Dark Energy Survey Year 1 Results: Cosmological Constraints from Galaxy Clustering and Weak Lensing
We present cosmological results from a combined analysis of galaxy clustering and weak gravitational lensing, using 1321 deg of imaging data from the first year of the Dark Energy Survey (DES Y1). We combine three two-point functions: (i) the cosmic shear correlation function of 26 million source galaxies in four redshift bins, (ii) the galaxy angular autocorrelation function of 650,000 luminous red galaxies in five redshift bins, and (iii) the galaxy-shear cross-correlation of luminous red galaxy positions and source galaxy shears. To demonstrate the robustness of these results, we use independent pairs of galaxy shape, photometric redshift estimation and validation, and likelihood analysis pipelines. To prevent confirmation bias, the bulk of the analysis was carried out while “blind” to the true results; we describe an extensive suite of systematics checks performed and passed during this blinded phase. The data are modeled in flat CDM and CDM cosmologies, marginalizing over 20 nuisance parameters, varying 6 (for CDM) or 7 (for CDM) cosmological parameters including the neutrino mass density and including the 457 457 element analytic covariance matrix. We find consistent cosmological results from these three two-point functions, and from their combination obtain and for CDM; for CDM, we find , , and at 68% CL. The precision of these DES Y1 results rivals that from the Planck cosmic microwave background measurements, allowing a comparison of structure in the very early and late Universe on equal terms. Although the DES Y1 best-fit values for and are lower than the central values from Planck for both CDM and CDM, the Bayes factor indicates that the DES Y1 and Planck data sets are consistent with each other in the context of CDM. Combining DES Y1 with Planck, Baryonic Acoustic Oscillation measurements from SDSS, 6dF, and BOSS, and type Ia supernovae from the Joint Lightcurve Analysis (JLA) dataset, we derive very tight constraints on cosmological parameters: and in CDM, and in CDM. Upcoming DES analyses will provide more stringent tests of the CDM model and extensions such as a time-varying equation of state of dark energy or modified gravity.
The discovery of cosmic acceleration Riess et al. (1998); Perlmutter et al. (1999) established the Cosmological Constant () Einstein (1917) + Cold Dark Matter (CDM) model as the standard cosmological paradigm that explains a wide variety of phenomena, from the origin and evolution of large-scale structure to the current epoch of accelerated expansion Lahav and Liddle (2014); Mortonson et al. (2013). The successes of CDM, however, must be balanced by its apparent implausibility: three new entities beyond the Standard Model of particle physics — one that drove an early epoch of inflation; another that serves as dark matter; and a third that is driving the current epoch of acceleration — are required, none of them easily connected to the rest of physics Frieman et al. (2008). Ongoing and planned cosmic surveys are designed to test CDM and more generally to shed light on the mechanism driving the current epoch of acceleration, be it the vacuum energy associated with the cosmological constant, another form of dark energy, a modification of General Relativity, or something more drastic.
The Dark Energy Survey (DES
The spatial distribution of galaxies in the Universe, and its temporal evolution, carry important information about the physics of the early Universe, as well as details of structure evolution in the late Universe, thereby testing some of the most precise predictions of CDM. Indeed, measurements of the galaxy two-point correlation function, the lowest-order statistic describing the galaxy spatial distribution, provided early evidence for the CDM model Blumenthal et al. (1984); Maddox et al. (1990); Baugh (1996); Maddox et al. (1996); Eisenstein and Zaldarriaga (2001); Collins et al. (1992); Szapudi and Gaztanaga (1998); Huterer et al. (2001); Saunders et al. (2000); Hamilton and Tegmark (2002); Cole et al. (2005); Tegmark et al. (2006). The data–model comparison in this case depends upon uncertainty in the galaxy bias Kaiser (1984), the relation between the galaxy spatial distribution and the theoretically predicted matter distribution.
In addition to galaxy clustering, weak gravitational lensing has become one of the principal probes of cosmology. While the interpretation of galaxy clustering is complicated by galaxy bias, weak lensing provides direct measurement of the mass distribution via cosmic shear, the correlation of the apparent shapes of pairs of galaxies induced by foreground large-scale structure. Further information on the galaxy bias is provided by galaxy–galaxy lensing, the cross-correlation of lens galaxy positions and source galaxy shapes.
The shape distortions produced by gravitational lensing, while cosmologically informative, are extremely difficult to measure, since the induced source galaxy ellipticities are at the percent level, and a number of systematic effects can obscure the signal. Indeed, the first detections of weak lensing were made by cross-correlating observed shapes of source galaxies with massive foreground lenses Tyson et al. (1990); Brainerd et al. (1996). A watershed moment came in the year 2000 when four research groups nearly simultaneously announced the first detections of cosmic shear Bacon et al. (2000); Kaiser et al. (2000); van Waerbeke et al. (2000); Wittman et al. (2000). While these and subsequent weak lensing measurements are also consistent with CDM, only recently have they begun to provide competitive constraints on cosmological parameters Jarvis et al. (2006); Massey et al. (2007); Schrabback et al. (2010); Lin et al. (2012); Heymans et al. (2013); Huff et al. (2014); Jee et al. (2016); Hildebrandt et al. (2017). Galaxy–galaxy lensing measurements have also matured to the point where their combination with galaxy clustering breaks degeneracies between the cosmological parameters and bias, thereby helping to constrain dark energy Brainerd et al. (1996); Fischer et al. (2000); Sheldon et al. (2004); Leauthaud et al. (2012); Mandelbaum et al. (2006); Johnston et al. (2007); Cacciato et al. (2009); Mandelbaum et al. (2013); Choi et al. (2012); Velander et al. (2014); Clampitt et al. (2017); Leauthaud et al. (2017); Kwan et al. (2017). The combination of galaxy clustering, cosmic shear, and galaxy–galaxy lensing measurements powerfully constrains structure formation in the late universe. As for cosmological analyses of samples of galaxy clusters (see Allen et al., 2011, for a review), redshift space distortions in the clustering of galaxies (Alam et al., 2016, and references therein) and other measurements of late-time structure, a primary test is whether these are consistent, in the framework of CDM, with measurements from cosmic microwave background (CMB) experiments that are chiefly sensitive to early-universe physics Hinshaw et al. (2013); Ade et al. (2014, 2016); Calabrese et al. (2017).
The main purpose of this paper is to combine the information from galaxy clustering and weak lensing, using the galaxy and shear correlation functions as well as the galaxy-shear cross-correlation. It has been recognized for more than a decade that such a combination contains a tremendous amount of complementary information, as it is remarkably resilient to the presence of nuisance parameters that describe systematic errors and non-cosmological information Hu and Jain (2004); Bernstein (2009); Joachimi and Bridle (2010); Nicola et al. (2016). It is perhaps simplest to see that the combined analysis could separately solve for galaxy bias and the cosmological parameters; however, it can also internally solve for (or, self-calibrate Huterer et al. (2006)) the systematics associated with photometric redshifts Zhang et al. (2010); Park et al. (2016); Samuroff et al. (2017), intrinsic alignment Zhang (2010), and a wide variety of other effects Joachimi and Bridle (2010). Such a combined analysis has recently been executed by combining the KiDS 450 deg weak lensing survey with two different spectroscopic galaxy surveys van Uitert et al. (2017); Joudaki et al. (2017). While these multi-probe analyses still rely heavily on prior information about the nuisance parameters, obtained through a wide variety of physical tests and simulations, this approach does significantly mitigate potential biases due to systematic errors and will likely become even more important as statistical errors continue to drop. The multi-probe analyses also extract more precise information about cosmology from the data than any single measurement could.
Previously, the DES collaboration analyzed data from the Science Verification (SV) period, which covered 139 deg, carrying out several pathfinding analyses of galaxy clustering and gravitational lensing, along with numerous others Melchior et al. (2015); Vikram et al. (2015); Chang et al. (2015); Becker et al. (2016); DES Collaboration (2016a); Crocce et al. (2016); Gruen et al. (2016); MacCrann et al. (2017); Prat et al. (2016); Clerkin et al. (2017); Chang et al. (2016); Melchior et al. (2016); Kacprzak et al. (2016); Kwan et al. (2017); Clampitt et al. (2017); Sanchez et al. (2017); DES Collaboration (2016b). The DES Y1 data set analyzed here covers about ten times more area, albeit shallower, and provides 650,000 lens galaxies and the shapes of 26 million source galaxies, each of them divided into redshift bins. The lens sample comprises bright, red-sequence galaxies, which have secure photometric redshift (photo-) estimates. We measure three two-point functions from these data: (i) , the angular correlation function of the lens galaxies; (ii) , the correlation of the tangential shear of sources with lens galaxy positions; and (iii) , the correlation functions of different components of the ellipticities of the source galaxies. We use these measurements only on large angular scales, for which we have verified that a relatively simple model describes the data, although even with this restriction we must introduce twenty parameters to capture astrophysical and measurement-related systematic uncertainties.
This paper is built upon, and uses tools and results from, eleven other papers:
Ref. Krause et al. (2017), which describes the theory and parameter-fitting methodologies, including the binning and modeling of all the two point functions, the marginalization of astrophysical and measurement related uncertainties, and the ways in which we calculate the covariance matrix and obtain the ensuing parameter constraints;
Ref. MacCrann et al. (2017), which applies this methodology to image simulations generated to mimic many aspects of the Y1 data sets;
a description of the process by which the value-added galaxy catalog (Y1 Gold) is created from the data and the tests on it to ensure its robustness Drlica-Wagner et al. (2017);
a shape catalog paper, which presents the two shape catalogs generated using two independent techniques and the many tests carried out to ensure that residual systematic errors in the inferred shear estimates are sufficiently small for Y1 analyses Zuntz et al. (2017);
Ref. Hoyle et al. (2017), which describes how the redshift distributions of galaxies in these shape catalogs are estimated from their photometry, including a validation of these estimates by means of COSMOS multi-band photometry;
Ref. Troxel et al. (2017), which measures and derives cosmological constraints from the cosmic shear signal in the DES Y1 data and also addresses the question of whether DES lensing data are consistent with lensing results from other surveys;
Ref. Prat et al. (2017), which describes galaxy–galaxy lensing results, including a wide variety of tests for systematic contamination and a cross-check on the redshift distributions of source galaxies using the scaling of the lensing signal with redshift;
Ref. Elvin-Poole et al. (2017), which describes the galaxy clustering statistics, including a series of tests for systematic contamination. This paper also describes updates to the redMaGiC algorithm used to select our lens galaxies and to estimate their photometric redshifts.
Armed with the above results, this paper presents the most stringent cosmological constraints from a galaxy imaging survey to date and, combined with external data, the most stringent constraints overall.
One of the guiding principles of the methods developed in these papers is redundancy: we use two independent shape measurement methods that are independently calibrated, several photometric redshift estimation and validation techniques, and two independent codes for predicting our signals and performing a likelihood analysis. Comparison of these, as described in the above papers, has been an important part of the verification of each step of our analysis.
The plan of the paper is as follows. §II gives an overview of the data used in the analysis, while §III presents the two-point statistics that contain the relevant information about cosmological parameters. §IV describes the methodology used to compare these statistics to theory, thereby extracting cosmological results. We validated our methodology while remaining blinded to the results of the analyses; this process is described in §V, and some of the tests that convinced us to unblind are recounted in Appendix A. §VI presents the cosmological results from these three probes as measured by DES in the context of two models, CDM and CDM, while §VII compares DES results with those from other experiments, offering one of the most powerful tests to date of CDM. Then, we combine DES with external data sets with which it is consistent to produce the tightest constraints yet on cosmological parameters. Finally, we conclude in §VIII. Appendix B presents further evidence of the robustness of our results.
DES uses the 570-megapixel Dark Energy Camera (DECam Flaugher et al. (2015)), built by the collaboration and deployed on the Cerro Tololo Inter-American Observatory (CTIO) 4m Blanco telescope in Chile, to image the South Galactic Cap in the filters. In this paper, we analyze DECam images taken from August 31, 2013 to February 9, 2014 (“DES Year 1” or Y1), covering 1786 square degrees in griz after coaddition and before masking Drlica-Wagner et al. (2017). The data were processed through the DES Data Management (DESDM) system (Desai et al., 2012; Sevilla et al., 2011; Mohr et al., 2008; Morganson et al., 2017), which detrends and calibrates the raw DES images, combines individual exposures to create coadded images, and detects and catalogs astrophysical objects. Further vetting and subselection of the DESDM data products was performed by Drlica-Wagner et al. (2017) to produce a high-quality object catalog (Y1 Gold) augmented by several ancillary data products including a star/galaxy separator. With up to 4 exposures per filter per field in Y1, and individual exposures of 90 sec and exposures of 45 sec, the characteristic 10 limiting magnitude for galaxies is , , , , and Drlica-Wagner et al. (2017). Additional analyses produced catalogs of red galaxies, photometric-redshift estimates, and galaxy shape estimates, as described below.
As noted in §I, we use two samples of galaxies in the current analysis: lens galaxies, for the angular clustering measurement, and source galaxies, whose shapes we estimate and correlate with each other (“cosmic shear”). The tangential shear is measured for the source galaxies about the positions of the lens galaxies (galaxy–galaxy lensing).
ii.1 Lens Galaxies
We rely on redMaGiC galaxies for all galaxy clustering measurements Elvin-Poole et al. (2017) and as the lens population for the galaxy–galaxy lensing analysis Prat et al. (2017). They have the advantage of being easily identifiable, relatively strongly clustered, and of having relatively small photometric-redshift errors; they are selected using a simple algorithm Rozo et al. (2016):
Fit every galaxy in the survey to a red-sequence template and compute the corresponding best-fit redshift .
Evaluate the goodness-of-fit of the red-sequence template and the galaxy luminosity, using the assigned photometric redshift.
Include the galaxy in the redMaGiC catalog if and only if it is bright and the red-sequence template is a good fit .
In practice, we do not specify but instead demand that the resulting galaxy sample have a constant comoving density as a function of redshift. Consequently, redMaGiC galaxy selection depends upon only two parameters: the selected luminosity threshold, , and the comoving density, , of the sample. Of course, not all combinations of parameters are possible: brighter galaxy samples must necessarily be less dense.
Three separate redMaGiC samples were generated from the Y1 data, referred to
as the high-density, high-luminosity, and higher-luminosity samples. The
corresponding luminosity thresholds
The clustering properties of these galaxies are an essential part of this combined analysis, so great care is taken in Elvin-Poole et al. (2017) to ensure that the galaxy maps are not contaminated by systematic effects. This requires the shallowest or otherwise irregular or patchy regions of the total 1786 deg Y1 area to be masked, leaving a contiguous 1321 deg as the area for the analysis, the region called “SPT” in Drlica-Wagner et al. (2017). The mask derived for the lens sample is also applied to the source sample.
ii.2 Source Galaxies
Gravitational lensing shear is estimated from the statistical alignment of shapes of source galaxies, which are selected from the Y1 Gold catalog Drlica-Wagner et al. (2017). In DES Y1, we measure galaxy shapes and calibrate those measurements by two independent and different algorithms, metacalibration and im3shape, as described in Zuntz et al. (2017).
metacalibration Huff and Mandelbaum (2017); Sheldon and Huff (2017) measures shapes by simultaneously fitting a 2D Gaussian model for each galaxy to the pixel data for all available -, -, and -band exposures, convolving with the point-spread functions (PSF) appropriate to each exposure. This procedure is repeated on versions of these images that are artificially sheared, i.e. de-convolved, distorted by a shear operator, and re-convolved by a symmetrized version of the PSF. By means of these, the response of the shape measurement to gravitational shear is measured from the images themselves, an approach encoded in metacalibration.
metacalibration also includes an algorithm for calibration of shear-dependent selection effects of galaxies, which could bias shear statistics at the few percent level otherwise, by measuring on both unsheared and sheared images all those galaxy properties that are used to select, bin and weight galaxies in the catalog. Details of the practical application of these corrections to our lensing estimators are given in Sheldon and Huff (2017); Zuntz et al. (2017); Troxel et al. (2017); Prat et al. (2017).
im3shape estimates a galaxy shape by determining the maximum likelihood set of parameters from fitting either a bulge or a disc model to each object’s -band observations Zuntz et al. (2013). The maximum likelihood fit, like the Gaussian fit with metacalibration, provides only a biased estimator of shear. For im3shape, this bias is calibrated using a large suite of image simulations that resemble the DES Y1 data set closely Zuntz et al. (2017); Samuroff et al. (2017).
Potential biases in the inferred shears are quantified by multiplicative shear-calibration parameters in each source redshift bin , such that the measured shear . The are free parameters in the cosmological inferences, using prior constraints on each as determined from the extensive systematic-error analyses in Zuntz et al. (2017). These shear-calibration priors are listed in Table 1. The overall metacalibration calibration is accurate at the level of percent. This uncertainty is dominated by the impact of neighboring galaxies on shape estimates. For tomographic measurements, the widths of the overall prior is increased to yield a per-bin uncertainty in , to account conservatively for possible correlations of between bins (see appendices of Zuntz et al., 2017; Hoyle et al., 2017). This yields the 2.3 percent prior per redshift bin shown in Table 1. The im3shape prior is determined with 2.5 percent uncertainty for the overall sample (increased to a 3.5 percent prior per redshift bin), introduced mostly by imperfections in the image simulations.
In both catalogs, we have applied conservative cuts, for instance on signal-to-noise ratio and size, that reduce the number of galaxies with shape estimates relative to the Y1 Gold input catalog significantly. For metacalibration, we obtain 35 million galaxy shape estimates down to an -band magnitude of . Of these, 26 million are inside the restricted area and redshift bins of this analysis. Since its calibration is more secure, and its number density is higher than that of im3shape, we use the metacalibration catalog for our fiducial analysis.
|flat (0.1, 0.9)|
|flat (0.87, 1.07)|
|flat (0.03, 0.07)|
|flat (0.55, 0.91)|
|Lens Galaxy Bias|
|flat (0.8, 3.0)|
|Lens photo- shift (red sequence)|
|Source photo- shift|
Redshift probability distributions are also required for source galaxies in cosmological inferences. For each source galaxy, the probability density that it is at redshift , , is obtained using a modified version of the BPZ algorithm Coe et al. (2006), as detailed in Hoyle et al. (2017). Source galaxies are placed in one of four redshift bins, , based upon the mean of their distributions. As described in Hoyle et al. (2017), Troxel et al. (2017) and Prat et al. (2017), in the case of metacalibration these bin assignments are based upon photo- estimates derived using photometric measurements made by the metacalibration pipeline in order to allow for correction of selection effects.
We denote by an initial estimate of the redshift distribution of the galaxies in bin produced by randomly drawing a redshift from the probability distribution of each galaxy assigned to the bin, and then bin all these redshifts into a histogram. For this step, we use a BPZ estimate based on the optimal flux measurements from the multi-epoch multi-object fitting procedure (MOF) described in Drlica-Wagner et al. (2017).
For both the source and the lens galaxies, uncertainties in the redshift distribution are quantified by assuming that the true redshift distribution in bin is a shifted version of the photometrically derived distribution:
with the being free parameters in the cosmological analyses. Prior constraints on these shift parameters are derived in two ways.
First, we constrain from a matched sample of galaxies in the COSMOS field, as detailed in Hoyle et al. (2017). Reliable redshift estimates for nearly all DES-selectable galaxies in the COSMOS field are available from 30-band imaging Laigle et al. (2016). We select and weight a sample of COSMOS galaxies representative of the DES sample with successful shape measurements based on their color, magnitude, and pre-seeing size. The mean redshift of this COSMOS sample is our estimate of the true mean redshift of the DES source sample, with statistical and systematic uncertainties detailed in Hoyle et al. (2017). The sample variance in the best-fit from the small COSMOS field is reduced, but not eliminated, by reweighting the COSMOS galaxies to match the multiband flux distribution of the DES source sample.
Second, the of both lens and source samples are further constrained by the angular cross-correlation of each with a distinct sample of galaxies with well-determined redshifts. The for the three lowest-redshift lens galaxy samples are constrained by cross-correlation of redMaGiC with spectroscopic redshifts Cawthon et al. (2017) obtained in the overlap of DES Y1 with Stripe 82 of the Sloan Digital Sky Survey. The for the three lowest-redshift source galaxy bins are constrained by cross-correlating the sources with the redMaGiC sample, since the redMaGiC photometric redshifts are much more accurate and precise than those of the sources Gatti et al. (2017)Davis et al. (2017). The limit of the redMaGiC sample precludes use of cross-correlation to constrain , so its prior is determined solely by the reweighted COSMOS galaxies.
For the first three source bins, both methods yield an estimate of , and the two estimates are compatible, so we combine them to obtain a joint constraint. The priors derived for both lens and source redshifts are listed in Table 1. The resulting estimated redshift distributions are shown in Figure 1.
Ref. Hoyle et al. (2017) and Figure 20 in Appendix B demonstrate that, at the accuracy attainable in DES Y1, the precise shapes of the functions have negligible impact on the inferred cosmology as long as the mean redshifts of every bin, parametrized by the , are allowed to vary. As a consequence, the cosmological inferences are insensitive to the choice of photometric redshift algorithm used to establish the initial of the bins.
Iii Two-point Measurements
We measure three sets of two-point statistics: the auto-correlation of the positions of the redMaGiC lens galaxies, the cross-correlation of the lens positions with the shear of the source galaxies, and the two-point correlation of the source galaxy shear field. Each of the three classes of statistics is measured using treecorr Jarvis et al. (2004) in all pairs of redshift bins of the galaxy samples and in 20 log-spaced bins of angular separation although we exclude some of the scales and cross-correlations from our fiducial data vector (see IV). Figures 2 and 3 show these measurements and our best-fit CDM model.
iii.1 Galaxy Clustering:
The inhomogeneous distribution of matter in the Universe is traced by galaxies. The overabundance of pairs at angular separation above that expected in a random distribution, , is one of the simplest measurements of galaxy clustering. It quantifies the strength and scale dependence of the clustering of galaxies, which in turn reflects the clustering of matter.
The upper panel of Figure 2 shows the angular correlation function of the redMaGiC galaxies in the five lens redshift bins described above. As described in Elvin-Poole et al. (2017), these correlation functions were computed after quantifying and correcting for spurious clustering induced by each of multiple observational variables. Figure 2 shows the data with the error bars set equal to the square root of the diagonal elements of the covariance matrix, but we note that data points in nearby angular bins are highly correlated. Indeed, as can be seen in Figure 5 of Krause et al. (2017), in the lowest redshift bins the correlation coefficient between almost all angular bins is close to unity; at higher redshift, the measurements are highly correlated only over the adjacent few angular bins. The solid curve in Figure 2 shows the best-fit prediction from CDM after fitting to all three two-point functions. In principle, we could also use the angular cross-correlations between galaxies in different redshift bins in the analysis, but the amount of information in these cross-bin two-point functions is quite small and would require substantially enlarging the covariance matrix, so we use only the auto-correlations.
iii.2 Galaxy–galaxy lensing:
The shapes of background source galaxies are distorted by the mass associated with foreground lenses. The characteristic distortion is a tangential shear, with the source galaxy ellipticities oriented perpendicular to the line connecting the foreground and background galaxies. This shear, , is sensitive to the mass associated with the foreground galaxies. On scales much larger than the sizes of parent halos of the galaxies, it is proportional to the lens galaxy bias parameters in each lens bin which quantifies the relative clumping of matter and galaxies. The lower panels of Figure 2 show the measurements of galaxy–galaxy lensing in all pairs of lens-source tomographic bins, including the model prediction for our best-fit parameters. The plots include bin pairs for which the lenses are nominally behind the sources (those towards the upper right), so might be expected to have zero signal. Although the signals for these bins are expected to be small, they can still be useful in constraining the intrinsic alignment parameters in our model (see, e.g., Troxel and Ishak (2014)).
In Prat et al. (2017), we carried out a number of null tests to ensure the robustness of these measurements, none of which showed evidence for significant systematic uncertainties besides the ones characterized by the nuisance parameters in this analysis. The model fits the data well. Even the fits that appear quite bad are misleading because of the highly off-diagonal covariance matrix. For the nine data points in the 3–1 bin, for example, while would be 30 if the off-diagonal elements were ignored.
iii.3 Cosmic shear:
The two-point statistics that quantify correlations between the shapes of galaxies are more complex, because they are the products of the components of a spin-2 tensor. Therefore, a pair of two-point functions are used to capture the relevant information: and are the sum and difference of the products of the tangential and cross components of the shear, measured with respect to the line connecting each galaxy pair. For more details, see Troxel et al. (2017) or earlier work in Refs Miralda-Escude (1991); Kaiser (1992, 1998); Kamionkowski et al. (1998); Hui (1999); Bartelmann and Schneider (2001); Refregier (2003); Hoekstra and Jain (2008). Figure 3 shows these functions for different pairs of tomographic bins.
To extract cosmological information from these two-point functions, we construct a model that depends upon both cosmological parameters and astrophysical and observational nuisance parameters. The cosmological parameters govern the expansion history as well as the evolution and scale dependence of the matter clustering amplitude (as quantified, e.g., by the power spectrum). The nuisance parameters account for uncertainties in photometric redshifts, shear calibration, the bias between galaxies and mass, and the contribution of intrinsic alignment to the shear spectra. §IV.2 will enumerate these parameters, and our priors on them are listed in Table 1. Here, we describe how the two-point functions presented in §III are computed in the model.
The lens galaxies are assumed to trace the mass distribution with a simple linear biasing model.Although this need not be true in general, the validity of this assumption over the scales used in this analysis was demonstrated in Krause et al. (2017), Prat et al. (2017), and MacCrann et al. (2017). The measured angular correlation function of the galaxies is thus related to the matter correlation function by a simple factor of in each redshift bin . The theoretical prediction for in bin depends upon the galaxy redshift distribution of that bin according to
where the speed of light has been set to one; is the comoving distance to that redshift (in a flat universe, which is assumed throughout); is the linear redMaGiC bias in redshift bin ; is the Bessel function of order zero; is the redshift distribution of redMaGiC galaxies in the bin normalized so that the integral over is equal to unity; is the Hubble expansion rate at redshift ; and is the 3D matter power spectrum at wavenumber (which, in this Limber approximation, is set equal to ) and at the cosmic time associated with redshift . The expansion rate, comoving distance, and power spectrum all depend upon the cosmological parameters, and the redshift distribution depends implicitly upon the shift parameter introduced in Eq. (II.1). Thus, the angular correlation function in a given redshift bin depends upon eight parameters in CDM.
The expression in Eq. (IV.1) and the ones in Eqs. (IV.2) and (IV.4) use the “flat-sky” approximation, while the corresponding expressions in Krause et al. (2017) use the more accurate expression that sums over Legendre polynomials. However, we show there that the differences between these two expressions are negligible over the scales of interest.
The model power spectrum here is the fully nonlinear power spectrum in CDM or CDM, which we estimate on a grid of by first running CAMB Lewis et al. (2000) or CLASS Lesgourgues (2011) to obtain the linear spectrum and then HALOFIT Takahashi et al. (2012) for the nonlinear spectrum. The smallest angular separations for which the galaxy two-point function measurements are used in the cosmological inference, indicated by the boundaries of the shaded regions in the upper panels of Figure 2, correspond to a comoving scale of 8 Mpc; this scale is chosen such that modeling uncertainties in the non-linear regime cause negligible impact on the cosmological parameters relative to their statistical errors, as shown in Krause et al. (2017) and Troxel et al. (2017).
As described in §VI of Krause et al. (2017), we include the impact of neutrino bias Villaescusa-Navarro et al. (2014); Biagetti et al. (2014); LoVerde (2014) when computing the angular correlation function of galaxies. For Y1 data, this effect is below statistical uncertainties, but it is computationally simple to implement and will be relevant for upcoming analyses.
We model the tangential shear as we modeled the angular correlation function, since it is also a two-point function: the correlation of lens galaxy positions in bin with source galaxy shear in bin . On large scales, it can be expressed as an integral over the power spectrum, this time with only one factor of bias,
where is the multiplicative shear bias, is the 2nd-order Bessel function, and the lensing efficiency function is given by
with the source galaxy redshift distribution. Because both the source and lens redshift distributions impact the signal, the shift parameters and are implicit, as are all the cosmological parameters. The shear signal also depends upon intrinsic alignments of the source shapes with the tidal fields surrounding the lens galaxies; details of our model for this effect (along with an examination of more complex models) are given in Krause et al. (2017) and in Troxel et al. (2017). The smallest angular separations for which the galaxy–galaxy lensing measurements are used in the cosmological inference, indicated by the boundaries of the shaded regions in the lower panels of Figure 2, correspond to a comoving scale of 12 Mpc; as above, this scale is chosen such that the model uncertainties in the non-linear regime cause insignificant changes to the cosmological parameters relative to the statistical uncertainties, as derived in Krause et al. (2017) and verified in MacCrann et al. (2017).
The cosmic shear signal is independent of galaxy bias but shares the same general form as the other sets of two-point functions. The theoretical predictions for these shear-shear two-point functions are
where the efficiency functions are defined above, and and are the Bessel functions for and . Intrinsic alignment affects the cosmic shear signal, especially the low-redshift bins, and are modeled as in Krause et al. (2017). Baryons affect the matter power spectrum on small scales, and the cosmic shear signal is potentially sensitive to these uncertain baryonic effects; we restrict our analysis to the unshaded, large-scale regions shown in Figure 3 to reduce uncertainty in these effects below our measurement errors, following the analysis in Troxel et al. (2017).
iv.2 Parameterization and Priors
We use these measurements from the DES Y1 data to estimate cosmological parameters in the context of two cosmological models, CDM and CDM. CDM contains three energy densities in units of the critical density: the matter, baryon, and massive neutrino energy densities, and . The energy density in massive neutrinos is a free parameter but is often fixed in cosmological analyses to either zero or to a value corresponding to the minimum allowed neutrino mass of 0.06 eV from oscillation experiments Patrignani et al. (2016). We think it is more appropriate to vary this unknown parameter, and we do so throughout the paper (except in §VII.4, where we show that this does not affect our qualitative conclusions). Since most other survey analyses have fixed , our results for the remaining parameters will differ slightly from theirs, even when using their data.
CDM has three additional free parameters: the Hubble parameter, , and the amplitude and spectral index of the primordial scalar density perturbations, and . This model is based on inflation, which fairly generically predicts a flat universe. Further when curvature is allowed to vary in CDM, it is constrained by a number of experiments to be very close to zero. Therefore, although we plan to study the impact of curvature in future work, in this paper we assume the universe is spatially flat, with . It is common to replace with the RMS amplitude of mass fluctuations on 8 Mpc scale in linear theory, , which can be derived from the aforementioned parameters. Instead of , in this work we will focus primarily on the related parameter
since is better constrained than and is largely uncorrelated with in the DES parameter posterior.
We also consider the possibility that the dark energy is not a cosmological constant. Within this CDM model, the dark energy equation of state parameter, (not to be confused with the angular correlation function ), is taken as an additional free parameter instead of being fixed at as in CDM. CDM thus contains 7 cosmological parameters. In future analyses of larger DES data sets, we anticipate constraining more extended cosmological models, e.g., those in which is allowed to vary in time.
In addition to the cosmological parameters, our model for the data contains 20 nuisance parameters, as indicated in the lower portions of Table 1. These are the nine shift parameters, , for the source and lens redshift bins, the five redMaGiC bias parameters, , the four multiplicative shear biases, , and two parameters, and , that parametrize the intrinsic alignment model.
Table 1 presents the priors we impose on the cosmological and nuisance parameters in the analysis. For the cosmological parameters, we generally adopt wide, flat priors that conservatively span the range of values well beyond the uncertainties reported by recent experiments. As an example, although there are currently potentially conflicting measurements of , we choose the lower end of the prior to be 10 below the lower central value from the Planck cosmic microwave background measurement Ade et al. (2016) and the upper end to be 10 above the higher central value from local measurements Riess et al. (2016). In the case of CDM, we impose a physical upper bound of , as that is required to obtain cosmic acceleration. As another example, the lower bound of the prior on the massive neutrino density, , in Table 1 corresponds to the experimental lower limit on the sum of neutrino masses from oscillation experiments.
For the astrophysical parameters , , and that are not well constrained by other analyses, we also adopt conservatively wide, flat priors. For all of these relatively uninformative priors, the guiding principle is that they should not impact our final results, and in particular that the tails of the posterior parameter distributions should not lie close to the edges of the priors. For the remaining nuisance parameters, and , we adopt Gaussian priors that result from the comprehensive analyses described in Refs. Hoyle et al. (2017); Cawthon et al. (2017); Zuntz et al. (2017); Gatti et al. (2017); Davis et al. (2017). The prior and posterior distributions of these parameters are plotted in Appendix A in Figure 19.
In evaluating the likelihood function (§IV.3), the parameters with Gaussian priors are allowed to vary over a range roughly five times wider than the prior; for example, the parameter that accounts for a possible shift in the furthest lens redshift bin, , is constrained in Cawthon et al. (2017) to have a 1- uncertainty of 0.01, so it is allowed to vary over . These sampling ranges conservatively cover the parameter values of interest while avoiding computational problems associated with exploring parameter ranges that are overly broad. Furthermore, overly broad parameter ranges would distort the computation of the Bayesian evidence, which would be problematic as we will use Bayes factors to assess the consistency of the different two-point function measurements, consistency with external data sets, and the need to introduce additional parameters (such as ) into the analysis. We have verified that our results below are insensitive to the ranges chosen.
iv.3 Likelihood Analysis
For each data set, we sample the likelihood, assumed to be Gaussian, in the many-dimensional parameter space:
where is the full set of parameters, are the measured two-point function data presented in Figures 2 and 3, and are the theoretical predictions as given in Eqs. (IV.1, IV.2, IV.4). The likelihood depends upon the covariance matrix that describes how the measurement in each angular and redshift bin is correlated with every other measurement. Since the DES data vector contains elements, the covariance is a symmetric matrix. We generate the covariance matrices using CosmoLike Krause and Eifler (2016), which computes the relevant four-point functions in the halo model, as described in Krause et al. (2017). We also describe there how the CosmoLike-generated covariance matrix is tested with simulations.
Eq. (IV.6) leaves out the in the prefactor
Parallel pipelines, CosmoSIS
iv.4 Tests on Simulations
The collaboration has produced a number of realistic mock catalogs for the DES Y1 data set, based upon two different cosmological -body simulations (Buzzard Busha et al. (2013), MICE Fosalba et al. (2015)), which were analyzed as described in MacCrann et al. (2017). We applied all the steps of the analysis on the simulations, from measuring the relevant two-point functions to extracting cosmological parameters. In the case of simulations, the true cosmology is known, and MacCrann et al. (2017) demonstrates that the analysis pipelines we use here do indeed recover the correct cosmological parameters.
V Blinding and Validation
The small statistical uncertainties afforded by the Y1 data set present an opportunity to obtain improved precision on cosmological parameters, but also a challenge to avoid confirmation biases. To preclude such biases, we followed the guiding principle that decisions on whether the data analysis has been successful should not be based upon whether the inferred cosmological parameters agreed with our previous expectations. We remained blind to the cosmological parameters implied by the data until after the analysis procedure and estimates of uncertainties on various measurement and astrophysical nuisance parameters were frozen.
To implement this principle, we first transformed the ellipticities in the shear catalogs according to where is a fixed blind random number between 0.9 and 1.1. Second, we avoided plotting the measured values and theoretical predictions in the same figure (including simulation outputs as “theory”). Third, when running codes that derived cosmological parameter constraints from observed statistics, we shifted the resulting parameter values to obscure the best-fit values and/or omitted axis labels on any plots.
These measures were all kept in place until the following criteria were satisfied:
All non-cosmological systematics tests of the shear measurements were passed, as described in Zuntz et al. (2017), and the priors on the multiplicative biases were finalized.
We checked that the CDM constraints (on, e.g., ) from the two different cosmic shear pipelines im3shape and metacalibration agreed. The pipelines were not tuned in any way to force agreement.
CDM constraints were stable when dropping the smallest angular bins for metacalibration cosmic shear data.
Small-scale metacalibration galaxy–galaxy lensing data were consistent between source bins (shear-ratio test, as described in §6 of Prat et al. (2017)). We note that while this test is performed in the nominal CDM model, it is close to insensitive to cosmological parameters, and therefore does not introduce confirmation bias.
Once the above tests were satisfied, we unblinded the shear catalogs but kept cosmological parameter values blinded while carrying out the following checks, details of which can be found in Appendix A:
Consistent results were obtained from the two theory/inference pipelines, CosmoSIS and CosmoLike.
Parameter posteriors did not impinge on the edges of the sampling ranges and were in agreement with associated priors for all parameters.
Consistent results on all cosmological parameters were obtained with the two shear measurement pipelines, metacalibration and im3shape.
Consistent results on the cosmological parameters were obtained when we dropped the smallest-angular-scale components of the data vector, reducing our susceptibility to baryonic effects and departures from linear galaxy biasing. This test uses the combination of the three two-point functions (as opposed to from shear only as in test 5).
An acceptable goodness-of-fit value () was found between the data and the model produced by the best-fitting parameters. This assured us that the data were consistent with some point in the model space that we are constraining, while not yet revealing which part of parameter space that is.
Parameters inferred from cosmic shear () were consistent with those inferred from the combination of galaxy–galaxy lensing () and galaxy clustering ().
Once these tests were satisfied, we unblinded the parameter inferences. The following minor changes to the analysis procedures or priors were made after the unblinding: as planned before unblinding, we re-ran the MCMC chains with a new covariance matrix calculated at the best-fit parameters of the original analysis. This did not noticeably change the constraints (see Figure 21 in Appendix B), as expected from our earlier tests on simulated data Krause et al. (2017). We also agreed before unblinding that we would implement two changes after unblinding: small changes to the photo- priors referred to in the footnote to Table 1, and fixing a bug in im3shape object blacklisting that affected of the footprint.
All of the above tests passed, most with reassuringly unremarkable results; more details are given in Appendix A.
For test 11, we calculated the () value of the 457 data points used in the analysis using the full covariance matrix. In CDM, the model used to fit the data has 26 free parameters, so the number of degrees of freedom is . The model is calculated at the best-fit parameter values of the posterior distribution (i.e. the point from the posterior sample with lowest ). Given the uncertainty on the estimates of the covariance matrix, the formal probabilities of a distribution are not applicable. We agreed to unblind as long as was less than 605 (). The best-fit value passes this test, with Considering the fact that 13 of the free parameters are nuisance parameters with tight Gaussian priors, we will use giving
The best-fit models for the three two-point functions are over-plotted on the data in Figures 2 and 3, from which it is apparent that the is not dominated by conspicuous outliers. Figure 4 offers confirmation of this, in the form of a histogram of the differences between the best-fit theory and the data in units of the standard deviation of individual data points. The three probes show similar values of : for for 227 data points; for for 187 data points; and for for 43 data points. A finer division into each of the 45 individual 2-point functions shows no significant concentration of in particular bin pairs. We also find that removing all data at scales yields for 277 data points (), not a significant reduction, and also yields no significant shift in best-fit parameters. Thus, we find that no particular piece of our data vector dominates our result.
The value for the full -point data vector passed our unblinding criterion, even though it would be unacceptable ( for ) if we were expecting adherence to the distribution. We argue that the -value of the distribution should be treated with caution since it may not be a robust statistic here. We can expect deviations from a strict distribution due to a number of factors: potential non-Gaussian error distributions, which will have less impact on our likelihood estimates than on the -value; the effect of priors and marginalized systematic parameters; and uncertainties in our estimation of the covariance matrix. For example, if we increase the diagonal elements of the covariance matrix by a factor 1.1, we obtain a and the -value rises to easily acceptable. We expect such a change in the covariance to, at most, increase the size of the uncertainty we obtain on cosmological parameters by 5%. Based on these observations, we believe that a pessimistic view of our result is that it indicates that our error bars are underestimated by (since multiplying the whole covariance matrix by 1.2 would clearly obtain an acceptable ).
Finally, for test (12), we examined several measures of consistency between (i) cosmic shear and (ii) in CDM. As an initial test, we computed the mean of the 1D posterior distribution of each of the cosmological parameters and measured the shift between (i) and (ii). We then divided this difference by the expected standard deviation of this difference (taking into account the estimated correlation between the and inferences). For all parameters, these differences had absolute value , indicating consistency well within measurement error.
As a second consistency check, we compared the posteriors for the nuisance parameters from cosmic shear to those from clustering plus galaxy–galaxy lensing, and they agreed well. We found no evidence that any of the nuisance parameters pushes against the edge of its prior or that the nuisance parameters for cosmic shear and are pushed to significantly different values. The only mild exceptions are modest shifts in the intrinsic alignment parameter, , as well as in the second source redshift bin, . The full set of posteriors on all 20 nuisance parameters for metacalibration is shown in Figure 19 in Appendix A.
As a final test of consistency between the two sets of two-point-function measurements, we use the Bayes factor (also called the “evidence ratio”). The Bayes factor is used for discriminating between two hypotheses, and is the ratio of the Bayesian evidences, (the probability of observing dataset, , given hypothesis ) for each hypothesis. An example of such a hypothesis is that dataset can be described by a model , in which case the Bayesian evidence is
where are the parameters of model .
For two hypotheses and , the Bayes factor is given by
where the second equality follows from Bayes’ theorem and clarifies the meaning of the Bayes factor: if we have equal a priori belief in and (i.e., ), the Bayes factor is the ratio of the posterior probability of to the posterior probability of . The Bayes factor can be interpreted in terms of odds, i.e., it implies is favored over with odds (or disfavoured if ). We will adopt the widely used Jeffreys scale Jeffreys (1961) for interpreting Bayes factors: and are respectively considered substantial and strong evidence for over . Conversely, is strongly favored over if , and there is substantial evidence for if .
We follow Marshall et al. (2006) by applying this formalism as a test for consistency between cosmological probes. In this case, the null hypothesis, , is that the two datasets were measured from the same universe and therefore share the same model parameters. Two probes would be judged discrepant if they strongly favour the alternative hypothesis, , that they are measured from two different universes with different model parameters. So the appropriate Bayes factor for judging consistency of two datasets, and , is
where is the model, e.g., CDM or CDM. The numerator is the evidence for both datasets when model is fit to both datasets simultaneously. The denominator is the evidence for both datasets when model M is fit to both datasets individually, and therefore each dataset determines its own parameter posteriors.
Before the data were unblinded, we decided that we would combine results from these two sets of two-point functions if the Bayes factor defined in Eq. (V.3) did not suggest strong evidence for inconsistency. According to the Jeffreys scale, our condition to combine is therefore that (since would imply strong evidence for inconsistency). We find a Bayes factor of , an indication that DES Y1 cosmic shear and galaxy clustering plus galaxy–galaxy lensing are consistent with one another in the context of CDM.
The DES Y1 data were thus validated as internally consistent and robust to our assumptions before we gained any knowledge of the cosmological parameter values that they imply. Any comparisons to external data were, of course, made after the data were unblinded.
Vi DES Y1 Results: Parameter Constraints
We first consider the CDM model with six cosmological parameters. The DES data are most sensitive to two cosmological parameters, and as defined in Eq. (IV.5), so for the most part we focus on constraints on these parameters.
Given the demonstrated consistency of cosmic shear with clustering plus galaxy–galaxy lensing in the context of CDM as noted above, we proceed to combine the constraints from all three probes. Figure 5 shows the constraints on and (bottom panel), and on and the less degenerate parameter (top panel). Constraints from cosmic shear, galaxy clustering + galaxy–galaxy lensing, and their combination are shown in these two-dimensional subspaces after marginalizing over the 24 other parameters. The combined results lead to constraints
The value of is slightly lower than that inferred from either cosmic shear or clustering plus galaxy–galaxy lensing separately. In general, when projecting down onto a small subspace, this can occur. In this particular case, we get a glimpse of why by noting from the bottom panel of Figure 5 that the degeneracy directions of shear differ slightly in the plane from , with the two converging on lower values of . We present the resulting marginalized constraints on the cosmological parameters in the top rows of Table 2.
|CDM||DES Y1 3x2|
|CDM||Planck (No Lensing)|
|CDM||DES Y1 + Planck (No Lensing)|
|CDM||DES Y1 + JLA + BAO|
|CDM||Planck + JLA + BAO|
DES Y1 +
Planck + JLA + BAO
|CDM||DES Y1 3x2|
|CDM||Planck (No Lensing)|
|CDM||DES Y1 + Planck (No Lensing)|
|CDM||Planck + JLA + BAO|
DES Y1 +
Planck + JLA + BAO
The results shown in Figure 5, along with previous analyses such as that using KiDS + GAMA data van Uitert et al. (2017), are an important step forward in the capability of combined probes from optical surveys to constrain cosmological parameters. These combined constraints transform what has, for the past decade, been a one-dimensional constraint on (which appears banana-shaped in the plane) into tight constraints on both of these important cosmological parameters. Figure 6 shows the DES Y1 constraints on and along with some previous results and in combination with external data sets, as will be discussed below. The sizes of these parameter error bars from the combined DES Y1 probes are comparable to those from the cosmic microwave background (CMB) obtained by Planck.
In addition to the cosmological parameters, these probes constrain important astrophysical parameters. The intrinsic alignment (IA) signal is modeled to scale as ; while the data do not constrain the power law well (), they are sensitive to the amplitude of the signal:
Further strengthening evidence from the recent combined probes analysis of KiDS van Uitert et al. (2017); Joudaki et al. (2017), this result is the strongest evidence to date of IA in a broadly inclusive galaxy sample; previously, significant IA measurements have come from selections of massive elliptical galaxies, usually with spectroscopic redshifts (e.g. Singh et al. (2015)). The ability of DES data to produce such a result without spectroscopic redshifts demonstrates the power of this combined analysis and emphasizes the importance of modeling IA in the pursuit of accurate cosmology from weak lensing. We are able to rule out at 99.36% CL with DES alone and at 99.89% CL with the full combination of DES and external data sets. The mean value of is nearly the same when combining with external data sets, suggesting that IA self-calibration has been effective. Interestingly, the measured amplitude agrees well with a prediction made by assuming that only red galaxies contribute to the IA signal, and then extrapolating the IA amplitude measured from spectroscopic samples of luminous galaxies using a realistic luminosity function and red galaxy fraction Krause et al. (2017). Our measurement extends the diversity of galaxies with evidence of IA, allowing more precise predictions for the behavior of the expected IA signal.
The biases of the redMaGiC galaxy samples in the five lens bins are shown in Figure 7 along with the results with fixed cosmology obtained in Prat et al. (2017) and Elvin-Poole et al. (2017). Most interesting is the constraining power: even when varying a full set of cosmological parameters (including , which is quite degenerate with bias when using galaxy clustering only) and 15 other nuisance parameters, the combined probes in DES Y1 constrain bias at the ten percent level.
A variety of theoretical alternatives to the cosmological constant have been proposed Frieman et al. (2008). For example, it could be that the cosmological constant vanishes and that another degree of freedom, e.g., a very light scalar field, is driving the current epoch of accelerated expansion. Here we restrict our analysis to the simplest class of phenomenological alternatives, models in which the dark energy density is not constant, but rather evolves over cosmic history with a constant equation of state parameter, . We constrain by adding it as a seventh cosmological parameter. Here, too, DES obtains interesting constraints on only a subset of the seven cosmological parameters, so we show the constraints on the three-dimensional subspace spanned by , , and . Figure 8 shows the constraints in this 3D space from cosmic shear and from galaxy–galaxy lensing + galaxy clustering. These two sets of probes agree with one another. The consistency in the three-dimensional subspace shown in Figure 8, along with the tests in the previous subsection, is sufficient to combine the two sets of probes. The Bayes factor in this case was equal to 0.6. The combined constraint from all three two-point functions is also shown in Figure 8.
The marginalized 68% CL constraints on and on the other two cosmological parameters tightly constrained by DES, and , are shown in Figure 9 and given numerically in Table 2. In the next section, we revisit the question of how consistent the DES Y1 results are with other experiments. The marginalized constraint on from all three DES Y1 probes is
Finally, if one ignores any intuition or prejudice about the mechanism driving cosmic acceleration, studying CDM translates into adding an additional parameter to describe the data. From a Bayesian point of view, the question of whether CDM is more likely than CDM can again be addressed by computing the Bayes factor. Here the two models being compared are simpler: CDM and CDM. The Bayes factor is
Values of less than unity would imply CDM is favored, while those greater than one argue that the introduction of the additional parameter is warranted. The Bayes factor is for DES Y1, so although CDM is slightly favored, there is no compelling evidence to favor or disfavor an additional parameter .
It is important to note that, although our result in Eq. (VI.3) is compatible with CDM, the most stringent test of the model from DES Y1 is not this parameter, but rather the constraints on the parameters in the model shown in Figure 5 as compared with constraints on those parameters from the CMB measurements of the universe at high redshift. We turn next to that comparison.
Vii Comparison with external data
We next explore the cosmological implications of comparison and combination of
DES Y1 results with other experiments’ constraints. For the CMB, we take
constraints from Planck Ade et al. (2016). In the first subsection below, we use
only the temperature and polarization auto- and cross-spectra from
Planck, omitting the information due to lensing of the CMB that is contained
in the four-point function. The latter depends on structure and distances at
late times, and we wish in this subsection to segregate late-time information
from early-Universe observables. We use the joint TT, EE, BB and TE likelihood
for between 2 and 29 and the TT likelihood for between 30 and
2508 (commonly referred to as TT+lowP), provided by
We use measured angular diameter distances from the Baryon Acoustic Oscillation (BAO) feature by the 6dF Galaxy Survey (Beutler et al., 2011), the SDSS Data Release 7 Main Galaxy Sample (Ross et al., 2015), and BOSS Data Release 12 (Alam et al., 2016), in each case extracting only the BAO constraints. These BAO distances are all measured relative to the physical BAO scale corresponding to the sound horizon distance ; therefore, dependence of on cosmological parameters must be included when determining the likelihood of any cosmological model (see Alam et al. (2016) for details). We also use measures of luminosity distances from observations of distant Type Ia supernovae (SNe) via the Joint Lightcurve Analysis (JLA) data from Betoule et al. (2014).
This set of BAO and SNe experiments has been shown to be consistent with the CDM and CDM constraints from the CMB Hinshaw et al. (2013); Ade et al. (2016), so we can therefore sensibly merge this suite of experiments—BAO, SNe, and Planck—with the DES Y1 results to obtain unprecedented precision on the cosmological parameters. We do not include information about direct measurements of the Hubble constant because those are in tension with this bundle of experiments Bernal et al. (2016).
vii.1 High redshift vs. low redshift in Cdm
The CMB measures the state of the Universe when it was 380,000 years old, while DES measures the matter distribution in the Universe roughly ten billion years later. Therefore, one obvious question that we can address is: Is the CDM prediction for clustering today, with all cosmological parameters determined by Planck, consistent with what DES observes? This question, which has of course been addressed by previous surveys (e.g., Heymans et al. (2013); Hildebrandt et al. (2017); van Uitert et al. (2017); Joudaki et al. (2017)), is so compelling because (i) of the vast differences in the epochs and conditions measured; (ii) the predictions for the DES Y1 values of and have no free parameters in CDM once the recombination-era parameters are fixed; and (iii) those predictions for what DES should observe are very precise, with and determined by the CMB to within a few percent. We saw above that and are constrained by DES Y1 at the few-percent level, so the stage is set for the most stringent test yet of CDM growth predictions. Tension between these two sets of constraints might imply the breakdown of CDM.
Figure 10 compares the low- constraints for CDM from all three DES Y1 probes with the constraints from the Planck anisotropy data. Note that the Planck contours are shifted slightly and widened significantly from those in Figure 18 of Ade et al. (2016), because we are marginalizing over the unknown sum of the neutrino masses. We have verified that when the sum of the neutrino masses is fixed as Ade et al. (2016) assumed in their fiducial analysis, we recover the constraints shown in their Figure 18.
The two-dimensional constraints shown in Figure 10 visually hint at tension between the Planck CDM prediction for RMS mass fluctuations and the matter density of the present-day Universe and the direct determination by DES. The 1D marginal constraints differ by more than 1 in both and , as shown in Figure 6. The KiDS survey Hildebrandt et al. (2017); van Uitert et al. (2017) also reports lower than Planck at marginal significance.
However, a more quantitative measure of consistency in the full 26-parameter space is the Bayes factor defined in Eq. (V.3). As mentioned above, a Bayes factor below 0.1 suggests strong inconsistency and one above 10 suggests strong evidence for consistency. The Bayes factor here is , indicating “substantial” evidence for consistency on the Jeffreys scale, so any inconsistency apparent in Figure 10 is not statistically significant according to this metric. In order to test the sensitivity of this conclusion to the priors used in our analysis, we halve the width of the prior ranges on all cosmological parameters (the parameters in the first section of Table 1). For this case we find , demonstrating that our conclusion that there is no evidence for inconsistency is robust even to a dramatic change in the prior volume. The Bayes factor in Eq. (V.3) compares the hypothesis that two datasets can be fit by the same set of model parameters (the null hypothesis), to the hypothesis that they are each allowed an independent set of the model parameters (the alternative hypothesis). The alternative hypothesis is naturally penalized in the Bayes factor since the model requires an extra parameters. We also test an alternative hypothesis where only and are allowed to be constrained independently by the two datasets; in this case we are introducing only two extra parameters with respect to the null hypothesis. For this case, we find , which again indicates that there is no evidence for inconsistency between the datasets.
We therefore combine the two data sets, resulting in the red contours in Figure 10. This quantitative conclusion that the high– and low– redshift data sets are consistent can even be gleaned by viewing Figure 10 in a slightly different way: if the true parameters lie within the red contours, it is not unlikely for two independent experiments to return the blue and green contour regions.
Figure 11 takes the high- vs. low- comparison a step further by combining DES Y1 with results from BAO experiments and Type Ia supernovae. While these even tighter low-redshift constraints continue to favor slightly lower values of and than Planck, the Bayes factor rises to i.e. near Jeffrey’s threshold for “strongly” favoring consistency of DES Y1+BAO+JLA with Planck. The addition of BAO and SNe to DES Y1 thus strengthens the agreement between high- and low- measures within the CDM model.
The goal of this subsection is to test the CDM prediction for clustering in DES, so we defer the issue of parameter determination to the next subsections. However, there is one aspect of the CMB measurements combined with DES that is worth mentioning here. DES data do not constrain the Hubble constant directly. However, as shown in Figure 12, the DES CDM constraint on combined with Planck’s measurement of leads to a shift in the inference of the Hubble constant (in the direction of local measurements Riess et al. (2016)). Since is lower in DES, the inferred value of moves up. As shown in the figure and quantitatively in Table 2, the shift is greater than . As shown in Table 2, this shift in the value of persists as more data sets are added in.
vii.2 Cosmological Parameters in Cdm
To obtain the most stringent cosmological constraints, we now compare DES Y1 with the bundle of BAO, Planck, and JLA that have been shown to be consistent with one another Ade et al. (2016). Here “Planck” includes the data from the four-point function of the CMB, which captures the effect of lensing due to large-scale structure at late times. Figure 13 shows the constraints in the – plane from this bundle of data sets and from DES Y1, in the CDM model. Here the apparent consistency of the data sets is borne out by the Bayes factor for dataset consistency (Eq. V.3):
Combining all of these leads to the tightest constraints yet on CDM parameters, shown in Table 2. Highlighting some of these: at 68% C.L., the combination of DES with these external data sets yields
This value is about 1 lower than the value without DES Y1, with comparable error bars. The clustering amplitude is also constrained at the percent level:
Note that fortuitously, because is so close to , the difference in the central values of and is negligible. The combined result is about 1 lower than the inference without DES, and the constraints are tighter by about 20%.
As mentioned above, the lower value of leads to a higher value of the Hubble constant:
with neutrino mass varied.
Figure 14 shows the results in the extended CDM parameter space using Planck alone, and DES alone, combined, and with the addition of BAO+SNe. As discussed in Ade et al. (2016), the constraints on the dark energy equation of state from Planck alone are misleading. They stem from the measurement of the distance to the last scattering surface, and that distance (in a flat universe) depends upon the Hubble constant as well, so there is a strong degeneracy. The low values of seen in Figure 14 from Planck alone correspond to very large values of , ruled out by local distance indicators. Since DES is not sensitive to the Hubble constant, it does not break this degeneracy. Additionally, the Bayes factor in Eq. (VI.4) that quantifies whether adding the extra parameter is warranted is . Therefore, opening up the dark energy equation of state is not favored on a formal level for the DES+Planck combination. Finally, the Bayes factor for combining DES and Planck (no lensing) in CDM is equal to 0.18, which we identified earlier as “substantial” evidence in favor of the hypothesis that these two data sets are not consistent in the context of this model. These factors degrade the legitimacy of the value returned by the DES+Planck combination.
The addition of BAO, SNe, and Planck lensing data to the DES+Planck combination yields the red contours in Figure 14, shifting the solution substantially along the Planck degeneracy direction, demonstrating (i) the problems mentioned above with the DES+Planck (no lensing) combination and (ii) that these problems are resolved when other data sets are introduced that restrict the Hubble parameter to reasonable values. The Bayes factor for combination of Planck with the low- suite of DES+BAO+SNe in the CDM model is substantially more supportive of the combination of experiments than the case for Planck and DES alone. The DES+Planck+BAO+SNe solution shows good consistency in the –– subspace and yields our final constraint on the dark energy equation of state:
DES Y1 reduces the width of the allowed 68% region by ten percent. The evidence ratio for this full combination of data sets, disfavoring the introduction of as a free parameter.
vii.4 Neutrino Mass
The lower power observed in DES (relative to Planck) has implications for the constraint on the sum of the neutrino masses, as shown in Figure 15. The current most stringent constraint comes from the cosmic microwave background and Lyman-alpha forest YÃ¨che et al. (2017). The experiments considered here (DES, JLA, BAO) represent an independent set so offer an alternative method for measuring the clustering of matter as a function of scale and redshift, which is one of the key drivers of the neutrino constraints. The 95% C.L. upper limit on the sum of the neutrino masses in CDM becomes less constraining:
Adding in DES Y1 loosens the constraint by close to 20% (from 0.245 eV). This is consistent with our finding that the clustering amplitude in DES Y1 is slightly lower than expected in CDM informed by Planck. The three ways of reducing the clustering amplitude are to reduce , reduce , or increase the sum of the neutrino masses. The best fit cosmology moves all three of these parameters slightly in the direction of less clustering in the present day Universe.
We may, conversely, be concerned about the effect of priors on on the cosmological inferences in this paper. The results for DES Y1 and Planck depicted in Figure 10 in CDM were obtained when varying the sum of the neutrino masses. Neutrinos have mass Fukuda et al. (1998) and the sum of the masses of the three light neutrinos is indeed unknown, so this parameter does need to be varied. However, many previous analyses have either set the sum to zero or to the minimum value allowed by oscillation experiments ( eV), so it is of interest to see if fixing neutrino mass alters any of our conclusions. In particular: does this alter the level of agreement between low- and high-redshift probes in CDM? Figure 16 shows the extreme case of fixing the neutrino masses to the lowest value allowed by oscillation data: both the DES and Planck constraints in the plane change. The Planck contours shrink toward the low-