing the lensed luminosities of background galaxies with the luminosity function of an undistorted field. Under the assumption that these galaxies are a random sample of luminosity space, this method is not limited by clustering noise. We use photometric redshift information to estimate galaxy distance and intrinsic luminosity. Knowing the redshift distribution of the background population allows us to lift the mass/background degeneracy common to lensing analysis. In this paper we use nine filters observed over 12 h with the Calar Alto 3.5-m telescope to determine the redshifts of 1000 galaxies in the field of Abell 1689. Using a complete sample of 146 background galaxies we measure the cluster mass profile. We find that the total projected mass interior to 0.25  h ⁻¹ Mpc is M _2D(<0.25  h ⁻¹ Mpc)=(0.48±0.16)×10¹⁵  h ⁻¹ M_⊙, where our error budget includes uncertainties from the photometric redshift determination, the uncertainty in the offset calibration and finite sampling. This result is in good agreement with that found by number-count and shear-based methods and provides a new and independent method to determine cluster masses.     

Large-scale bias in the Universe — II. Redshift-space bispectrum

The determination of the density parameter Ω₀ from the large-scale distribution of galaxies is one of the major goals of modern cosmology. However, if galaxies are biased tracers of the underlying mass distribution, linear perturbation theory leads to a degeneracy between Ω₀ and the linear bias parameter b , and the density parameter cannot be estimated. In Matarrese, Verde &38; Heavens we developed a method based on second-order perturbation theory to use the bispectrum to lift this degeneracy by measuring the bias parameter in an Ω₀-independent way. The formalism was developed assuming that one has perfect information on the positions of galaxies in three dimensions. In galaxy redshift surveys, the three-dimensional information is imperfect, because of the contaminating effects of peculiar velocities, and the resulting clustering pattern in redshift space is distorted. In this paper we combine second-order perturbation theory with a model for collapsed, virialized structures, to extend the method to redshift space, and demonstrate that the method should be successful in determining with reasonable accuracy the bias parameter from state-of-the-art surveys such as the Anglo-Australian 2 degree Field Survey and the Sloan Digital Sky Survey.     

Unbiased estimates of galaxy scaling relations from photometric redshift surveys

Many physical properties of galaxies correlate with one another, and these correlations are often used to constrain galaxy formation models. Such correlations include the colour–magnitude relation, the luminosity–size relation, the fundamental plane, etc. However, the transformation from observable (e.g. angular size, apparent brightness) to physical quantity (physical size, luminosity) is often distance dependent. Noise in the distance estimate will lead to biased estimates of these correlations, thus compromising the ability of photometric redshift surveys to constrain galaxy formation models. We describe two methods which can remove this bias. One is a generalization of the V _max method, and the other is a maximum-likelihood approach. We illustrate their effectiveness by studying the size–luminosity relation in a mock catalogue, although both methods can be applied to other scaling relations as well. We show that if one simply uses photometric redshifts one obtains a biased relation; our methods correct for this bias and recover the true relation.     

The redshift-space two-point correlation function of ELAIS-S1 galaxies

We investigate the clustering properties of galaxies in the recently completed ELAIS-S1 redshift survey through their spatial two-point autocorrelation function. We used a subsample of the ELAIS-S1 catalogue covering approximately 4 deg² and consisting of 148 objects selected at 15 μm with a flux >0.5 mJy and a redshift   z < 0.5  . We detected a positive signal in the correlation function that in the range of separations  1–10  h ⁻¹ Mpc  is well approximated by a power law with a slope  γ= 1.4 ± 0.25  and a correlation length   s ₀= 5.4 ± 1.2  h ⁻¹ Mpc  , at the 90 per cent significance level. This result is in good agreement with the redshift-space correlation function measured in more local samples of mid-infrared-selected galaxies such as the IRAS Point Source Catalog (PSC z ) redshift survey. This suggests a lack of significant clustering evolution of infrared-selected objects out to   z = 0.5  that is further confirmed by the consistency found between the correlation functions measured in a local  ( z < 0.2)  and a distant  (0.2 < z < 0.5)  subsample of ELAIS-S1 galaxies. We also confirm that optically selected galaxies in the local redshift surveys, especially those of the SDSS sample, are significantly more clustered than infrared objects.     

A reliable cluster detection technique using photometric redshifts: introducing the 2TecX algorithm

We present a new cluster detection algorithm designed for finding high-redshift clusters using optical/infrared imaging data. The algorithm has two main characteristics. First, it utilizes each galaxy's full redshift probability function, instead of an estimate of the photometric redshift based on the peak of the probability function and an associated Gaussian error. Second, it identifies cluster candidates through cross-checking the results of two substantially different selection techniques (the name 2TecX representing the cross-check of the two techniques). These are adaptations of the Voronoi Tesselations and Friends-Of-Friends methods. Monte Carlo simulations of mock catalogues show that cross-checking the cluster candidates found by the two techniques significantly reduces the detection of spurious sources. Furthermore, we examine the selection effects and relative strengths and weaknesses of either method. The simulations also allow us to fine-tune the algorithm's parameters, and define completeness and mass limit as a function of redshift. We demonstrate that the algorithm isolates high-redshift clusters at a high level of efficiency and low contamination.