Dark Energy Survey Year 3 Results: Covariance Modelling and its … · 2020. 12. 23. ·...

DES-2019-0466FERMILAB-PUB-20-663-E-SCD-V

MNRAS 000, 000–000 (0000) Preprint 2 August 2021 Compiled using MNRAS LATEX style file v3.0

Dark Energy Survey Year 3 Results: Covariance Modellingand its Impact on Parameter Estimation and Quality of Fit

O. Friedrich,1,2 F. Andrade-Oliveira,3,4 H. Camacho,3,4 O. Alves,5,3 R. Rosenfeld,6,4 J. Sanchez,7X. Fang,8 T. F. Eifler,8,9 E. Krause,8 C. Chang,10,11 Y. Omori,12 A. Amon,12 E. Baxter,13

J. Elvin-Poole,14,15 D. Huterer,5 A. Porredon,14,16,17 J. Prat,10 V. Terra,3 A. Troja,6,4 A. Alarcon,18

K. Bechtol,19 G. M. Bernstein,20 R. Buchs,21 A. Campos,22 A. Carnero Rosell,23,24 M. Car-rasco Kind,25,26 R. Cawthon,19 A. Choi,14 J. Cordero,27 M. Crocce,16,17 C. Davis,12 J. DeRose,28,29

H. T. Diehl,7 S. Dodelson,22 C. Doux,20 A. Drlica-Wagner,10,7,11 F. Elsner,30 S. Everett,29

P. Fosalba,16,17 M. Gatti,31 G. Giannini,31 D. Gruen,32,12,21 R. A. Gruendl,25,26 I. Harrison,27

W. G. Hartley,33 B. Jain,20 M. Jarvis,20 N. MacCrann,34 J. McCullough,12 J. Muir,12 J. Myles,32

S. Pandey,20 M. Raveri,11 A. Roodman,12,21 M. Rodriguez-Monroy,35 E. S. Rykoff,12,21 S. Samuroff,22

C. Sánchez,20 L. F. Secco,20 I. Sevilla-Noarbe,35 E. Sheldon,36 M. A. Troxel,37 N. Weaverdyck,5B. Yanny,7 M. Aguena,38,4 S. Avila,39 D. Bacon,40 E. Bertin,41,42 S. Bhargava,43 D. Brooks,30

D. L. Burke,12,21 J. Carretero,31 M. Costanzi,44,45 L. N. da Costa,4,46 M. E. S. Pereira,5J. De Vicente,35 S. Desai,47 A. E. Evrard,48,5 I. Ferrero,49 J. Frieman,7,11 J. García-Bellido,39

E. Gaztanaga,16,17 D. W. Gerdes,48,5 T. Giannantonio,50,1 J. Gschwend,4,46 G. Gutierrez,7S. R. Hinton,51 D. L. Hollowood,29 K. Honscheid,14,15 D. J. James,52 K. Kuehn,53,54 O. Lahav,30

M. Lima,38,4 M. A. G. Maia,4,46 F. Menanteau,25,26 R. Miquel,55,31 R. Morgan,19 A. Palmese,7,11

F. Paz-Chinchón,50,26 A. A. Plazas,56 E. Sanchez,35 V. Scarpine,7 S. Serrano,16,17 M. Soares-Santos,5M. Smith,57 E. Suchyta,58 G. Tarle,5 D. Thomas,40 C. To,32,12,21 T. N. Varga,59,60 J. Weller,59,60 andR.D. Wilkinson43

(DES Collaboration)

2 August 2021

ABSTRACTWe describe and test the fiducial covariance matrix model for the combined 2-pointfunction analysis of the Dark Energy Survey Year 3 (DES-Y3) dataset. Using a varietyof new ansatzes for covariance modelling and testing we validate the assumptions andapproximations of this model. These include the assumption of Gaussian likelihood,the trispectrum contribution to the covariance, the impact of evaluating the modelat a wrong set of parameters, the impact of masking and survey geometry, deviationsfrom Poissonian shot-noise, galaxy weighting schemes and other, sub-dominant effects.We find that our covariance model is robust and that its approximations have littleimpact on goodness-of-fit and parameter estimation. The largest impact on best-fitfigure-of-merit arises from the so-called fsky approximation for dealing with finite sur-vey area, which on average increases the χ2 between maximum posterior model andmeasurement by 3.7% (∆χ2 ≈ 18.9). Standard methods to go beyond this approxima-tion fail for DES-Y3, but we derive an approximate scheme to deal with these features.For parameter estimation, our ignorance of the exact parameters at which to evaluateour covariance model causes the dominant effect. We find that it increases the scatterof maximum posterior values for Ωm and σ8 by about 3% and for the dark energyequation of state parameter by about 5%.

Key words: cosmology: observations, large-scale structure of Universe

† E-mail: [email protected]

© 0000 The Authors

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1 INTRODUCTION

Our understanding of the Universe has become much moreaccurate in the past decades due to a massive amount ofobservational data collected through different probes, suchas the cosmic microwave background (CMB; see e.g. PlanckCollaboration 2020), Big Bang Nucleosynthesis (BBN; seee.g. Fields et al. 2020), type IA supernovae (see e.g. Riess2017; Smith et al. 2020), number counts of clusters of galax-ies (see e.g. Mantz et al. 2014; Costanzi et al. 2019; Ab-bott et al. 2020), the correlation of galaxy positions, andthat of their measured shape (see e.g. Abbott et al. 2018;Heymans et al. 2020). From the study of that data a stan-dard cosmological model has emerged characterized by asmall number of parameters (see e.g. Frieman et al. 2008;Peebles 2012; Blandford et al. 2020). Current spectroscopicand photometric surveys of galaxies such as the ExtendedBaryon Oscillation Spectroscopic Survey (eBOSS1) and ear-lier phases of the Sloan Digital Sky Survey (SDSS), the Hy-per Suprime-Cam Subaru Strategic Program (HSC-SSP2),the Kilo-Degree Survey (KiDS3) and the Dark Energy Sur-vey (DES4) have become instrumental in testing this stan-dard model at a new front: the growth of density pertur-bations in the late-time Universe. And future surveys, suchas the Dark Energy Spectroscopic Instrument (DESI5), theVera Rubin Observatory Legacy Survey of Space and Time(LSST6), Euclid7 and the Nancy Grace Roman Space Tele-scope 8 will push this test to a precision exceeding that pro-vided by other cosmological probes.

An important part of this program is the Dark En-ergy Survey, a state-of-the-art galaxy survey that completedits six-year observational campaign in January 2019 (Diehlet al. 2019) collecting data on position, color and shape formore than 300 million galaxies. This makes DES the mostsensitive and comprehensive photometric galaxy survey everperformed. The main cosmological analyses of the first year(Y1) of DES data have been concluded (Abbott et al. 2018;Abbott et al. 2019b) and analyses of the first three years ofdata (Y3) are under way. The study of the large-scale struc-ture (LSS) of the Universe based on the DES-Y3 data sethas the potential to become the most stringent test of ourunderstanding of cosmological physics to date.

To achieve this goal the DES team is comparing differ-ent theoretical models characterized by a range of cosmolog-ical parameters to the measured statistics of the LSS in or-der to determine the model and range of parameters that arein best agreement with the data. The statistics of the LSSconsidered in the main DES-Y3 analysis are 2-point corre-lation functions of the galaxy density field (galaxy cluster-ing), the weak gravitational lensing field (cosmic shear) andthe cross-correlation functions between these fields (galaxy-galaxy lensing) in real space and measured in different red-shift bins. These three types of 2-point correlation functions

1 www.sdss.org/surveys/eboss2 hsc.mtk.nao.ac.jp/ssp3 kids.strw.leidenuniv.nl4 www.darkenergysurvey.org5 www.desi.lbl.gov6 www.lsst.org7 www.euclid-ec.org8 nasa.gov/content/goddard/nancy-grace-roman-space-telescope

are combined into one data vector - the so-called 3x2pt datavector.

A key ingredient in analyzing these statistics is a modelfor the likelihood of a cosmological model given the mea-sured correlation functions. Under the assumption of Gaus-sian statistical uncertainties (which is to be validated) thislikelihood is completely characterized by the covariance ma-trix that describes how correlated the uncertainties of dif-ferent data points in the 3x2pt data vector are. Validatingthe quality of the covariance model for the DES-Y3 2-pointanalyses is the main focus of this paper.

There are several methods to estimate covariance ma-trices that can roughly be divided into four main categories:covariance estimation from the data itself (e.g.through jack-knife or sub-sampling methods, cf. Norberg et al. 2009;Friedrich et al. 2016), covariance estimation from a suiteof simulations (e.g. Hartlap et al. 2007; Dodelson & Schnei-der 2013; Taylor et al. 2013; Percival et al. 2014; Taylor &Joachimi 2014; Sellentin & Heavens 2017; Joachimi 2017;Avila et al. 2018; Shirasaki et al. 2019), theoretical covari-ance modelling (e.g. Schneider et al. 2002; Eifler et al. 2009;Krause et al. 2017) or hybrid methods combining both sim-ulations and theoretical covariance models (e.g. Pope & Sza-pudi 2008; Friedrich & Eifler 2018; Hall & Taylor 2019).

For the DES-Y3 3x2pt analysis we adopt a theoreti-cal covariance model as our fiducial covariance matrix. Thisfiducial covariance model is based on a halo model and in-cludes a dominant Gaussian component, a non-Gaussiancomponent (trispectrum and super-sample covariance), red-shift space distortions, curved sky formalism, finite angularbin width, non-Limber computation for the clustering part,Gaussian shape noise, Poissonian shot noise and fsky ap-proximation to treat the finite DES-Y3 survey footprint (al-though taking into account the exact survey geometry whencomputing sampling noise contributions to the covariance).In order to assess the accuracy of that model, we study theimpact of several approximations and assumptions that gointo it (and into 2-point function covariance models in gen-eral):

• the Gaussian likelihood assumption, i.e. whether knowledgeof the covariance is sufficient to calculate the likelihood;• robustness with respect to the modelling of the non-Gaussian covariance contributions, i.e. contributions fromthe trispectrum and super sample covariance;• treatment of the fact that 2-point functions are measuredin finite angular bins;• cosmology dependence of the covariance model;• random point shot-noise;• the assumption of Poissonian shot-noise;• survey geometry and the fsky approximation;• other covariance modelling details such as flat sky vs.curved sky calculations, Limber approximation and redshiftspace distortions.

We generate different types of mock data and/or analyt-ical estimates to determine how each of these effects im-pacts the quality of the fit between measurements of the3x2pt data vector and maximum posterior models (quan-tified by the distribution of χ2 between the two). We alsoshow how they impact cosmological parameter constraintsderived from measurements of the 3x2pt data vector. Formost of these tests we employ a linearized Gaussian like-

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DES Y3: Covariance validation 3

lihood framework which allows us to analytically quantifythe impact of covariance errors on the χ2 distribution andparameter constraints. This is complemented by a set of log-normal simulations and importance sampling techniques toquickly assess large numbers of mock (non-linear) likelihoodanalyses.

This paper is part of a larger release of scientific resultsfrom year-3 data of the Dark Energy Survey and our anal-ysis is informed by the (in some cases preliminary) analysischoices of the other DES-Y3 studies. In addition to carvingout the most stringent constraints on cosmological parame-ters from late-time 2-point statistics of galaxy density andcosmic shear yet, the year-3 analysis of the DES collabora-tion is introducing and testing numerous methodological in-novations that pave the way for future experiments. Detailsof the DESY3 galaxy catalogs and the photometric estima-tion of their redshift distribution are presented by Sevilla-Noarbe et al. (2020); Hartley et al. (2020); Everett et al.(2020); Myles et al. (2020); Gatti et al. (2020a); Cawthonet al. (2020); Buchs et al. (2019); Cordero et al. (2020). Themeasurements of galaxy shapes and the calibration of thesemeasurements for the purpose of cosmic weak gravitationallensing analyses are detailed by Gatti et al. (2020b); Jarviset al. (2020); MacCrann et al. (2020). Krause et al. (2020)develop and test the theoretical modelling pipeline of theDES-Y3 3x2pt analysis, Pandey et al. (2020) outline howgalaxy bias is incorporated in this pipeline, DeRose et al.(2020) validate this pipeline with the help of simulated dataand Muir et al. (2020) describe how we have blinded ouranalysis to focus our efforts on model independent valida-tion criteria and reduce the chance for confirmation bias.The DESY3 methodology to sample high-dimensional like-lihoods and to characterize external and internal tensions isoutlined by Lemos et al. (2020); Doux et al. (2020). Mea-surements of cosmic shear 2-point correlation functions andanalyses thereof are presented by Amon et al. (2020); Seccoet al. (2020), the measurement and analysis of galaxy clus-tering 2-point statistics is carried out by Rodríguez-Monroyet al. (2020) and 2-point cross-correlations between galaxydensity and cosmic shear (galaxy-galaxy lensing) are mea-sured and analysed by Prat et al. (2020), with additionalanalyses of lensing magnification and shear ratios carriedout by Elvin-Poole et al. (2020); Sánchez et al. (2020) andresults for an alternative lens galaxy sample presented byPorredon et al. (2020, prep). Finally, in DES Collaborationet al. (2020) we present our cosmological analysis of the full3x2pt data vector.

Our paper is structured as follows. We start by pre-senting a discussion of our validation strategy in Section 2,where we also summarize our main findings before plunginginto the details in the remaining of the paper. In Section3 we review the modelling and structure of the 3x2pt datavector. Section 4 describes our fiducial covariance model aswell as two alternatives to it that are used to validate sev-eral modelling assumptions. In Section 5 we describe ourlinearized likelihood formalism and derive analytically howdifferent covariance matrices impact parameter constraintsand maximum posterior χ2 within that formalism (includingthe presence of nuisance parameters and allowing for Gaus-sian priors on these parameters). In Section 6 we presentthe details of each step in our validation strategy followedby a short Section 7 presenting a simple test to corroborate

some of the results from the linearized framework. We con-clude with a discussion of our results in Section 8. Sevenappendices describe in more detail some results used in thiswork.

2 COVARIANCE VALIDATION STRATEGYAND SUMMARY OF THE RESULTS

How should one validate the quality of a covariance model(and the associated likelihood model) for the purpose of con-straining cosmological model parameters from a measuredstatistic? A straightforward answer seems to be that oneshould run a large number of accurate cosmological simula-tions, then measure and analyse the statistic at hand in eachof the simulated data sets and test whether the true param-eters of the simulations are located within the, say, 68.3%quantile of the inferred parameter constraints in 68.3% ofthe simulations. There are however at least 2 problems withsuch an approach.

The first one is a conceptual problem. Consider aBayesian analysis of a measured statistic ξ with a modelξ[π] that is parametrised by model parameters π. For eachvalue of π the statistical uncertainties in the measurementξ will have some distribution

L(π|ξ) ≡ p(ξ|π) (1)

which is also called the likelihood of the parameters given thedata. If this function is known, then a Bayesian analysis willassign a posterior probability distribution to the parametersas

p(π|ξ) =1

N L(π|ξ) pr(π) . (2)

Here pr(π) is a prior probability distribution thatparametrises prior knowledge from other experiments (ortheoretical constraints) and the normalisation constant Nis fixed by demanding that p(π|ξ) be a probability distri-bution. The 68.3% confidence region for the parameters πwould then e.g. be stated as a volume V68.3% in parameterspace that contains 68.3% of the probability. To unambigu-ously define that volume one can e.g. impose the additionalcondition that

minπ∈V68.3%

p(π|ξ) > maxπ/∈V68.3%

p(π|ξ) (3)

or, more frequently, one would directly define one dimen-sional intervals that satisfy the above conditions for themarginalised posterior distributions on the individual pa-rameter axes. Unfortunately, if one performs such an analy-sis many times one is not guaranteed that the true parame-ters (e.g. of a simulation) are located within V68.3% in 68.3%of the times. This has recently been referred to as prior vol-ume effect (this issue is discussed in, e.g. Raveri & Hu (2019)and Abbott et al. (2019a)). One may argue that a Bayesianposterior should not be interpreted in terms of frequenciesbut that doesn’t help for the task of validating this posterioron the basis of a large number of simulated data sets. 9

9 In order to deal with the prior volume effect Joachimi et al.(2020) proposed to report parameter constraints through whatthey call projected joint highest posterior density. This topic willbe addressed in a separate DES paper (Raveri et al. - in prep.).

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4 DES Collaboration

Another, more practical problem is the fact that it isnot (yet) feasible to generate enough sufficiently accuratemock data sets to validate covariance matrices of large datavectors with high precision. We recall that for the DES-Y1 analysis, a total of 18 realistic simulated data sets wereavailable to validate the inference pipeline (MacCrann et al.2018). At the same time, the main reason why N-body sim-ulations would be required to test the accuracy of covari-ance (and likelihood) models is to capture contributions tothe covariance coming from the trispectrum (connected 4-point function) of the cosmic density field. But for DES-likeanalyses it has been shown that this contribution is negli-gible (see e.g. Krause et al. (2017); Barreira et al. (2018)).The reason for this is twofold: first, very small scales (wherethe trispectrum contribution to the covariance would mattermost) are often cut off from analyses because on these scalesalready the modelling of the data vector, ξ[π], is inaccurate.And secondly, on small scales the covariance matrix is oftendominated by effects coming from sparse sampling such asshot noise and shape noise. These covariance contributionsare typically easy to model (although one has to be carefulwhen estimating effective number densities and shape-noisedispersions or when estimating the number of galaxy pairs inthe presence of complex survey footprints, see Troxel et al.2018a; Troxel et al. 2018b).

As a result of the considerations above we base our co-variance validation strategy mostly on the use of a linearizedlikelihood (where the model ξ[π] is linear in the parame-ters π). In this framework the Bayesian likelihood allowsfor an interpretation in terms of frequencies - both for totaland marginalised constraints. Also, this allows us to per-form large numbers of simulated likelihood analyses very ef-ficiently, without the need to run computationally expensiveMarkov Chain Monte Carlo (MCMC) codes. In addition, anyleading order deviation from a linearized likelihood will benext-to-leading order for the purpose of studying the impactof covariance errors (i.e. errors on errors) on our analysis.

Within the linearized likelihood formalism we confirmthe findings of Krause et al. (2017); Barreira et al. (2018)for the DES-Y3 setup: both super-sample covariance andtrispectrum have a negligible impact on our analysis. Thisallows us to estimate the impact of other assumptions in ourcovariance and likelihood model either analytically or by themeans of simplified mock data such as lognormal simulations(as opposed to full N-body simulations, cf. Section 4.3).

We summarize our main findings in Figure 1 and Table1 for the busy reader. For the combined data vector of theDES-Y3 two-point function analysis (the 3x2pt data vector,see details in Section 3) the left panel of Figure 1 showsthe impact of different assumptions in our likelihood modelon the mean and scatter of χ2 between maximum posteriormodel and measurements. To obtain the maximum posteriormodel we are fitting for all the 28 parameters listed in Table3 within the linearized likelihood framework described inSection 5.1. Since we assume Gaussian priors on 13 nuisanceparameters, the effective number of parameters in that fitwill be between 28 and 15. Within the linearized likelihoodapproach we find that with a perfect covariance model theaverage χ2 is expected to be about 507.6, i.e. the effectivenumber of degrees of freedom in the fit is Nparam,eff ≈ 23.4.The right panel of Figure 1 shows the equivalent resultswhen cosmic shear correlation functions are excluded from

the data vector (the 2x2pt data vector). The green points inboth panels denote effects that have been already accountedfor in the previous year-1 analysis of DES.

What stands out in our analysis is the large effect offinite angular bin sizes on the cosmic variance and mixedterms of our covariance model (cf. Section 4 for this termi-nology, where we also show that it is unavoidable to takeinto account finite bin width in the pure shot noise andshape noise terms of the covariance). In DES-Y1 this hasbeen dealt with in an approximate manner, by computingthe covariance model for a very fine angular binning andthan re-summing the matrix to obtain a coarser binning(Krause et al. 2017). This time we incorporate the exacttreatment of finite angular bin size for all the three two-point functions into our fiducial covariance model (cf. Sec-tion 4). The blue points in Figure 1 denote improvementsthat have been made in the year-3 analysis compared to theyear-1 covariance model. And the red points are estimates ofeffects that are not taken into account in the fiducial DES-Y3 likelihood - either because they are negligible, or becausean exact treatment is unfeasible (cf. Section 6 for details).Adding these effects in quadrature, our results suggest thatthe maximum posterior χ2 of the DES-Y3 3x2pt analysisshould be on average ≈ 4% (∆χ2 ≈ 20.3) higher than ex-pected if the exact covariance matrix of our data vector wasknown.

Table 1 summarizes the offsets in χ2 displayed in theleft panel of Figure 1 and also shows how parameter con-straints based on the 3x2pt data vector are impacted byassumptions of our covariance and likelihood model. We dis-tinguish two effects here: the scatter of a maximum poste-rior parameter π (which we denote by σ[π]) and the widthof posterior constraints inferred from our likelihood model(which we denote by σπ). For our tests of likelihood non-Gaussianity we state the changes in the difference betweenthe fiducial parameter values and the upper (high) and lower(low) boundaries of the 68.3% quantile with respect to thestandard deviation of the Gaussian likelihood. For our testsof the impact of covariance cosmology, we show the mean ofall σπ obtained from our 100 different covariances and alsoindicate the scatter of these σπ values.

The effect that has the dominant impact on parameterconstraints is that of evaluating the covariance model at a17 set of parameters that do not represent the exact cos-mology of the Universe. When computing the covariance at100 different cosmologies that were randomly drawn froma Monte Carlo Markov chain (run around a fiducial modeldata vector, see Section 6.8 for details) we find that the dif-ferences between these covariances introduce an additionalscatter in maximum posterior parameter values. This scat-ter increases by about 3% for Ωm and σ8 and by about 5%for the dark energy equation of state parameter w. Thisincreased scatter is in fact the dominant effect, since thewidth of the derived parameter constraints hardly changesbetween the different covariance matrices. Note especiallythat re-running the analysis with a covariance updated tothe best-fit parameters does not mitigate this effect.

In Figure 2 we take the two effects that had the largestimpacts on χ2 and show the resulting mismatch betweenscatter of maximum posterior values and width of the in-ferred contours for a wider range of parameters. All of our

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results take into account marginalisation over nuisance pa-rameters (and all other parameters).

Our reason for exclusively investigating the impact ofcovariance errors on χ2 and parameter constraints is thatthose are the two measures by which our final (on-shot) dataanalysis will be interpreted and judged10. In the remainderof this paper we detail how the above results were obtained.

3 THE 3X2-POINT DATA VECTOR

The combined 3x2pt data vector of the DES-Y3 analysisconsists of measurements of the following 2-point correla-tions:

• the angular 2-point correlation function w(θ) of galaxy den-sity contrast measured for luminous red galaxies in 5 dif-ferent redshift bins (see e.g. Rodríguez-Monroy et al. 2020;Cawthon et al. 2020, as well as other relevant referencesgiven in Section 1),• the auto- and cross-correlation functions ξ+(θ) and ξ−(θ)between the galaxy shapes of 4 redshift bins of source galax-ies (see e.g. Amon et al. 2020; Secco et al. 2020; Myles et al.2020; Gatti et al. 2020a) ,• the tangential shear γ(θ) imprinted on source galaxy shapesaround positions of foreground redMaGiC galaxies (see e.g.Prat et al. 2020).

At the time of writing this paper the exact choices for red-shift intervals and angular bins considered for each 2-pointfunction are still being determined by a careful study of theirimpact on the robustness of DES-Y3 parameter constraints(Krause et al. 2020; DES Collaboration et al. 2020). Forthe purposes of testing the modelling of the covariance ma-trix we will use the most recent but possibly not final DES-Y3 analysis choices. We do not expect that our tests andconclusions will change in a significant manner with furtherupdated analysis choices. We assume that each of the corre-lation functions are measured in 20 logarithmically spacedangular bins between θmin = 2.5′ and θmax = 250′. Someof these bins in some of the measured 2-point functions arebeing cut from the analysis to ensure unbiased cosmologicalresults, resulting in a total of 531 data points when usingthe preliminary DES-Y3 scale cuts.

Our starting point of modelling the different 2-pointfunctions in the 3x2pt data vector is the 3D nonlinear matterpower spectrum P (k, z) at a given wavenumber k and red-shift z. We obtain it by using either of the Boltzmann solversCLASS 11 or CAMB 12 to calculate the linear power spec-trum and the HALOFIT fitting formula (Smith et al. 2003)in its updated version (Takahashi et al. 2012) to turn thisinto the late time nonlinear power spectrum. From this 3Dpower spectrum the angular power spectra required for ourthree 2-point functions (cosmic shear (κκ), galaxy-galaxylensing (δgκ) and galaxy-galaxy clustering (δgδg)) in theLimber approximation are given by (e.g. Krause et al. 2017;

10 Alternatively, one could investigate the distribution of p-values(or probability to exceed, cf. Hall & Taylor 2019) as opposed tothe distribution of χ2.11 www.class-code.net12 camb.info

Limber 1953):

Cijκκ(`) =

∫dχqiκ(χ)qjκ(χ)

χ2P

(`+ 1

2

χ, z(χ)

), (4)

Cijδgκ(`) =

∫dχqiδ

(`+ 1

2χ, χ)qjκ(χ)

χ2P

(`+ 1

2

χ, z(χ)

), (5)

Cijδgδg (`) =

∫dχqiδ

(`+ 1

2χ, χ)qjδ

(`+ 1

2χ, χ)

χ2P

(`+ 1

2

χ, z(χ)

),

(6)

where χ is the comoving radial distance, i and j denote dif-ferent combinations of pairs of redshift bins and the lensingefficiency qiκ and the radial weight function for clustering qiδare given by

qiκ(χ) =3H2

0 Ωmχ

2a(χ)

∫ χh

χ

dχ′(χ′ − χχ

)niκ(z(χ′))

dz

dχ′,

qiδ(k, χ) = bi(k, z(χ)) niδ(z(χ))dz

dχ. (7)

Here H0 is the Hubble parameter today, Ωm the ratio of to-day’s matter density to today’s critical density of the Uni-verse, a(χ) is the Universe’s scale factor at comoving dis-tance χ and bi(k, z) is a scale and redshift dependent galaxybias. Furthermore, niκ,g(z) denote the redshift distributionsof the different DES-Y3 redshift bins of source and lensgalaxies respectively, normalised such that∫dz niκ,g(z) = 1 . (8)

Note that on large angular scales the DES-Y3 analysis doesnot make use of the Limber approximation for galaxy clus-tering but instead employs the method derived in Fang et al.(2020a).

The above angular power spectra are now related to thereal space correlation functions w(θ), γt(θ) and ξ±(θ) as

wi(θ) =∑`

2`+ 1

4πP`(cos θ)Ciiδgδg (`) ,

γijt (θ) =∑`

2`+ 1

4π

P 2` (cos θ)

`(`+ 1)Cijδgκ(`) ,

ξij± (θ) =∑`>2

2`+ 1

4π

2(G+`,2(x)±G−`,2(x))

`2(`+ 1)2Cijκκ(`) .

(9)

Here P` are the Legendre polynomials of order `, Pm` arethe associated Legendre polynomials, x = cos θ and thefunctions G+,−

`,2 (x) are given in appendix A (see also Steb-bins 1996). Note that we only consider the auto-correlationswi(θ) for each tomographic bin since in the Y1 analysis itwas shown that the cross correlations do not carry signifi-cant information (Elvin-Poole et al. 2018).

The above relations between angular power spectra andreal space correlation functions can all be written in the form

ξAB(θ) =∞∑`=0

2`+ 1

4πFAB` (θ)CAB` . (10)

MNRAS 000, 000–000 (0000)

6 DES Collaboration

460 480 500 520 540 5602

finite angular binwidth in cosmic

variance termand mixed term

connected4pt function

curvedsky

non-Limberand RSD

non-Gaussianlikelihood

covariancecosmology

random pointshot-noise

non-Poissonshot-noise

masking andsurvey geometry

in cosmicvariance term

and mixed term

Gaussian vs halomodel

log-normal vs halomodel

fsky skewness

5 × fsky skewness

under estimates 2 by 18%

fsky

beyond fsky

3x2pt , Ndata = 531accounted for inDESY1 covarianceaccounted for inDESY3 covarianceestimating effects thatare unaccounted for

250 260 270 280 290 300 310 3202

finite angular binwidth in cosmic

variance termand mixed term

connected4pt function

curvedsky

non-Limberand RSD

non-Gaussianlikelihood

covariancecosmology

random pointshot-noise


masking andsurvey geometry

in cosmicvariance term

and mixed term

Gaussian vs halomodel

log-normal vs halomodel

fsky skewness

5 × fsky skewness

under estimates 2 by 30%

fsky

beyond fsky

2x2pt , Ndata = 302accounted for inDESY1 covarianceaccounted for inDESY3 covarianceestimating effects thatare unaccounted for

Figure 1. Impact of different covariance modelling choices on χ2 between measured 3x2pt (left panel) and 2x2pt (right panel) datavectors and maximum posterior models. The dashed vertical lines and error bars indicate the 1σ fluctuations expected in χ2. See maintext for details.

This is particularly useful when deriving covariance expres-sions and when performing averages over finite bins in theangular scale θ. To achieve the latter, one can simply deriveanalytic averages of the functions FAB` (θ). Both of thesepoints will be considered in the next sections.

For most of our tests we consider the 3x2pt data vec-tor and its covariance matrix at the fiducial cosmology de-scribed in section 5, where we also show the Gaussian priorsassumed on some of these parameters when assessing the im-pact of covariance modelling on parameter constraints andmaximum posterior χ2.

4 COVARIANCE MATRICES FOR THE 3×2PTDATA VECTOR

The covariance matrix of measurements of cosmological 2-point statistics typically contains three contributions (cf.Krause & Eifler 2017; Krause et al. 2017),

C = CG + CnG + CSSC . (11)

Here, CG is the contribution to the covariance that wouldbe present if the cosmic matter density and cosmic shearfields where pure Gaussian random fields (see also Schnei-der et al. 2002; Crocce et al. 2011), CnG are contributionsinvolving the connected 4-point function of these fields (thetrispectrum) and CSSC is the so-called super-sample covari-ance contribution resulting from the fact that any surveyonly observes a finite volume of the Universe and that the

MNRAS 000, 000–000 (0000)


Effect 〈χ2〉 σ[χ2] σ[Ωm] σΩm σ[σ8] σσ8 σ[w] σw

Fiducial 507.6 31.8 0.0509 0.0509 0.0975 0.0975 0.244 0.244

angularbin width

402.1 26.0 +0.8% +7.4% +0.8% +8.3% +1.0% +7.4%

connected4-pointfunction

507.6 31.8 +0.1% -0.8% +0.1% -0.9% +0.1% -0.8%

curved sky 507.7 31.8 +0.0% -0.0% +0.0% -0.0% +0.0% -0.0%

non-Limber &RSD

511.4 32.1 +0.1% -0.6% +0.1% -0.6% +0.3% -1.4%

non-Gauss.likelihood

- 32.6 +0.8% (low)-0.9% (high)

- +0.4% (low)-0.4% (high)

- +0.5% (low)+0.05% (high)

-

covariancecosmology

508.6 32.4 +2.9% +(0.1± 0.06)% +2.8% +(0.1± 0.05)% +4.7% +(0.1± 0.06)%

randompointshot-noise

511.3 32.0 +0.0% -0.5% +0.0% -0.6% +0.0% -0.2%


515.0 32.3 +0.0% -0.7% +0.0% -0.8% +0.0% -0.6%

maskingand surveygeometry

526.5 33.8 +0.6% -0.8% +0.7% -0.3% +0.3% -1.3%

Table 1. Summary of the impact of the different effects tested here on the distribution of χ2 between measurement and maximumposterior model, on the scatter σ[π] of maximum posterior parameters π and on the standard deviations σπ on these parameters inferredfrom the likelihood. See text for details.

mean density in that volume is subject to fluctuations dueto long wavelenght modes (Takada & Hu 2013; Schaan et al.2014).

In the fiducial DES-Y3 analysis we model all of thesecovariance contributions analytically. This fiducial model isdescribed in Section 4.1. In Section 4.2 we describe an alter-native model for the non-Gaussian covariance contributionsthat is used to test the robustness of our analysis with re-spect to the modelling of the trispectrum contribution. Fi-nally, Section 4.3 describes a set of log-normal simulations(Xavier et al. 2016) and the covariance matrix of the 3x2ptdata vector estimated from them. These simulations alsoallow us to test the accuracy of our Gaussian likelihood as-sumption and the treatment of masking and finite surveyarea in our fiducial covariance model.

4.1 Fiducial DES-Y3 Covariance

In our fiducial covariance matrix, we model the non-Gaussian covariance contributions CnG and CSSC using ahalo model combined with leading-order perturbation the-

ory to approximate the trispectrum of the cosmic densityfield and to compute the mode coupling between scaleslarger than the considered survey volume with scales insidethat volume. These calculations are carried out using theCosmoCov code package (Fang et al. 2020a) based on theCosmoLike framework (Krause & Eifler 2017). Our mod-elling of these contributions has not changed with respectto the year-1 analysis of DES and we refer the reader toKrause et al. (2017) as well as to the CosmoLike papers fordetails. However, the modelling of the Gaussian contributionhas changed as described in the following.

4.1.1 Gaussian covariance

Our modelling of the Gaussian covariance part has changedwith respect to the year-1 analysis in the following ways:

MNRAS 000, 000–000 (0000)

8 DES Collaboration

m b 8 ns h100 w0 b1 b2 b3 b4 b5 AIA, 1 AIA, 2 AIA, 3 AIA, 40.950

0.975

1.000

1.025

1.050

1.075

1.100

1.125

1.150

best

-fit s

catte

r / li

kelih

ood

widt

h super-Poisson vs Poissonbeyond fsky vs fskyFLASK vs fsky

Figure 2. Impact of covariance errors on the ratio of the standard deviation of maximum posterior parameters to the width of theposterior derived from the erroneous covariance. Green triangles shot the effect caused by non-Poissonian shot-noise and orange circlesshow the effect caused by the fsky approximation (cf. appendix C for our beyond-fsky treatment). These ratios have been calculatedpurely on the base of different analytic covariance models and within the linearized likelihood framework discussed in Section 5.1. Wealso show the ratio of maximum posterior parameter scatter observed from the 197 FLASK simulations to the statistical uncertaintiesexpected from a log-normal covariance matrix matching the FLASK configuration. Within the statistical uncertainties, these ratios areconsistent with 1.

• we use (and present for the first time13) analytic expressionfor the angular bin averaging of the functions FAB` (θ) (cf.Equation 10) for all 4 types of two point functions present inour data vector (see Section 6.3, this is especially relevant forthe sampling-noise contribution to the covariance, cf. Troxelet al. 2018b);

• we account for redshift space distortions (RSD) and alsouse a non-Limber calculation to obtain the galaxy-galaxyclustering power spectrum Cδgδg (`) (see Section 6.5);

• we do not make use of the flat-sky approximation anymore(see Section 6.4).

To derive expressions for the Gaussian covariance part, letus first consider an all-sky survey. If a 2-point function mea-surement ξAB(θ) could be obtained from data on the en-tire sky, then for most types of 2-point correlations it wouldbe related to power spectrum measurements CAB` from aspherical harmonics decomposition of the same all-sky datathrough Equation 10, i.e.

ξAB(θ) =

∞∑`=0

2`+ 1

4πFAB` (θ) CAB` . (12)

A notable exception to this are the cosmic shear 2-pointfunctions ξ± which obtain contributions from both the so-called E-mode and B-mode power spectra (Schneider et al.2002). For these functions equation (12) in the curved sky

13 We have shared our results with Fang et al. (2020a) who haveused them for their covariance calculations.

formalism becomes

ξij± (θ)

=∑`>2

2`+ 1

4π

2(G+`,2(x)±G−`,2(x))

`2(`+ 1)2

(CE,ijγγ (`)± CB,ijγγ (`)

),

(13)

where in the absence of shape-measurement systematics(and ignoring post-Born corrections) 〈CE,ijγγ (`)〉 = Cijκκ(`)

and 〈CB,ijγγ (`)〉 = 0.

Since this is a linear equation in C(`)’s, the covarianceof two different 2-point function measurements ξAB and ξCD

at two different angular scales θ1 and θ2 would be given interms of the covariance of the corresponding power spectrummeasurements by

Cov[ξAB(θ1), ξCD(θ2)

]=∑`1,`2

(2`1 + 1)(2`2 + 1)

(4π)2×

FAB`1 (cos θ1)FCD`2 (cos θ2) Cov[CAB`1 , CCD`2

].

(14)

Again, for ξAB(θ) = ξ±(θ) one would have to use CAB` =CEγγ(`)± CBγγ(`) in this sum.

For the auto-power spectrum of galaxy density contrastin one of our redshift bins the harmonic space covariancewould be (Crocce et al. 2011)

Cov[Ciiδgδg (`1), Ciiδgδg (`2)] =2δ`1`2

(2`1 + 1)

(Ciiδgδg (`1) +

1

ng

)2

.

(15)

Here ng is the number density of the galaxies and δ`1`2 isthe Kronecker symbol. To account for partial-sky surveys(such as DES) we simply divide this expression (and similarones for the other 2-point functions) by the observed sky

MNRAS 000, 000–000 (0000)


0.8

1.0

1.2

z-bin 1reduced " 2" = 1.1p-value = 33.1%

ratio of covariances for + ( ) (diagonal)Gauss/CosmoLikelog-normal/CosmoLikeFLASK/log-normal

0.8

1.0

1.2


0.7

0.8

0.9

1.0

1.1

1.2


101 102

[arcmin]

0.7

0.8

0.9

1.0

1.1

1.2


0.8

1.0

1.2


ratio of covariances for w( ) (diagonal)Gauss/CosmoLikelog-normal/CosmoLikeFLASK/log-normal

0.7

0.8

0.9

1.0

1.1


0.7

0.8

0.9

1.0

1.1


0.8

0.9

1.0


101 102

[arcmin]

0.8

1.0

1.2

1.4


Figure 3. Ratio of the diagonal elements of the different covariance matrices introduced in this section with respect to each other. Theleft panel compares the variances of measurements of ξ+(θ) while the right panel compares the variances of measurements of w(θ). Togive a sense of the goodness of fit between the covariance estimated from FLASK and our fiducial analytic matrix, we treat the diagonalelements of the FLASK covariance as a multivariate Gaussian whose covariance can be inferred from the properties of the Wishartdistribution (Taylor et al. 2013). The low p-value for the highest redshift bin of w(θ) most likely results from our incomplete treatmentof the survey mask (cf. discussion in Section 6 and appendix C).

MNRAS 000, 000–000 (0000)

10 DES Collaboration

Cov[Cijgg(`1), Cklgg(`2)] =

δ`1`2

[(Cikgg(`1) + δik

nig

)(Cjlgg(`1) +

δjl

njg

)+(Cilgg(`1) + δil

nig

)(Cjkgg (`1) +

δjk

njg

)](2`1 + 1)fsky

(16)

Cov[CE,ijγγ (`1), CE,klγγ (`2)] =

δ`1`2

[(Cikκκ(`1) +

δikσ2ε,i

nis

)(Cjlκκ(`1) +

δjlσ2ε,j

njs

)+

(Cilκκ(`1) +

δilσ2ε,i

nis

)(Cjkκκ(`1) +

δjkσ2ε,j

njs

)](2`1 + 1)fsky

(17)

Cov[CB,ijγγ (`1), CB,klγγ (`2)] =

δ`1`2

[δikσ

2ε,i

nis

δjlσ2ε,j

njs

+δilσ

2ε,i

nis

δjkσ2ε,j

njs

](2`1 + 1)fsky

(18)

Cov[Cijgκ(`1), Cklgκ(`2)] =

δ`1`2

[(Cikgg(`1) + δik

nig

)(Cjlκκ(`1) +

δjlσ2ε,j

njs

)+ Cilgκ(`1)Ckjgκ(`1)

](2`1 + 1)fsky

(19)

Cov[Cijgg(`1), CE,klγγ (`2)] =δ`1`2

[Cikgκ(`1)Cjlgκ(`1) + Cilgκ(`1)Cjkgκ(`1)

](2`1 + 1)fsky

(20)

Cov[Cijgg(`1), Cklgκ(`2)] =

δ`1`2

[(Cikgg(`1) +

δikσ2ε,i

nis

)Cjlgκ(`1) + Cilgκ(`1)

(Cjkgg (`1) +

δjk

njg

)](2`1 + 1)fsky

(21)

Cov[Cijgκ(`1), CE,klγγ (`2)] =

δ`1`2

[Cikgκ(`1)

(Cjlκκ(`1) +

δjlσ2ε,j

njs

)+ Cilgκ(`1)

(Cjkκκ(`1) +

δjkσ2ε,j

njs

)](2`1 + 1)fsky

(22)

Cov[Cijgg(`1), CB,klγγ (`2)] = 0 (as are all other covariances with only one CBγγ) . (23)

MNRAS 000, 000–000 (0000)


At this point let us introduce the following nomenclature:we will denote the terms that contain two power spectra ascosmic variance contribution to the covariance, the termsthat contain no power spectrum at all as the sampling noisecontributions (or shape noise and shot noise contributions)and the terms that contain contribution from one powerspectrum and a sampling noise as the mixed terms. We testthe accuracy of the fsky-approximation that results in Equa-tions (16-23) in Section 6.6 by comparing it to more accurateexpressions.

4.2 Analytic lognormal covariance model

To test the robustness of the CosmoLike covariance we alsoemploy an alternative model for the connected 4-point func-tion part of the covariance - the lognormal model. Hilbertet al. (2011) originally derived this as a model for the co-variance of cosmic shear correlation function, assuming thatthat the lensing convergence κ can be written in terms of aGaussian random field n as (see also Xavier et al. 2016)

κ = λ(en+µ − 1

), (24)

where it is assumed that 〈n〉 = 0. For given values λ >0 and µ the power spectrum of n can be chosen such asto reproduce a desired 2-point correlation function ξκ (seeXavier et al. (2016) for caveats). Furthermore, for any givenvalue λ > 0 one can choose µ such that 〈κ〉 = 0. This makesλ the only free parameter of the lognormal covariance model.Hilbert et al. (2011) show that this model leads to a numberof correction terms to the Gaussian covariance model, andidentify the most dominant of these terms to be

CLN[ξκ(θ1), ξκ(θ2)]

≈ CG[ξκ(θ1), ξκ(θ2)] +4 ξκ(θ1)ξκ(θ2)

ASλ2VarS(κ) . (25)

Here AS is the area of the considered survey footprintand VarS(κ) is the variance of κ when averaged over thefootprint. We generalise this to the covariance of 2-pointcorrelations ξAB and ξCD between arbitrary scalar fieldsδA, δB , δC , δD as

CLN[ξAB(θ1), ξCD(θ2)]− CG[ξAB(θ1), ξCD(θ2)]

≈ ξAB(θ1)ξCD(θ2)

AS

CovS(δA, δC)

λAλC+

CovS(δA, δD)

λAλD+

+CovS(δB , δC)

λBλC+

CovS(δB , δD)

λBλD

. (26)

Here, CovS(δA, δC) is the covariance of δA and δC af-ter the two fields have been averaged over the entire sur-vey footprint (and likewise for the other terms appearingabove). Following Hilbert et al. (2011) we use this expres-sion even when considering non-scalar fields (i.e. the shearfield) by replacing ξXY (θ) by the appropriate 2-point func-tions ξ+(θ), ξ−(θ), γt(θ) (or w(θ), for the scalar galaxy den-sity contrast).

To choose the parameters λX (also called the lognormalshift parameters, cf.Xavier et al. 2016) we follow a proceduresimilar to the one outlined in Friedrich et al. (2018). Thereit is shown how the value of λX can be adjusted in order tomatch the re-scaled cumulant

S3(ϑ) ≡ 〈δX(ϑ)3〉〈δX(ϑ)2〉2 (27)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6z

0

1

2

3

4

5

6

7

8

n(z)

source galaxieslens galaxies

Figure 4. Redshift distributions of lens galaxies (shaded regions)and source galaxies (solid lines) in our fiducial test configuration.

of the random field δX smoothed with a top-hat filter ofangular radius ϑ to the value of S3 predicted by leading-order perturbation theory for that same smoothing scale.Since the focus in our paper is the covariance matrix of 2-point statistics (hence a 4-point function), we modify theirmethod to match instead the value of reduced fourth ordercumulant

S4(ϑ) ≡ 〈δX(ϑ)4〉 − 3〈δX(ϑ)2〉2

〈δX(ϑ)2〉3 . (28)

The field δX here will be either projections of the 3D matterdensity contrast along the line-of-sight distribution of ourlens galaxies or the lensing convergence fields correspond-ing to our 4 source redshift bins. The smoothing scale ϑ atwhich we use the λX to match S4 to its perturbation theoryvalue is chosen such that it corresponds to about 10Mpc/hat the mean redshift of the line-of-sight projection kernelscorresponding to the different δX . This is approximately thescale at which Friedrich et al. (2018) found the shifted log-normal model to be a good approximation of the overallPDF of density fluctuations in N-body simulations (cf. theirfigure 5).

Our results are shown in Table 2 , where we presentthe number density, galaxy bias (relevant for lenses only),shape-noise dispersion (per shear component; relevant forsources only) and the lognormal shift parameters obtainedfrom the procedure described above. Note that for the sourcegalaxy samples, the relevant line-of-sight projection kernelused to derive the shift parameter is the lensing kernel (andnot the redshift distribution of the source galaxies). For thelens galaxies, all shift parameters come out to be > 1. As aconsequence there will be pixels with negative density in ourlognormal simulations. However, the fraction of such pixels is< 0.01 for all runs and all bins and setting δg = −1 in thesebins has an unnoticeable effect on the statistics measured inthese maps (e.g. for bin 4, which is affected most, the stan-dard deviation of δg changes by 0.053%). Note further thatat the time of completing the simulation runs presented inSection 4.3, the DES Y3 shear catalog and redshift distribu-tion were not finalized. As a consequence, the shape noisedispersion values used for simulations differ from the valuesin this table.

MNRAS 000, 000–000 (0000)


z-bin ng [arcmin−2] bias σε log-normal shift

lenses 1 0.0221 1.7 − 1.089lenses 2 0.0381 1.7 − 1.106

lenses 3 0.0583 1.7 − 1.047

lenses 4 0.0295 2.0 − 1.252lenses 5 0.0251 2.0 − 1.177

sources 1 1.7971 − 0.2724 0.00453sources 2 1.5521 − 0.2724 0.00885

sources 3 1.5967 − 0.2724 0.01918

sources 4 1.0979 − 0.2724 0.03287

Table 2. Number density, galaxy bias (relevant for lenses only),shape-noise dispersion (per shear component; relevant for sourcesonly) and the lognormal shift parameters obtained from the pro-cedure described in Section 4.2.

Figure 5. Validation of FLASK simulations. Each panel showsthe absolute difference of three 2-point correlations measuredon FLASK realizations and the predicted correlation functionsfrom input C(`)s normalized to the statistical error given by thestandard deviation along FLASK realizations (∆X/σX , whereX = w, γt, ξ+, ξ−). Gray dots are single realizations and bluedots its mean.

4.3 Lognormal covariance from simulations

We also produce a test DES-Y3 covariance matrix from a setof simulations. We use the publicly available code FLASK(Full sky Lognormal Astro fields Simulation Kit) (Xavieret al. 2016) 14 to generate 800 DES-Y3 footprint sky mapsof density, convergence and shear healpix maps (Górski et al.2005) with NSIDE=8192, as well as galaxy positions cata-logs, used to reproduce the DES-Y3 properties. FLASK isable to quickly produce tomographic correlated simulationsof clustering and weak lensing lognormal fields based on theDES-Y3 lens and sources samples. The lognormal distribu-tion of cosmological fields has been shown to be a good ap-proximation (Coles & Jones 1991; Wild et al. 2005; Clerkinet al. 2017) but much less computationally expensive to gen-erate than full N-body simulations.

As input for the simulations, we used a set of autoand cross correlated power spectrum and the lognormalfield shift parameters. The theoretical input power spec-trum was generated using CosmoLike, and the lognormal

14 http://www.astro.iag.usp.br/ flask/

Figure 6. FLASK (lower diagonal) vs. CosmoLike halo model(upper diagonal) correlation matrix.

shifts are the ones listed in Table 2. In order to repro-duce the properties of shear fields, we added the shape-noise term by sampling each pixel of the simulated mapsto match the correspondent shape-noise dispersion σε andnumber density ng of the tomographic bin. At the timeof completing the simulation runs, the DES Y3 shear cat-alog and redshift distribution were not finalized. For thisreason, the values used in the simulations are slightly dif-feent from the values in Table 2. For the simulations,we set the number density for the five tomographic lensbins as 0.0227, 0.0392, 0.0583, 0.0451, 0.0278 (arcmin−2).The shape-noise dispersion values for the four tomographicbins of sources were set to 0.27049, 0.33212, 0.32537, 0.35037.The cosmology adopted for the theoretical power spectra isset as Ωm = 0.3, σ8 = 0.82355, ns = 0.97, Ωb = 0.048,h0 = 0.69, and Ωνh

20 = 0.00083.

We use the publicly available code TreeCorr15 (Jarviset al. 2004) to measure the 3x2 point correlation measure-ments for 200 DES-Y3 realizations. For all measurements,we used 20 log-spaced angular separation bins on scales be-tween 2.5 and 250 arcmin. We set the bin_slop TreeCorrparameter to zero, essentially setting all estimators to brute-force computation. In Figure 5 we show the validation of themeasurements comparing with the theoretical input.

We will use the FLASK covariance mainly to estimatethe impact of the survey geometry.

4.4 Comparisons among covariances

Here we present some comparisons between the different co-variance matrices. In Figure 3 we show the ratio of the diag-

15 https://github.com/rmjarvis/TreeCorr

MNRAS 000, 000–000 (0000)

https://github.com/rmjarvis/TreeCorr


onal elements of the different covariance matrices introducedin this section displaying both the variances of the measure-ments of ξ+(θ) of w(θ).

In Figure 6 we compare the covariance matrices ob-tained from the FLASK simulations and the analytical halomodel covariance.

5 IMPACT OF COVARIANCE ERRORS ON ALINEARIZED GAUSSIAN LIKELIHOOD

As discussed above a full assessment of the impact of usingdifferent covariance matrices to parameter estimation be-comes unfeasible due to the computational demand of run-ning a large number of MCMC chains. Since the covariancematrices studied in this work differ by subdominant effectswe do not expect large modifications in the results of theestimation of the parameters. Therefore we will bypass thisdifficulty by using a linearized approximation of the modeldata vector as a function of the parameters. The measureddata is assumed to be a Gaussian multivariate variable char-acterized by a covariance matrix and a given prior matrix.This approach is called the Gaussian linear model (Seeharset al. 2014, 2016; Raveri & Hu 2019).

Within this approach we study the following impacts ofdifferent covariances:

• error in the parameter estimation, characterized by thewidth of the contours;• the scatter of the best fit (maximum posteriors) parame-ters;• change in the maximum posterior χ2 value;• error in the maximum posterior χ2 value.

In the remainder of this section we detail this method.

5.1 Linearized likelihoods

To speed up our simulated likelihood analyses, we employa linearized model of the data vector ξ (e.g. the DES-Y33x2-point function data vector). This can be considered alinear Taylor expansion of our full model around a fiducialset of parameters π0 which is summarized in Table 3. In thisapproximation our model data vector becomes

ξ(π) = ξ(π0) +∑α

(πα − π0α)

∂ξ(π)

∂πα

∣∣∣∣π=π0

(29)

where the sum is over all components πα of the parametervector π (we will use latin indices for the components ofthe data vector and greek indices for the components of theparameter vector). Given a 2-point function measurement ξand abbreviating

ξ0 = ξ(π0)

δξ = ξ − ξ0

δπ = π − π0

∂αξ =∂ξ(π)

∂πα

∣∣∣∣π=π0

Table 3. Fiducial cosmology and standard deviation of Gaussianparameter priors used in our mock likelihood analyses. AIA,i is theintrinsic alignment amplitude in the ith source redshift bin, mi isthe multiplicative shear bias and ∆zs,i parametrizes systematicshifts in the photometric redshift distribution of that bin. ∆zl,iparametrizes systematic shifts in the photometric redshift distri-bution of the ith lens redshift bin. The Gaussian priors we choosefor the parameters follow the analysis choices of Abbott et al.(2018) and we assume infinite flat priors for all other parameters.

Parameter Fiducial value σprior

CosmologyΩm 0.3 -σ8 0.82355 -h100 0.69 -ns 0.97 -w0 -1 -Ωb 0.048 -Ων 0.001743 -ΩΛ 1− Ωm − Ων

b1 1.7 -b2 1.7 -b3 1.7 -b4 2.0 -b5 2.0 -

∆zl,1 0.0 0.04

∆zl,2 0.0 0.04∆zl,3 0.0 0.04

∆zl,4 0.0 0.04

∆zl,5 0.0 0.04

∆zs,1 0.0 0.08

∆zs,2 0.0 0.08∆zs,3 0.0 0.08

∆zs,4 0.0 0.08

AIA,1 0.0 -AIA,2 0.0 -AIA,3 0.0 -AIA,4 0.0 -

m1 0.0 0.03m2 0.0 0.03

m3 0.0 0.03

m4 0.0 0.03

our figure of merit χ2 as a function of the parameters be-comes in the linearized approximation

χ2[δπ] =

(δξ −

∑α

δπα∂αξ

)TC−1

(δξ −

∑α

δπα∂αξ

)

+(π − πprior

)TP(π − πprior

). (30)

Here we have allowed for a Gaussian prior with covariancematrix P−1 and central value πprior. To find the deviationδπMP = πMP − π0 from our fiducial parameters that min-imizes this function (the maximum posterior value of theparameters is denoted by πMP) we have to solve

∂χ2

∂(δπβ)

∣∣∣∣δπ=δπMP

= 0 . (31)

MNRAS 000, 000–000 (0000)


Defining a vector x such that xβ = δξTC−1∂βξ as well asthe Fisher matrix Fαβ = ∂βξ

TC−1∂αξ this becomes

(F + P) δπMP = x + P (πprior − π0)

⇒ πMP = π0 + (F + P)−1x + (F + P)−1P (πprior − π0) .(32)

We now want to consider the situation when a modelcovariance matrix Cmod is used to calculate the likelihood inequation (30 which is different from the true covariance ma-trix Ctrue of the statistical uncertainties in the data vectorξ. In that case our linearized likelihood will be a Gaussiancentered around πMP and with parameter covariance matrix

Cπ,like = (Fmod + P)−1 , (33)

where Fmod,αβ = ∂βξTC−1

mod∂αξ is the Fisher matrix calcu-lated from the model covariance.

The actual covariance matrix of πMP includes twosources of noise. First, statistical uncertainties in the mea-surement ξ which are described by the covariance matrixCtrue and are represented by the first term in equation(32) that is proportional to x. And secondly, statistical un-certainties in our choice of the prior center which are de-scribed by the prior covariance matrix P−1 and are repre-sented by the second term in equation (32) that is propor-tional to πprior. The latter term has the covariance matrix(Fmod + P)−1P(Fmod + P)−1 (because the covariance ma-trix of πprior is P−1). Hence, the total covariance matrix ofπMP can be written as

(Cπ,MP)αβ ≡ Cov[πMPα , πMP

β ] =

= (Fmod + P)−1P(Fmod + P)−1 +

+∑κ,λ

(Fmod + P)−1ακ (Fmod + P)−1

λβ ×

×∑i,k

∂κξi (C−1modCtrueC

−1mod)ik ∂λξk . (34)

For Cmod = Ctrue it is easy to see that this parameter co-variance Cπ,MP equals the covariance Cπ,like that describesthe shape of our likelihood (as it should).

5.2 Impact on the width of the likelihood andscatter of best fit parameters

We can use the above findings to study the impact of differ-ent effects in covariance modelling on parameter constraints.If a covariance matrix C1 contains a noise contribution thatis missing in another covariance matrix C2, then we quan-tify the difference between these matrices by considering twoeffects:

• Width of likelihood contours:

Denoting the Fisher matrices obtained from C1 or C2 as F1

and F2 respectively, the width of likelihood contours drawnfrom the different covariances are given by

Cπ,like, 1 = (F1 + P)−1

Cπ,like, 2 = (F2 + P)−1 . (35)

Hence, if the difference C1 − C2 = E represents noise

contributions missing from (or miss-estimated in C2), thena comparison of Cπ,like, 1 and Cπ,like, 2 quantifies theimpact of this on the width of parameter contours.

• Scatter in the center of likelihood contours:

If the data vector ξ had C1 as its true covariance matrix butC2 would be used to derive the maximum posterior param-eters πMP from it, then the maximum posterior parametercovariance would be given by

(Cπ,MP, 2)αβ = (F2 + P)−1 P (F2 + P)−1 +

+∑κ,λ

(F2 + P)−1ακ (F2 + P)−1

λβ ×

×∑i,k

∂κξi (C−12 C1 C−1

2 )ik ∂λξk . (36)

If the difference C1−C2 = E represents noise contributionsmissing from (or miss-estimated in C2), then a comparisonof Cπ,MP, 2 and Cπ,MP, 1 ≡ Cπ,like, 1 quantifies the impactof this on the scatter in the location of parameter contours.

An inaccurate covariance model will in general have adifferent impact on the width and the location of parame-ter contours. Hence, in order to quantify the importance ofdifferent effects in covariance modelling for parameter esti-mation, we compare both the pair Cπ,like, 1 / Cπ,like, 2 andthe pair Cπ,MP, 1 / Cπ,MP, 2.

5.3 Distribution of χ2 when fitting for parameters

Within the linearized likelihood model developed in theprevious section we now investigate how errors in the co-variance model impact the distribution of χ2

MP betweenmeasured data vector ξ and a maximum posterior modelξMP = ξ(πMP),

χ2MP = (ξ − ξMP)TC−1(ξ − ξMP) . (37)

We start with the case that

(i) the true covariance C of ξ is known(ii) no parameter priors are used when determining the best

fitting model ξMP

(iii) the true expectation value ξ ≡ 〈ξ〉 lies within our pa-rameter space. I.e. there are parameters πtrue such thatξ(πtrue) = ξ .

We will show that, as expected, in this case χ2MP should

follow a χ2-distribution with Ndata−Nparam degrees of free-dom.

Using equations (29) and (32) (and setting again δξ ≡ξ−ξ0) one can see that the maximum posterior data vectoris given by

ξMP = ξ0 +∑αβ

∂αξ (F−1)αβ(δξTC−1∂βξ

)= ξ +

∑αβ

∂αξ (F−1)αβ(

(ξ − ξ)TC−1∂βξ)

= ξ +∑αβ

∑kl

∂αξ (F−1)αβ (ξk − ξk)(C−1)

kl∂βξl

≡ ξ + P · (ξ − ξ) . (38)

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Here, the second line follows from the fact that ξ = 〈ξ〉 =〈ξMP〉 and we have defined the matrix

Pij =∑αβ

∂αξi∑l

(F−1)αβ(C−1)

lj∂βξl . (39)

It can be shown that P is an idempotent matrix (P2 = P)and furthermore that

Trace (P) = Nparam

C−1PC = PT . (40)

The residual between the measurement ξ and the best fittingmodel ξMP can be written in terms of P as

ξ − ξMP = (ξ − ξ)− (ξMP − ξ)

= (1−P) · (ξ − ξ) . (41)

Hence, the covariance matrix of ξ − ξMP is given by

CP ≡ 〈(ξ − ξMP)T (ξ − ξMP)〉 = (1−P)C (1−P)T (42)

This makes it straightforward to find the expectation value

〈χ2MP〉 =〈(ξ − ξMP)TC−1(ξ − ξMP)〉

= Trace(CP C−1)

=∑jk

Ckj(C−1)

jk−∑k

Pkk

= Ndata −Nparam . (43)

Similarly, the variance of χ2MP can be shown to be

Var(χ2MP) = 〈(χ2

MP)2〉 − 〈χ2MP〉2

= 2 Trace([

CP C−1]2)= 2(Ndata −Nparam) . (44)

So far, we have only re-derived textbook results (Anderson2003). Now how do 〈χ2

MP〉 and Var(χ2MP) change if the co-

variance model Cmod we use to find the best fitting modelξMP and to compute χ2

MP is different from the true covari-ance matrix C of ξ?

Following similar steps as Eqs. (38) and (39) one canshow that

ξMP = ξ + Pmod · (ξ − ξ) (45)

where

(Pmod)ij =∑αβ

∂αξi∑l

(F−1mod)αβ

(C−1

mod

)lj∂βξl (46)

and where the Fisher matrix Fmod is computed from themodel covariance Cmod. Equation 45 especially shows thatξMP is still an unbiased estimator of ξ even when Cmod 6= C.When deriving the moments of χ2

MP we will still come acrossexpectation values like (cf. Equation 43)

〈(ξi − ξi)(ξj − ξj)〉 ≡ (C)ij 6= (Cmod)ij . (47)

Hence the expectation value and variance of χ2MP are given

by

〈χ2MP〉 = Trace

(CPmodC−1

mod

)(48)

Var(χ2MP) = 2 Trace

([CPmodC−1

mod

]2), (49)

where

CPmod = (1−Pmod)C (1−Pmod)T (50)

Now we are left to investigate how Equations 48 and 49change when a Gaussian parameter prior P is included inthe likelihood function (cf. Equation 30). A complication inthis case is, that now ξMP is not necessarily an unbiased es-timate of ξ anymore. This is because in Equation 30 we havecentered our prior around the model parameters πprior whichmay be different from the true parameters πtrue. Insertingthe full expression for the maximum posterior parameters(Equation 32) into our linearized model we now get

ξMP = ξ0 + Pmod · (ξ − ξ0) + ζ (51)

with

(Pmod)ij =∑αβ

∂αξi∑l

(Fmod + P)−1αβ

(C−1

mod

)lj∂βξl

ζ =∑α

[(Fmod + P)−1P (πprior − π0)

]α∂αξ

(52)

The residual between ξ and ξMP hence becomes

ξ − ξMP = (1−Pmod) · (ξ − ξ0)− ζ . (53)

Treating the prior center πprior again as a random vectorcentered around πtrue, ζ also becomes a random vector withcovariance

(Cζ)ij ≡ Cov[ζi, ζj ]

=∑αβγδ

∂αξi (Fmod + P)−1αβ Pβγ (Fmod + P)−1

γδ ∂δξj .

(54)

Hence, along lines similar to the case without a prior, wecan write the moments of χ2

MP for a given model covarianceas

〈χ2MP〉 = Trace

(CPmod + Cζ C−1

mod

)(55)

Var(χ2MP) = 2 Trace

([CPmod + Cζ C−1

mod

]2). (56)

Notice that in the absence of priors Cζ = 0 and for the truecovariance C we recover equations (43) and (44) as expected.Equations (55) and (56) are used to produce our main resultshown in Figure 1 for different covariance matrices.

6 EXPLORING DIFFERENT EFFECTS INTHE COVARIANCE MODELLING

Our main goal is to study the impact of including differ-ent effects in the covariance modelling on the estimationof parameters. Several covariance matrices were generatedand tested under different assumptions and approximations.The main results were already shown in Section 2. We nowpresent the details of each step in the validation strategythat was outlined in Section 5.

6.1 Gaussian likelihood assumption

A basic assumption of our framework of testing differentcovariance matrices is that the likelihood function of thedata is Gaussian. One simple reason of why the samplingdistribution of the correlation functions can not be an exactmultivariate Gaussian is that this violates the positivity con-straint of the power spectrum (Schneider & Hartlap 2009).

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350 400 450 500 550 600 6502

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

p(2 )

Gaussian distribution

log-normal distributionskewness from fsky

log-normal distribution5 × skewness from fsky

0.5 0.6 0.7 0.8 0.9 1.0ML 8

0

1

2

3

4

5

6

7

p(8)

expected distribution forGaussian likelihood

log-normal distributionskewness from fsky

log-normal distribution5 × skewness from fsky

Figure 7. Top panel: Distribution of χ2 when drawing 3x2ptdata vectors from a Gaussian distribution (blue histogram), froma shifted log-normal distribution where the skewness of each datapoint was computed in the fsky approximation (red histogram)and when assuming that the skewness of the data points is 5 timesthat of the fsky approximation (green histogram). Bottom panel:Distribution of maximum posterior σ8 when fitting the linearizedmodel to Section 5.1 Gaussian realisations of our fiducial datavector , to lognormal realisations of our fiducial data vector (bluehistogram) and to lognormal realisations with 5 times the skew-ness of the fsky approximation employed on Section 6.1 (orangehistogram).

There are also other reasons described below. The purposeof this Subsection is to assess the impact of non-Gaussianityof the likelihood of 2-point functions in the parameter esti-mation. In this sense checking this basic assumption is a testof the whole framework and is different from the robustnesstests for the covariance matrix modelling described in theremaining Subsections of this Section.

The impact of a non-Gaussian likelihood in parameterestimation of weak lensing correlation functions has beenrecently studied in Lin et al. (2020) where no significantbiases were found in one-dimensional posteriors of Ωm andσ8 between the multivariate Gaussian likelihood model andmore complex non-Gaussian likelihood models. Also in Sel-lentin et al. (2018) the skewed distributions of weak lensingshear correlation functions are used to derive an analyticalexpression for a non-Gaussian likelihood.

We first consider a full-sky survey such that each of our

2-point function estimators ξAB(θ) is a harmonic transformof a harmonic space estimator CAB` (cf. Equation 12), i.e.

ξAB(θ) =

∞∑`=0

2`+ 1

4πFAB` (θ) CAB` . (57)

Each CAB` is given in terms of the spherical harmonics co-efficients a`m, b`m of two Gaussian random fields as

CAB` =1

2`+ 1

∑m=−`

a`mb∗`m . (58)

The product of two Gaussian random variables does notfollow a Gaussian distribution. Therefore, in principle onewould not expect CAB` (and consequently ξAB(θ)) to havea Gaussian likelihood. However, at small scales, i.e. at highmultipoles ` , the sum of the random variables a`mb∗`m inEquation 58 will approach a Gaussian distribution by meansof the central limit theorem, since there is a large number(2` + 1) of independent modes. It should be pointed outthat at these small scales the galaxy density and shear fieldscharacterized by a`m and b`m are themselves non-Gaussiandue to the non-linear evolution of gravity.

It is hence our working hypothesis that non-Gaussianityof CAB` only matters at the largest scales (small `’s) whereboth a`m and b`m can be considered Gaussian random vari-ables but not their product. In the full-sky case it can thenbe shown that the second and third central moments of CAB`are given by

〈(CAB` − CAB`

)2

〉 =

[(CAB`

)2+ CAA` CBB`

]2`+ 1

(59)

〈(CAB` − CAB`

)3

〉 =2[(CAB`

)3+ 3CAA` CBB` CAB`

](2`+ 1)2

. (60)

If only a fraction fsky of the sky is being observed, thesemoments get divided by fsky and f2

sky respectively.Assuming different multipoles to be uncorrelated, the

corresponding moments of ξAB(θ) can be computed as

〈(ξAB(θ)− ξAB(θ)

)2

〉

=

∞∑`=0

(2`+ 1

4πFAB` (θ)

)2

〈(CAB` − CAB`

)2

〉 (61)


)3

〉

=

∞∑`=0

(2`+ 1

4πFAB` (θ)

)3

〈(CAB` − CAB`

)3

〉 . (62)

Equation 61 is of course nothing but the diagonal of thecovariance matrix (cf. Equation 14).

The dominant effect of the non-Gaussianity of the C`’sis a positive skewness in the distribution of our data vectors(Sellentin et al. 2018). To estimate its impact on our param-eter constraints, we approximate the entire distribution ofour 3x2pt data vector by a multivariate lognormal distribu-tion. The covariance of our data vector and the skewness ofeach data point as given by Equation 62 are sufficient to fixthe parameters of a shifted log-normal distribution. We havealready discussed this in Sections 4.2 and 4.3, though witha conceptual difference: in that section we describe how toconfigure log-normal simulations of the cosmic density field,

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while here we assume measurements of the 3x2-point func-tions to have a multivariate log-normal distribution. To beexplicit, we fix the shift parameters λ(θ) that enter the log-normal PDF of the measurements ξAB(θ) in the differentangular bins (cf. Equation 24 for the definition of λ) via theequation


)3

〉 =

3〈(ξAB(θ)− ξAB(θ)

)2

〉2

λ(θ)+〈(ξAB(θ)− ξAB(θ)

)2

〉3

λ(θ)3, (63)

which relates the 2nd and 3rd central moments of log-normalrandom variables (Hilbert et al. 2011).

In the top panel of Figure 7 we show the impact ofthis non-Gaussianity on the distribution of maximum pos-terior χ2. For that figure we generated 300, 000 random re-alisations of our fiducial data vector from a multi-variateGaussian distribution, 300, 000 random realisations of thatdata vector from a multi-variate lognormal distribution and300, 000 random realisations from another lognormal distri-bution, whose skewness in each data point was increased bya factor of 5. For each of these random realisations we an-alytically determined the maximum posterior model withinthe linearized likelihood formalism of Section 5.1 and thencomputed the χ2 between that model and the random re-alisation. The blue histogram in the top panel of Figure7 shows the distribution of these χ2 values for the Gaus-sian random realisations and the red histogram correspondsto the log-normal random realisations. The two histogramsare almost identical. Hence, within the fsky-approximationemployed above non-Gaussianity in the likelihood does notseem to affect our analysis. And even in the extreme sce-nario of enhancing the skewness of the data vector by afactor of 5 (green histogram) the increase in the scatter ofχ2 remains smaller than about 3% of the average χ2 - whichstill wouldn’t dominate over the other effects discussed insubsequent sections (cf. Figure 1).

The impact of non-Gaussianity on the likelihood be-comes even more negligible when directly considering thedistribution of maximum posterior parameters. We demon-strate this in the bottom panel of Figure 7 for the best-fitvalues of σ8 but find similar results for our other key cosmo-logical parameters Ωm and w0. Therefore, we conclude thatit is safe to assume a Gaussian distribution for the statisticaluncertainties of the DES-Y3 2-point function measurements.

6.2 Modelling of connected 4-point function incovariance

The connected 4-point contribution to the covariance is thepart that is most challenging to model analytically (Schnei-der et al. 2002; Hilbert et al. 2011; Sato et al. 2011; Takada &Hu 2013). This contribution is most relevant at small scalesand turns out to be a small one for current large-scale struc-ture analyses (Krause et al. 2017; Barreira et al. 2018). Thisis for two reasons: 1) such analyses typically cut away theirsmallest scales because of uncertainties in the modelling oftheir data vectors and 2) at small scales the covariance ma-trix is often dominated by shape noise and shot noise whichare believed to be well understood.

We test whether the non-Gaussian covariance parts (by

which we mean both the connected 4-point function andsuper-sample covariance) are a relevant contribution to ourerror budget by either

• replacing the non-Gaussian contributions from the fiducialhalo model with the lognormal covariance described in Sec-tion 4.2• or setting it to zero, i.e. using a Gaussian covariance matrixonly.

Figure 1 and Table 1 show that neither of these changes hasa significant impact on the distribution of χ2 and our param-eter constraints. Assuming that our halo model and lognor-mal recipes do not underestimate the non-Gaussian covari-ance parts by orders of magnitude (see e.g. Sato et al. (2009);Hilbert et al. (2011) for justifications of this assumption) thisdemonstrates that we are insensitive to the exact modellingof these contributions. At the same time, we want to stressthat this finding holds for the specific scale cuts, redshiftdistributions and tracer densities of the DESY3 3x2pt anal-ysis and cannot necessarily be generalized to other analysissetups.

6.3 Exact angular bin averaging

Equation 14 holds when measuring the 2-point correlationfunctions in infinitesimally small bins around the angularscales θ1 and θ2. This is unfeasible in practice and in factalso leads to divergent covariance matrices. This can for ex-ample be seen for the galaxy clustering correlation func-tions, where the constant term proportional to 1/n2

g in theharmonic space covariance gives a contribution to the realspace covariance of

1

4π2n2gfsky

limN→∞

N∑`=1

(2`+ 1)

2P` (cos θ)2

→ 1

4π2n2gfsky

δD(cos θ − cos θ)

(=∞) .

The reason for this divergence is simply the fact that thenumber of galaxy pairs found in an infinitesimal bin van-ishes, leading to infinite shot-noise. This problem disappearswhen considering finite angular bins.

To analytically average over a finite angular bin[θmin, θmax], we assume that the number of galaxy pairs withangular separation θ is proportional to sin θ (correspondingto a uniform distribution of galaxies on the sky). We thenreplace the functions FAB` (θ) in Equations 9 and 10 by

FAB` (θ)→ 1

cos θmin − cos θmax

∫ θmax

θmin

dθ sin θ FAB` (θ) .

(64)

For the galaxy clustering correlation function w(θ) this leadsto

P`(cos θ)→[P`+1(x)− P`−1(x)]cos θmin

cos θmax

(2`+ 1)(cos θmin − cos θmax). (65)

The corresponding expressions for the galaxy-galaxylensing correlation function γt(θ) and for the cosmic shearcorrelation functions ξ± are presented (together with deriva-tions of all the bin averaged expressions) in appendix B.

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We show below how the bin averaging solves the prob-lem of diverging diagonal values of the covariance for w(θ).This can be seen from

∑`

([P`+1 (x)− P`−1 (x)]cos θmin

cos θmax

)2

2(2`+ 1)fsky(ngAbin)2

=

∫ cos θmin

cos θmax

dx1

∫ cos θmin

cos θmax

dx2

∑`

2`+ 1

2

P` (x1)P` (x2)

fsky(ngAbin)2

=

∫ cos θmin

cos θmax

dx1

∫ cos θmin

cos θmax

dx2δD(x1 − x2)

fsky(ngAbin)2

=cos θmin − cos θmax

fsky(ngAbin)2

=2

AsurveyAbinn2g

=1

Npair, (66)

where Asurvey = 4πfsky is the total survey area, Abin =2π (cos θmin − cos θmax) is the bin area and Npair the totalnumber of galaxy pairs used to estimate w. The above ex-pression is the usual formula for the shot-noise part of thereal space covariance.

The impact of the exact angular bin averaging for thenoise and mixed terms in the Gaussian part of the covari-ance matrix is included for all 4 types of two point functionspresent in the DES-Y3 data vector and the DES-Y3 fiducialcovariance.

6.4 Flat vs. Curved sky

For the Y1 analysis it was shown that the flat-sky approxi-mation was valid for the galaxy-galaxy shear and shear-shear2-point correlation function (Krause et al. 2017). In Y3 thefiducial covariance computes the full sky correlations, seeequations (12) and (13). We show in Fig. (1) that the ef-fect of including curved sky results has negligible impact onthe χ2 distribution. Table 1 shows that this is also true forparameter constraints.

6.5 RSD and Limber approximation and redshiftspace distortion effects

The modelling of the angular power spectrum of two tracersinvolves a projection from the three dimensional powerspectrum that requires integrals with integrands containingthe product of two spherical Bessel functions, which arehighly oscillatory. The inclusion of redshift space distortion(RSD) effects in a simple linear modelling (Kaiser 1987)involves the computation of those integrals with derivativesof the Bessel functions. These integrals are notoriouslydifficult to perform numerically and it is usual to apply theso-called Limber approximation (Limber 1953; LoVerde &Afshordi 2008). An efficient computation of these integralswithout resorting to the Limber approximation was recentlyimplemented in the case of the angular power spectrumfor galaxy clustering in Fang et al. (2020b). We use theirapproach to study the impact of taking into account bothnon-Limber computations and RSD effects in the covariancematrix. Figure 1 and Table 1 show that not taking theseeffects into account leads to an increase in average χ2 of

about 0.5% and an underestimation of uncertainties in keycosmological parameters by 0.6% to 1.4%.

6.6 Effect of the mask geometry

The analytical covariance models described in Section 4make use of the so called fsky approximation, i.e. they takethe covariance of an all-sky survey and divide this by thesky fraction of DES-Y3 to approximate the covariance of ourpartial sky data. In appendix C we show how to go beyondthis approximation. First, we note there that the covari-ance of the 2-point function measurements between pairs ofscalar random fields (δa, δb) and (δc, δd) within angular bins[θab− , θ

ab+ ] and [θcd− , θ

cd+ ] is given by

Covξab[θab− , θ

ab+ ], ξcd[θcd− , θ

cd+ ] Nab

pair[θab− , θ

ab+ ] Ncd

pair[θcd− , θ

cd+ ]

nanbncnd

= (2π)2∑`1 `2

[P`1+1(x)− P`1−1(x)]θab−θab+

[P`2+1(x)− P`2−1(x)]θcd−θcd+·

CovCab`1 , C

cd`2

. (67)

Here P` are again the Legendre polynomials and the an-gular bin averaging was already evaluated. The factorNab

pair[θab− , θ

ab+ ] (resp. Ncd

pair[θcd− , θ

cd+ ]) is the average number of

pairs of random points that uniformly sample the footprintwith densities na, nb (resp. nc, nd) and whose separationfalls into the angular bin [θab− , θ

ab+ ] (resp. [θcd− , θ

cd+ ]). Hence,

these factors describe how the exact survey geometry sup-presses the number of pairs of positions in the bins [θab− , θ

ab+ ]

and [θcd− , θcd+ ] with respect to the fsky approximation. And

finally, CovCab`1 , Ccd`2 is the covariance of pseudo-C` esti-

mates of the power spectra between the fields (δa, δb) and(δc, δd) (see appendix C for more details). Note that thefull survey footprint modifies the covariance with respectto the fsky approximation used in Section 3 both throughthe factors Nab

pair[θab− , θ

ab+ ]/nanb, Ncd

pair[θcd− , θ

cd+ ]/ncnd and by

changing CovCab`1 , Ccd`2 compared to Equations 16-23.

One can determine the factors Nabpair[θ

ab− , θ

ab+ ]/nanb and

Ncdpair[θ

cd− , θ

cd+ ]/ncnd either by counting pairs in a set of ran-

dom points that trace the survey footprint homogeneouslyor they can be calculated analytically from the power spec-trum of the survey mask (see our appendix C as well asTroxel et al. 2018b). This will generally lead to an enhance-ment of statistical uncertainties (i.e. of the covariance ma-trix) with respect to the fsky approximation. To calculateCovCab`1 , C

cd`2 one could e.g. follow approximations made

by Efstathiou (2004). We slightly modify their arguments inappendix C to arrive at

CovCab`1 , C

cd`2

≈

1

4

(Cac`1 C

bd`2

+ Cac`2 Cbd`1

+ Cac`1 Cbd`1

+ Cac`2 Cbd`2

(2`1 + 1)(2`2 + 1)+

+Cad`1 C

bc`2

+ Cad`2 Cbc`1

+ Cad`1 Cbc`1

+ Cad`2 Cbc`2

(2`1 + 1)(2`2 + 1)

)M`1`2 . (68)

Here the mode coupling matrixM`1`2 again depends on thepower spectrum of the survey mask and is also detailed inappendix C. Note that in order to keep our notation brief, wehave assumed that the power spectra Cac`1 etc. in the above

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equation include both the underlying cosmological powerspectra and contributions to the power spectra from sam-pling noise, such as shape-noise and shot-noise.

In practice, Equation 68 and the approximations pro-posed by Efstathiou (2004) yield very similar results andthey are both valid on scales `1, `2 which are much smallerthan the typical scales of the mask W . Unfortunately, theDES-Y3 analysis mask has features and holes over a largerange of scales. Hence, the angular scales of interest in the3x2pt analysis are never strictly smaller than the scales ofour mask. Hence, Equation 68 is not sufficiently accurate inour case and in fact significantly overestimates our statisticaluncertainties. In Figure 8 we explain a simple scheme thatcan be used to correct for this. To motivate this procedure,consider how one would compute the Gaussian covariancemodel directly from the real space 2-point correlation func-tions, i.e. without taking the detour to Fourier space thatwas used in Section 3. For the covariance of ξab[θab− , θab+ ] andξcd[θcd− , θ

cd+ ] this would amount to integration over all pairs

of locations within our survey mask that fall into the angu-lar bins [θab− , θ


cd+ ]. Schematically, this leads to

terms of the form

Cov ∝∫(ab)∈mask,bin

dΩadΩb∫

(cd)∈mask,bin

dΩcdΩdξac(θac)ξbd(θbd) + . . . .

(69)

Here Ωa . . .Ωd are fours locations inside the survey masksuch that the distance between Ωa and Ωb lies inside theangular bin [θab− , θ

ab+ ] and the distance between Ωc and Ωd

lies inside the angular bin [θcd− , θcd+ ]. Now the approxima-

tion of Efstathiou (2004) assumes that the 2-point functionsξac(θ), ξbd(θ) are negligible on scales θ comparable to thesmalles features in the mask. Schematically this amounts tomaking the approximation∫

dΩa . . . W (Ωa)ξac(θac)

≈ ξac∫

dΩa . . . W (Ωa)δ2Dirac(Ωa − Ωc) (70)

where ξac is a suitable average of the 2-point function overdifferent scales. Our understanding of the approximation ofEfstathiou (2004) via Equation 70 is based on findings thatwe present in appendix D. This approximation fails whenthe mask contains features (e.g. holes) on scales where the2-point function has not yet decayed. Assuming that suchsmall scales holes cover a fraction of fmask of a more coarseversion of the footprint, then this can roughly be correctedfor with a multiplicative factor, i.e. by instead using theapproximation∫


≈ fmask ξac

∫dΩa . . . W (Ωa)δ2

Dirac(Ωa − Ωc) . (71)

The right panel of Figure 8 visualizes this for the mixedterms in the covariance, where one of the correlation func-tions ξac or ξbd is due to sampling noise such as shape-noiseor shot-noise and is hence exactly proportional to a Diracdelta function. In that case, the integration is over pairs thatshare one end point. Now the approximation made e.g. in

Efstathiou (2004) or by our Equation 68 assumes that alsothe correlation function between the other two end pointseffectively acts as a delta function with respect to the small-est scale features in the survey mask (cf. Equation 70). Wefind that this is not the case for the DES-Y3 mask and thatit contains features on all scales relevant to our analysis. Butas indicated in Equation 71, one can approximately correctfor this by multiplying the mixed terms in the covarianceby the fraction fmask of the coarser survey geometry that iscovered by small scale holes in the mask. This can be con-sidered a next-to-leading order correction to our Equation68.

By applying Equation 71 twice one can see that thecosmic variance terms (terms where neither of the 2-pointfunctions ξac or ξbd are exactly proportional to delta func-tions) can be corrected by multiplication with f2

mask. To im-plement this correction in practice we draw circles within theDES-Y3 survey footprint with radii ranging from 5arcmin to20arcmin and measure the masking fraction in these circles.We find that this fraction is ≈ 90% across the consideredscales. Multiplying the mixed terms in the covariance bythat fraction and the cosmic variance terms by the squareof that fraction (together with using Equation 68) we indeedfind significant improvement of the maximum posterior χ2

obtained for the FLASK simulations - as is shown in the leftpanel of Figure 8 (as well as in Figure 1).

In Figure 9 we use our FLASK measurements togetherwith the technique of precision matrix expansion (PME,from inverse covariance = precision matrix Friedrich & Ei-fler 2018) and perform a consistency of the modelling ansatzdescribed above by investigating the impact of masking onindividual covariance terms. We find both with the PMEmethods and with our analytic ansatz that masking effectsare most impactful in the covariance terms that depend onshape-noise of the weak lensing source galaxies (i.e. in whatwe called mixed terms in Section 3). This also agrees withthe findings of Joachimi et al. (2020) and it further mo-tivates the modelling of masking effects that we have de-scribed here. Nevertheless, we do not elevate this modellingansatz to our fiducial covariance model because its motiva-tion remains rather heuristic. But we consider it a realisticestimate for the error made by the fsky approximation andcan hence use it to estimate the impact of that approxima-tion on parameter constraints. In Figure 2 we have alreadyshown that this impact below the 1% level, i.e. we under-estimate the scatter of maximum posterior parameters byless than 1% when making the fsky approximation in ourfiducial covariance model.

Note that Kilbinger & Schneider (2004); Sato et al.(2011); Shirasaki et al. (2019); Philcox & Eisenstein (2019)have devised and promoted an alternative method to correctfor masking, which amounts to direct Monte-Carlo integra-tion of expressions like Equation 69. Given the large areaof DES-Y3 and and its numerous combinations of redshiftbins, we did not find this to be feasible.

6.7 Non-Poissonian shot noise

In the Poissonian limit and in a complete region of the skythe power spectrum of shot-noise is scale-independent and

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250 300 350 400 4502

0.000

0.005

0.010

0.015

0.020

p(2 )

FLASK 2x2pt 2: impact of fsky approximation2 distribution

Ndof = 3022 from fsky

approximation2 from new

beyond fskyapproximation

(θ3, φ3) = (θ4, φ4)

(θ1, φ1) (θ2, φ2)w(θ12)

1/ngal Covariance mixed term ~ ∫ⅾΩ1 w(θ12)/ngal mask(θ1, φ1)

Small scaleholes in

survey mask

Figure 8. The impact of masking on the DES-Y3 covariance. The blue histogram in the right panel shows the distribution of χ2 obtainedfrom our FLASK data vectors when using the fsky approximation. We restrict this figure to the 2x2pt function part of the data vectorsince it is this part that suffers the most from masking effects (cf. Figure 1). Ansatzes in the CMB literature (e.g. Efstathiou 2004) arenot sufficient to correct for this, because the DES footprint has features down to very small scales. In the main text we have motivateda possible way to correct for these small scale masking features and the orange histogram in the left panel shows that this ansatz indeedsignificantly improves the χ2 obtained from our FLASK measurements. The sketch in the right panel visualises how small scale featuresin the mask lead to an overestimation in the covariance when using common ways to treat the impact of survey geometry on the 2-pointfunction covariance (see main text for explanation).

given by

Ncomplete` =

1

n, (72)

where n is the galaxy density per steradian. As noted in theprevious subsection, in galaxy surveys not every region ofthe sky is fully accessible, i.e., the presence of bright stars,satellite trails, etc. lead to artificial changes in the measuredgalaxy density. These density changes can potentially mod-ify the observed galaxy power spectrum and bias any cos-mological analyses derived from them, and thus, they areavoided by removing certain regions of the sky where arti-facts may be found. These regions are usually smaller thanthe resolution of the (pixelated) survey mask used to de-termine whether a region of the sky is within the footprintor not, since it is computationally expensive to increase theresolution. This this can be described through a fractionalmask Wi = 1/fi, where i is a given pixel of the mask fiis the fractional area of the pixel unaffected by the pres-ence of artifacts. If we compute the galaxy overdensity asδg,i = Ni/(NWi) − 1, with N =

∑iNi/

∑iWi, the mean

number of sources per pixel we can estimate the Poissoniannoise power spectrum as (Nicola et al. 2020)

N` = ΩpixW

N, (73)

where w is the mean of the weights wi across the footprint,and Ωpix is the area of the pixels from the mask in stera-dians. In the case where all the pixels in the footprint arefully complete we recover Equation (72) since n = N/Ωpix,and N =

∑iNi/

∑i 1. However, in the case that any of

the pixels of the mask are not fully complete we obtain anincreased shot-noise contribution by a factor W > 1 (since0 6 fi 6 1).

In previous studies the DES-Y1 lens galaxies were

shown to prefer a super-Poissonian variance (Friedrich et al.2018; Gruen et al. 2018) which might be a consequenceof their complex selection criteria or due to the nature oftheir formation and evolution (see e.g. Baldauf et al. 2013;Dvornik et al. 2018). This super-Poissonian variance leadsto an enhance in shot-noise. In order to test for this ef-fect, we proceeded to estimate the angular power spectrum,C` ≈ C`,galaxies +N` + δC`, of DES-Y1 redmagic galaxiesselected in Elvin-Poole et al. (2018) using NaMaster (Alonsoet al. 2019), where N` is the shot noise contribution fromequation (73), and δC` is the excess power which can bedue to a number of factors (variations in completeness notcaptured by the mask, super-Poissonian shot noise, observa-tional systematics, etc.). We also compute the power spec-trum, C`,rnd of a random field with the same number ofobjects as the galaxy sample considered, and the probabil-ity of populating a pixel i is proportional to its weight, Wi.We find that C`,rnd is statistically consistent with N`. Wethen compute the ratio:

r` =C` − C`,rnd

N`≈ C`,galaxies

N`+δC`N`

. (74)

In Figure 10 we show r` compared to the theoretical expec-tation for C`,galaxies/N` = C`,th/N`, where C`,th is the the-oretical power spectrum computed using the best-fit param-eters found in Elvin-Poole et al. (2018). We also allow for a20% variation in the linear galaxy bias, which is much largerthan the uncertainty found in Elvin-Poole et al. (2018). Wefind that there is an excess power at ` > 3000 that cannotbe explained by an excess galaxy clustering (i.e., a largerthan measured linear bias or a non-linear bias component).We identify this excess (between 2% and 6%) with δC`

N`in

equation (74).This excess will translate into an extra shot-noise-like

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fiducialcovariance

correction toshape-noise free

terms

correction tofull covariance

302.5

305.0

307.5

310.0

312.5

315.0

317.5

number of data points

2 of FLASK sims using different covariances (no fitting!)estimate using1st order PMEanalytic estimate

Figure 9. The method of precision matrix expansion (Friedrich& Eifler 2018) allows us to estimate the impact of individual co-variance terms on χ2 even when only few simulated measurementsare available. The orange squares show the average χ2 betweenour FLASK measurements and their mean (rescaled by a factor ofNFLASK/(NFLASK − 1) to account for the correlation of individ-ual measurements and mean) when using either no PME at all orwhen using PME estimates from shape-noise free sims or from thefull sims. The blue dots show the corresponding χ2 values whenusing the heuristically motivated analytical treatment of maskingand survey geometry presented in the main text. The grey dashedline represents the number of data points and should be the av-erage χ2 if we had a perfect covariance model (note that for thiscomparison we have not performed any parameter fitting).

Bin number α

1 1.042± 0.0022 1.069± 0.0033 1.072± 0.003

4 1.057± 0.0035 1.021± 0.001

Table 4. Best-fit values of αn to correct for the excess shot-noisewith the DES-Y1 redmagic galaxies.

contribution to the covariance matrix (Philcox et al. 2020).The way we include this is by fitting a correction to thenumber density αn such that the excess power is compatiblewith zero. In order to do so we minimize the following χ2:

χ2(αn) =∑`

(C` − C`,thαnN`

− C`,rndN`

)2(∆C`,thαnN`

)−2

(75)

where ∆C`,th is varied in the range 1.22C`,th − 0.82C`,th(so we are allowing for 20% uncertainty in the bias for thefit). The resulting values for αn can be found in Table 4. InFigure 1, Figure 2 and Table 1 one can see that depletingthe lens galaxy densities in our fiducial covariance modelby these factors has a negligible effect on both maximumposterior χ2 and parameters constraints.

0 1000 2000 3000 4000 5000 6000 7000 8000

0.000

0.025

0.050

0.075

0.100

0.125

0.150

0.175

0.200

(CC

,rnd

)/N

bin 4bin 5

Figure 10. Measured ratio r` =C`−C`,rnd

N`(crosses) compared

to predicted contribution of the galaxy power spectra over theshot noise (solid line) for the fiducial parameters at Elvin-Pooleet al. (2018) allowing for a 20% uncertainty in the galaxy bias(shaded regions) for 2 redshift bins (bin 4 in blue and bin 5 inorange). Horizontal dashed lines are just to guide the eye. If theshot-noise were to be completely Poissonian, the measured andpredicted ratios would agree, however we find an excess between2% and 6%.

6.8 Cosmology dependence of the covariancemodel

In order to evaluate our covariance model, we choose aparticular set of cosmological parameters. We do not varythese parameters when sampling our parameter posteriorand this may impact the width of our parameter constraints(Hamimeche & Lewis 2008; Eifler et al. 2009; White & Pad-manabhan 2015; Kalus et al. 2016). Our main reason for notsampling the covariance model along with the data model isthat computing a covariance matrix is computationally toocostly for this to be feasible. Recently, Carron (2013) havealso indicated that it may indeed be incorrect to vary thecovariance cosmology when running MCMC chains.

It is only after running the MCMC chains that we canrecompute the covariance at our best-fit parameters and re-derive our parameter constraints - repeating this process un-til our constraints have converged (cf. Abbott et al. 2018, forthe application of this procedure in DES Y1 data). There-fore, the cosmology at which we compute our covariance isexpected to be off from the best-fit cosmology. In this sub-section, we investigate how χ2, as well as cosmological pa-rameter constraints, shift when computing the covariance atcosmologies that are randomly drawn from the DESY3-likeposterior.

We test the robustness of our constraints against thechoice of cosmological parameters at which we evaluate thecovariance model by taking a set of 100 different cosmologiesdrawn randomly from the simulated DES Y3 3x2pt poste-rior and generating 100 lognormal covariance matrices. Us-ing each of these covariances, we estimate posteriors for agiven realization of simulated DES Y3 3x2pt data with noisedrawn from a fiducial lognormal covariance.

Since it is prohibitively expensive to perform simulatedanalyses running MCMC chains for each covariance matrix,we use the technique of importance sampling. That allows usto quickly evaluate how these different likelihood modeling

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choices impact the derived parameter constraints withoutrepeatedly running expensive sampling algorithms. In ourimportance sampling pipeline, we take a fiducial analysisas a proposal distribution, re-evaluate the likelihoods usingthe alternative covariance matrix, and compute importanceweights as:

wi =L(πi|ξ,Calt)

L(πi|ξ,Cfid), (76)

where Cfid is the fiducial covariance in the analysis and Calt

the alternative one. If the changes induced by the new co-variance matrix in the posterior are not too large, the re-weighted samples represent the target distribution (i.e., theposterior for the alternative covariance matrix). So we have:

Ep[f(Xi)] =∑i

pif(Xi) =∑i

qipiqif(Xi) = Eq[wif(Xi)],

(77)

for a function f(Xi) of the posterior samples. Here, pi is theprobability of Xi under the target distribution p and qi isthe probability of of Xi under the proposal distribution q(see e.g. Owen (2013) and MacKay (2002)).

To diagnose the performance of our importance sam-pling estimates, we use the Effective Sample Size (ESS):

ESS =〈wi〉2

〈w2i 〉Nsamples (78)

where Nsamples is the total number of posterior samples usedin the estimation. The ESS as defined above quantifies thestatistical power of the sample set after the re-weightingprocess (assuming uncorrelated samples). It is equal to theoriginal sample size re-scaled by the ratio of the variancesunder each of the distributions (Martino et al. 2017), suchthat the error of the mean of a quantity x with standarddeviation σx under the target distribution can be estimatedas σx/

√ESS. Additionally, since our proposal distribution is

itself a weighted sample set, we incorporate both the originaland the importance weights in our ESS estimate.

Using the fiducial lognormal covariance matrix we runthe nested sampling algorithm MultiNest (Feroz & Hobson2008; Feroz et al. 2009, 2019), and perform the importancesampling procedure to estimate parameters using each ofthe 100 covariance matrices randomly sampled in parame-ter space. The (S8,Ωm) contours can be seen in Fig. (11).The effective sample sizes for the importance sampled esti-mates range from 16446 to 18329 (implying a standard errorof the mean within 0.78% of the standard deviation for allcases), and the contours show good statistics. As the impactof covariance cosmology is barely noticeable for this rangeof tested parameters, we repeat the analysis for a few moreextreme (and unlikely) cosmologies in appendix F.

These results all confirm that we can safely neglect theimpact of the choice of covariance cosmology in DES Y33x2pt analysis. One caveat of this conclusion is that wehave indeed only varied cosmological parameters (includinggalaxy bias parameters) but not nuisance parameters (mul-tiplicative shear bias, photometric redshift uncertainties) orparameters that describe intrinsic alignment. However, theDES-Y3 shear and photo-z calibration yield tight Gaussian

0.26 0.30 0.34 0.38

m

0.80

0.82

0.84

0.86

S 80.80 0.85

S8

Figure 11. (S8,Ωm) constraints for a given noisy realization ofthe DES Y3 3x2pt data vector analyzed using 100 log-normalcovariance matrices, each computed from a different cosmologydrawn from a simulated DES Y3 3x2pt posterior. The 100 con-tours are superimposed in the plot, showing very small changein constraints. The points indicate the cosmologies at which thecovariances were evaluated.

priors on the corresponding nuisance parameters. And in-trinsic alignment is relevant only on small angular scaleswhere the covariance matrix is dominated by sampling noisecontributions. Hence, we do not expect the results of this sec-tion to change significantly had all parameters been varied.

6.9 Random point shot-noise

We also consider the effect of additional shot-noise in themeasurements of galaxy clustering resulting from the use offinite numbers of random points. The Landy-Szalay estima-tor (Landy & Szalay 1993) is estimating the galaxy clus-tering correlation function inside an angular bin [θ1, θ2] as

w[θ1, θ2] =DD[θ1, θ2]− 2DR[θ1, θ2] +RR[θ1, θ2]

RR[θ1, θ2], (79)

where DD[θ1, θ2] is the number of galaxy pairs found withinthe angular bin, RR[θ1, θ2] is the (normalised) number ofpairs of random points that uniformly samples the surveyfootprint and DR[θ1, θ2] the (normalised) number of galaxy-random-point pairs within the angular bin. If the numberdensity of random points nr is much larger than the numberdensity of the galaxies ng (as is recommended for reducesampling noise) then both RR and DR must be rescaled byfactors of (ng/nr)

2 and (ng/nr) respectively.We stress that the Landy-Szalay estimator was devised

at a time of very limited computational resources, where itwas prohibitively costly to measure galaxy pair in a largenumber of random points. Hence, it was vital to minimize

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random point shot-noise. Nowadays, footprint geometriesof photometric surveys are typically characterised by highresolution healpix maps. The most straightforward way tocalculate galaxy clustering correlation function is to simplyassign a value of galaxy density contrast to each of these pix-els and then measure the scalar auto-correlation function ofthe unmasked pixels. This way, one is avoiding random pointshot-noise completely.

Nevertheless, it is still very common to measure w(θ) bymeans of Equation 79. So we also tested what impact a finitenumber of random points would have on our analysis. To doso we extended expressions of Cabré & Gaztañaga (2009,see their appendix A) to the case where the same randompoints are used to estimate w(θ) in each of our redshift binsand also to subtract shear around random points from ourgalaxy-galaxy lensing correlation functions. Note that thiscauses a noise contribution to the 2-point function measure-ments that is correlated among different redshift bins. Weassumed a random point density of 1.36/arcmin2, which ismore that 20 times larger than the density of our most denselens galaxy sample. From Table 1 it can be seen that notaccounting for the random point shot-noise in the covarianceleads to an increase in average χ2 of . 1% and to an un-derestimation of parameter uncertainties by ≈ 0.5%. Hence,this effect can be ignored for our analysis.

6.10 Effective densities and effective shape-noise

We are closing this section by spelling out an aspect of co-variance modelling that may seem straightforward but whichhas repeatedly came up in covariace discussions.

If the tracer galaxies used to estimate 2-point corre-lation functions are weighted according to some weightingscheme, then this may change the effective number densi-ties and the effective shape noise that should be used whenevaluating the covariance expressions in Section 4. In thefollowing we will derive how this can be done for each of the2-point functions in the DESY3 3x2pt data vector.

6.10.1 Galaxy clustering

We start with the galaxy clustering correlation functionw(θ). We assume a weighting scheme that is aimed at cor-recting for non-cosmological density fluctuations resultingfrom spatially varying observing conditions (as e.g. in Elvin-Poole et al. 2018). This means that the weights assigned toeach galaxy in fact sample a weight map that spans the en-tire footprint.

Instead of measuring w(θ) from the weighted galaxies bymeans of, say, the Landy-Szalay estimator (Landy & Szalay1993) it will be more convenient to think of the galaxy den-sity contrast as a pixelized field on the sky. Further more, wewill assume that the weight map has been normalised suchthat 〈w〉 = 1 (which can always be done without changingthe outcome of the weighted measurement). Consider pixeli with galaxy count Ng,i and weight wi. If ng is the averagegalaxy density of the unweighted sample, then by taking ex-pectation values with respect to many Poissonian shot-noiserealisations (and hence ignoring fluctuations of the underly-

ing matter density field) we get

〈Ng,i〉 =Apixngwi

(80)

Var(Ng,i) =Apixngwi

(81)

Var(wiNg,i) = wiApixng (82)

Var

(wiNg,iApixng

− 1

)≡ Var(δg,i)

=wi

Apixng, (83)

where Apix is the area of each pixel and the second to lastline serves as definition of δg,i and needs the fact that wedemanded 〈w〉 = 1. Note that these equation are only validfor an ensemble of observations that shares the same weightmaps and differs only in their shot-noise realisations.

From the set of all pixels we can now estimate w(θ)within a finite angular bin [θ1, θ2] as

w[θ1, θ2] =

∑pxls i>j ∆ij

[θ1,θ2] δg,i δg,j∑pxls i>j ∆ij

[θ1,θ2]

, (84)

where the symbol ∆ij[θ1,θ2] in the double sum over all pixels

is 1 when the distance of the pixel pair i, j is within [θ1, θ2]and 0 otherwise. Note that we assume an enumeration ofthe pixels and that we demand i > j in the sum in order tonot count any pair of pixels twice.

If shot-noise is the only source of noise, then it isstraight forward to calculate the variance of this measure-ment as

Var(w[θ1, θ2]) =

∑pxls i>j ∆ij

[θ1,θ2] 〈δ2g,i〉〈δ2

g,j〉[∑pxls i>j ∆ij

[θ1,θ2]

]2=

1

(Apixng)2

∑pxls i>j ∆ij

[θ1,θ2] wi wj[∑pxls i>j ∆ij

[θ1,θ2]

]2=

1

Npair,g[θ1, θ2]

∑pxls i>j ∆ij

[θ1,θ2] wi wj∑pxls i>j ∆ij

[θ1,θ2]

.

(85)

In the last line, Npair,g[θ1, θ2] is the number of unweightedgalaxy pairs within the angular bin [θ1, θ2] . Note that in thepresence of clustering, this should be calculated from a setof random points instead of from the actual galaxy catalog.

The first factor on the right side of Equation 85 is whatthe shot-noise variance of w should be in the absence of aweighting scheme. The second term is a 2-point functionof the weight map itself. If the weight map has a white-noise power spectrum, then this factor will be close to 1in any angular bin that doesn’t include angular distancesof 0. This means that at large enough scales the last lineof Equation 85 looks like the covariance for plain Poissonianshot-noise without any notion of an effective number density.This maybe surprising, but it stems from the fact that theweighting scheme we assumed does not simply multiply thegalaxy density contrast field. Instead it reverses an alreadyexisting depletion of galaxy density from non-cosmologicaldensity fluctuations.

In conclusion, the effective number density that should

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be used to compute the covariance of w[θ1, θ2] is

ng,eff [θ1, θ2] = ng

√√√√ ∑pxls i>j ∆ij

[θ1,θ2]∑pxls i>j ∆ij

[θ1,θ2] wi wj. (86)

6.10.2 Galaxy-galaxy lensing

We move on to consider the galaxy-galaxy lensing corre-lation function γt[θ1, θ2]. We assume that the lens galaxysample comes with weights derived from a weight map wl asin the previous subsection while each source galaxy j has aweight wsj which does not come from an entire weight mapbut is instead the result of the individual quality of shape-measurement for this galaxy. A measurement of γt can beconstructed as

γt[θ1, θ2] =

∑pxl i, source j ∆ij

[θ1,θ2] δl,i εt,j→i wsj∑

pxl i, source j ∆ij[θ1,θ2] w

sj

. (87)

Here, δl,i is the galaxy density contrast of the lenses definedin analogy to the previous subsection, εt,j→i is the tangen-tial component of the shear of source j with respect to lensgalaxy i and wj is the weight of source galaxy j. Note thatdue to our definition of the lens galaxy density contrast thisestimator already includes subtraction of shear around ran-dom points.

If shot-noise and shape-noise are the only sources ofnoise, then it can be readily shown that

Var(γt[θ1, θ2])

=


[θ1,θ2] 〈δ2l,i〉〈(εt,j→i wsj )2〉[∑


sj

]2≈ 1

Npair,ls[θ1, θ2]


[θ1,θ2] 〈(εt,j→i wsj )

2〉∑pxl i, sources j ∆ij

[θ1,θ2] wsj

≈ 1

2Npair,ls[θ1, θ2]

∑source j〈|εj |

2 (wsj )2〉

Ns(88)

(only with 〈wsj 〉 = 1 = 〈wlj〉 !).

Here Npair,ls[θ1, θ2] is the number of unweighted lens-sourcepairs in the angular bin [θ1, θ2], Ns is the total number ofsource galaxies and εj = ε1,j + iε2,j is the complex intrinsicellipticity of source galaxy j.

Note that the final expression in Equation 88 explic-itly allows for the possibility that the source weights wsjare correlated with the intrinsic ellipticities εj of the sourcegalaxies. One can interpret Equation 88 as

Var(γt[θ1, θ2]) =σ2ε,eff

Npair,ls[θ1, θ2](89)

with the effective dispersion of intrinsic ellipticity per shearcomponent given by

σ2ε,eff =

1

2

∑source j |εj |

2 (wsj )2

Ns. (90)

One subtlety here is that the above derivation requires〈wsj 〉 = 1. The above expressions mus be modified is thisis not the case or when taking into account responses Rjof a shape catalog generated with metacalibation (Sheldon& Huff 2017). We detail what to do in the latter case inappendix G.

6.10.3 cosmic shear

For cosmic shear we follow Schneider et al. (2002) and con-struct a measurement of ξ+ from a set of sources as

ξ+[θ1, θ2] =

∑i>j ∆ij

[θ1,θ2] wiwj (ε1,iε1,j + ε2,iε2,j)∑i>j ∆ij

[θ1,θ2] wiwj. (91)

If shape-noise is the only source of noise and if the intrinsicellipticities of galaxies are not correlated with their weights,then the variance of ξ+ is given by

Var(ξ+[θ1, θ2])

=

∑i>j ∆ij

[θ1,θ2] 〈ε21,iw

2i 〉〈ε21,jw2

j 〉+ 〈ε22,iw2i 〉〈ε22,jw2

j 〉[∑i>j ∆ij

[θ1,θ2] wiwj]2

≈2σ4

ε,eff

Npair[θ1, θ2]. (92)

Here Npair[θ1, θ2] is the number of source galaxy pairs inthe bin [θ1, θ2] and we have replaced each expectation value〈ε21/2,iw2

i 〉 by σ2ε,eff from Equation 90. Note that we again

assumed 〈wj〉 = 1 and that this may require re-scaling ofboth weights and σε when using shape measurements frommetacalibration.

6.10.4 Testing validity of effective shape noise

To test the validity of our expression for effective shape-noisein Equation 90 we run a sub-sample covariance estimatoron our data (see e.g. Friedrich et al. 2016). In particular,we divide all of our source and lens galaxy samples into200 randomly chosen sub-samples and measure the galaxy-galaxy lensing correlation function of each source-lens bincombination. As a result we obtain 200 measurements of γtin each source-lens bin combination. Since we employ com-pletely random sub-sampling, i.e. without any regard for e.g.a division of our footprint into sub-regions, the sample co-variance of these 200 measurements will almost exclusivelybe dominated by shape-noise and shot-noise. This is evenmore so, because the lens and source densities of the sub-samples are very low.

In Figure 12 we show the ratio of the variances of the200 galaxy-galaxy lensing measurements γt in the differentlens-source bin combinations to Equation 89. Assuming thatthe sub-sample covariances follow a Wishart distribution wefind that these ratios are perfectly consistent with 1. Thisindicates that Equation 90 indeed yields an accurate effec-tive shape-noise dispersion, and that one should indeed usethe plain density of lens galaxies (as opposed to any notionof effective density) when evaluating covariance expressions.

7 A SIMPLE χ2 TEST

In this short section we present a simple χ2 test that doesnot rely on the linearized framework. However, it has the dis-advantage of not addressing the impact on the estimation ofparameters. Here we generate a large number of “contami-nated" data vectors (we use 1, 000) by a Gaussian samplingof a given covariance matrix that includes different effectsand to compute a χ2 distribution from these data vectors us-ing a fiducial covariance matrix. The resulting shifts in the

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0.8

1.0

1.2source bin 1

0.8

1.0

1.2source bin 2

0.8

1.0

1.2source bin 3

0 20 40 60 80 100angular bin in gg-lensing 2pt functions

0.8

1.0

1.2source bin 4

sam

ple

cov.

/ (

2 ,eff/

Npa

ir)

Figure 12. Ratio between the sample variance of γt measured in 200 randomly selected sub-samples of the DESY3 lens and sourcescatalogs and Equation 89 for the shape-noise contribution to the covariance (again using Equation 90 to calculate the effective shape-noisedispersion σε,eff). Each row displays the variances measured for a different source redshift bin and vertical dashed lines separate pointsbelonging to different lens redshift bins (1-5 from left to right). Assuming that the covariances estimates have a Wishart distribution wecalculate the covariance matrix of these ratios (cf. Taylor et al. 2013) and find that they are consistent with 1 (both for the cosmic shearand galaxy-galaxy lensing variances).

mean value of χ2 and their standard deviations give anotherbenchmark for the importance of the different effects consid-ered here. We show the results of this test in Figure 13. Notethat the relative increases in χ2 follow closely what we ob-tained within the linearized likelihood framework in Figure1. This indicates that the dominant way in which covari-ance errors cause χ2 offsets is not through the altered scat-ter of maximum posterior parameter locations but simplythrough using an erroneous inverse covariance when com-puting χ2. That also justifies our usage of the linearizedlikelihood framework since any impact of non-linear param-eter dependencies on parameter fitting can be expected tobe even less relevant then linear fitting in the first place.

8 DISCUSSIONS AND CONCLUSIONS

In this paper we have presented the fiducial covariancemodel of the DES-Y3 joint analysis of cosmic shear, galaxy-galaxy lensing and galaxy clustering correlation functions(the 3x2pt analysis). We then investigated how the assump-tions and approximations of that model (including the as-sumption of Gaussian statistical uncertainties) impact thedistribution of maximum posterior χ2 and maximum poste-rior estimates of cosmological parameters.

The fiducial covariance matrix of the DES-Y3 3x2ptanalysis uses the formalism of Krause & Eifler (2017) to

model super-sample covariance as well as the trispectrumcontribution to the covariance. The model for the Gaussiancovariance part (i.e. the contributions from the disconnected4-point function) correctly takes into account sky curvatureand includes analytical averaging over the finite angular binsin which the 2-point functions are measured. Furthermore,the galaxy clustering power spectra that enter our calcu-lation of the Gaussian covariance part are computed usingthe non-Limber formalism of Fang et al. (2020b) and also in-clude modelling of redshift space distortions. The finite sur-vey area of DES-Y3 is incorporated in the covariance modelvia the fsky approximation (except in the pure shape-noiseand shot-noise terms where we follow Troxel et al. 2018b).

In order to perform our validation tests for the DES-Y3covariance matrix we developed a plethora of new modellingansatzes and testing strategies which are applicable in gen-eral. These new techniques are:

• We have motivated and devised a way of drawing realisa-tions of the 3x2pt data vector from a non-Gaussian distribu-tion in order to test the accuracy of our Gaussian likelihoodassumption.• We have derived analytic expressions for angular bin av-eraging of all four types of 2-point correlation functions in-cluded in the 3x2pt vector (ξ+, ξ−, γt, w). These expressionscorrectly account for sky curvature. To the best of our knowl-edge, an analytic treatment of bin averaging for cosmic shear

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Figure 13. χ2 tests taking into account with different effects.Colors follow the scheme of Figure 1.

2-point function has not been presented before (though wehave shared our results with Fang et al. 2020a who haveused them for the fiducial covariance computations).• We have extended the lognormal analytical model for thecovariance of cosmic shear 2-point function of Hilbert et al.(2011) to the other 2-point functions present in the 3x2ptdata vector.• Within a linearized likelihood formalism we have analyt-ically derived how covariance model inaccuracies influencethe distribution of maximum-posterior χ2 and of maximum-posterior parameter estimates. The results we presented alsoallow for the possibility of including the Gaussian priors oncertain model parameters and can be used to analytically es-timate the impact of covariance errors on cosmological like-lihood analyses.• By fitting an effective number density to the high-` plateauof galaxy clustering C` measurements we have estimatedhow much the assumption of Poissonian shot-noise influ-ences our likelihood analysis. This is similar in spirit to theRASCALC technique presented by Philcox et al. (2020), andwe agree with those authors that non-Poissonian shot noisecan be viewed as an effective description of how short-scalenon-linearities in galaxy clustering influence the covariance.• We calculated covariance matrices for 100 different sets ofcosmological and nuisance parameters randomly drawn froma simulated likelihood chain. This allowed us to investigatewhether calculating our covariance model at a reasonable,but wrong point in parameter space significantly impacts ouranalysis. This was done both within our linearised likelihoodframework and by using importance sampling to quicklyevaluate the 100 non-linear likelihoods.• We have derived how the 2-point correlation function of

weight maps influence the covariance of galaxy clustering 2-point function measurements. In that context we have alsofound that traditional ways of deriving an effective numberdensity for a given set of galaxy weights are erroneous whenthose weights are aimed at undoing a suspected depletionof galaxy density (e.g. due to variations in observing condi-tions).• We have derived an expression for the effective dispersion ofintrinsic source shapes for the situation when source galaxyweights are correlated with galaxy ellipticity. We have alsoshown how metacalibation responses (Sheldon & Huff 2017)enter that expression for effective shape-noise.• We have described a clean sub-sample covariance estima-tion scheme that directly measures the sampling noise con-tributions to the covariance from a given data set. We thenused the resulting covariance estimates to test the validityof our assumed effective shape noise values.• We have employed the hybrid covariance estimation tech-nique PME (Friedrich & Eifler 2018) to efficiently evaluatethe importance of individual contributions to the covariancefrom only a limited set of simulated data (in our case: 200realisations of the 3x2pt data vector including shape-noiseand 100 realisations without shape noise).• We have devised a treatment of survey geometry in covari-ance modelling that improves upon existing approximations(of e.g. Efstathiou 2004) and we have demonstrated howto carry those approximations from harmonic space to realspace.

Using these results we perform several tests for the fidu-cial DES-Y3 3x2pt covariance matrix and likelihood model,with the following conclusions:

• The assumption of Gaussian statistical uncertainties is suf-ficiently accurate (cf. Section 6.1). Hence, knowledge of thecovariance of the 3x2pt data vector is sufficient to modelour statistical uncertainties. The main assumption made toarrive at this conclusion is that non-Gaussian error bars areprimarily a large-scale problem and that at small scales thenumber of modes present within the DES-Y3 survey volumeconverges to a Gaussian distribution via the central limittheorem.• The non-Gaussian part of the covariance has a negligibleimpact on both maximum posterior χ2 and parameter con-straints (cf. Section 6.2). This statement is not a generalone but only holds for the specific DES-Y3 3x2pt analysissetup. The main assumption made to arrive at that conclu-sion is that the CosmoLike model (Krause et al. 2016) or thelog-normal model (Hilbert et al. 2011) for the non-Gaussiancovariance do not vastly underestimate the true covariance.Given the results of Sato et al. (2009); Hilbert et al. (2011)we find this a safe assumption.• Of all covariance modelling assumptions investigated inthis paper the fsky approximation (made in the mixed termand cosmic variance term of our covariance model) has thelargest effect on maximum posterior χ2. On average it in-creases χ2 between measurement and maximum posteriormodel by about 3.7% (∆χ2 ≈ 18.9) for the 3x2pt data vec-tors and by about 5.7% (∆χ2 ≈ 16.0) for the 2x2pt datavector (cf. Table 1).• However, neither fsky approximation nor any other covari-ance modelling detail tested in this paper (cf. Table 1 ; withthe exception of finite bin width, see next point) has any

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significant impact on the location and width of constraintson the parameters Ωm, σ8, w.• The only exception to this statement is finite angular binwidth which - if not taken into account in the mixed termand cosmic variance term of the covariance model - signifi-cantly increases the scatter of maximum posterior param-eters (without increasing the inferred constraints accord-ingly). However, finite bin width has been taken into accountin the past in an approximate manner - see e.g. Krause et al.(2017).• The fact that we do not know the true cosmological pa-rameters of the Universe forces us to evaluate our covariancemodel at a wrong set of parameters. Even when iterativelyadjusting those parameters to the maximum posterior pa-rameters of the analysis, the parameters of the covariancemodel will scatter around the ’true’ cosmological / nuisanceparameters. We consider this an irreducible covariance errorand find that it increases the maximum posterior scatter ofΩm and σ8 by about 3% and that of the dark energy equa-tion of state parameter w by about 5% (cf. Section 6.8 andTable 1). At the same time, we find this effect to have anegligible impact on maximum posterior χ2.

In summary, we have shown that our fiducial covarianceand likelihood model underestimates the scatter of maxi-mum posterior parameters by about 3-5%, which is mostlycaused by uncertainty in the set of cosmological and nui-sance parameters at which we evaluate that model. On av-erage, the χ2 between maximum posterior model and mea-surement of the 3x2pt data vector will be ∼ 4% higher thanexpected with perfect knowledge of the covariance matrix.This is mainly caused by our use of the fsky approxima-tion. We have devised an improved treatment of the fullsurvey geometry (cf. Section 6.6) but for the reason men-tioned above this was only used to test the impact of thefsky approximation on parameter constraints.

Given the small impact that we estimated from the un-accounted effects in the covariance modelling, we concludethat the fiducial covariance model is adequate to be usedin the 3x2pt DES-Y3 analysis. While our validation of thiscovariance model has been carried out with a preliminaryset of scale cuts and redshift distributions, we don’t expectqualitative changes for the final DES-Y3 analysis setup.

While the DESY3 specific outcomes of our study cannot straightforwardly be transferred to other surveys andanalyses, our methodological innovations will be useful toolsin the covariance and likelihood validation of future experi-ments.

ACKNOWLEDGEMENTS

OF gratefully acknowledges support by the Kavli Founda-tion and the International Newton Trust through a Newton-Kavli-Junior Fellowship and by Churchill College Cambridgethrough a postdoctoral By-Fellowship. The authors thankHenrique Xavier for very helpful discussions about mocksimulations and the FLASK code. This research was par-tially supported by the Laboratório Interinstitucional de e-Astronomia (LIneA), the Brazilian Funding agency CNPq,the INCT of the e-Universe and the Sao Paulo State Re-search Agency (FAPESP). The authors acknowledge theuse of computational resources from LIneA, the Center for

Scientific Computing (NCC/GridUNESP) of the Sao PauloState University (UNESP), and from the National Labora-tory for Scientific Computing (LNCC/MCTI, Brazil), wherethe SDumont supercomputer (sdumont.lncc.br) was used.

This paper has gone through internal review by the DEScollaboration. Funding for the DES Projects has been pro-vided by the U.S. Department of Energy, the U.S. NationalScience Foundation, the Ministry of Science and Educationof Spain, the Science and Technology Facilities Council ofthe United Kingdom, the Higher Education Funding Councilfor England, the National Center for Supercomputing Appli-cations at the University of Illinois at Urbana-Champaign,the Kavli Institute of Cosmological Physics at the Univer-sity of Chicago, the Center for Cosmology and Astro-ParticlePhysics at the Ohio State University, the Mitchell Institutefor Fundamental Physics and Astronomy at Texas A&MUniversity, Financiadora de Estudos e Projetos, FundaçãoCarlos Chagas Filho de Amparo à Pesquisa do Estado do Riode Janeiro, Conselho Nacional de Desenvolvimento Cientí-fico e Tecnológico and the Ministério da Ciência, Tecnologiae Inovação, the Deutsche Forschungsgemeinschaft and theCollaborating Institutions in the Dark Energy Survey.

The Collaborating Institutions are Argonne NationalLaboratory, the University of California at Santa Cruz,the University of Cambridge, Centro de InvestigacionesEnergéticas, Medioambientales y Tecnológicas-Madrid, theUniversity of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Ei-dgenössische Technische Hochschule (ETH) Zürich, FermiNational Accelerator Laboratory, the University of Illi-nois at Urbana-Champaign, the Institut de Ciències del’Espai (IEEC/CSIC), the Institut de Física d’Altes Ener-gies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universität München and the associated Ex-cellence Cluster Universe, the University of Michigan, theNational Optical Astronomy Observatory, the University ofNottingham, The Ohio State University, the University ofPennsylvania, the University of Portsmouth, SLAC NationalAccelerator Laboratory, Stanford University, the Universityof Sussex, Texas A&M University, and the OzDES Member-ship Consortium.

Based in part on observations at Cerro Tololo Inter-American Observatory at NSF’s NOIRLab (NOIRLab Prop.ID 2012B-0001; PI: J. Frieman), which is managed bythe Association of Universities for Research in Astronomy(AURA) under a cooperative agreement with the NationalScience Foundation.

The DES data management system is supported bythe National Science Foundation under Grant NumbersAST-1138766 and AST-1536171. The DES participants fromSpanish institutions are partially supported by MINECOunder grants AYA2015-71825, ESP2015-66861, FPA2015-68048, SEV-2016-0588, SEV-2016-0597, and MDM-2015-0509, some of which include ERDF funds from the Euro-pean Union. IFAE is partially funded by the CERCA pro-gram of the Generalitat de Catalunya. Research leading tothese results has received funding from the European Re-search Council under the European Union’s Seventh Frame-work Program (FP7/2007-2013) including ERC grant agree-ments 240672, 291329, and 306478. We acknowledge supportfrom the Brazilian Instituto Nacional de Ciência e Tecnolo-gia (INCT) e-Universe (CNPq grant 465376/2014-2).

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This manuscript has been authored by Fermi ResearchAlliance, LLC under Contract No. DE-AC02-07CH11359with the U.S. Department of Energy, Office of Science, Officeof High Energy Physics.

This work made use of the software packages GetDist(Lewis 2019), ChainConsumer (Hinton 2016), matplotlib(Hunter 2007), and numpy (Harris et al. 2020).

We would like to thank the anonymous journal refereefor their helpful comments.

DATA AVAILABILITY

The DES-Y3 3x2pt covariance matrix and likelihoods will bemade public upon publication of our final data analysis. C++and python tools to configure FLASK as described in Sec-tion 4.3 are available at https://github.com/OliverFHD/CosMomentum . Tools to compute Gaussian and halomodelcovariance is available at https://github.com/CosmoLike/CosmoCov .

APPENDIX A: CURVED SKY FORMALISM

The angular clustering correlation function of galaxies w(θ)is given in terms of the galaxy clustering power spectrumCgg` as

w(θ) =∑`

2`+ 1

4πP` (cos θ)

(Cgg` +

1

n

), (A1)

where n is the galaxy density per steradian. The term pro-portional to 1

nis usually ommited (cf. Ross et al. (2011))

since it sums up to16

1

2πn

∑`

2`+ 1

2P` (cos θ) =

1

2πnδD(cos θ − 1)

=δD(θ)

2πn sin θ, (A2)

which has to be interpreted as a 2-dimensional Dirac deltafunction on the sphere.

According to de Putter & Takada (2010) (see also Steb-bins 1996) the galaxy-galaxy lensing correlation functionγt(θ) is given in terms of the galaxy-convergence cross-powerspectrum Cgκ` as

γt(θ) =∑`

2`+ 1

4π

P 2` (cos θ)

`(`+ 1)Cgκ` , (A3)

where Pm` are the associated Legendre Polynomials.Finally, he cosmic shear correlation functions ξ±(θ) are

given by

ξ±(θ) =∑`>2

2`+ 1

4π

2(G+`,2(x)±G−`,2(x))

`2(`+ 1)2CE` ,

(A4)

where x = cos θ, CE` is the E-mode power spectrum of shearand we assume B-modes to vanish. The functions G±`,2(x)

16 from∑`

2`+12P`(x)P`(y) = δD(x− y) - see N. Bronstein &

A. Semendjajew (1979) for this and other properties of Legendrepolynomials.

are defined in eq. 4.1817 of Stebbins (1996). Eq. A4 can beexpressed in terms of associated Legendre polynomials byusing eq. 4.19 of Stebbins (1996), which gives

G+`,2(x)±G−`,2(x) = P 2

` (x)

4− `± 2x(`− 1)

1− x2− `(`− 1)

2

+P 2

`−1(x)(`+ 2)(x∓ 2)

1− x2. (A5)

In appendix B we show how to obtain A4 from the notationin Stebbins (1996).

It can be seen from above that each of the considered2-point correlation functions can be written in terms of thecorresponding power spectra as

ξA(θ) =∑`

2`+ 1

4πFA` (cos θ) CA` . (A6)

APPENDIX B: AVERAGING CORRELATIONFUNCTIONS OVER FINITE BINS

The area average can be performed for γtby replacingP 2` (cos θ) with∫ θmax

θmindθ sin θ P 2

` (cos θ)

cos θmin − cos θmax=

∫ cos θmin

cos θmaxdx P 2

` (x)


=

∫ cos θmin

cos θmaxdx (1− x2) d2

dx2P` (x)

cos θmin − cos θmax.

(B1)

Using integration by parts and various recursion relations ofLegendre polynomials (cf. N. Bronstein & A. Semendjajew1979), this becomes:∫ θmax

θmindθ sin θ P 2

` (cos θ)


=1


(`+

2

2`+ 1

)[P`−1(x)]cos θmin

cos θmax

+ (2− `) [xP`(x)]cos θmincos θmax

− 2

2`+ 1[P`+1(x)]cos θmin

cos θmax

. (B2)

In his equation 4.26 Stebbins (1996) defines the shearcorrelation function Cγ(θ, φ1, φ2) which, by inspection ofequation 4.27 and figure 2 of this work, translates into theshear correlation functions ξ±(θ) as

ξ±(θ) = Cγ(θ, 0, 0)± Cγ(θ, π/4, π/4) . (B3)

This way one directly arives at our expression for the shearcorrelation functions, equation A4.

To account for finite bin width in equation A4 and inthe covariance of ξ± one has to perform the area-weightedbin average of the functions

(G+`,2(cos θ)±G−`,2(cos θ)

). To

do so one can insert the relations∫ x2

x1

dxx P 2

` (x)

1− x2=

[x

dP`(x)

dx

]x2x1

− [P`(x)]x2x1∫ x2

x1

dxP 2` (x)

1− x2=

[dP`(x)

dx

]x2x1

(B4)

17 Note that a factor of 1/i sin(θ) is missing in the second line ofthis equation.

MNRAS 000, 000–000 (0000)

https://github.com/OliverFHD/CosMomentum

https://github.com/OliverFHD/CosMomentum

https://github.com/CosmoLike/CosmoCov

https://github.com/CosmoLike/CosmoCov


into equation A5. In summary this means one has to ex-change the functions

(G+`,2(cosθ)±G−`,2(cosθ)

)as follows:

∫ cos θmin

cos θmaxdx

(G+`,2(x)±G−`,2(x)

)cos θmin − cos θmax

=

−(`(`− 1)

2

)(`+

2

2`+ 1

)[P`−1(x)]cos θmin

cos θmax

− `(`− 1)(2− `)2

[xP`(x)]cos θmincos θmax

+`(`− 1)

2`+ 1[P`+1(x)]cos θmin

cos θmax

+ (4− `)[

dP`(x)

dx

]cos θmin

cos θmax

+ (`+ 2)

[x

dP`−1(x)

dx

]cos θmin

cos θmax

− [P`−1(x)]cos θmincos θmax

± 2(`− 1)

[x

dP`(x)

dx

]cos θmin

cos θmax

− [P`(x)]cos θmincos θmax

∓ 2(`+ 2)

[dP`−1(x)

dx

]cos θmin

cos θmax

1

cos θmin − cos θmax.

(B5)

These expressions can be very efficiently calculated and pre-tabulated e.g. with the help of the gnu scientific library(Galassi et al. 2009).

APPENDIX C: MASKING IN REAL SPACECOVARIANCES

DES observations don’t cover the entire sky, but are locatedwithin a survey mask, described by a function W (n) whichis = 1 if we have observed the sky at location n and zerootherwise18. In this appendix we derive how this maskingchanges the covariance of any measured 2-point statistics.

We start by computing, how many galaxy pairs we ex-pect to find within our mask (assuming that galaxies do notcluster). If n1, n2 are the number densities of 2 differenttracer samples, then the expected number of pairs dNpair(θ)with angular separation within [θ, θ + dθ] is given by (cf.

18 A map of the mask will come with a finally resolution, in whichcase the fractional values 0 < W < 1 of the mask will describethe completeness of observations within the map resolution.

Troxel et al. 2018b)

dNpair(θ)

n1n2

= dθ

∫dΩ1dΩ2 W (n1) W (n2) δD(arccos[n1 · n2]− θ)

= dθ

∫dΩ1dΩ2 W (n1) W (n2)

√1− x2

12δD(x12 − cos θ)

= dθ sin θ

∫dΩ1dΩ2 W (n1) W (n2) δD(x12 − cos θ)

= dθ sin θ∑`

2`+ 1

2P`(cos θ)

∫dΩ1dΩ2 W (n1) W (n2)P`(x12) .

(C1)

Using the fact that

P`(n · m) =4π

2`+ 1

∑m=−`

Y`m(n)Y ∗`m(m) (C2)

(⇒ ξ(n · m) =

∑`

2`+ 1

4πC` P`(n · m) =

∑`m

C`Y`m(n)Y ∗`m(m)

)(C3)

this becomes

dNpair(θ)

n1n2

= 2π sin θ dθ∑`m

P`(cos θ) ∗

∗(∫

dΩ1 W (n1) Y ∗`m(n1)

)(∫dΩ2 W (n2) Y`m(n2)

)= 2π sin θ dθ

∑`m

P`(cos θ) |W`m|2

= 2π sin θ dθ∑`

(2`+ 1)P`(cos θ) CW`

= 8π2 sin θ dθ ξW (θ) , (C4)

where in the last steps we have defined the angular powerspectrum of the mask through (2` + 1)CW` =

∑m |W`m|2

as well as the angular 2-point function of the mask, ξW (θ).The total number of galaxy pairs in a finite angular bin[θmin, θmax] is then

Npair[θmin, θmax] = (8π2n1n2)

∫ θmax

θmin

dθ sin θ ξW (θ)

= (8π2n1n2)∑`

[P`+1(x)− P`−1(x)]θminθmax

4πCW` .

(C5)

The 2-point correlation function of 2 scalar random fields,ξab(θ) = 〈δa(na)δb(nb)|na · nb = cos θ〉, is in practice esti-

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mated within a finite angular bin as

ξab[θmin, θmax] · Npair[θmin, θmax]

nanb

=

∫ θmax

θmin

dθ sin θ

∫dΩadΩb W (na)W (nb)∗

∗ δD(xab − cos θ) δa(na)δb(nb)

= 2π∑`m

∫ θmax

θmin

dθ sin θ P`(θ) ∗

∗∫

dΩadΩb W (na) W (nb) Y`m(na) Y ∗`m(nb) δa(na)δb(nb)

= 2π∑`


∗

∗ 1

2`+ 1

∑m

∫dΩa W (na)δa(na)Y`m(na) ∗

∗∫

dΩb W (nb) Y ∗`m(nb)δb(nb) . (C6)

From the last line it can be seen that ξab - apart from thenormalisation by Npair[θmin, θmax]/nanb - is exactly the an-gular space counter part of the Pseudo-Cells estimator in thecorresponding harmonic space (cf. Efstathiou 2004). Whythis is the case can be understood most easily in the limitof an infinitesimal angular bin. In this limit

Npair(θ) ∝ sin θ ξW (θ)

Npair(θ) · 〈ξab(θ)〉 ∝ sin θ ξW (θ) ξab(θ) , (C7)

where the second line shows that the convolution of maskand signal power spectrum in harmonic space (Efstathiou2004) becomes a simple multiplication in angular space. Es-pecially, normalisation by Npair(θ) is the angular analog ofmultiplication with the inverse mode-coupling matrix thatappears in harmonic space.

Let Cab` be the pseudo-C` estimator we identified inEquation C6, i.e. we write

ξab[θmin, θmax] · Npair[θmin, θmax]

nanb

= 2π∑`


Cab` (C8)

with

Cab` ≡1

2`+ 1

∑m

(Wδa)`m (Wδb)∗`m . (C9)

Then the covariance of the 2-point function measurementsbetween the fields δa&δb and δc&δd within angular bins[θab− , θ


cd+ ] is given by

Covξab[θab− , θ

ab+ ], ξcd[θcd− , θ

cd+ ] Nab

pair[θab− , θ

ab+ ] Ncd

pair[θcd− , θ

cd+ ]

nanbncnd

= (2π)2∑`1 `2

[P`1+1(x)− P`1−1(x)]θab−θab+

[P`2+1(x)− P`2−1(x)]θcd−θcd+·

CovCab`1 , C

cd`2

. (C10)

Assuming that δa, δb, δc, δd are Gaussian random fields anddefining the symbols

W`1`2m1m2 ≡∫

dΩ W (n) Y`1m1(n) Y ∗`2m2(n) (C11)

it is straight forward to show that CovCab`1 , C

cd`2

is given

by (cf. Efstathiou 2004)

CovCab`1 , C

cd`2

=

1

(2`1 + 1)(2`2 + 1)

∑m1 m2

∫dΩadΩbdΩcdΩd ∗

∗ W (na)W (nb)W (nc)W (nd) ∗∗ Y`1m1(na)Y ∗`1m1

(nb)Y`2m2(nc)Y∗`2m2

(nd) ∗

∗ξac(θac)ξbd(θbd) + ξad(θad)ξbc(θbc)

=

1

(2`1 + 1)(2`2 + 1)

∑m1 m2

∑`3 m3

∑`4 m4

(Cac`3 C

bd`4 + Cad`3 C

bc`4

)∗

∗ W`1`3m1m3W`3`2m3m2W`2`4m2m4W`4`1m4m1

(C12)

At this point e.g. Efstathiou (2004); Varshalovich et al.(1988) follow with the approximation

≈ 1

2

Cac`1 Cbd`2

+ Cac`2 Cbd`1

+ Cad`1 Cbc`2

+ Cad`2 Cbc`1

(2`1 + 1)(2`2 + 1)∗

∗∑m1m2

∑`3m3

∑`4m4

W`1`3m1m3W`3`2m3m2W`2`4m2m4W`4`1m4m1 .

(C13)

We however find that for fixed values of `1 and `2 the ex-pression∑m1 m2 m3 m4

W`1`3m1m3W`3`2m3m2W`2`4m2m4W`4`1m4m1

(C14)

has four maxima at [`3 = `1, `4 = `2] , [`3 = `2, `4 = `1] ,[`3 = `1, `4 = `1] and at [`3 = `2, `4 = `2] and that the areaaround each of these maxima contributes a similar amountto the sums over `3 and `4. Hence, we opt instead for theapproximation

CovCab`1 , C

cd`2

≈

1

4

(Cac`1 C

bd`2

+ Cac`2 Cbd`1

+ Cac`1 Cbd`1

+ Cac`2 Cbd`2

(2`1 + 1)(2`2 + 1)+

+Cad`1 C

bc`2

+ Cad`2 Cbc`1

+ Cad`1 Cbc`1

+ Cad`2 Cbc`2

(2`1 + 1)(2`2 + 1)

)∗

∗∑

m1,m2,m3,m4

W`1`3m1m3W`3`2m3m2W`2`4m2m4W`4`1m4m1 .

(C15)

In practice, both approximations yield very similar resultsand they are valid on scales `1, `2 which are much smallerthan the typical scales of the maskW (Efstathiou 2004; Var-shalovich et al. 1988). Unfortunately, the DES-Y3 analysismask has features and holes over a large range of scales.Hence, the angular scales of interest in the 3x2pt analy-sis are never strictly smaller than the scales of our mask.Hence, Equation C15 is not sufficiently accurate in our caseand infact significantly overestimates our covariance matrix.In Figure 8 we explain a simple scheme that can be used tocorrect for this: Calculating the covariance of ξab[θab− andθab+ ], ξcd[θcd− , θ

cd+ ] within the Gaussian covariance model (see

also Section 3) requires integration over all pairs of loca-tions within our survey mask that fall into the angular bins

MNRAS 000, 000–000 (0000)


[θab− , θab+ ] and [θcd− , θ

cd+ ]. Schematically, the covariance then

depends on expressions of the form

Cov ∝∫(ab)∈mask,bin

dΩadΩb∫

(cd)∈mask,bin

dΩadΩbξac(θac)ξbd(θbd) + . . . .

(C16)

Figure 8 visualizes this for the mixed terms in the covari-ance, where one of the correlation functions ξac or ξbd isdue to sampling noise such as shape-noise of shot-noise andhence is proportional to a Dirac delta function. In that case,the integration is over pairs that share one end point. Nowthe approximation made e.g. in Efstathiou (2004) or by ourEquation C15 assumes that also the correlation function be-tween the other two end points effectively acts as a deltafunction - at least with respect to the smallest scale featuresin the survey mask. We find that this is not the case forthe DES-Y3 mask and that it contains features on all scalesrelevant to our analysis. But Figure 8 also indicates a sim-ple way to fix this: approximating the integrand over thedistance of the two remaining endpoints by a delta functionroughly overestimates the integral by a factor equal to oneover the fraction of small scale hole in the survey footprintcompared to the average scale at which the 2-point functionbetween the 2 end points decays. And multiplying the themixed terms in the covariance by this fraction can serve asa next-to-leading order correction to our Equation C15. Bysimilar arguments one can deduce that the cosmic varianceterms (terms where none of the end points must be joined)can be corrected by performing this multiplication twice.

To implement this correction we draw circles within theDES-Y3 survey footprint with radii ranging from 5arcmin to20arcmin and measure the masking fraction in these circles.We find that this fraction is ≈ 90% across the consideredscales. Multiplying the mixed terms in the covariance by thatfraction and the cosmic variance terms by the square of thatfraction (and using Equation C15) we indeed find significantimprovement of the maximum posterior χ2 obtained for theFLASK simulations (cf. lower panel of Figure 8 as well asFigure 1).

We end this appendix by further simplifying EquationC15. Using the completeness of the Y`m as well as the factthat W (n)2 = W (n) one can show that (Efstathiou 2004)∑

m1,m2,m3,m4

W`1`3m1m3W`3`2m3m2W`2`4m2m4W`4`1m4m1

= |W`1`2m1m2 |2 . (C17)

Then re-writing W`1`2m1m2 as

W`1`2m1m2 =

∫dΩ W (n) Y`1m1(n) Y ∗`2m2

(n)

=∑`m

W`m

∫dΩ Y`m(n) Y`1m1(n) Y ∗`2m2

(n)

= (−1)m2∑`m

W`m

√(2`+ 1)(2`1 + 1)(2`2 + 1)

4π∗

∗(` `1 `20 0 0

)(` `1 `2m m1 −m2

). (C18)

and using orthogonality properties of Wigner 3j symbols one

can see that∑m1m2

|W`1`2m1m2 |2

(2`1 + 1)(2`2 + 1)=∑`m

|W`m|2

4π

(` `1 `20 0 0

)2

=∑`

2`+ 1

4πCW`

(` `1 `20 0 0

)2

.

(C19)

⇒ CovCab`1 , C

cd`2

≈

1

4

(Cac`1 C

bd`2 + Cac`2 C

bd`1 + Cac`1 C

bd`1 + Cac`2 C

bd`2 +

+ Cad`1 Cbc`2 + Cad`2 C

bc`1 + Cad`1 C

bc`1 + Cad`2 C

bc`2

)∗

∗∑`

2`+ 1

4πCW`

(` `1 `20 0 0

)2

≡ 1

4

(Cac`1 C

bd`2 + Cac`2 C

bd`1 + Cac`1 C

bd`1 + Cac`2 C

bd`2 +

+ Cad`1 Cbc`2 + Cad`2 C

bc`1 + Cad`1 C

bc`1 + Cad`2 C

bc`2

)M`1`2 .

(C20)

For an efficient numerical evaluation of the above sumwe point out the following useful relation of Wigner-3j symbols (following from functions.wolfram.com/HypergeometricFunctions/ThreeJSymbol/):(

` `1 `20 0 0

)2/ (

`− 2 `1 `20 0 0

)2

=(`2 − `1 + `− 1)(`1 − `2 + `− 1)(`1 + `2 − `+ 2)(`1 + `2 + `)

(`2 − `1 + `)(`1 − `2 + `)(`1 + `2 − `+ 1)(`1 + `2 + `+ 1).

(C21)

APPENDIX D: MOTIVATION FOR OURRE-SCALING ANSATZ FOR MASKINGEFFECTS

In this appendix we make some of the arguments presentedin Section 6.6 more precise. We have stated there thatcalculating the covariance of ξab[θab− , θab+ ] and ξcd[θcd− , θ

cd+ ]

amounts to integration over all pairs of locations within oursurvey mask that fall into the angular bins [θab− , θ

ab+ ] and

[θcd− , θcd+ ]. Schematically, this leads to terms of the form

Cov =

1

N

∫(ab)∈mask,bin

dΩadΩb∫

(cd)∈mask,bin

dΩcdΩdξac(θac)ξbd(θbd)

+ . . . , (D1)

where compared to Equation 69 we have now explicitly in-cluded a normalisation factor which is proportional to theproduct of the number of pairs of locations within our maskthat fall into the angular bins [θab− , θ


cd+ ],

N ∝ Nabpair[θ

ab− , θ

ab+ ] Ncd

pair[θcd− , θ

cd+ ] . (D2)

We have argued in Section 6.6 that the approximation ofEfstathiou (2004) for evaluating the impact of masking on

MNRAS 000, 000–000 (0000)

functions.wolfram.com/HypergeometricFunctions/ThreeJSymbol/

functions.wolfram.com/HypergeometricFunctions/ThreeJSymbol/


the 2-point function covariance can roughly be understoodas making the replacements∫


≈ ξac∫

dΩa . . . W (Ωa)δ2Dirac(Ωa − Ωc) (D3)

and∫dΩb . . . W (Ωb)ξbd(θbd)

≈ ξbd∫

dΩb . . . W (Ωb)δ2Dirac(Ωb − Ωd) , (D4)

where ξac is the integral of ξac(θac) over the 2-dimensionalangular distance vector θac and ξbd is the integral of ξbd(θbd)over θbd.

Within these approximations the right side of EquationD1 can only be non-zero if the angular bins [θab− , θ

ab+ ] and

[θcd− , θcd+ ] are identical. If that is the case, then the covariance

becomes

Cov ≈ ξacξbd

N

∫(ab)∈mask,bin

dΩadΩb + . . . . (D5)

But the integral on the right side of this equation is nothingbut Nab

pair[θab− , θ

ab+ ], so we have

Covξac[θ−, θ+], ξbd[θ−, θ+]

∝ 1

Nabpair[θ−, θ+]

, (D6)

with proportionality coefficients that do not depend on thesurvey mask. So if our above understanding of the approx-imation proposed by Efstathiou (2004) is (at least approx-imately) correct, then its ratio with respect to the fsky ap-proximation (i.e. the approximation where one computes thecovariance of a full-sky survey and then re-scales it with thesky fraction fsky of the survey footprint) should be given bythe inverse ratio of the exact value of Nab

pair[θ−, θ+] to thefsky calculation

Nabpair,fsky [θ−, θ+] = 4π2(θ2

+ − θ2−)fskynanb . (D7)

In Figure D1 we show the ratio of Npair to Npair,fsky

(green dashed lines) and compare it to the ratios of differ-ent covariance terms when computed with either the fsky

approximation or the approximation of Efstathiou (2004)(cosmic variance term: blue dotted lines; mixed term: solidorange lines). The upper panel of the figure computes theseratios for the galaxy clustering 2-point function w(θ) in thefirst lens bin of our fiducial configuration while the lowerpanel considers the cosmic shear correlation function ξ+(θ)for our first source bins (other redshift bins and 2-point func-tions behave similarly). For w(θ) these different ratios in-deed closely agree with each other. The agreement betweenthe ratio of pair counts and the ratio of the different ap-proximations for the mixed terms is especially striking. It ismost likely caused by the fact that for the mixed covarianceterms one of the two replacements in Equations D3 and D4is actually exact. Even for ξ+(θ) the ratio of the differentapproximations for the mixed term is well described by theratio of pair counts on most scales. For the cosmic varianceof ξ+(θ) one can on the other hand observe a strong devi-ation. This does not necessarily signify a breakdown of ourarguments but may be caused by the fact that cosmic shearis a spin-2 field. The calculations of Efstathiou (2004) do

0.78

0.80

0.82

0.84

0.86

0.88

0.90

0.92 considering w( ) in 1st lens bin

CovCV, fsky / CovCV, Efstathiou

Covmixed, fsky / Covmixed, EfstathiouNpair, exact / Npair, fsky

101 102

[arcmin]

0.775

0.800

0.825

0.850

0.875

0.900

0.925

0.950considering + ( ) in 1st source bin

Figure D1. Ratio of exact galaxy pair countsNpair to pair countscomputed using Equation D7 (green dashed lines) compared tothe ratios of different covariance terms when computed with eitherthe fsky approximation or the approximation of Efstathiou (2004)(cosmic variance term: blue dotted lines; mixed term: solid orangelines).

in fact only hold for scalar fields and our extension of theirformulae to shear correlation functions is only approximate(see e.g. Challinor & Chon 2005, for more general calcula-tions). Since we have identified the mixed terms to carrythe strongest impact of masking on the total covariance, wedo not address this any further. Instead, we consider theagreement between pair count ratios and mixed term ratiosobserved in Figure D1 as sufficient justification for the re-scaling ansatz of the different covariance terms presented inSection 6.6.

MNRAS 000, 000–000 (0000)


APPENDIX E: PRECISION MATRIXEXPANSION TO INVESTIGATE THE IMPACTOF MASKING ON INDIVIDUAL COVARINCETERMS

In this appendix we briefly summarize the PME methodthat went into Figure 9. The covariance of the 2x2pt (i.e.non-cosmic-shear) part of our data vector has contributionsfrom shape-noise because of the presence of the mixed termdescribed in Section 4. To pinpoint further which parts of ouranalytic covariance contribute to the elevation in χ2 (and tofurther motivate our heuristic modelling ansatz for maskingeffect in the covariance presented in Section 6.6), we re-run100 of the FLASK simulations with shape-noise turned off. Wethen use the covariances estimated from the different FLASKruns to derive corrections to our covariance model. This canbe done - even with only a limited number of simulations -with the method of precision matrix expansion (PME) thatwas described by Friedrich & Eifler (2018). At the 1st ordertheir expansion estimates the precision matrix Ψ (i.e. theinverse covariance matrix) as

Ψ = C−1model −C−1

model(B−Bmodel)C−1model . (E1)

Here, the matrix Bmodel can be either the full covariancemodel, in which case B is the full covariance estimated fromFLASK or it could be the shape-noise free part of the covari-ance, in which case B will be the covariance estimated fromthe shape-noise free FLASK simulations. Friedrich & Eifler(2018) have also derived a 2nd order correction to EquationE1, but given the small magnitude of our observed χ2 eleva-tion we restrict ourselves to the 1st order expansions whichshould also reduce the noise of the PME. Note that Equa-tion E1 does not contain the inverse of any noisy matrix.This is why PME works well even in the presence of onlyfew numerical simulations (a benefit that is even furtherboosted because the matrix B can be chosen to representonly sub-parts of the covariance).

For each FLASK measurement of the 2x2pt data vec-tor we estimate the 1st order PME from the remaining 196FLASK data vectors (respectively from the ∼ 100 shape-noisefree data vectors). The average resulting χ2 values betweeneach data vector and the mean of all data vectors are dis-played in Figure 9 and compared to the χ2 values obtainedwhen applying the analytic masking corrections presented inSection 6.6 to either the shape-noise free covariance termsor the full covariance. The average χ2 when using the ana-lytic, best-guess covariance matrix is ≈ 318.8 for a total of302 data points in the 2x2pt data vector. This correspondsto a bias in χ2 of about 5.5%. The PME estimate of theinverse covariance manages to push this down to ≈ 307.9(≈ 304.8 with our analytic ansatz) hence decreasing the biasin χ2 to about 1.9% (< 1% for the analytic anasatz). If thePME correction term is computed with the shape-noise freeFLASK covariance, then the bias is only slightly reduces to〈χ2〉 ≈ 314.3 (≈ 316.6 with our analytic ansatz). Hence, theshape-noise dependent mixed terms in the covariance in-deed seem to be the main cause of our remaining χ2 offset.This was also found by Joachimi et al. (2020) for the latestanalysis of the Kilo Degree Survey. These mixed terms donot depent on the connected 4-point function of the densityfield (cf. Section 3) and the only approximation we makein their calculation is the treatment of our survey footprint

Figure F1. (S8,Ωm) constraints for a given noisy realizationof the DES Y3 3x2pt data vector analyzed using: a fiducial co-variance matrix (blue); a covariance matrix evaluated at σ8 =

σfiducial8 − 2σσ8 (green); a covariance matrix evaluated at σ8 =σfiducial8 +2σσ8 (red). The green and red posteriors were obtainedby importance sampling the fiducial samples.

through the fsky approximation. Hence, we follow Joachimiet al. (2020) in our conclusion that this approximation is themain driver of the residual errors in our covariance model.

APPENDIX F: IMPACT OF EXTREMECOSMOLOGIES ON PARAMETERCONSTRAINTS

To demonstrate that the importance sampling technique em-ployed in Section 6.8 indeed manages to capture even strongchanges in the likelihood, we repeat the tests presented therewith covariance matrices that drastically differ from ourfiducial covariance model. In particular we shift the valueof σ8 for which the covariance model is evaluated by ±2σof the marginalised σ8 constraints expected from DES-Y3.Note that is a radical change because it ignores parameterdegeneracies, i.e. such a shift of σ8 without changes in otherparameters would be detected at high significance. FigureF1 shows the likelihood contours in the S8-Ωm plane ob-tained from both our fiducial covariance and from impor-tance sampling with the altered covariance matrices. Onecan now clearly see a change in contour width. But as wehave show in Section 6.8, this effect is far less significant forrealistic parameter uncertainties in the covariance model.

MNRAS 000, 000–000 (0000)


APPENDIX G: EFFECTIVE SHAPE NOISEWHEN USING METACALIBRATION

In Section 6.10 we have considered how the sampling noisecontribution to covariance of the galaxy-galaxy lensing 2-point function can be expressed in terms of an effectiveshape-noise when each source galaxy is weighted by a cer-tain weight (e.g. weight wj for the jth galaxy, with theaverage weight 〈wj〉j = 1). The expressions derived therehave to change when taking into account responses Rj ofa shape catalog generated with metacalibation (Sheldon &Huff 2017). In that case a measurement of γt[θ1, θ2] becomes

γt[θ1, θ2] =


[θ1,θ2] δl,i εt,j→i wsj∑


sjRj

. (G1)

One can re-write this to conform with the derivations ofSection 6.10 by defining

γt[θ1, θ2] =


[θ1,θ2] δl,iεt,j→iRj

wsjRj∑pxl i, source j ∆ij

[θ1,θ2] wsjRj

≡∑

pxl i, source j ∆ij[θ1,θ2] δl,i εt,j→i w

sj∑


sj

. (G2)

Now the transformed weights wsj should be normalised to〈wsj 〉 = 1 and then be used together with the transformedellipticities εj to calculate σε,eff from Equation 90. This iswhat we have done for Figure 12.

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AUTHOR AFFILIATIONS1 Kavli Institute for Cosmology, University of Cambridge,Madingley Road, Cambridge CB3 0HA, UK2 Churchill College, University of Cambridge, CB3 0DSCambridge, UK3 Instituto de Física Teórica, Universidade EstadualPaulista, São Paulo, Brazil4 Laboratório Interinstitucional de e-Astronomia - LIneA,Rua Gal. José Cristino 77, Rio de Janeiro, RJ - 20921-400,Brazil5 Department of Physics, University of Michigan, AnnArbor, MI 48109, USA6 ICTP South American Institute for Fundamental Re-searchInstituto de Física Teórica, Universidade Estadual Paulista,São Paulo, Brazil7 Fermi National Accelerator Laboratory, P. O. Box 500,Batavia, IL 60510, USA8 Department of Astronomy/Steward Observatory, Uni-versity of Arizona, 933 North Cherry Avenue, Tucson, AZ85721-0065, USA9 Jet Propulsion Laboratory, California Institute of Tech-nology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA10 Department of Astronomy and Astrophysics, Universityof Chicago, Chicago, IL 60637, USA11 Kavli Institute for Cosmological Physics, University ofChicago, Chicago, IL 60637, USA12 Kavli Institute for Particle Astrophysics & Cosmology,P. O. Box 2450, Stanford University, Stanford, CA 94305,USA13 Department of Physics and Astronomy, Watanabe 416,2505 Correa Road, Honolulu, HI 9682214 Center for Cosmology and Astro-Particle Physics, TheOhio State University, Columbus, OH 43210, USA15 Department of Physics, The Ohio State University,Columbus, OH 43210, USA16 Institut d’Estudis Espacials de Catalunya (IEEC), 08034Barcelona, Spain17 Institute of Space Sciences (ICE, CSIC), Campus UAB,Carrer de Can Magrans, s/n, 08193 Barcelona, Spain18 Argonne National Laboratory, 9700 South Cass Avenue,Lemont, IL 60439, USA19 Physics Department, 2320 Chamberlin Hall, Universityof Wisconsin-Madison, 1150 University Avenue Madison,WI 53706-139020 Department of Physics and Astronomy, University ofPennsylvania, Philadelphia, PA 19104, USA21 SLAC National Accelerator Laboratory, Menlo Park, CA94025, USA

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22 Department of Physics, Carnegie Mellon University,Pittsburgh, Pennsylvania 15312, USA23 Instituto de Astrofisica de Canarias, E-38205 La Laguna,Tenerife, Spain24 Universidad de La Laguna, Dpto. Astrofísica, E-38206La Laguna, Tenerife, Spain25 Department of Astronomy, University of Illinois atUrbana-Champaign, 1002 W. Green Street, Urbana, IL61801, USA26 National Center for Supercomputing Applications, 1205West Clark St., Urbana, IL 61801, USA27 Jodrell Bank Center for Astrophysics, School of Physicsand Astronomy, University of Manchester, Oxford Road,Manchester, M13 9PL, UK28 Department of Astronomy, University of California,Berkeley, 501 Campbell Hall, Berkeley, CA 94720, USA29 Santa Cruz Institute for Particle Physics, Santa Cruz,CA 95064, USA30 Department of Physics & Astronomy, University CollegeLondon, Gower Street, London, WC1E 6BT, UK31 Institut de Física d’Altes Energies (IFAE), The BarcelonaInstitute of Science and Technology, Campus UAB, 08193Bellaterra (Barcelona) Spain32 Department of Physics, Stanford University, 382 ViaPueblo Mall, Stanford, CA 94305, USA33 Département de Physique Théorique and Center forAstroparticle Physics, Université de Genève, 24 quai ErnestAnsermet, CH-1211 Geneva, Switzerland34 Department of Applied Mathematics and TheoreticalPhysics, University of Cambridge, Cambridge CB3 0WA,UK35 Centro de Investigaciones Energéticas, Medioambientalesy Tecnológicas (CIEMAT), Madrid, Spain36 Brookhaven National Laboratory, Bldg 510, Upton, NY11973, USA37 Department of Physics, Duke University Durham, NC27708, USA38 Departamento de Física Matemática, Instituto de Física,Universidade de São Paulo, CP 66318, São Paulo, SP,05314-970, Brazil39 Instituto de Fisica Teorica UAM/CSIC, UniversidadAutonoma de Madrid, 28049 Madrid, Spain40 Institute of Cosmology and Gravitation, University ofPortsmouth, Portsmouth, PO1 3FX, UK41 CNRS, UMR 7095, Institut d’Astrophysique de Paris,F-75014, Paris, France42 Sorbonne Universités, UPMC Univ Paris 06, UMR 7095,Institut d’Astrophysique de Paris, F-75014, Paris, France43 Department of Physics and Astronomy, Pevensey Build-ing, University of Sussex, Brighton, BN1 9QH, UK44 INAF-Osservatorio Astronomico di Trieste, via G. B.Tiepolo 11, I-34143 Trieste, Italy45 Institute for Fundamental Physics of the Universe, ViaBeirut 2, 34014 Trieste, Italy46 Observatório Nacional, Rua Gal. José Cristino 77, Riode Janeiro, RJ - 20921-400, Brazil47 Department of Physics, IIT Hyderabad, Kandi, Telan-gana 502285, India48 Department of Astronomy, University of Michigan, AnnArbor, MI 48109, USA49 Institute of Theoretical Astrophysics, University of Oslo.P.O. Box 1029 Blindern, NO-0315 Oslo, Norway

50 Institute of Astronomy, University of Cambridge, Mad-ingley Road, Cambridge CB3 0HA, UK51 School of Mathematics and Physics, University ofQueensland, Brisbane, QLD 4072, Australia52 Center for Astrophysics | Harvard & Smithsonian, 60Garden Street, Cambridge, MA 02138, USA53 Australian Astronomical Optics, Macquarie University,North Ryde, NSW 2113, Australia54 Lowell Observatory, 1400 Mars Hill Rd, Flagstaff, AZ86001, USA55 Institució Catalana de Recerca i Estudis Avançats,E-08010 Barcelona, Spain56 Department of Astrophysical Sciences, Princeton Uni-versity, Peyton Hall, Princeton, NJ 08544, USA57 School of Physics and Astronomy, University ofSouthampton, Southampton, SO17 1BJ, UK58 Computer Science and Mathematics Division, Oak RidgeNational Laboratory, Oak Ridge, TN 3783159 Max Planck Institute for Extraterrestrial Physics,Giessenbachstrasse, 85748 Garching, Germany60 Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians Universität München, Scheinerstr. 1, 81679München, Germany

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