Density split statistics: Joint model of counts and lensing in cells

We present density split statistics, a framework that studies lensing and counts-in-cells as a function of foreground galaxy density, thereby providing a large-scale measurement of both 2-point and 3-point statistics. Our method extends our earlier work on trough lensing and is summarized as follows: given a foreground (low redshift) population of galaxies, we divide the sky into subareas of equal size but distinct galaxy density. We then measure lensing around uniformly spaced points separately in each of these subareas, as well as counts-in-cells statistics (CiC). The lensing signals trace the matter density contrast around regions of fixed galaxy density. Through the CiC measurements this can be related to the density profile around regions of fixed matter density. Together, these measurements constitute a powerful probe of cosmology, the skewness of the density field and the connection of galaxies and matter. In this paper we show how to model both the density split lensing signal and CiC from basic ingredients: a non-linear power spectrum, clustering hierarchy coefficients from perturbation theory and a parametric model for galaxy bias and shot-noise. Using N-body simulations, we demonstrate that this model is sufficiently accurate for a cosmological analysis on year 1 data from the Dark Energy Survey.

Figure 1

Top panel: splitting the lines of sight in one DES-Y1 like Buzzard simulation into 5 quantiles of galaxy density (color coding from cyan, most underdense, to red, most overdense). The map uses a 20 arcmin top-hat radius and redMaGiC galaxies with a redshift range of $0.2\lesssim z\lesssim 0.45$. Bottom left: histogram of redMaGiC galaxy counts in 20 arcmin radii (counts-in-cells). We show the mean histogram from 4 Buzzard realizations of DES-Y1 (black points), our model based on perturbation theory and cylindrical collapse (solid line) and a model that assumes the projected density contrast to be a Gaussian random field (dotted line). The color coding corresponds exactly to the density quantiles in the top panel. Bottom right: Lensing signals around random points split by the density quantile in which these points are located. We show the mean measurement from 4 Buzzard realisations (black points), our perturbation theory model (solid line) and a model that assumes projected density contrast and lensing convergence to be joint Gaussian random variables (dotted line). Color coding is the same as in the other panels. The asymmetry between the lensing signals around the most underdense and most overdense lines-of-sight indicates the skewness of the cosmic density PDF.

Figure 2

Redshift distributions of the tracer galaxy sample and the source samples of our N-body realizations of DES-Y1.

Figure 3

Left panel: Forecast of $1\sigma$ and $2\sigma$ constraints on ${\mathrm{\Omega }}_m$ and ${\sigma }_8$ achievable with density split lensing and counts-in-cells in DES Y1 data. The constraints are marginalized over ${\mathrm{\Omega }}_b$, $n_s$, $h_{100}$, redMaGiC galaxy bias $b$, and galaxy-matter correlation coefficient $r$. For the parameters ${\mathrm{\Omega }}_b$, $n_s$, $h_{100}$ we have assumed the same flat priors as used in the DES Y1 combined probes analysis presented in DES Collaboration [21]. Right panel: $\mathrm{\Delta }{S}_3/S_3$ measures relative deviations from our fiducial perturbation theory prediction of the scaling coefficient $S_3\equiv \frac{⟨{\delta }^3⟩}{⟨{\delta }^2{⟩}^2}$ [cf. Eq. (2.30) and Sec. 4 for details]. It can hence be thought of as the Bispectrum amplitude. We show $1\sigma$ constraints on this parameter achievable with density split lensing and counts-in-cells in DES data alone (solid lines) and using additional information on cosmology from Planck (dashed lines, no lensing). The sharp cutoff of the contours for low values of ${\mathrm{\Sigma }}_8$ is caused by the requirement that matter density and a shot-noise free galaxy density field must have a correlation coefficient $r\le 1$. All likelihoods are centered around our fiducial cosmology, i.e., the parameters describing the Buzzard simulations.

Figure 4

Forecast of posterior constraints on galaxy bias $b$ and galaxy-matter correlation coefficient $r$ achievable with density split lensing and counts-in-cells in DES Y1 data. The constraints are marginalized over ${\mathrm{\Omega }}_m$, ${\sigma }_8$, ${\mathrm{\Omega }}_b$, $n_s$, and $h_{100}$, where for the last three parameters we have assumed the same flat priors as used in the DES Y1 combined probes analysis presented in DES Collaboration [21].

Figure 5

The PDF of projected density contrast in Buzzard compared to several models for various smoothing scales (${\delta }_{m,T}$ is the projected density contrast ${\delta }_m$ smoothed by our top-hat radius ${\theta }_T$). In the upper panel of each plot the black line shows a histogram of ${\delta }_{m,T}$ measured in an all-sky map from Buzzard. The grey lines show histograms measured in 14 DES year1 shaped patches of that all-sky map. The blue lines show the PDF predicted by our tree-level computation of the cumulant generating function, the green lines show the PT-motivated log-normal model and the red lines show a Gaussian PDF with the same variance as the other two models. The bottom panels of each plot are showing the ratio of each PDF to the one measured in the Buzzard all-sky. For all aperture radii our halofit power spectrum is predicting a standard deviation of ${\delta }_{m,T}$ that is $\gtrsim 2\%$ higher than what we find in Buzzard (cf. Fig. 6). For ${\theta }_T=20^{\prime }$ and $30^{\prime }$ this is the dominant source of mismatch.

Figure 6

This figure shows the ratio of moments of ${\delta }_{m,T}$ measured in a Buzzard all-sky density map to the moments predicted by our model as a function of ${\theta }_T$. The error bars represent the standard deviations of the moments found in a set of 14 DES Y1-sized patches in the Buzzard map. At our fiducial radius ${\theta }_T=20^{\prime }$ we find a $\sim 2.4\%$ deviation between the variance of ${\delta }_{m,T}$ measured in Buzzard and that predicted by our model. This would correspond to a $\sim 1.2\%$ deviation in ${\sigma }_8$.

Figure 7

Average galaxy overdensity ${\delta }_{g,T}$ as a function of matter overdensity, ${\delta }_{m,T}$, for our simulated maps at different smoothing scales: 10 arcmin [left], 20 arcmin [middle], and 30 arcmin [right]. Solid lines show a linear bias model with the bias parameters obtained from maximizing the likelihood in Eq. (4.63) and the residual between the two are shown in the bottom panels. Note that the coefficient of linearity found with (4.63) ($\approx 1.54$) is almost identical to the value of the product $b·r$ determined with equation (4.54). To indicate the range of ${\delta }_{m,T}$ that is relevant to our computation, we also show the density PDFs of Fig. 5 as shaded regions. The error bars were derived from a jackknife approach.

Figure 8

Ratio of the variance in the galaxy count distribution to average galaxy counts in our simulated maps as a function of matter overdensity. Differently coloured solid lines show the result for each smoothing scale. The dashed black lines show the predictions of the 2-parametric shot-noise model described in Sec. 4c2. The blue dotted lines show the corresponding prediction of the alternative, 1-parametric model described in Sec. 4c1. The horizontal solid line shows the expectation if shot noise was purely Poissonian. The coloured regions show the 95% confidence limits derived from jackknife resampling.

Figure 9

For the fiducial smoothing radius of our data analysis presented in Gruen et al. [39], ${\theta }_T=20\mathrm{arcmin}$, we show the residuals between our baseline prediction of the counts-in-cells histogram and the average of measurements in 4 Buzzard realizations of DES Y1 data. Blue error bars represent the uncertainties we expect for DES Y1 while green error bars show the actual uncertainties of the mean measurement from our simulations.

Figure 10

To test our model for possible systematic deviations from N-body simulations, we try to recover the Buzzard cosmology in a simulated likelihood analysis. Top panel: $1\sigma$ and $2\sigma$ contours in the ${\mathrm{\Omega }}_m\text{−}{\sigma }_8$ plane from a likelihood computed around the mean of 4 shape noise free realisations of DES Y1 (but assuming the full covariance matrix for a single DES Y1). The green contours are marginalized over ${\mathrm{\Omega }}_b$, $n_s$, $h_{100}$, redMaGiC galaxy bias $b$, and galaxy-matter correlation coefficient $r$. For the parameters ${\mathrm{\Omega }}_b$, $n_s$, $h_{100}$ we have assumed the same flat priors as used in the DES Y1 combined probes analysis presented in DES Collaboration [21]. The red contours are marginalized only over $b$ and $r$ and the blue contours only vary ${\mathrm{\Omega }}_m$ and ${\sigma }_8$. Even when going to this small parameter space, our model agrees with Buzzard within $1\sigma$ errors of DES Y1. Bottom panel: Same contours but in the ${\mathrm{\Sigma }}_8-\mathrm{\Delta }{S}_3/S_3$ plane and varying one additional parameter, $\mathrm{\Delta }{S}_3/S_3$.

Figure 11

The PDF of projected density contrast ${\delta }_{m,T}$ in the Millennium Run (MR) compared to our model. In each plot, the black line shows a histogram of ${\delta }_{m,T}$ measured from 64 patches of $4\times 4{\mathrm{deg}}^2$ made from the MR by projecting the 3D density contrast with a constant selection function $q_l$ between $0.19\lesssim z\lesssim 0.43$, i.e. with a constant comoving density between those redshifts. The blue lines display the PDF predicted by our PT-motivated log-normal model, and the red lines show a Gaussian PDF with the same variance. The grey band is using the subsample covariance to estimate the error on the mean of all patches [89].

Figure 12

$1\sigma$ and $2\sigma$ contours in the ${\alpha }_0\text{−}{\alpha }_1$ plane from a likelihood computed around the mean of 4 shape noise free realisations of DES Y1 (but assuming the full covariance matrix for a single DES Y1). The blue contour only varies ${\alpha }_0$ and ${\alpha }_1$. The red contour marginalizes over ${\mathrm{\Omega }}_m$, ${\sigma }_8$ and galaxy bias $b$. The green contour additionally marginalizes over ${\mathrm{\Omega }}_b$, $n_s$, $h_{100}$, assuming the priors used by [21]. And the black contour also allows variation of the parameter $\mathrm{\Delta }{S}_3/S_3$. Dotted lines show the values of ${\alpha }_0$ and ${\alpha }_1$ that were found to describe our mock data best in Sec. 4c2.

Figure 13

In analogy to figure 10, we test whether our alternative shot-noise parametrization can recover the Buzzard cosmology in a simulated likelihood analysis. Top panel: $1\sigma$ and $2\sigma$ contours in the ${\mathrm{\Omega }}_m\text{−}{\sigma }_8$ plane from a likelihood computed around the mean of 4 shape noise free realisations of DES Y1 (but assuming the full covariance matrix for a single DES Y1). The green contours are marginalized over ${\mathrm{\Omega }}_b$, $n_s$, $h_{100}$, redMaGiC galaxy bias $b$ as well as the shot-noise parameters ${\alpha }_0$ and ${\alpha }_1$. For the parameters ${\mathrm{\Omega }}_b$, $n_s$, $h_{100}$ we have assumed the same flat priors as used in the DES Y1 combined probes analysis presented in Abbott et al. (in prep.). The red contours are marginalized only over bias and shot-noise parameters and the blue contours only vary ${\mathrm{\Omega }}_m$ and ${\sigma }_8$. Even when going to this small parameter space, our model agrees with Buzzard within $1\sigma$ errors of DES Y1. Bottom panel: Same contours but in the ${\mathrm{\Sigma }}_8-\mathrm{\Delta }{S}_3/S_3$ plane and varying one additional parameter, $\mathrm{\Delta }{S}_3/S_3$. Dotted lines show the true Buzzard cosmology and our fiducial value of $\mathrm{\Delta }{S}_3/S_3=0$.