Density split statistics: Cosmological constraints from counts and lensing in cells
in DES Y1 and SDSS
We derive cosmological constraints from the probability distribution function (PDF) of evolved large-scale matter density fluctuations. We do this by splitting lines of sight by density based on their count of tracer galaxies, and by measuring both gravitational shear around and counts-in-cells in overdense and underdense lines of sight, in Dark Energy Survey (DES) First Year and Sloan Digital Sky Survey (SDSS) data. Our analysis uses a perturbation theory model (Friedrich et al., 2017) and is validated using -body simulation realizations and log-normal mocks. It allows us to constrain cosmology, bias and stochasticity of galaxies w.r.t. matter density and, in addition, the skewness of the matter density field.
From a Bayesian model comparison, we find that the data weakly prefer a connection of galaxies and matter that is stochastic beyond Poisson fluctuations on arcmin angular smoothing scale. The two stochasticity models we fit yield DES constraints on the matter density and that are consistent with each other. These values also agree with the DES analysis of galaxy and shear two-point functions (3x2pt, DES Collaboration et al.) that only uses second moments of the PDF. Constraints on are model dependent ( and for the two stochasticity models), but consistent with each other and with the 3x2pt results if stochasticity is at the low end of the posterior range.
As an additional test of gravity, counts and lensing in cells allow to compare the skewness of the matter density PDF to its CDM prediction. We find no evidence of excess skewness in any model or data set, with better than 25 per cent relative precision in the skewness estimate from DES alone.
pacs:Valid PACS appear here
Measurements of the two-point correlation function of the evolved matter density field have provided competitive constraints on fundamental cosmological parameters. In combination with cosmic microwave background (CMB) and other geometric data, they are stringent tests of CDM predictions on the evolution of structure over cosmic time (Mandelbaum et al., 2013; Hildebrandt et al., 2017; van Uitert et al., 2017; Troxel et al., 2017; DES Collaboration et al., 2017). Ostensibly, larger studies of this kind are the primary goal of the upcoming ambitious ground-based and space-based surveys by Euclid, LSST and WFIRST.
On a given smoothing scale, one can describe a field locally by its PDF. An example of this is the PDF of fluctuations of the mean matter density inside spherical or cylindrical volumes. The variance, or second moment, of the PDF is measured by two-point statistics. For Gaussian distributions, this captures all the information in all the moments of the PDF. But when the field is non-Gaussian, the third moment (skewness) can take any value – therefore it contains information that is not contained in two-point statistics.
Unlike the primordial CMB, which is extremely close to a Gaussian random field, the density distribution in the evolved Universe has been driven away from Gaussianity by gravitational collapse. Third and higher order moments arise on any scale. Two-point measurements are therefore inherently very incomplete pictures of the matter density field. Even the full hierarchy of -point correlations ceases to fully describe its statistics (Coles & Jones, 1991; Carron, 2012; Carron & Neyrinck, 2012). This is unfortunate in two ways: (1) A lot of information on cosmology, and (2) a lot of opportunities to test additional, independent CMB predictions of the growth of structure beyond its variance, are lost by looking at two-point functions alone.
There are other reasons that make two-point correlations a somewhat blunt tool. Firstly, the information to be gained from galaxy auto- or cross-correlations must be related to the clustering of matter by a bias model, i.e. a description of how galaxies trace matter density. Yet the information on the bias model that is available from two-point functions alone is limited. The primary reason for the success of joint probes is that they can partially break these degeneracies. For instance, the joint analysis of galaxy clustering and galaxy-galaxy lensing can constrain two combinations of , galaxy bias, and galaxy stochasticity. As one pushes to smaller scales where a lot of the cosmological constraining power resides, a linear bias model without stochasticity is not sufficient. The resulting degeneracies thus largely annihilate the information that is gained. Secondly, the information on the variance of the matter density field can be used only if that variance can be modeled – on non-linear scales, complex physics that involve baryons and neutrinos begin to influence any moment of the matter density field. Using two-point functions, these complex physics can be constrained (Foreman et al., 2016; MacCrann et al., 2017b), although (for the same reason as for the bias model) only with limited discriminating power. If we could recover small scale information with models that can be trusted, the ability of presently and imminently available data sets to confirm or reduce tensions between the CMB and evolved power spectrum would immediately be boosted.
For these reasons, studies of the cosmic density PDF, which address these problems from a different and complementary direction, have gained interest over the last years. The full shape of the joint matter and galaxy density PDF depends on moments of the matter density field and parameters of the bias model that are degenerate in correlation function measurements. Numerical simulations (Takahashi et al., 2011; Pandey et al., 2013; Klypin et al., 2017), tree-level perturbation calculations (Bernardeau & Valageas, 2000; Valageas, 2002), and extensions of theory beyond that (Uhlemann et al., 2016), have been shown to provide accurate predictions for the matter density PDF. Parameter forecasts show that PDF measurements on data are promising (Codis et al., 2016; Liu et al., 2016; Patton et al., 2017) due to the complementary information, different degeneracies, and different dependence on observational systematics – factor-of-two improvements in constraining power can be achieved in joint measurements of PDF and two-point functions. While the galaxy count PDF alone can be used to break degeneracies of cosmological and bias parameters (Ross et al., 2008), gravitational lensing greatly complements this by measuring the actual matter density PDF. Practical application of shear PDF statistics to data has been made with DES (Clerkin et al., 2017; Chang et al., 2017), yet so far with limited use for quantitative constraints.
In this paper, we use the smoothed, joint, projected galaxy count and matter density PDF to constrain cosmological parameters and a galaxy bias model. Our basic concept is to (1) split the sky by the count of tracer galaxies in a top-hat aperture and extended redshift range into quantiles of density, and to (2) measure the gravitational shear around each of the quantiles to reconstruct the matter density PDF. These measurements are a generalization of trough lensing, introduced in Gruen et al. (2016) (see also Higuchi & Shirasaki (2016); Barreira et al. (2017); Cautun et al. (2017)). They are also closely related to the galaxy-matter aperture statistics of Simon et al. (2013). We make them on Dark Energy Survey Year 1 (DES Y1) and SDSS DR8 data.
The measurements are analyzed with a tree-level perturbation theory prediction for the joint statistical properties of lensing convergence and density contrast and galaxy bias models of varying complexity (cf. our companion paper Friedrich et al.). We use the analysis not just to provide an independent measurement of cosmological parameters, but also to confront the CDM prediction for the skewness of the matter density field with data, in a model independent test of structure formation. That is, we measure the asymmetry of the low and high density tails of the distribution of matter density in the Universe. Two-point statistics, which only measure the width of the matter density distribution, discard this information.
This paper is structured as follows. We describe the data we use and our measurement methodology in section II. Our modeling of these measurements, based on Friedrich et al. (2017), is summarized in section III. The covariance matrix we estimate is described in section IV. We combine measurements, model and covariance into an inference framework in section V. Results are presented in section VI, and we conclude in section VII. Several tests and technical aspects of this work are detailed in the appendix.
In the following section, we first describe our method of splitting lines of sight by density based on counts of a tracer galaxy sample (subsection II.1). The redMaGiC tracer catalogs we use to do this in DES and SDSS and their redshift distribution calibration are presented in subsubsection II.2.1 and subsubsection II.3.1. Details on our DES and SDSS source shape and photometric redshift catalogs are given in subsubsection II.2.2 and subsubsection II.3.2. The measurement of shear and counts-in-cells signals is described in subsection II.4.
ii.1 Splitting the sky by density
The basic idea of this study is to split the sky into lines of sight of different density.
To this end, we use a sample of foreground galaxies as tracers of the matter field (the redMaGiC galaxies at described in subsubsection II.2.1 and II.3.1). We count these galaxies within circular top-hat apertures with a range of radii , centered on a regular healpix (Górski et al., 2005) grid of (3.4 arcmin grid spacing).
We then assign each line of sight to one of five density quintiles by sorting all lines of sight by galaxy count. The 20 per-cent of lines of sight with the lowest galaxy count are what we will call the lowest density quintile 1 (or troughs, cf. Gruen et al. (2016)). The 20 per-cent of lines of sight with the highest galaxy count (quintile 5) we will denote as overdense lines of sight.
Compared to Gruen et al. (2016), we apply a more elaborate scheme of accounting for varying fractions of masked area within the respective survey region. A mask accompanying the redMaGiC (Elvin-Poole et al., 2017) catalog that we will use as our tracers (subsubsection II.2.1) indicate what fraction of the area inside each pixel in a healpix map is covered by DES Y1 Gold photometry (Drlica-Wagner et al., 2017) to sufficient depth for detecting redMaGiC galaxies out to at least . For each line of sight, we estimate the fraction of masked area within the corresponding top-hat aperture from the redMaGiC masks. Centers with more than a fraction of area within the aperture lost to masking are discarded.
The depth of SDSS is very uniform, with the redMaGiC sample being complete to everywhere. In this case, we use . Despite its greater overall depth, DES Y1 (Diehl et al., 2014) is generally more inhomogeneous than the final SDSS imaging data. Where the redMaGiC sample is not complete to , we remove all tracer galaxies and define the area to be fully masked. Due to the larger fraction of masked area, we use , above which we discard lines of sight.
To account for residual differences in we apply the following probabilistic scheme of quintile assignment. For each line of sight with masking fraction and raw tracer galaxy count , we define as a draw from a Binomial distribution with repetitions and success probability ,
This emulates the masking of a fixed fraction of area within each aperture. It preserves the expectation value of galaxy count in an aperture, regardless of its masking fraction. Under the assumption that galaxies or masked pixels do not cluster, and that galaxy count is not stochastic beyond Poissonian noise, it also preserves the full distribution of galaxy counts at fixed matter density (see Appendix A). The latter conditions are not true in practice, but tests of likelihood runs and the masked in the Buzzard simulations (see Appendix E and Friedrich et al. (2017)) indicate that this is not a major concern for our analysis.
We assign a line of sight to a density quintile based on many random realizations of . Different realizations of can cause different quintile assignments. To account for this, we define a weight , proportional to the number of times is in quintile . This weight is assigned to line of sight when measuring the signal, e.g. the mean tangential shear, of quintile .
Figure 1 shows the result of this quintile assignment procedure for the joint region covered by DES Y1 and SDSS.
ii.2 Dark Energy Survey Y1 data
The Dark Energy Survey data we use in this work is from the SPT region of the first year of science observations (Y1) performed between 31 August 2013 and 9 February 2014. Details of the data and photometric pipeline are described in Drlica-Wagner et al. (2017).
We use catalogs of luminous red galaxies (redMaGiC galaxies) as tracers of the foreground matter density field and galaxy shape and photometric redshift catalogs for measuring its gravitational shear signal, all of which are described briefly below and in detail in Elvin-Poole et al. (2017); Zuntz et al. (2017); Hoyle et al. (2017).
In all likelihood analyses run on data in this work, we propagate the three most relevant calibration uncertainties of these catalogs:
the multiplicative bias of the shear signal, characterized as ,
the bias in mean redshift of each source bin , characterized by a which we use to evaluate , where is the photometric estimate of the source redshift distribution, and
the bias in mean redshift of the tracer galaxy sample, characterized by a which we use to evaluate .
The derivation of priors on these calibration uncertainties is described or referenced in subsection V.2.
ii.2.1 Tracer catalog
The redMaGiC (Rozo et al., 2015) algorithm identifies a sample of red galaxies with constant comoving density and fixed luminosity threshold. This is done by fitting the DES photometry of each galaxy in the survey to find its maximum likelihood luminosity and redshift under the assumption of the redMaPPer (Rykoff et al., 2014) red sequence template. Galaxies are removed from the redMaGiC catalog if their fitted luminosity falls below a threshold ( for the high density run used in this work). The catalog is further pruned to retain a fixed number density of galaxies per comoving volume element, keeping those that are best fit (in terms of photometric ) by the red sequence template. The resulting galaxy density is in the case of the high density catalog.
This procedure was run on two different photometric measurements of DES Y1 galaxies, one with the SExtractor MAG_AUTO method and one performing a joint fit to the multi-epoch data of multiple overlapping objects (MOF). Potential correlation of the surface density of redMaGiC galaxies with observational systematics in DES Y1 have been extensively tested in Elvin-Poole et al. (2017) for both versions of the catalog. They found that in the redshift range used for the tracer galaxies, the MAG_AUTO version of the redMaGiC catalog shows smaller correlations with observational systematics.
We hence adopt MAG_AUTO redMaGiC with high density as our fiducial tracer catalog. In a trade-off of signal and noise, we choose as the tracer redshift range. We derive weights for the correction of redMaGiC density for the effect of systematics as in Elvin-Poole et al. (2017). We find significant correlations of redMaGiC density with band exposure time and seeing, and with band sky brightness. In the algorithm described in subsection II.1, we have applied these by dividing the fraction of good area in each pixel by the systematics weight that decorrelates redMaGiC density with these survey properties.
We do, however, test whether the choice of photometry pipeline (MAG_AUTO or MOF) and the choice of whether we apply the systematics weight in our density splitting procedure makes a difference to our analysis. These tests are detailed in Appendix C and show that the effect on the amplitude of our measured signals is negligible.
The redshift distribution of the tracer galaxy population, estimated by convolving the photometric redshift of each redMaGiC galaxy with its error estimate (Elvin-Poole et al., 2017), is shown as the grey contour in Figure 2. Note that due to scatter in photo- this extends beyond the redshift range inside which these galaxies were selected.
ii.2.2 Lensing source catalogs
Detailed descriptions and tests of the DES Y1 lensing source catalogs are presented in Zuntz et al. (2017), Troxel et al. (2017) and Prat et al. (2017), and the redshift distributions of source galaxies are estimated and calibrated in Hoyle et al. (2017); Davis et al. (2017a); Gatti et al. (2017). We only give a brief summary of the two independent shape catalogs from DES Y1 here.
The fiducial catalog with the larger number of source galaxies is based on the metacalibration method (Huff & Mandelbaum, 2017; Sheldon & Huff, 2017). In this scheme, a Gaussian, convolved with the individual exposure point-spread function, is fit jointly to all single-epoch , , and -band images of each galaxy. Galaxies are selected by the size and signal-to-noise ratio of the best fit, and the ellipticity of the Gaussian is used as an estimate of shear. Multiplicative biases in mean shear are caused by both the galaxy selection (selection bias) and the use of a maximum likelihood estimator with a simplified model (noise and model bias). In metacalibration, these are calibrated and removed using a repetition of the Gaussian fit on versions of the galaxy images that have been artificially sheared by a known amount.
As a second catalog, we use im3shape, which produces a maximum likelihood estimate of shear based on a bulge or disc fit to all DES Y1 band images of each galaxy. Multiplicative biases in these estimators are calibrated using realistic images simulations of DES Y1 (Zuntz et al., 2017; Samuroff et al., 2017).
Our estimator of tangential shear around overdense and underdense lines of sight, including the bias corrections, is described in subsubsection II.4.1. We use the galaxy selection criteria recommended in Zuntz et al. (2017). We split galaxies into redshift bins using the mean of the individual galaxy as estimated by BPZ (Benítez, 2000; Hoyle et al., 2017). The three source redshift bins we use are identical to the three highest redshift bins of Hoyle et al. (2017), i.e. with sources at mean . Their redshift distributions, as estimated by BPZ, are shown in Figure 2.
ii.3 SDSS DR8 data
ii.3.1 redMaGiC tracer catalog
The tracer population in SDSS is the redMaGiC (Rozo et al., 2015) high density sample, selected by SDSS photometry and cut to the same redshift range . Despite this similarity, we will not assume in this work that SDSS and DES redMaGiC galaxies are the exact same populations.
SDSS has the benefit of an overlapping sample of galaxies with spectroscopic redshifts. We use this to calibrate the mean of the redshift distribution with clustering redshifts, independent of the photometric estimate, in subsection F.1. We find no significant bias, yet marginalize over the uncertainty in the analyses presented herein (see subsection V.2).
ii.3.2 Lensing source catalogs
We use the shape and photometric redshift catalog of Sheldon et al. (2009) with minor modifications, identical to those in Clampitt & Jain (2015). We refer to these papers for details, but describe our source selection and priors on systematic uncertainties of shears and photometric redshifts below.
Due to the lower observational depth, the SDSS shape catalog peaks at much lower redshift than the one from DES Y1. The source redshift dependence of the trough lensing signal (cf. Figure 2) and complications arising from significant overlap of sources with the tracer redshift range lead us to only use sources with a mean redshift estimate of . We split these into four bins of and . Within each of these bins, each individual source is assigned a minimum-variance relative weight (cf. Clampitt & Jain (2015)) of
Due to the moderate signal-to-noise ratio and redshift range of sources, we combine the four source redshift bins into one for the purpose of our final data vector. In this, we apply an optimal relative weighting of the bins as follows.
The predicted amplitude of shear around our troughs at (see black line in Figure 2) scales with source redshift approximately as the amplitude of gravitational shear (see Equation II.4) due to a lens at . We use the value of estimated for and the stacked of each of these four bins to apply a relative weight of and to each of them. Because the number density of sources is steeply falling with source redshift in this range, the effective total relative weights of the four bins (equal to this times the sum of all source ) are and . We use these effective weights to combine the measured shear signal and the from each of the four bins into a single source sample.
As a calibration of the photometric estimate, the mean redshift of the sources is constrained by their angular cross-correlation with galaxies with known spectroscopic redshift (subsection F.2).
ii.4 Measured signals
Our data vector in this work contains two components, the modeling of which was extensively tested in Friedrich et al. (2017). In subsubsection II.4.1, we describe the measurement of gravitational shear signals around overdense and underdense lines of sight. Section II.4.2 details our measurement of mean counts-in-cells in each density quintile in the presence of masking.
All measurements are made in jackknife resamplings of the survey. The covariance model constructed in section IV can therefore be compared to an jackknife covariance. These were made based on 100 and 200 patches in the DES and SDSS footprint, respectively, defined by -means clustering111https://github.com/esheldon/kmeans_radec of the tracer galaxies.
The ellipticity of a galaxy is a pseudo-vector with two components, and that, for any lens position, can equivalently be described by a component tangential to a circle around a lens () and by a component rotated by relative to that ().
Gravitational shear due to any single lens only affects the mean component of for an ensemble of sources sampling a full annulus around the lens. As a function of angular separation from the lens, this effect is described by the tangential shear profile . For a single lens, the tangential shear profile is directly related to the azimuthally averaged, projected surface mass density of the lens, i.e. the projected mass per physical area, as
where, in a flat universe,
is the inverse of the critical surface mass density and is the co-moving distances to the deflector at redshift and the lensed source, respectively. For a set of lenses along the line of sight, the signal on any source is close to the sum of the effects of all lenses. One can still define a convergence related to mean gravitational shear as in Equation II.3, although it is no longer relatable to a uniformly weighted surface mass density (see Equation III.7).
The relation between tangential shear and measured tangential ellipticity is less straightforward and depends on the implementation of the selection and measurement of source ellipticities. For the two schemes used on DES Y1, the responsivity of observed ellipticity to applied tangential shear is calibrated very differently: for metacalibration, it is estimated from versions of the actual galaxy images sheared with image manipulation algorithms; for im3shape, it is estimated from realistic simulations of DES imaging data (and usually defined as ). Both types of calibration contain an explicit or implicit correction for selection biases, i.e. the shear dependence of the choice of whether to include a galaxy in the source sample.
For metacalibration, we define the estimator of mean tangential shear around lines of sight with probability to be in a given density quintile as
where is the ellipticity of source in the tangential direction around line of sight , the sums run over all lines of sight in the mask of the density-split sky and all sources in an angular bin around each line of sight, and the second term subtracts shear around random lines of sight. is the sum of shear and selection responsivity,
where superscripts on indicate an ellipticity measured on an image artificially sheared by in the same component and superscripts on indicate an average taken on an ensemble of source selected by quantities measured on an image artificially sheared by .
We note that this is identical to the methodology for DES Y1 galaxy-galaxy lensing employed in Prat et al. (2017), except that we estimate the responsivity separately for the source galaxies in each radial bin around the cluster, rather than as a global scalar. The scale dependence, however, is negligible – the metacalibration R is equal to within 0.5 per-cent for any two angular bins.
For im3shape, we use the source weights defined in Zuntz et al. (2017) to first measure the weighted mean of the source sample, then define the estimator for tangential shear as
The im3shape are defined with the calibration correction for additive bias already applied.
The discriminating power of density split lensing signals for cosmological parameters and parameters describing the connection of galaxies and matter is greatly improved by adding some degree of information of galaxy clustering or bias. Here, we use a very basic statistic, the mean tracer galaxy overdensity in our density quintiles, that was extensively tested in Friedrich et al. (2017) – other signals could significantly improve the constraining power in the future.
Operationally, we define the mean tracer galaxy overdensity in all quintiles as follows. We convert the raw tracer galaxy count within the aperture radius around each line of sight to a stochastically masked count with fixed masking fraction by a Bernoulli draw (subsection II.1 and Appendix A). We then order lines of sight by and take the mean of in each quintile of that list as . The mean tracer galaxy overdensity in quintile is
where the average in the denominator runs over all lines of sight.
We note that this does account for the fact, in a stochastic fashion, that a given line of sight can end up in different density quintiles depending on the realization on masking that decides the galaxy count.
Figure 5 shows the full distribution in both DES and SDSS, alongside a model evaluated at the maximum likelihood parameter values fit to the shear signal and mean tracer galaxy overdensity in quintiles. The model not only fits these mean overdensities, but also the full extremely well: absolute differences in probabilities of finding galaxies in a random line of sight, , are below and for any in DES and SDSS, respectively. The bias model used for the plot is one with two-parametric stochasticity (called in subsection III.3), although even a simpler model can reproduce the well.
In order to describe our signal as a function of
parameters that connect galaxy counts to the matter (over)density, and
we use the model developed and tested in Friedrich et al. (2017). We only briefly summarize it here, with an emphasis on required extensions for the use on observational data, and refer the reader to that paper for details.
Let be a unit vector on the sky. The signal we have to predict in this work is the shear profile around lines of sight that fall into a certain quintile of foreground tracer density. Also, our data vector includes the average tracer density contrast in each of those density quintiles. To model these two parts of our data vector, we have to consider the following fields on the sky:
: the line-of-sight density contrast underlying our tracer galaxies. Given the redshift distribution of our tracer sample, this is given by
where is co-moving distance and the projection kernel is given in terms of as
: the result of smoothing the field with a circular top-hat aperture .
: the number of tracer galaxies in the aperture around the line-of-sight
: the convergence inside an angular radius around the line-of-sight .
Because the Universe is isotropic, we will omit the dependence on , i.e. only consider a single line of sight.
As detailed in Friedrich et al. (2017), the density split lensing signal can be calculated from the convergence profile around lines of sight with a fixed value of . This profile can be computed as
where Bayes’ theorem can be used to express the PDF of at fixed as
Here is the overall PDF of , is the probability of finding in a line-of-sight with fixed and
From the convergence profile the corresponding shear profile can be computed as (cf. Friedrich et al.)
and the shear profile around a certain quintile of tracer density is given by the average of over the values occurring in that quintile (cf. Friedrich et al. for further details). Fundamentally, we therefore have to model
, the PDF of matter density smoothed inside our aperture,
, the expectation value of convergence inside an angular radius around a line-of-sight with given density contrast inside our aperture, and
, the probability of finding galaxies in an aperture, given its density contrast is .
We describe our approaches on each of these ingredients in the following subsections, and close with a description of how we account for biases in source redshift and shear estimates, and overlap of the source redshift distribution with the tracer redshift distribution.
In all these steps, in order to predict the non-linear 3D matter power spectrum, we use the Takahashi et al. (2012) halofit approximation with the Eisenstein & Hu (1998) transfer function with baryonic features, which is sufficiently accurate given our large scale binning.
iii.1 PDF of matter density contrast
The PDF can be computed from its cumulant generating function (CGF). This function can be derived at tree-level in perturbation theory with the help of the the cylindrical collapse model (Friedrich et al. (2017), see also pioneering work on the computation of the CGF in Bernardeau (1994); Bernardeau & Valageas (2000); Valageas (2002)).
The computations are numerically involved and, at least in our implementation, too slow for application in a likelihood analysis. We however show in Friedrich et al. that, on the scales used in this work, the perturbation theory computation of is well approximated by a log-normal distribution that matches the second and third moments and of the perturbation theory approach. We use this log-normal model (section 4.1.1 in Friedrich et al., 2017) for the smoothed, projected matter density field in this work.
iii.2 Mean convergence around apertures with fixed density contrast
We now turn to the convergence field , defined as
where the lensing efficiency is given by
is the line-of-sight density of the sources. As before, denote by the result of smoothing the convergence field over circles of angular radius .
As described in Friedrich et al. (2017) (see also Bernardeau & Valageas, 2000, and references therein), the expectation value of around lines of sight with fixed values of is mostly determined by the moments
as well as the mixed moments
In a similar way as for the projected density PDF, a full tree-level computation of can be replaced by a log-normal approximation that involves the above moments (cf. Friedrich et al. for details of this). We want to stress, that this does not mean that we employ a log-normal approximation for the joint PDF of and . E.g. Xavier et al. (2016) have shown that such an approximation can be inaccurate if the lensing kernel and the line-of-sight distribution of tracers have strongly different widths in co-moving distance. Rather, we model the convergence field as a sum of two fields, one of which is a log-normal random field and one of which is Gaussian and uncorrelated to . Also, unlike for a joint log-normal distribution, we allow the log-normal parameter of , i.e. the minimum allowed value of , to depend on the scale . In Friedrich et al. (2017) we have shown that this indeed gives a good approximation to the joint statistical properties of convergence and density contrast.
iii.3 Probability of galaxy counts in apertures with fixed density contrast
Finally, we need to model the probability of finding galaxies inside an aperture given the matter density contrast . As defined in Friedrich et al. (2017), we consider three models of increasing complexity. All of them assume bias to be linear, i.e. the mean count of galaxies to be proportional to the overdensity of matter in the large aperture volumes we consider. They differ, however, in their parametrization of stochasticity (Dekel & Lahav, 1999). We note that the latter may arise arise from non-linear biasing on scales smaller than our apertures or from truly non-Poissonian noise in galaxy density at fixed matter density that is present in subhalo distributions (Boylan-Kolchin et al., 2010; Mao et al., 2015; Jiang & van den Bosch, 2017).
In all equations below, denotes the mean count of tracer galaxies inside apertures after masking a fraction of area, and the generalized Poisson distribution that is also defined for non-integer arguments is
with the Gamma function .
Our three models are:
bias only: model – as in Gruen et al. (2016), one could assume to be a Poisson distribution of a non-stochastic tracer population with bias ,
bias and stochasticity: model – in this case, the galaxy count is assumed to be distributed as
where is an auxiliary galaxy density field with
The auxiliary field is correlated with the smoothed matter density field with a correlation coefficient . Setting reduces this to the model with no stochasticity.
bias and density dependent non-Poissonianity: model – because it introduces independent scatter, stochasticity with boosts the shot noise in galaxy count at fixed matter density; yet a dependence of this super-Poissonianity on matter density that may be present in the data need not be fully described by the model; to account for this, we use a more general model defined in Friedrich et al. (2017). Here,
We note that this model can be related to the halo count and occupation distributions (Fry et al., 2011). Our ansatz can be thought of as a model of Poisson-distributed haloes with redMaGiC galaxies in each one of them, similar to e.g. the relation of Poissonian photon and non-Poissonian electron shot noise in CCD detectors, described by a gain factor . It could similarly accommodate non-Poissonianity in halo counts (Neyrinck et al., 2014). We allow for to be different in higher and lower density regions, e.g. because more massive haloes might be more common in the former, by means of a linear dependence of on as
We note that a bias model without stochasticity is a common assumption made for the galaxy distribution on large scales (e.g. DES Collaboration et al., 2017). MacCrann et al. (2017a) show that in the Buzzard simulations, large scale stochasticity is present. From the combination of probes with different sensitivity to and , such as the three galaxy and convergence auto- and cross-correlation functions, the two parameters could be disentangled. Density split statistics, in addition, are sensitive to differences in higher moments of the galaxy and matter density field, and can test and, potentially, constrain, more complex models such as .
iii.4 Nuisance effects on data
In all runs on data, biases in the means of redshift distributions in DES and SDSS are accounted for at the level of the model: we marginalize over lens redshift and (multiple, in the case of a tomographic analysis) source redshift bias parameters by shifting the tracer galaxy and source galaxy redshift distributions accordingly before computing predictions for the signals. Likewise, we scale the predicted shear signal by to account for multiplicative shear biases .
A more complex issue arises from the clustering of sources with the overdense and anti-correlation of sources with the underdense lines of sight. This is a common problem in cluster lensing or galaxy-galaxy lensing, accounted for by so-called boost factors (Sheldon et al., 2004; Mandelbaum et al., 2005).
In the case of density split lensing, we apply the assumption of linear bias to predict the radius dependence of boost factors and their effect, given the non-thin lenses, on our model predictions. For a given tracer redshift distribution and a the matter field at redshift , the angular clustering of quintile with matter can be calculated with the same formalism as the convergence in subsection III.2. Assuming a linear bias of source galaxies , their redshift distribution at separation from quintile changes due to clustering to
The lowest redshift bin in DES Y1 or the Buzzard simulations and the sources in SDSS have sufficiently strong overlap with the lens redshift distribution that we include this effect in the modeling and marginalize over in the analysis. This means that we use a different source redshift distribution for predicting each point of the density split, radially binned shear signal data vector. While is in reality a function of , one can very accurately describe the de-boosting of the lensing signals by an effective because the radial profile shape of the shear signal is almost independent of source redshift.
We note that in this derivation we neglect a second, but likely subdominant effect: the source redshift dependence of the probability of failing to include a source in the DES shape catalogs due to blending, that might cause a similar density dependence of source .
A potential spurious signal is due to intrinsic alignment of physical source galaxy shapes with the underdense or overdense lines of sight due to gravitational interactions (see Troxel & Ishak (2015); Joachimi et al. (2015) for a review). For cross-correlations between the positions of object and gravitational shear, such as counts and lensing in cells, intrinsic alignments affect only the signal from source galaxies that are physically associated with the lensing objects, i.e. if redshift distributions of source galaxies and lensing objects overlap. This is the case primarily in the lowest redshift bin for DES Y1. Hence test (4) in Sect. V.4, which demonstrates the robustness of the results to removing the lowest source redshift bin from the data vector, indicates that the current analysis is at most weakly affected by intrinsic alignments. This is in agreement with our expectation that the tidal alignment of galaxies with the comparatively small mean over- and underdensities of our density quintiles is small at separation.
On small scales, baryonic effects can modify the matter power spectrum from its dark matter only prediction, primarily by affecting overdense regions. For our statistic, this could be absorbed by the bias model on scales smaller than the top-hat aperture . The shear signal is used on scales larger than only, and parameter constraints are robust to a more conservative scale cut of (subsection V.4). We hence do not expect a significant impact of baryonic effects on the parameter constraints at the accuracy level of the current analysis, but note that these effects require further study for future, more constraining analyses.
In order to interpret our measurements, we need an accurate description of their covariance. We construct this covariance from a large number of mock realizations of our data vectors. In that, we make use of the fact that the noise in our measurements can be separated into two components: a contribution from shape noise and a contribution from large scale structure and shot noise in the galaxy catalog. This approach is similar to the one of Murata et al. (2017).
In the following, we describe how we measure these contributions, and how we combine them into a covariance matrix.
We assume in all following analyses that the signals measured in SDSS and DES Y1 are uncorrelated, justified by the fact that the survey footprints (using only the contiguous SPT region of DES) are well separated.
iv.1 Shape noise
The primary contribution to the shape we measure for any individual galaxy in our survey is the sum of its intrinsic shape and measurement noise, not the weak gravitational shear that distorts the galaxy image.
Because of this dominance of the noise over the signal, and because the intrinsic shapes of neighboring galaxies are almost uncorrelated, we can measure shape noise by rotating each galaxy in our shape catalog by an independent random angle. The shear signal around our actual underdense and overdense lines of sight as measured from these rotated source catalogs represents a random realization of the shape noise (cf., e.g., Kacprzak et al. (2016), for a similar technique for shear peak statistics, Sánchez et al. (2017) for void lensing, and Murata et al. (2017) for cluster lensing).
In measuring the signals on the rotated catalogs, we take care to use the same methodology as for the measurements on data. That means we use each randomly rotated source catalog for cross-correlation with all of the maps contributing to our data vector. As on the data, we subtract the mean shears measured around random points, for which we simply use the centers of all healpix pixels that are used as lines of sight in any density quintile. The subtraction of shear around random points considerably reduces shape noise on large scales (see Figure 6, and refer to Singh et al. (2017) for a detailed study of the effect).
iv.2 Cosmic variance and shot noise
Two additional effects cause our signal to deviate from its expectation value:
The cosmic density field present in our survey volume is a random realization. This is true both for the volume in which our tracer galaxies are located (and in which the signal of troughs and overdense lines of sight originates) and for the redshift range along the line of sight in between us and the source galaxies that is not contained in that volume. This causes there to be noise in the true convergence around the lines of sight we identify, and in the counts-in-cells distribution.
In a given realization of the matter density field, tracer galaxies could be placed differently (for instance, according to Poisson noise around their expectation value in any given volume). Which of these possible galaxy catalogs is realized causes there to be a different true shear signal around what we identify as troughs and overdense lines of sight, and a different counts-in-cells distribution.
On the scales we care about in this work, we can measure the sum of both contributions to the covariance, to good approximation, from log-normal simulations of the related matter and convergence fields and Poissonian realizations of the tracer galaxy catalog. We do this by generating a large number of realizations of these fields and catalogs with flask (Xavier et al., 2016).
We note that this part of the covariance is dependent on cosmology and the parameters describing the connection of galaxies and matter. For the covariance in this work, we will assume the settings of the Buzzard simulations, namely a fiducial flat CDM cosmology with , , , and km s with .
For the matter and associated galaxy field in the tracer redshift range, we use the power spectrum with a linear bias of , a redshift distribution, and a mean density of the tracer galaxy population as in the Buzzard-v1.1 suite of simulations. We assume Poissonianity of the galaxy count at fixed density, i.e. the model (subsection III.3). Note that the relation of galaxies and matter in the Buzzard simulations (MacCrann et al., 2017a) and, potentially, the actual Universe is more complex than that. We ensure, using mock likelihood runs on Buzzard, that this does not mean our covariance from the log-normal mocks is significantly underestimated (see Appendix E). To set the log-normal parameter of the projected matter field (i.e., the minimum allowed value of ), we use the methodology of Friedrich et al. (2017). In the Buzzard cosmology and at a top-hat smoothing scale of 20 arcmin, this yields . Details are given in Appendix B.
For the source redshift distributions of the simulated convergence fields, we use those estimated for the source samples in our data. We separate the convergence field into two parts: a component correlated with the matter field that our tracer galaxies populate, and an uncorrelated component in front and behind of that volume along the line of sight. The correlated component is modeled as a log-normal field with auto- and cross-power spectrum and set to match perturbation theory predictions of , , and (Friedrich et al. (2017) and Appendix B). For the three source redshift bins of the DES data vector, this leads to ; for SDSS, the source is 0.0094.
We apply the same mask to the tracer galaxies as in our data (or in our simulations, for the mock analysis described in Appendix E), and the same prescription for splitting the survey into lines of sight of different density.
We then measure tangential shear signals, as in our data, yet on the noiseless shear maps output by flask at resolution. In order to sample the density fields as in our data, we use the sum of weights of sources in our actual shear catalogs situated in a pixel as the weight of the shear signal in that pixel. We do this both for the correlated and the uncorrelated part of the convergence field (see above) and co-add the two signals. In addition, as in our data, we measure the mean tracer galaxy overdensity in our density quintiles.
On scales much smaller than the aperture radius , a checkerboard pattern in the off-diagonal shape noise covariance is apparent (see Figure 7). We find that this is due to an interference of the healpix grid we use to sample the density field and the angular binning scheme for our shear signal – for adjacent healpix pixels, sources move from one angular bin to the next and their intrinsic shape orientation changes relative to the pixel centers. Since these effects are present in the data as well (as seen from the jackknife covariance) and only significant on angular scales below our scale cut, we do not attempt to address them further.
iv.3 Constructing the covariance matrix
We create 1000 realizations of both the shape noise and the large scale structure and shot noise contributions to the covariance. Despite this relatively large number, there is noise in our estimated covariance matrix. When inverting the covariance matrix to calculate values and run a likelihood analysis, this noise has two consequences.
First, the inverse of a noisy estimate of the covariance matrix is a biased estimate of the inverse covariance matrix. We follow the correction described in Hartlap et al. (2007) to correct for this effect, i.e. we multiply the calculated with the inverse of our estimated covariance matrix by a factor
The number of entries in our data vector is at most in the fiducial DES analysis and we use realizations of the log-normal field to estimate the covariance, which means is 0.78 or larger for all our likelihood runs. We confirm, using independent log-normal mock realizations of our data vector, that the inverse covariance matrix re-scaled such does lead to a consistent distribution of residuals (Appendix D).
Second, the noise in the inverse variance leads to additional scatter in the best fit we find (Dodelson & Schneider, 2013; Sellentin & Heavens, 2016; Friedrich & Eifler, 2017). Under the assumption that the model is linear in all parameters within the range probed, this can be compensated by multiplying with a factor
where is the number of free parameters in the model.
These corrections are only appropriate for a monolithic covariance estimated from a fixed number of independent realizations. From the previous subsections, we can get independent, unbiased estimates of the two contributions, and . The sum of would be an unbiased and less noisy estimate of the total covariance. But to apply the above corrections, we need to resort to co-adding shape noise and cosmic variance realizations before estimating the full covariance matrix.
In addition to the 960 realizations used to estimate the covariance, we use 40 independent realizations to confirm that our prediction indeed matches the mean signal measured from the log-normal simulations at the expected . This is a test of both the numerical scheme employed by FLASK and the implementation of the analytical calculations of Friedrich et al. (2017). We find that the two are in excellent agreement, except for a small offset of the predicted and measured counts-in-cells statistic. The mean tracer galaxy overdensities (subsubsection II.4.2) we measure in log-normal mocks are offset from the predictions at most at the level. We hypothesize that this is due to resolution effects of the simulations, but cannot exclude that similar effects could also present in the data.222We confirm, however, that the mean tracer galaxy overdensities in our data are well fit by the model at its maximum likelihood parameters. To compensate for this, we boost the variance of each of the four counts-in-cells entries in our data vector by . Using this covariance and (but not ) as defined above, the mean of all realizations with no shape noise matches the predicted signal at the true input parameters at total with 208 degrees of freedom, proving the numerical accuracy of the prediction code at a sufficient level. Additional tests of our likelihood pipeline run on the 40 log-normal realizations are shown in Appendix D.
iv.4 Comparison with jackknife covariances
From jackknife resamplings of our data, we can internally estimate the covariance matrix. While more care would have to be taken for applying this estimate of the covariance matrix in a likelihood analysis (Friedrich et al., 2016), it does provide confirmation of our scheme to compare the jackknife estimate to the covariance estimated above.
Figure 7 shows the shape noise and cosmic variance + shot noise components of the covariance matrix and compares their sum to the jackknife covariance, for the shear signal around underdense lines of sight in SDSS. The same for the full density and source-redshift tomographic covariance matrix of DES, including counts-in-cells, is displayed in Figure 8.
We compare our data to model predictions in a Bayesian fashion, i.e. we sample the posterior distribution of model parameters with a Monte Carlo Markov Chain (MCMC) run on the likelihood
For our fiducial likelihood analysis, we remove the following parts of the full data vector:
lensing and counts-in-cells signal for any aperture radii other than – on smaller smoothing scales, small but significant deviations of our model and measurements in -body simulations appear (Friedrich et al., 2017). Smoothing on larger scales than 20’ yields signals with errors that are highly correlated to the 20’ measurements, thus adding little independent information.
lensing signal on scales smaller than – small but significant deviations of our model and measurements in -body simulations are present on scales smaller than the aperture radius . The lensing signal on these scales has low signal-to-noise ratio. In addition, shape noise in adjacent small-scale bins is anticorrelated, visible as the checkerboard pattern in the lower left of Figure 7. This is due to interference of the radial binning scheme with the healpix grid of lines of sight: when we measure the contribution of a source galaxy to the shear signal around two adjacent lines of sight, its intrinsic orientation relative to a line of sight and its distance from the line of sight change coherently. While the effect is consistently seen in jackknife and model covariance, it makes these small scale lensing signals numerically redundant. This leaves 17 angular bins in each shear profile.
signal for quintile 3 – the signals we use are not linearly independent between all quintiles; we therefore discard the signal in the median quintile, which is close to zero by construction anyway.
Therefore, in all of the following, unless otherwise noted, contains the shear signals measured at and the relative overdensity of tracer galaxy count for the lower two and upper two quintiles of galaxy count, measured in apertures. For the source tomographic DES Y1 analysis, these are 208 entries (72 for SDSS).
The precision matrix is estimated as detailed in section IV.
In the following subsections, we describe our choice of parametrization, the nuisance parameters and associated priors, and the consistency tests we perform before unblinding the estimated cosmological parameters.
v.1 Cosmological parameters
Since this is our first cosmological analysis of counts and lensing in cells, we choose to only vary a minimal set of cosmological parameters, adopting fixed priors for ones that the density split lensing and counts signal is not very sensitive to. For the fiducial run of our likelihood, we validate this approach by marginalizing over these parameters with informative external priors.
All likelihoods assume a flat CDM cosmology. The main parameters we wish to constrain are the matter density in units of the critical density , and the amplitude of structure in the present day Universe, parametrized as the RMS of overdensity fluctuations on Mpc scale, .
In an alternate run of our likelihoods, we will also leave free the parameter that describes the skewness of the matter density field when smoothed over the given aperture and redshift range,
was first defined by Peebles (1980), who derived a perturbation theory prediction for the unsmoothed matter density field, and later generalized to top-hat smoothed fields and higher orders (Fry, 1984; Juszkiewicz et al., 1993; Bernardeau, 1994). Perturbation theory predicts also the smoothed to be almost independent of and and to only vary slowly with redshift or scale. A skewness that is inconsistent with these predictions could be caused either by non-Gaussian initial density fluctuations (although CMB limits set tight constraints on these Planck Collaboration et al. (2016)) or by physics beyond gravity that affect collapse either in the overdense or underdense regime.
We assume wide, flat priors for , and, in the likelihood runs that vary it, that do not limit the range sampled by the likelihoods. We fix the Baryon density , the spectral index of primordial density fluctuations , and a dimensionless Hubble parameter , equal to the values used in the Buzzard simulations and consistent with best constraints. For the transfer function of primordial to initial matter power spectrum, we assume a radiation density . The evolution of expansion and growth of structure in the late universe assumes only matter and a cosmological constant.
An overview of these choices is given in Table 1.
v.2 Nuisance parameters
In our likelihoods, we apply three different models to describes the distribution of redMaGiC galaxy count inside an aperture at given mean matter overdensity , . Details of this are described in subsection III.3, and sampling ranges for the parameters , , or , designed to span any physically sensible configurations, are listed in Table 1.
Similar to previous cosmological constraints derived from DES Y1 data, we assume and always marginalize over nuisance parameters describing photometric redshift and shear biases in our measurements. The nuisance parameter for the redshift bias of redMaGiC sources in the redshift range that is constrained from cross-correlations with a sample of galaxies with spectroscopic redshifts as in Cawthon et al. (2017). Specifics of this are described in Appendix F.1.
The photometric redshift biases and multiplicative shear biases of source galaxies are described by two parameters in each redshift bin. Priors on the redshift biases are taken from the combination of the redshift distributions of a matched sample of galaxies in the COSMOS survey and angular cross-correlation with redMaGiC galaxies (Davis et al., 2017b; Davis et al., 2017a; Gatti et al., 2017) as described in detail in Hoyle et al. (2017). The priors on multiplicative shear bias in DES Y1 are described in detail in Zuntz et al. (2017). Both of these priors are widened in our analysis to account for their potential correlation between bins (see appendices of Hoyle et al., 2017; Zuntz et al., 2017), conservatively assuming comparable signal-to-noise ratio in each bin.
Multiplicative bias in an independent SDSS shear catalog that is consistent with the one we use (Simet et al., 2017) was investigated in detail in Mandelbaum et al. (2013). We assume a Gaussian uncertainty of for the multiplicative shear bias in the SDSS catalog in this work.
These priors are also summarized in Table 1.
|flat (0.1, 0.9)|
|flat (0.2, 1.6)|
|fixed to PT / flat|
|flat (0.8, 2.5)|
|flat (0, 1)|
|flat (0.1, 3.0)|
|flat (-1.0, 4.0)|
|Tracer galaxy photo- shift|
|Source photo- shift|
v.3 Sampling and Evidence
To sample the posterior likelihoods efficiently, we employ both the emcee (Foreman-Mackey et al., 2013) and the MultiNest (Feroz et al., 2009) algorithm. The latter has the advantage of also estimating Bayesian evidences ,
where are the parameters of the model.
Knowing the evidence of two models and allows comparing them with the Bayes factor, . If the latter ratio exceeds 3.2 or 10, the evidence for model 1 over model 2 can be considered substantial or strong in the nomenclature of Jeffreys (1961).
v.4 Blinding and tests
Since most of the tests in this paper were performed after the scaling factors of the initial, blinded shear catalogs had been revealed (Zuntz et al., 2017), we primarily rely on parameter level blinding. This means that we do not compare measurements on data to predictions in a known cosmology before the following tests are passed:
Log-normal simulations show values of data vs. model at the input set of parameters consistent with a fiducial distribution with the appropriate number of d.o.f.. Likelihood runs on mock data have a coverage within expectations (i.e., the input cosmology lies within the confidence interval the expected fraction of times). Results are unremarkable and described in Appendix D.
Bayesian model comparisons run on log-normal simulations without stochasticity do not provide evidence for more complex models. We find that this requirement is met, both for the model of stochasticity and models with free skewness , in Appendix D.
21 independent Buzzard -body realizations of our data vector give consistent relative to the model evaluated at the input cosmology and independently measured nuisance parameters. Their coverage in likelihood runs, i.e. the number of times the input cosmology is within derived confidence limits, is within expectations only for the model of bias and stochasticity (Appendix E). The fact that the most complex bias model is required may be particular to these mock galaxy catalogs, which may have different relations to matter density than real redMaGiC galaxies. We still take this as evidence that the most general stochasticity model, unless disfavored by the data, needs to be considered in our analysis.
Likelihood runs on -body realizations are insensitive to replacing true source redshift distributions with source redshift distributions estimated from BPZ and marginalizing over uncertainties. Results: we find that the mean shifts in cosmological parameters are at or below the ten per cent level of their statistical uncertainty, and that the statistical uncertainty increases by less than five per cent due to marginalization over , both tested with the model for the galaxy-matter connection.
Once these tests are successful, we continue to make tests on likelihood analyses run on the data itself. To ensure that these do not introduce experimenter bias, before looking at any chains we shift all cosmological parameters by a constant unknown vector, uniformly distributed between and standard deviations of the parameters as found from -body simulations. We then proceed with the following tests, the results of which are shown in Figure 9:
Cosmological constraints from the data are consistent between the fiducial metacalibration and additional im3shape measurements. For models including galaxy stochasticity (lower two panels of Figure 9) this is the case to a fraction of the statistical uncertainty. For the only model (top panel), im3shape constraints on are offset by . Accounting for the fact that shape noise is largely uncorrelated between the two catalogs, is is possible that this is simply a statistical fluctuation. We note, however, that DES Collaboration et al. (2017) found a similar discrepancy, likely attributed to the multiplicative bias or source redshift calibration of the im3shape highest source redshift bin.
Cosmological constraints are robust to removing the lowest or highest source redshift bin from the data vector. This is the case for all models, indicating that the calibration of metacalibration catalogs is consistent between bins.
Cosmological constraints are robust to replacing the source redshift distributions estimated by BPZ by ones directly estimated from COSMOS. Again, this is the case for all models, with a noticeable but insignificant offset in the only model.
Cosmological constraints are robust to cutting scales smaller than or larger than ’ from the shear signal. Removal of large scale shear information shifts in the model by approximately . Given the cosmic variance in large-scale modes and the unremarkable result of all over variants of the scale cut test, this does not pose a significant issue.
Cosmological constraints are robust to not correcting for clustering of the overlapping source redshift bins with the matter distributions around overdense and underdense lines of sight. While this is not necessary a null test – it could be possible that we need to account for the effect – we find that marginalizing over neither significantly widens nor shifts the contours in either model.
Cosmological constraints are consistent between DES and SDSS. We note that these are completely independent data sets, i.e. have no cross-covariance, and thus we a priori expect larger offsets between the two than in the other tests. We find that constraints on are very similar and is offset by , both consistent with these expectations.
In addition, we confirm that the central value of the nuisance parameter priors (multiplicative shear bias, tracer and source galaxy redshift biases as defined in Zuntz et al. (2017); Hoyle et al. (2017), subsection F.1 and subsection F.2) is within the confidence interval of the posteriors for both DES and SDSS.
Only after unblinding do we test whether the model at its maximum likelihood parameters is a good fit to the data. For the tomographic data vector of DES Y1, there are 208 elements fit with 13 parameters in the model. Because the model is not linear in the parameters, the number of degrees of freedom and expectation value for the distribution is not known precisely (Andrae et al., 2010), but likely between and . Its standard deviation is . The data vectors for the two shear pipelines have a and , respectively, both consistent with expectations for multivariate Gaussian noise around a signal that is correctly described by our model. The only and models give equally acceptable fits. For the SDSS single source bin data vector with 72 entries and 9 parameters in the model, we find , and equally acceptable results for the other models.
We also perform a run of the fiducial DES Y1 data vector with a full cosmological model that marginalizes, in addition, over baryon density , spectral index of primordial density fluctuations and Hubble parameter with the flat priors also used in DES Collaboration et al. (2017). We find that this does not shift or increase the uncertainty on the reported parameters at a discernible level in any of the models for the connection of galaxies and matter.
Vi Cosmological constraints
|DES Collaboration et al. (2017)|
We perform likelihood analyses, i.e. we determine the probability of finding our fiducial data vectors (section II) as a function of the parameters of our model (section III) and given their covariance (section IV). We use models of different complexity for the connection of galaxies and matter density – one with linear bias only (), one adding stochasticity (), and one allowing for density dependence of stochasticity () (for details see subsection III.3). Our philosophy, decided with parameters still blinded, will be to compare these models via their Bayesian evidence (subsection V.3) and report the results for models that are supported by the data.
Figure 10 shows constraints on the matter density , the amplitude of late-time structure formation , and galaxy bias of the redMaGiC tracer galaxies, when marginalizing over all remaining model parameters. Confidence limits are summarized in Table 2. The stochastic models, favored by the data (see next subsection), constrain the matter density consistently as for the model and for the model from DES data. The degeneracy directions of - for the two stochasticity models are different, thus leading to a higher central value of for and a lower for .
Bias models and cosmology are thus interdependent: a prior, even a weak one, on the values of stochasticity parameters would significantly improve these cosmological constraints – if galaxies have less stochasticity, the relevant regions of the yellow and orange contours in Figure 10 are closer to the black region. Likewise, external information on cosmological parameters allows to constrain bias parameters and, potentially, even choose between the bias models. If the true cosmology is and , both the and the model are off, while the model is consistent.
For a sense of how these results compare to two-point function measurements, Figure 10 contains constraints from the three tomographic auto- and cross-correlation functions of DES redMaGiC galaxy positions and source galaxy shapes (3x2pt, DES Collaboration et al., 2017).333We use the version of the likelihood that does not vary neutrino mass, as in our counts and lensing in cells analysis. The galaxy bias parameter plotted is the mean bias of redMaGiC galaxies in the first two bins ( and ), weighted 1:2, which is not quite the same as the bias of our single lens bin. The 3x2pt contours are tighter than the constraints that counts and lensing in cells yield. This is due to the wide freedom on stochasticity parameters and models that we have allowed: if we could fix the stochasticity (such as in the black contour with ), the smaller scale density PDF measurements would yield highly competitive cosmological constraints. It is clear from this and the different degeneracies that a joint analysis would result in improved constraints – yet we are lacking a covariance matrix and inference pipeline to perform this at this point. Prima facie, the 3x2pt constraint is consistent with any of the bias models, and indicates a relatively small stochasticity, i.e. a point in parameter space close to where the black, yellow and green contours intersect.
Finally, we compare the results from our DES Y1 and SDSS analysis. Within their mutual uncertainty, the two independent data sets provide consistent measurements of cosmological parameters. It is less clear whether the bias model of redMaGiC galaxies in SDSS and DES is identical, a question we turn to in the following subsection.
vi.1 Results on bias and stochasticity
The Bayes factors for the stochastic models, i.e. the ratio of their evidence over the evidence of the only model, are 3.6 () and 2.5 (). This means that there is substantial evidence, as defined by the Jeffreys scale, for stochasticity in the count of redMaGiC galaxies at fixed projected matter density within 20’ apertures and with a redshift range of . Similar observations are made in SDSS, with Bayes factors 2.8 and 4.5 for the introduction of the stochastic models. The data thus prefers a model with stochasticity, but at an odds ratio of , the preference is not very conclusive.
The DES constraint on is . In likelihood runs of the model on log-normal mocks with no stochasticity (Appendix D), we find smaller central values for than this in 3 out of 40 independent realizations.
We note that this finding is not in conflict with the non-detection of stochasticity in the DES Collaboration et al. (2017) 3x2pt analysis, and the associated explicit tests for consistency of the clustering and galaxy-galaxy lensing constraints on bias (Elvin-Poole et al., 2017; Prat et al., 2017). Those analyses use significantly larger scales ( and in the lowest lens redshift bins, corresponding to comoving Mpc for clustering, and comoving Mpc for galaxy-galaxy lensing), on which stochasticity, if present, is expected to be small. Our statistic in sensitive to stochasticity on scales smaller or equal to the radius of the apertures inside which we count tracer galaxies. Physically, this corresponds to comoving Mpc in the redshift range. Uncertainty as to whether the non-stochastic bias model would be sufficient on these smaller scales was a primary reason for the conservative 3x2pt scale cuts (Krause et al., 2017).
Figure 11 shows constraints on , bias and stochasticity of the tracer galaxies, in both DES and SDSS. The deviation of from unity is at the level. Individual parameter constraints are consistent, while there is a hint for a lower value of in SDSS at fixed cosmology. Note how the primary degeneracy of is not with bias, but with stochasticity – even a mildly informative prior on would significantly lower uncertainty on . For cosmological constraints close to those of the 3x2pt analysis ( for the run with fixed mass), the counts and lensing in cells data is well fit by small stochasticity.
For the more complex model of density dependent stochasticity, which is not significantly preferred to by the data anywhere, the situation is qualitatively similar. The SDSS data does not constrain these parameters very well, especially (cf. Table 2), but there is an indication of super-Poissonian scatter in galaxy count at fixed matter density, the amplitude of which increases with density, broadly consistent with the effect of a single stochasticity parameter (Friedrich et al., 2017, their figure 6).
It is difficult to compare this tentative detection of stochasticity on the comoving Mpc aperture scale to the literature. Various works have found levels of stochasticity that are broadly consistent, using a range of samples and scales in numerical simulations (e.g. Somerville et al., 2001; Bonoli & Pen, 2009) and data (e.g. Hoekstra et al., 2002; Wild et al., 2005; Wang et al., 2007; Simon et al., 2007; Swanson et al., 2008; Leauthaud et al., 2017). The comparison of low- galaxy clustering and galaxy-galaxy lensing in DES SV on scales above comoving Mpc provided similar hints of (Crocce et al. (2016); Prat et al. (2016), see also Giannantonio et al. (2016)). Even those studies that found no evidence for do not exclude a mild stochasticity on the relevant scales within their uncertainties (Jullo et al., 2012; Comparat et al., 2013). Most of these studies use two-point correlations, which means their results on stochasticity would have to be transformed to aperture statistics using a numerical model or simulations.
Note that we do not attempt to combine the DES and SDSS results because, without more detailed study, it is not certain that the redMaGiC samples trace the exact same galaxy populations. A larger stochasticity of redMaGiC galaxies in SDSS, if at all significant, could also be due to correlations of the redMaGiC density with SDSS observational systematics that, unlike in the case of DES (Elvin-Poole et al., 2017), has not been removed.
vi.2 Test for excess skewness of matter density
As described in subsection V.1, we can allow for the skewness of the projected, smoothed matter density field, , to be a free parameter in our likelihood, rather than predicting it from perturbation theory.
We first test whether the introduction of this additional parameter to our model is justified by the data. The Bayes factor of the extended models with as a free parameter, relative to any of the three models for the connection of galaxies and matter with fixed , is smaller than unity, both on DES and on SDSS runs. This indicates no evidence that such an extension is required.