Dark Energy Survey year 1 results: Galaxy clustering for combined probes

We measure the clustering of DES year 1 galaxies that are intended to be combined with weak lensing samples in order to produce precise cosmological constraints from the joint analysis of large-scale structure and lensing correlations. Two-point correlation functions are measured for a sample of $6.6\times {10}^5$ luminous red galaxies selected using the redMaGiC algorithm over an area of 1321 square degrees, in the redshift range $0.15<z<0.9$, split into five tomographic redshift bins. The sample has a mean redshift uncertainty of ${\sigma }_z/(1+z)=0.017$. We quantify and correct spurious correlations induced by spatially variable survey properties, testing their impact on the clustering measurements and covariance. We demonstrate the sample’s robustness by testing for stellar contamination, for potential biases that could arise from the systematic correction, and for the consistency between the two-point auto- and cross-correlation functions. We show that the corrections we apply have a significant impact on the resultant measurement of cosmological parameters, but that the results are robust against arbitrary choices in the correction method. We find the linear galaxy bias in each redshift bin in a fiducial cosmology to be $b({\sigma }_8/0.81){|}_{z=0.24}=1.40\pm 0.07$, $b({\sigma }_8/0.81){|}_{z=0.38}=1.60\pm 0.05$, $b({\sigma }_8/0.81){|}_{z=0.53}=1.60\pm 0.04$ for galaxies with luminosities $L/L_{*}>0.5$, $b({\sigma }_8/0.81){|}_{z=0.68}=1.93\pm 0.04$ for $L/L_{*}>1$ and $b({\sigma }_8/0.81){|}_{z=0.83}=1.98\pm 0.07$ for $L/L_{*}>1.5$, broadly consistent with expectations for the redshift and luminosity dependence of the bias of red galaxies. We show these measurements to be consistent with the linear bias obtained from tangential shear measurements.

Figure 1

Galaxy distribution of the redMaGiC Y1 sample used in this analysis. The fluctuations represent the raw counts, without any of the corrections derived in this analysis. We have restricted the analysis to the contiguous region shown in the figure. The area is 1321 square degrees.

Figure 2

Redshift distribution of the combined redMaGiC sample in 5 redshift bins. They are calculated by stacking Gaussian PDFs with mean equal to the redMaGiC redshift prediction and standard deviation equal to the redMaGiC redshift error. Each curve is normalized so that the area of each curve matches the number of galaxies in its redshift bin.

Figure 3

Correlations of volume-limited redMaGiC galaxy number density with seeing FWHM and exposure time before any survey property (SP; see text for more details) cuts (illustrated with the red vertical lines) were applied to the mask. In the absence of systematic correlations, the results obtained from these samples are expected to be consistent with no trend (the reference green dashed line). The cuts removed regions with $i$-band FWHM $>4.5$ pixels and $i$-band exposure time $>500s$ as these showed correlations that differed significantly from the mean ($>20\%$) or were not well fit by a monotonic function. No SP weights were used in this figure.

Figure 4

Galaxy number density divided by the mean number density across the footprint for each redshift bin, split by the number density of stars. The points with error-bars display the results for our $3\mathrm{\Delta }{\chi }^2(68)$ weighted sample, the cyan curves display the results without these weights. For the weighted sample, the ${\chi }^2$ of the line $N_{\mathrm{gal}}/⟨N_{\mathrm{gal}}⟩=1$ with the data points shown for each bin is 24.9, 16.0, 13.1, 6.6 and 10.9 with $N_{\mathrm{d}.\mathrm{o}.\mathrm{f}}=10$. The $\mathrm{\Delta }{\chi }^2$ between the null signal and a linear best fit is 0.99, 0.95, 0.24, 0.013, and 0.082. This does not meet either of the $\mathrm{\Delta }{\chi }^2$ thresholds used in this analysis. We, therefore, see no evidence for stellar contamination or obscuration in this sample.

Figure 5

Galaxy number density in the highest redshift bin, $0.75<z<0.9$, as a function of two example SP maps, FWHM $r$-band and FWHM $i$-band. The black points correspond to the $3\mathrm{\Delta }{\chi }^2(68)$ weighted sample, the cyan line is the unweighted sample. In this redshift bin, the SP maps used in the $3\mathrm{\Delta }{\chi }^2(68)$ weights were Airmass $i$ and FWHM $r$. The left panel demonstrates the effect of the weights on the FWHM $r$ correlation. The right panel demonstrates that correlations with SP maps that were not included in the weights are still reduced due to correlations among the SP maps. The full set of SP correlations for the maps in Table 3 are shown in Appendix pp1.

Figure 6

The significance of each systematic correlation. The significance is calculated by comparing the $\mathrm{\Delta }{\chi }^2$ measured on the data to the distribution in the mock realizations. We find the 68th percentile $\mathrm{\Delta }{\chi }^2$ value, labeling it $\mathrm{\Delta }{\chi }^2(68)$, for each map obtained from the mock realizations. We quote the significance for the relationship obtained on the data as $\mathrm{\Delta }{\chi }^2/\mathrm{\Delta }{\chi }^2(68)$. Weights are applied for the SP map with the largest significance, with the caveat that we do not correct for both depth and the components of depth (e.g., exposure time, PSF FWHM) in the same band. For example, in the bin $0.15<z<0.3$, correcting for $r$-band depth (the most significant contaminant) did not remove all the $r$-band correlations with $\mathrm{\Delta }{\chi }^2/\mathrm{\Delta }{\chi }^2(68)>2$, so is not included in the final $2\mathrm{\Delta }{\chi }^2/\mathrm{\Delta }{\chi }^2(68)$ weights. This is repeated iteratively until all maps are below a threshold significance, shown here for thresholds of $2\mathrm{\Delta }{\chi }^2/\mathrm{\Delta }{\chi }^2(68)$ and $3\mathrm{\Delta }{\chi }^2/\mathrm{\Delta }{\chi }^2(68)$. The $x$ axis is shown in order of decreasing significance for the unweighted sample. The labels in bold are the SP maps included in the $2\mathrm{\Delta }{\chi }^2/\mathrm{\Delta }{\chi }^2(68)$ weights. In the second redshift bin, $0.3<z<0.45$, the $3\mathrm{\Delta }{\chi }^2/\mathrm{\Delta }{\chi }^2(68)$ and $2\mathrm{\Delta }{\chi }^2/\mathrm{\Delta }{\chi }^2(68)$ weights are the same because correcting for only $g$-band depth removes all correlations with $\mathrm{\Delta }{\chi }^2/\mathrm{\Delta }{\chi }^2(68)>2$.

Figure 7

Two-point correlation functions for the fiducial analysis in each of the 5 redshift bins. These panels show the autocorrelation used in Y1COSMO and the galaxy bias measurements presented in this work. A correction for correlations with survey properties is applied according to the methodology in Sec. 5. The grey dashed line is the correlation function calculated without the SP weights. The black points use the $2\mathrm{\Delta }{\chi }^2(68)$ weights. We show correlations down to $\theta ={2.5}^{\prime }$ to highlight the goodness of the fit towards small scales, but data points within grey shaded regions have not been used in bias constraints or the galaxy clustering part of Y1COSMO. That scale cut has been set in co-moving coordinates at $8\mathrm{Mpc}{h}^{-1}$. The solid red curve is the best-fit model using only the $w(\theta )$ autocorrelations at fixed cosmology, using $\mathrm{\Delta }{z}^i$ priors from [29]. The solid blue curve is the best-fit model from the full cosmological analysis in Y1COSMO.

Figure 8

Constraints on the ratio, $r$, of galaxy bias measured on $w(\theta )$ and measured from the galaxy-galaxy lensing signal (see [30] denoted Y1GGL in the text) in each redshift bin. The histograms show the posterior distributions of $r^i$ from an MCMC fit for each $z$ in $i$. The bottom-right panel displays the individual measurements for each bin (purple for our $w(\theta )$ measurements and orange for those obtained in Y1GGL). All cosmological parameters were fixed at the DES Y1COSMO posterior mean values, and all nuisance parameters were varied as in Y1COSMO. The constraints were calculated using the full Y1COSMO covariance matrix, so the covariance between the two probes has been taken into account. We see no significant evidence for $r\ne 1$ within the errors.

Figure 9

Parameter constraints showing the impact of the SP weights, varying ${\mathrm{\Omega }}_m$, 5 linear bias parameters $b^i$, and 5 nuisance parameters $\mathrm{\Delta }{z}^i$. Contours are drawn at 68% and 95% confidence level. These constraints use the same $\mathrm{\Delta }{z}^i$ priors as Y1COSMO. The blue contour shows the constraints on $w(\theta )$ calculated with no SP weights. The gray and red contours use SP weights removing all $2\mathrm{\Delta }{\chi }^2(68)$ and $3\mathrm{\Delta }{\chi }^2(68)$ correlations respectively. In this parameter space, ignoring the correlations with survey properties would have significantly biased the constraints from $w(\theta )$. As expected, the best fit when using the $2\mathrm{\Delta }{\chi }^2(68)$ weights is at smaller values of $b^i$ than the $3\mathrm{\Delta }{\chi }^2(68)$ weights, although the difference is not significant compared to the size of the contour.

Figure 10

Parameter constraints showing the impact of the estimator bias, $w_{\mathrm{est}}$ and false correction bias $w_{\text{false}}$. The fiducial data vector and was calculated using the $2\mathrm{\Delta }{\chi }^2(68)$ weights on the data. The $w_{\mathrm{est}}$ and $w_{\text{false}}$ were measured on Gaussian mock surveys using a $2\mathrm{\Delta }{\chi }^2(68)$ threshold significance. We see no evidence for significant bias in the $b^i$, ${\mathrm{\Omega }}_m$ plane. These constraints use the same $\mathrm{\Delta }{z}^i$ priors as Y1COSMO.

Figure 11

Parameter constraints showing the impact of the systematics correction on the covariance. Both contours use the fiducial theory data vector. The blue contour uses the covariance from mock surveys with no contamination added (labeled “cov: no sys”). The gray contour uses the covariance determined from mock surveys with the $2\mathrm{\Delta }{\chi }^2(68)$ contaminations added (labeled “cov: $2\sigma$ sys”). These constraints use the same $\mathrm{\Delta }{z}^i$ priors as Y1COSMO.

Figure 12

Maps of potential sources of systematics. Shown here for $i$-band only. Maps in other bands show fluctuations on similar scales. Each SP map is shown at $N_{\mathrm{side}}=1024$. The stellar density map is shown at $N_{\mathrm{side}}=512$.

Figure 13

Galaxy number density as a function of different SP maps. We show here only the correlations with SP maps used in the $2\mathrm{\Delta }{\chi }^2(68)$ weights calculation. The cyan line is the correlation of the sample without weights. The black points show the correlation after correction with the $2\mathrm{\Delta }{\chi }^2(68)$ weights. The error bars were calculated by measuring the same correlation on the Gaussian mock surveys described in Sec. 4b. The significance of these correlations are shown in Fig. 6.

Figure 14

The two-point cross correlations between redshift bins. These measurements are expected to be nonzero, with a significance related to the degree of overlap in the $n(z)$ displayed in Fig. 2. The numbers in each panel correspond to the redshift bins used in the cross-correlation, Adjacent bins are shown on the diagonal. The error-bars were calculated using the log-normal mock surveys used for Y1COSMO covariance validation [20]. The solid red curve is the best-fit model from the autocorrelation using only $w(\theta )$ at fixed cosmology. The solid blue curve (mostly under the red) is the best-fit model from the full cosmological analysis in Y1COSMO. For many of the cross-correlation panels, these predictions are indistinguishable. These measurements were not used in any of the parameter fits in this work or in Y1COSMO.