Galaxy clustering with photometric surveys using PDF redshift information

Abstract

Photometric surveys produce large-area maps of the galaxy distribution, but with less accurate redshift information than is obtained from spectroscopic methods. Modern photometric redshift (photo-z) algorithms use galaxy magnitudes, or colours, that are obtained through multiband imaging to produce a probability density function (PDF) for each galaxy in the map. We used simulated data to study the effect of using different photo-z estimators to assign galaxies to redshift bins in order to compare their effects on angular clustering and galaxy bias measurements. We found that if we use the entire PDF, rather than a single-point (mean or mode) estimate, the deviations are less biased, especially when using narrow redshift bins. When the redshift bin widths are Δz = 0.1, the use of the entire PDF reduces the typical measurement bias from 5 per cent, when using single point estimates, to 3 per cent.

1 INTRODUCTION

The analysis of the three-dimensional distribution of galaxies has become one of the major probes used to understand the history of the Universe and the growth of matter perturbations. Spectroscopic surveys such as WiggleZ¹ (Drinkwater et al. 2010), BOSS² (Dawson et al. 2013) and VVDS³ (Le Fèvre et al. 2004) have obtained precise maps of this distribution and many studies have increased our knowledge about the expansion history of the Universe and the growth of structures. However, targeting galaxies and obtaining spectra is a slow and costly process; therefore, past and current spectroscopic surveys have been limited to relatively low redshift and a reduced number of galaxies with respect to photometric surveys.

Multiband imaging of wide areas of the sky is complementary to spectroscopic surveys. These photometric surveys, such as the Canada–France–Hawaii Telescope Legacy Survey (CFHTLS),⁴ Dark Energy Survey (DES;⁵ Dark Energy Survey Collaboration 2005) and the Large Synoptic Survey Telescope (LSST;⁶ Ivezić et al. 2008) enable lower accuracy redshift estimation from the colours of millions of galaxies without being affected by spectroscopic selection effects.

Photometric redshifts (photo-z) are estimated by using multiband photometry as inputs to one or more different techniques that map galaxy photometric properties into a redshift. These techniques can broadly be classified into two categories. The first is known as template-based methods (Benitez 2000; Ilbert et al. 2006), in which a set of calibrated galaxy spectral energy distributions (SEDs) is fit to the photometric data to find the one that best represents the observed fluxes. The second category use a spectroscopic training set and machine learning algorithms, such as artificial neural networks (Collister & Lahav 2004), boosted decision trees (Gerdes et al. 2010), or prediction trees and random forests (Carrasco Kind & Brunner 2013), to generate a photo-z PDF estimate.

Probability density functions (PDFs)of various astronomical measurements have been used in cosmological analyses, for example, luminosity functions (Sheth 2007), weak lensing (Mandelbaum et al. 2008), cluster identification (van Breukelen & Clewley 2009), the real-space clustering of quasars (Myers, White & Ball 2009) and tomographic magnification (Morrison et al. 2012). However, a systematic analysis of the use of photometric redshift PDFs in galaxy clustering has not been performed, mostly due to the lack of reliable PDF estimation and its computational cost.

In this work, we study how the angular clustering of galaxies depends on the chosen photo-z estimate and on the photo-z bin width, by using realistic simulations to compare clustering measurements based on photometric and spectroscopic redshifts. We also show how to include the full probability density when estimating the angular correlation functions, and we study the impact of using different photo-z PDF statistics (e.g. mean, mode and median) on estimating the galaxy bias. In Section 2, we describe the methodology followed in the paper. We describe the angular clustering results and the fitting to galaxy bias in Section 3, and we discuss these results in Section 4. Finally, we summarize the main conclusions in Section 5.

2 METHODOLOGY

The standard approach to analyse galaxy clustering in photometric surveys begins with subdividing the catalogue into subsamples selected by ‘top-hat’ photo-z redshift bins. The photo-z value used to determine if a galaxy is in one or another bin is usually a point estimate of the photo-z PDF. In this paper, we quantify how angular clustering analyses are affected by the choice of the specific photometric redshift estimator, including one that uses the full PDF information instead of photo-z point estimates. We address this by measuring the clustering of a subsample of the DES-BCC Aardvark simulation mock galaxy catalogue (Busha et al. 2013). We compared the clustering measurements given by different photo-z estimators and when considering different redshift bin widths. As a specific test, we quantify these differences by fitting theoretical galaxy correlation functions in order to estimate the linear galaxy bias (Kaiser 1984), as a metric to evaluate in which photo-z estimator is more reliable (Coupon et al. 2012; Crocce et al. 2015; Soltan & Chodorowski 2015).

2.1 Simulation data

The mock galaxy catalogue considered here is the Aardvark v1.0 catalogue from the Blind Cosmology Challenge (BCC) simulations, developed for the DES. The catalogue is created from three ΛCDM N-body dark matter simulations, with sizes of 1050 , 2600 and 4000 Mpc h⁻¹ and 1400³, 2048³ and 2048³ particles, respectively. They were created using Gadget-2 (Springel 2005) and initial conditions given by camb (Lewis, Challinor & Lasenby 2000) and 2LPT (Crocce, Pueblas & Scoccimarro 2006). The algorithm that populates the dark matter haloes with galaxies, ADDGALS (Busha et al. 2013), follows a prescription based on SubHalo Abundance Matching (SHAM) techniques (Conroy, Wechsler & Kravtsov 2006; Busha et al. 2013; Reddick et al. 2013). The final catalogue is complete down to r < 25 and covers 1/4 of the sky. Galaxy properties such as colour or luminosity are assigned by matching a spectroscopic training sample from the SDSS DR6 value added catalogue (Blanton et al. 2005) at low redshift. This training is extrapolated to higher redshift matching the colour distribution to SDSS DR8 (Aihara et al. 2011) and DEEP2 (Newman et al. 2013) photometric data. Then, the output catalogue includes DES colours and errors for each galaxy of the catalogue. These catalogues have been compared with real data by the DES collaboration (Chang at al. 2015; Leistedt et al. 2015).

The full BCC-Aardvark-v1.0c catalogue covers 10 313 deg² to the full DES depth, and includes a total of 1.36 × 10⁹ galaxies. The simulated catalogue is stored in files according to healpix⁷ (Górski et al. 2005) pixels of n_side = 8. We chose a contiguous area of the simulation by using 24 pixels, which corresponds to an area of about 1200 deg² on the sphere in order to have a significant sampling of small scales. For our study, we have selected the galaxy sample according to a magnitude limited cut of g < 24. This cut corresponds to a selection in the g-band of signal-to-noise greater than 20 in the simulation, which incorporates the DES observed photometric error. The total number of galaxies in the catalogue after applying the magnitude cut is around 30 million galaxies.

2.2 Photometric redshift code

We have used the publicly available code tpz⁸ (Carrasco Kind & Brunner 2013) to estimate the galaxy redshift probability distributions. tpz is a parallel code that estimates photo-z PDFs using prediction trees and random forests. A prediction tree is constructed by splitting the data in recursive branches until a convergence criterion is reached. In order to construct more robust PDFs, the code uses the random forest technique in which N_T bootstrap samples of the training set are created and prediction trees are generated for the N_T samples. In order to include the measurement errors, e.g. magnitude errors, N_R training samples are created by perturbing the training set according to the errors of the measurement variables. Finally, the PDF of each galaxy in the sample is created by combining the prediction trees. tpz was one of the algorithms used in the DES Science Verification Data (Sánchez et al. 2014), and produced one of the best performances for that data set.

We have considered 10⁵ galaxies as a training set, up to the full depth and a cut in the magnitude errors avoiding extremely large values in order to use all the available magnitude-redshift information from the simulation, and we, therefore, use less than 1 per cent of the available data for this purpose. The training set galaxies were confined to a region of 54 deg². The test data used for the main analysis of this paper was directly selected from the simulation with no cuts on magnitude errors. The effect of the redshift selection of galaxies for the training set in the results is shown in Section 3.5.1.

As defined in Carrasco Kind & Brunner (2013), the concentration of individual galaxy PDFs, p(z), is output by tpz as a PDF concentration parameter called zConf. This parameter is defined as the integrated probability between z_phz ± σ_TPZ(1 + z_phz), where z_phz is the photometric redshift, and it measures the narrowness of the PDF. In this case, we selected σ_TPZ = 0.075, which is similar to the 1 − σ confidence interval of the PDFs. We can select different quality cuts by using this parameter, which is related to the BPZ ODDS parameter (Benitez 2000).

tpz is a particular method of the mlz framework. mlz is code that computes photometric redshift PDFs using machine learning techniques. It incorporates a Bayesian combination of techniques that estimate photometric redshift PDFs, including both template based methods and unsupervised machine learning algorithms (Carrasco Kind & Brunner 2014b), and also enables an efficient storage of the PDFs by using a sparse representation basis (Carrasco Kind & Brunner 2014a). For simplicity, we only used tpz for the photo-z in this paper, which is justified by the excellent results produced by tpz on the DES Science Verification Data (Sánchez et al. 2014), and by the fact that we want to study the impact of using photo-z PDFs in clustering as produced by a single technique (to simplify the resulting analysis). tpz has been used, together with other codes listed in Sánchez et al. (2014), in several DES Science Verification Data studies (Bonnett et al. 2015; Crocce et al. 2015; Giannantonio et al. 2016; Dark Energy Survey Collaboration 2015).

2.3 Survey configuration: photo-z binning

Galaxy clustering analyses in photometric surveys are usually done by measuring angular correlations of galaxy samples selected in different redshift bins. We divide the full redshift range, which in this paper we restrict to the range 0.2 < z < 1.4, into N_z redshift bins of width Δz in order to reduce the extent of the projection of radial information for 2D clustering analysis. As shown in Asorey et al. (2012) and Eriksen & Gaztañaga (2015), the optimal photometric redshift bins are given by shells of about twice the size of the photo-z standard deviation. In this paper, we consider different configurations: Δz = {0.1, 0.15, 0.2, 0.3} in order to study the evolution of photometric clustering with bin width. The true redshift distribution of galaxies, n(z), is shown in Fig. 1, together with the redshift distribution obtained by stacking tpz PDFs. We also show (red) the n(z) of the spectroscopic training set. For this paper, we were only interested in a patch of the sky that covers 1200 deg², which allows us to measure the small scale clustering and to study how it depends on the different photo-z statistical quantities.

2.4 Photo-z estimators per galaxy

2.4.1 Single point statistics

Once we have computed photo-z PDFs with tpz, we estimate single statistical summary quantities. In this study, we focus on the mode redshift, |$\hat{z}$|⁠, and the mean redshift, |$\bar{z}$|⁠.

We also evaluated the median redshift, but we decided not to include it in the final results as it is similar to the other point estimates; thus, for clarity we decided to reduce the analysis to the chosen single point estimates. We created a catalogue for each redshift bin considered by selecting the galaxies with the single point estimate in the range covered by the redshift bin.

2.4.2 Photo-z weights

In Fig. 4, we present a graphic example of the different photometric redshift estimators that we use. We intentionally selected a PDF with a most frequent (mode) redshift, given by equation (3) within the photometric bin 0.5 < z < 0.8, but where the mean, defined in equation (2) is in a different redshift bin. In blue, we show the portion of the PDF between 0.5 < z < 0.8 that corresponds to the weight of the galaxy in that redshift bin. Of course, the PDF displayed in Fig. 4 is an extreme case. For this particular PDF, zConf = 0.37, while near z = 0.6 the mean PDF quality parameter is zConf ∼ 0.95. In Appendix A, we discuss the statistical properties of the mean and the mode and overall quality of the photometric sample PDFs.

2.5 Two point angular correlation function estimators

2.5.1 Pixel based estimator

2.5.2 Direct pair counting estimator

2.6 Theoretical modelling

We used camb (Lewis et al. 2000) to obtain the linear power spectrum with halofit (Smith et al. 2003; Takahashi et al. 2012) in order to include non-linearities at small scales. We Fourier transform this angular power spectrum in order to compute the 3D angular correlations required by equation (10). For this paper, we considered a flat ΛCDM model driven by the simulation cosmological parameters when computing the theoretical correlation function. We also included linear redshift space distortions as a series of multipoles following (Kaiser 1987; Hamilton 1992).

3 RESULTS

3.1 Comparison between direct pair counting and pixel-based estimators

As shown in Wang et al. (2013), pixel-based and point-to-point pair count methods yield similar results for 2-point angular clustering. However, this previous work only considered unweighted pair counts in any given bin i.e. weights of f_z = 1. In Fig. 5, we extend this previous result to compare the results for both pair count and pixel-based methods when considering galaxy weights in the redshift bin 1.0 < z < 1.2 and a threshold p_threshold = 0.1. As shown in this figure, over the angular range 0.1 < θ < 1.0, we find a good agreement. The number of jackknife regions is different in the two cases, however, being N_JK = 32 for the point-to-point case and N_JK = 384 when considering the pixel-based estimator. We opted to use the computationally simpler pixel-based estimator in the rest of the paper.

3.2 Clustering amplitude

We now focus our analysis on the relative amplitude between the angular clustering signal using different photo-z selection criteria and the true redshift clustering. This allows us to directly study how the different statistical representations of photometric redshift change the measurement signal.

In the analysis, we divide the redshift range 0.2 < z < 1.4 into different numbers of bins in order to consider different bin configurations. As a result, a comparison between individual redshift bins for different bin configurations may consider different redshift regions for different configuration. For this reason, we present clustering results for bins with different widths in symmetrical manner about a given central redshift. As discussed previously, we restrict our analysis to four different bin widths. In Fig. 6, we present the ratios of the photometric redshift clustering with respect to the true redshift clustering, for different statistical estimators and a redshift bin centred at z = 1.

In the left-hand panels, we show that the results for a broad bin of Δz = 0.3 in the redshift range 0.85 < z < 1.15. As expected, for all algorithms, the amplitude of the clustering of the photometric sample is smaller than when using the true redshifts, since the errors on a photometric redshift estimate will suppress this inherent clustering. Notice that the clustering measurements when using the mode, |$\hat{z}$|⁠, and mean, |$\bar{z}$|⁠, are similar. Any small differences are due to the fact that each selection produces a different n(z) when individual PDFs are not symmetric, like the one shown in Fig. 4.

In the bottom panels, we show the clustering ratios with respect to the true clustering when using PDF weights with different probabilenlrgity thresholds. This allows us to both clean our sample, as if using a quality cut, and sample more narrow redshift bins. The thresholds considered are p_threshold = 0.0, 0.1. The signal depends on the cut on the selection of weighted galaxies. The stronger the cut, the cleaner the sample and the clustering amplitude increases, as well as the intrinsic n(z). But this may also bias our results as we may be changing the average galaxy types in the sample, as discussed in Martí et al. (2014). Here, we considered a low threshold that is non-negligible to compare with the full PDF case. To aid in the comparison with the point estimate photo-z selection, we also present the ratio for |$\hat{z}$| with the red dashed line.

We consider narrower bin configurations in order to test what happens as we approach the intrinsic photo-z dispersion error, summarized in Fig. 2. We see in the case when Δz = 0.2 that the amplitude of the photometric samples clustering decreases with respect to the true redshift clustering. The angular clustering signal is proportional to n(z)², as explained by equation (10), which in the top-hat case means that it is inversely proportional to (Δz)². Photometric samples distribution in true redshift are broader than the top-hat bin, and therefore, the signal amplitude is smaller. The bin considered in this case for galaxy selection is 0.9 < z < 1.1. We see that the differences between the mode and the mean are bigger than in the previous case with Δz = 0.3. This may be a result of the fact that when we consider bins much bigger than the intrinsic separations, and the differences between photo-z single statistic estimators are smaller. As we extend the comparison to smaller widths, Δz = 0.15 (0.925 < z < 1.075), the ratio between the angular clustering signal for the photometric samples and the sample with true redshift becomes smaller. This is in agreement with the trend we saw before, as w(θ) ∝ (1/Δz)² for true redshift clustering, while the photo-z dispersion keeps the corresponding signal diluted in the radial direction.

The case with Δz = 0.1 (0.95 < z < 1.05) shows the same trend than the previous cases with bigger bin widths. The ratio of the photo-z signals to the true redshift signal continues to decrease. The results with mean and mode estimators tend to converge as the bin width approaches the photo-z dispersion error.

We found that the evolution of the clustering amplitude with bin width evolves differently for the different estimators. We show in Fig. 7 the clustering amplitude evolution at θ = 0.1 degrees. For photometric redshift estimators, the clustering signal increases by about 50 per cent when increasing the numbers of bins by a factor of 4. Therefore, increasing the number of bins beyond the limit in which the bin width is comparable with the photo-z error is not an efficient process for any photo-z estimator, at least from the point of view of a clustering measurement, especially for bins with a width smaller than Δz = 0.15, which is twice the mean photo-z error, as shown in Fig. 2.

3.3 Bias measurement

3.3.1 PDF redshift distributions

The total redshift range considered is 0.2 < z < 1.4, and all bins have the same width for each configuration. The angular range considered was set to cover the comoving coordinates range 10 h⁻¹Mpc < r < 60 h⁻¹Mpc, which corresponds to different angular ranges in each redshift bin. The minimum scale was selected by testing at which scale the linear growth model for the spatial correlation departs from a non-linear model. Notice that this is a conservative cut when compared with the cuts used in (Crocce et al. 2015). This corresponds to θ_min = 0.8 deg at the lowest redshift and θ_min = 0.19 deg at highest redshift bin.

The comparison between the different photo-z selection methods is done by comparing each galaxy bias measurement with the true result, which is determined by using the true redshift distribution of the selected galaxies in order to do a fair comparison. First, we show, in Fig. 8, the galaxy bias measurement made by using different photo-z statistics and the bias measurement done by selecting galaxies according to the spectroscopic redshift. Notice that the spectroscopic sample in each bin is different than the photometric samples considered, but since this is accounted for in the bias measurement, we can study how the photo-z statistic measurements compare with the spectroscopic one.

In panel a of Fig. 8, we show that the evolution of galaxy bias for the broad Δz = 0.3 bin configuration. We only show results for the true redshift, z_s, the mode redshift, |$\hat{z}$|⁠, the mean redshift, |$\bar{z}$|⁠, and the PDF weighted samples for clarity. We do not show here the results when applying photometric redshift quality cuts. The measurements are similar and the slightly different values for the different estimators are within the statistical error bars. This is reasonable as we are considering a broad redshift bins in this panel, and the differences between different photometric samples redshift distributions are thus small.

The same trend is observed when considering N_z = 6 redshift bins, as shown in panel b on Fig. 8. This case corresponds to bins with Δz = 0.2 width, which is larger than twice the photo-z dispersion, and therefore, photometric redshift effects are still not the biggest issue. The evolution of linear galaxy bias with redshift resembles the results of Crocce et al. (2015) for a MICECATv2.0 (Carretero et al. 2015) sample, as the ratio b_g(z = 1.1)/b_g(z = 0.3) is similar for both simulations.

We show in the lower-left plot in Fig. 8 the measurement of the bias for the different redshift estimators when considering N_z = 8 redshift bins. In this case, we begin to observe bigger differences between the case when using photo-z and the case when the bias was obtained by using spectroscopic redshifts, especially when compared to the previous cases that used larger bin widths.

Finally, we show in the bottom-right panel of Fig. 8 the bias evolution when using 12 bins of width Δz = 0.1. The differences between the photo-z galaxy bias results and the spectroscopic bias measurement are larger than for the previous cases with broader bin configurations. The closest result to the spectroscopic results value of the bias is obtained when using the full PDF information (p_threshold = 0), especially at intermediate redshifts.

We show in Fig. 9 the measurement bias between the method that uses PDF stacking to estimate n(z) and the true value, given by the n(z) measured directly from the true redshifts from the simulation of the photometric sample. When considering full PDF information, we weighted the stacked PDFs and the corresponding true redshifts by the corresponding PDF weight. We show in the top-left panel the relative differences when considering Δz = 0.3 for the three methods, finding small deviations with respect to the true results with minor differences between the different selection techniques. These observed differences exist because the PDF stacking technique is not perfectly reconstructing the true n(z) of the population sample in the tomographic bins. In this case, the differences are small because the bin width is broad and photo-z systematics in the n(z) are smaller.

When we decrease the bin width to Δz = 0.2, the differences grow, as shown in the top-right panel of Fig. 9. The three methods are still producing similar results, and because the bins are still too broad, the relative bias is zero, within the error bars. When the configuration changes to bins with widths Δz = 0.15, differences start to become more apparent and the PDF weighting method begins to differ from the single point estimate estimators. For the narrowest bin width configuration considered, Δz = 0.1, the differences at intermediate redshifts are larger than 5 per cent for single point estimators, whereas for PDF weighted galaxy samples, these differences are around 3 per cent. We include a table in Appendix B that presents all galaxy bias measurements and the relative differences with the true results.

In order to summarize and quantify these results, we show in Fig. 10 the mean value of the mean absolute deviation between each selection method and the true result for each bin width. We found that for the largest bin widths, the differences are around 1 per cent and are similar for the three photo-z selection statistics: mean, mode and PDF weighting. However, for the narrower bins, the deviation when we consider summary statistics is around 5 per cent, while it is 3 per cent when using the photo-z PDF galaxy weighting method.

3.4 Reducing the redshift bin catalogue size

Using full PDF information in galaxy clustering produces less-biased measurements than point estimate photo-z methods, but it also increases the size of each redshift bin galaxy sample. We also studied how a Monte Carlo sampling of the PDF, in order to define a point estimate that encloses more of the PDF than the mean or the mode, or applying a threshold cut based on the amount of PDF in each bin compares to the full PDF inclusion method.

3.4.1 Monte Carlo sampling redshift

We extended the previous analysis to include a Monte Carlo sampling redshift, z_MC, which assigns a redshift value based on the cumulative distribution function for each galaxy. We make our previous galaxy bias measurement in the different redshift bins according to z_MC. In Fig. 11, we show the best-fitting galaxy bias for galaxies selected according to z_MC in N_z = 8 redshift bins of Δz = 0.15. We observe that in this case the results are similar to the results given by PDF weights (for example, the results from panel c of Fig. 8), both when using PDF sampling or the true redshift distributions. This is expected, as we are using the probabilistic information to determine the Monte Carlo sampling redshifts.

3.4.2 Quality cuts

The effect of sparse PDFs with multiple peaks can introduce significant noise into our PDF weighting scheme. Although it is not the main interest of this paper, we considered a case in which we applied a threshold cut f_z > (p_threshold = 0.1) in order to select galaxies in the different bins. The effect is a combination of a quality cut and a cut on galaxies that are not in the bin but whose tails are inside the bin, which produces bigger catalogues in each tomographic bin. In Fig. 12, we present a comparison between a photo-z sample selected according to full photo-z PDFs for a configuration with bin width Δz = 0.15 and a sample selected by applying a threshold to the photo-z PDF weights of f_z > 0.1. We found that the results are similar, supporting the idea of applying threshold cuts to reduce the size of the galaxy density in each pixel in the map, although cuts to a sample have to be applied carefully in order to avoid introducing selection biases to the sample, see e.g. Martí et al. (2014). We observe this effect in Fig. 12, as the true results for both samples are not exactly equivalent. A detailed study of using quality cuts from PDF information is outside the scope of this paper.

3.5 Systematics

3.5.1 Training set sample variance

Since we use one galaxy sample from one particular pixel of the simulation for our photo-z training set, we wanted to demonstrate that the choice of this one pixel did not bias our results. As a result, we compared the spectroscopic n(z) for the galaxies in our training sample with the mean spectroscopic n(z) from 10 randomly selected galaxy samples from the entire area, to demonstrate that our results were not dependent on the choice of a particular pixel. We are not, in this paper, exploring the more traditional concept of photometric redshift ‘sample variance’ as discussed, for example, in Cunha et al. (2012). We show in Fig. 13 the relative difference between theoretical angular correlations, computed by using equation (10), when we used the training set n(z) (red line in Fig. 1) and the mean n(z) of 10 different random samples extracted from the catalogue with the same number of galaxies as the training set, which is similar to the blue solid line in Fig. 1. We found relative differences smaller than 2 per cent over the redshift range for all angles, 0.1 < θ < 1. This implies a relative difference smaller than 1 per cent for the galaxy bias, which is lower than the differences observed for our different photo-z statistics.

3.5.2 True redshift distribution reconstruction

As observed in Section 3.3, the main difference between the photometric redshift and the true redshift galaxy bias measurement is a result of the failure to recover the true redshift distribution, n(z). As an example, in Fig. 14, we show the difference between the true redshift distribution (blue line) and the PDF stacking PDF (green dashed line) for galaxies selected with mean redshift within 0.65 < z < 0.8. The red shadowed region shows the range between n(z) created by stacking Gaussian PDFs with standard deviations in the range σ_gauss = 0.03– 0.1. We see that the difference between the measured n(z) is within the accuracy of the photo-z catalogue, shown in Fig. 2. We also show, for comparison, the true redshift distribution from the weighted sample and the true weighted redshift distribution. We explore this result in more detail in Appendix C, where we look at the differences between the n(z) obtained from stacking the photo-z PDF of galaxies selected in redshift shells according to their mean redshift and the true distribution of the same sample when using both different bin configurations and different redshift ranges.

4 DISCUSSION

In this paper, we have studied how the angular galaxy clustering obtained from photometric populations depends on the different statistical estimators used to assign galaxies to specific redshift bins. The primary estimators that we have considered are the mean and the mode of a galaxy's photo-z PDF. We found differences between the different estimators, in part, since they produce different galaxy samples in each top-hat photometric redshift bin. As a result, the clustering signal is different when using either the mean or the mode.

We also included the full PDF information in our clustering analysis by weighting each galaxy according to the integrated probability that the galaxy actually resided within each redshift bin. This clustering signal is smaller than the clustering signal from single point estimates samples. If we apply a threshold cut of p_threshold = 0.1, the clustering amplitude increases. This is explained by the fact that when we consider a larger threshold the corresponding n(z) is narrower than when considering all galaxies with non-negligible weights in a redshift bin. However, we also may be sampling different type of galaxies, since we are only selecting galaxies with higher probability to lie within the bin.

We extended the comparison between the different photo-z statistical representations to a cosmological parameter estimation analysis by measuring the linear galaxy bias in different redshift bins. We find that, in general, the photo-z estimators produce similar results, especially when considering broad bins. We find that there is a relative bias with respect to the true galaxy bias results, since the PDF stacking redshift distributions in each bin differ from the true redshift distributions. For narrow bins, the selection method given by PDF weights produces less biased differences with respect to the true results. The mean deviation for a bin configuration with width Δz = 0.1 is 3 per cent when using PDF weights, while it is 5 per cent when using summary or single point estimate statistics. Thus, the use of photo-z PDF weights to select galaxies in tomographic redshift bins in order to measure the galaxy clustering in a photometric survey produces more robust results than using single point estimates. We can use the methodology presented in this paper to calibrate the effect of assigning galaxies to photo-z bins to ensure that the model parameters from simulations mimic the real data catalogues. This also applies to other photo-z methods that estimate PDFs (Sánchez et al. 2014; Bonnett et al. 2015; Leistedt et al. 2015) as they will have similar behaviours.

Creating maps with PDF weights involves much larger data sets than catalogues of galaxies selected only by redshift. One way to reduce the amount of data is to apply a cautious PDF quality cut by using a threshold when considering PDF weights. We found similar results to the full PDF results, although any cut on a sample has to be carefully tested. Another way to reduce the size of the catalogues, while still retaining a certain level of the PDF information, is by using Monte Carlo sampling point estimates. We found that the Monte Carlo sampling estimators produce similar results to our PDF weight results.

The effect of choosing different photometric redshift training samples from the simulation on the calculation of the galaxy bias measurement is smaller than 1 per cent, which is lower than the effects due to the different photo-z statistics used in this paper. Likewise, the differences between stacking photo-z PDFs to compute the redshift distribution and the true redshift distribution are also within the photo-z errors.

5 CONCLUSIONS

With photometric surveys, we can accumulate much larger galaxy samples in less time than with spectroscopic surveys. However, the lack of true redshifts restricts the quality of any radial information on such a survey, as photometric redshift are produced from multiband imaging.

Therefore, we need to set a statistical definition of a photometric redshift in order to identify which tomographic redshift bin contains a given galaxy. The search for an optimal definition is the main goal of this paper. The core analysis of this paper consisted of defining a new photo-z selection method that includes the full photo-z PDF information by weighting each galaxy in the redshift bin with the probability that the galaxy lies in that bin, and to compare this result with methods based on single statistical estimates such as the mean or the mode of the photo-z PDF.

We found, using mock galaxy catalogues and a machine learning photo-z code, that if we use single point statistics, like the mode or the mean, there is an offset on the galaxy bias measurements. These bias measurements are obtained either by measuring photometric redshift distributions by stacking the individual photo-z PDFs or from the true redshift distribution of the same galaxies. This shift must be taken into account when considering similar large-scale structure analyses that leverage galaxies drawn from photometric surveys. This corrective effect can be estimated by applying a similar method to measure the offset in the determination of the cosmological measurement of interest by using simulations in similar conditions to the expected photometric data. In our case, we used the galaxy bias as the metric to test different photo-z statistics, and we found that, for single point statistics, the cumulative deviation is a 5 per cent for a bin configuration with width Δz = 0.1.

Our results are closer to the ground truth if we weight the contribution of each galaxy to a photo-z bin according to the amount of their photo-z PDF in each redshift bin. This approach, on the other hand, produces a difference of 3 per cent in the Δz = 0.1. Therefore, and especially for narrow photometric top-hat bins, PDF weighting is more optimal than simply using summary statistic photo-z.

JA and JJT acknowledge support from US Department of Energy grant no., DE-SC0009932. ISN would like to thank the Spanish Ministry of Economy and Competitiveness (MINECO) for funding support through grant FPA2013-47986-C3-2-P. RJB acknowledges support from the National Science Foundationgrant no. AST-1313415. RJB has been supported in part by the Center for Advanced Studies at the University of Illinois. This work also used resources from the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant no. OCI-1053575. We want to acknowledge M. Becker and R. Wechsler for their helpful guidance in properly using the simulation catalogue. We thank all the useful discussions with DES members and especially those with G. Bernstein, M. Crocce, E. Gaztañaga, W. Hartley, K. Honscheid, A. Kim, A. Ross, E. Sanchez and C. Sanchez.

Funding for the DES Projects has been provided by the US Department of Energy, the US National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the UK, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Científico e Tecnológico and the Ministério da Ciência, Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey.

The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH) Zürich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciències de l'Espai (IEEC/CSIC), the Institut de Física d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universität München and the associated Excellence Cluster Universe, the University of Michigan, the National Optical Astronomy Observatory, the University of Nottingham, The Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex and Texas A&M University.

The DES data management system is supported by the National Science Foundation under grant no. AST-1138766. The DES participants from Spanish institutions are partially supported by MINECO under grants AYA2012-39559, ESP2013-48274, FPA2013-47986 and Centro de Excelencia Severo Ochoa SEV-2012-0234. Research leading to these results has received funding from the European Research Council under the European Unions Seventh Framework Programme (FP7/2007-2013) including ERC grant agreements 240672, 291329 and 306478.

This paper has gone through internal review by the DES collaboration. The DES publication number for this article is DES-2015-0139. The Fermilab preprint number is FERMILAB-PUB-15-571.

REFERENCES

APPENDIX A: ERROR DISTRIBUTION

In order to check the robustness of the galaxy photo-z PDFs that we used in the paper, we estimated the distribution of the photometric standardized error of the photo-z BCC galaxies used in the paper, (z_phot − z_true/σ). The standard deviation, σ, is given by equation (1). In Fig. A1, we show the results using the mean redshifts. We observe that the simple error estimate is close to the unbiased estimate (μ = 0, σ_G = 1). We can also consider the mode redshift, as shown in Fig. A2, where we see that the distribution of the modes tends to be more concentrated than the distribution of the means.

We also tested how photo-z are distributed according to the confidence intervals by estimating the number of galaxies with photo-z inside 1 − σ and 2 − σ levels, which is shown in Table A1. We see that the distribution of mean values in confidence intervals is close to the expected 68 per cent and 95 per cent distributions. When considering the mode, the values are more concentrated as we are considering the peaks of each individual PDF.

APPENDIX B: GALAXY BIAS RESULTS

In tables B1, B2, B3 and B4, we show the galaxy bias fits for the different bin configurations and the three photometric redshift methods: mean, mode and PDF, used in this paper to select galaxies in tomographic redshift bins. In each case, we stack the galaxy photo-x PDFs to compute the redshift distribution. We also present the goodness of fit for each fit and the relative difference with the appropriate true measurement.

APPENDIX C: TRUE REDSHIFT DISTRIBUTIONS

As shown in Section 3.3, the galaxy bias measurements obtained from n(z) given by PDF stacking are different than the true measurements. This is caused by the difference between the true redshift distribution of the photo-z galaxy sample and the PDF stacking n(z). As an example, we show in Fig. C1 the differences between the true redshift n(z) and the PDF stacking n(z) for galaxies selected according to the mean redshift for a given set of top-hat redshift bins at low redshift. We note that the tails of the PDF stacking n(z) are longer than the true redshift distribution. This disagreement is expected and has been observed when using template based and machine learning algorithms that incorporate PDFs. We expanded the comparison to intermediate redshift (Fig. C2) and high redshift (Fig. C3), observing similar differences. For comparison, we also show the redshift distribution for the PDF weighted galaxies, when stacking PDFs or true redshifts.