des y1 results: splitting growth and geometry to test cdm · 2020. 10. 14. · year 1 (y1) galaxy...

DES-2019-0520FERMILAB-PUB-20-529-AE

DES Y1 results: Splitting growth and geometry to test ΛCDM

J. Muir,1, ∗ E. Baxter,2 V. Miranda,3 C. Doux,4 A. Ferté,5 C. D. Leonard,6 D. Huterer,7 B. Jain,4 P. Lemos,8

M. Raveri,9 S. Nadathur,10 A. Campos,11, 12 A. Chen,7 S. Dodelson,11 J. Elvin-Poole,13, 14 S. Lee,15

L. F. Secco,4 M. A. Troxel,15 N. Weaverdyck,7 J. Zuntz,16 D. Brout,17, 18 A. Choi,13 M. Crocce,19, 20

T. M. Davis,21 D. Gruen,22, 1, 23 E. Krause,3 C. Lidman,24, 25 N. MacCrann,13, 14 A. Möller,26 J. Prat,27

A. J. Ross,13 M. Sako,4 S. Samuroff,11 C. Sánchez,4 D. Scolnic,15 B. Zhang,25 T. M. C. Abbott,28

M. Aguena,29, 30 S. Allam,31 J. Annis,31 S. Avila,32 D. Bacon,10 E. Bertin,33, 34 S. Bhargava,35 S. L. Bridle,36

D. Brooks,8 D. L. Burke,1, 23 A. Carnero Rosell,37, 38 M. Carrasco Kind,39, 40 J. Carretero,41 R. Cawthon,42

M. Costanzi,43, 44 L. N. da Costa,30, 45 M. E. S. Pereira,7 S. Desai,46 H. T. Diehl,31 J. P. Dietrich,47 P. Doel,8

J. Estrada,31 S. Everett,48 A. E. Evrard,49, 7 I. Ferrero,50 B. Flaugher,31 J. Frieman,31, 9 J. Garćıa-Bellido,32

T. Giannantonio,51, 52 R. A. Gruendl,39, 40 J. Gschwend,30, 45 G. Gutierrez,31 S. R. Hinton,21 D. L. Hollowood,48

K. Honscheid,13, 14 B. Hoyle,47, 53, 54 D. J. James,55 T. Jeltema,48 K. Kuehn,56, 57 N. Kuropatkin,31

O. Lahav,8 M. Lima,29, 30 M. A. G. Maia,30, 45 F. Menanteau,39, 40 R. Miquel,58, 41 R. Morgan,42 J. Myles,22

A. Palmese,31, 9 F. Paz-Chinchón,51, 40 A. A. Plazas,59 A. K. Romer,35 A. Roodman,1, 23 E. Sanchez,60

V. Scarpine,31 S. Serrano,19, 20 I. Sevilla-Noarbe,60 M. Smith,61 E. Suchyta,62 M. E. C. Swanson,40 G. Tarle,7

D. Thomas,10 C. To,22, 1, 23 D. L. Tucker,31 T. N. Varga,53, 54 J. Weller,53, 54 and R.D. Wilkinson35

(DES Collaboration)1Kavli Institute for Particle Astrophysics & Cosmology,

P. O. Box 2450, Stanford University, Stanford, CA 94305, USA2Department of Physics and Astronomy, Watanabe 416, 2505 Correa Road, Honolulu, HI 96822

3Department of Astronomy/Steward Observatory, University of Arizona,933 North Cherry Avenue, Tucson, AZ 85721-0065, USA

4Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA5Jet Propulsion Laboratory, California Institute of Technology,

4800 Oak Grove Dr., Pasadena, CA 91109, USA6School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK

7Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA8Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK

9Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA10Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK

11Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15312, USA12Instituto de F́ısica Teórica, Universidade Estadual Paulista, São Paulo, Brazil

13Center for Cosmology and Astro-Particle Physics,The Ohio State University, Columbus, OH 43210, USA

14Department of Physics, The Ohio State University, Columbus, OH 43210, USA15Department of Physics, Duke University Durham, NC 27708, USA

16Institute for Astronomy, University of Edinburgh, Edinburgh EH9 3HJ, UK17Center for Astrophysics, Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

18NASA Einstein Fellow19Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain

20Institute of Space Sciences (ICE, CSIC), Campus UAB,Carrer de Can Magrans, s/n, 08193 Barcelona, Spain

21School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia22Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA

23SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA24Centre for Gravitational Astrophysics, College of Science,The Australian National University, ACT 2601, Australia25The Research School of Astronomy and Astrophysics,Australian National University, ACT 2601, Australia

26Université Clermont Auvergne, CNRS/IN2P3, LPC, F-63000 Clermont-Ferrand, France27Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637, USA

28Cerro Tololo Inter-American Observatory, NSF’s National Optical-InfraredAstronomy Research Laboratory, Casilla 603, La Serena, Chile

29Departamento de F́ısica Matemática, Instituto de F́ısica,Universidade de São Paulo, CP 66318, São Paulo, SP, 05314-970, Brazil

30Laboratório Interinstitucional de e-Astronomia - LIneA,Rua Gal. José Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil

31Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA32Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, 28049 Madrid, Spain

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33CNRS, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France34Sorbonne Universités, UPMC Univ Paris 06, UMR 7095,Institut d’Astrophysique de Paris, F-75014, Paris, France

35Department of Physics and Astronomy, Pevensey Building, University of Sussex, Brighton, BN1 9QH, UK36Jodrell Bank Center for Astrophysics, School of Physics and Astronomy,

University of Manchester, Oxford Road, Manchester, M13 9PL, UK37Instituto de Astrofisica de Canarias, E-38205 La Laguna, Tenerife, Spain

38Universidad de La Laguna, Dpto. Astrof́ısica, E-38206 La Laguna, Tenerife, Spain39Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 W. Green Street, Urbana, IL 61801, USA

40National Center for Supercomputing Applications, 1205 West Clark St., Urbana, IL 61801, USA41Institut de F́ısica d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology,

Campus UAB, 08193 Bellaterra (Barcelona) Spain42Physics Department, 2320 Chamberlin Hall, University of Wisconsin-Madison,

1150 University Avenue Madison, WI 53706-139043INAF-Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143 Trieste, Italy44Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy

45Observatório Nacional, Rua Gal. José Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil46Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India

47Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679 Munich, Germany48Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA

49Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA50Institute of Theoretical Astrophysics, University of Oslo. P.O. Box 1029 Blindern, NO-0315 Oslo, Norway

51Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK52Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK

53Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse, 85748 Garching, Germany54Universitäts-Sternwarte, Fakultät für Physik, Ludwig-MaximiliansUniversität München, Scheinerstr. 1, 81679 München, Germany

55Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA56Australian Astronomical Optics, Macquarie University, North Ryde, NSW 2113, Australia

57Lowell Observatory, 1400 Mars Hill Rd, Flagstaff, AZ 86001, USA58Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona, Spain

59Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ 08544, USA60Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain

61School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK62Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831

(Dated: February 1, 2021)

We analyze Dark Energy Survey (DES) data to constrain a cosmological model where a subsetof parameters — focusing on Ωm — are split into versions associated with structure growth (e.g.Ωgrowm ) and expansion history (e.g. Ω

geom ). Once the parameters have been specified for the ΛCDM

cosmological model, which includes general relativity as a theory of gravity, it uniquely predictsthe evolution of both geometry (distances) and the growth of structure over cosmic time. Anyinconsistency between measurements of geometry and growth could therefore indicate a breakdownof that model. Our growth-geometry split approach therefore serves as both a (largely) model-independent test for beyond-ΛCDM physics, and as a means to characterize how DES observablesprovide cosmological information. We analyze the same multi-probe DES data as Ref. [1]: DESYear 1 (Y1) galaxy clustering and weak lensing, which are sensitive to both growth and geometry,as well as Y1 BAO and Y3 supernovae, which probe geometry. We additionally include externalgeometric information from BOSS DR12 BAO and a compressed Planck 2015 likelihood, and externalgrowth information from BOSS DR12 RSD. We find no significant disagreement with Ωgrowm = Ω

geom .

When DES and external data are analyzed separately, degeneracies with neutrino mass and intrinsicalignments limit our ability to measure Ωgrowm , but combining DES with external data allows us toconstrain both growth and geometric quantities. We also consider a parameterization where we splitboth Ωm and w, but find that even our most constraining data combination is unable to separatelyconstrain Ωgrowm and w

grow. Relative to ΛCDM, splitting growth and geometry weakens bounds onσ8 but does not alter constraints on h.

∗ Corresponding author: [email protected]

I. INTRODUCTION

One of the major goals of modern cosmology is to bet-ter understand the nature of the dark energy that drivesthe Universe’s accelerating expansion. Though the sim-

mailto:[email protected]

3

plest model for dark energy, a cosmological constant Λ, isin agreement with nearly all observations to date, thereexist a number of viable alternative models which explainthe observed acceleration by introducing new fields or byextending general relativity via some form of modifiedgravity [2, 3]. Because there is no single most favoredtheoretical alternative, observational studies of dark en-ergy largely consist of searches for tensions with the pre-dictions of a minimal cosmological model, ΛCDM, whichconsists of a cosmological constant description of dark en-ergy (Λ), cold dark matter (CDM), and general relativityas the theory of gravity.

A tension that has attracted significant attention isone between constraints on the amplitude of matter den-sity fluctuations σ8 made by low redshift measurements,e.g. by the Dark Energy Survey (DES), and by Planckmeasurements of the Cosmic Microwave Background(CMB). This comparison is often phrased in terms of

S8 ≡ σ8√

Ωm/0.3, the parameter combination most con-strained by weak lensing analyses. Though the DES andPlanck results are not in tension according to the sta-tistical metrics used in the original DES Year 1 analy-sis [4] (note that this is a topic of some discussion [5]),the DES constraints prefer slightly lower σ8 than thosefrom Planck. This offset is in a direction consistent withother lensing results [6–14], and has been demonstratedto be independent [15] of the much-discussed tension be-tween CMB and local SNe measurements of the Hubbleconstant H0 [16–18]. In fact, of the numerous theoret-ical studies focused on alleviating the H0 tension, mosthave found a joint resolution of the σ8 and H0 tensionschallenging, as discussed in e.g. Refs. [19? –21]. Indepen-dent CMB measurements from ACT and WMAP give σ8constraints consistent with those from Planck [23], whileconstraints based solely on reconstructed Planck CMBlensing maps are consistent with σ8 constraints from bothDES and measurements of CMB temperature and polar-ization [24, 25].

These tensions are interesting because mismatchedconstraints from low and high-redshift probes could in-dicate a need to extend our cosmological model beyondΛCDM. Of course, it is also possible that these offsetscould be caused by systematic errors or a statistical fluke.Given this, it is important to examine how different ob-servables contribute to the σ8 (and H0) tension, as wellas what classes of model extensions have the potential toalleviate them.

With this goal in mind, we perform a consistency testbetween geometric measurements of expansion historyand measurements of the growth of large scale structure.The motivation for this test is similar to that of the early-vs. late-Universe (Planck vs. DES) comparison: we wantto check for agreement between two classes of cosmolog-ical observables that have been split in a physically mo-tivated way. More ambitiously, we can also view thisanalysis as a search for signs of beyond-ΛCDM physics.The growth-geometry split is motivated in particular bythe fact that modified gravity models have been shown

to generically break the consistency between expansionand structure growth expected in ΛCDM [26–30].

Our analysis focuses on data from the Dark EnergySurvey (DES). DES is an imaging survey conducted be-tween 2013-2019 which mapped galaxy positions andshapes over a 5000 deg2 area and performed a super-nova survey in a smaller 27 deg2 region. This large sur-vey volume and access to multiple observables make DESa powerful tool for constraining both expansion historyand structure growth. Constraints on cosmological pa-rameters from the first year of DES data (Y1) have beenpublished for the combined analysis of galaxy clusteringand weak lensing [4, 31], for the baryonic acoustic oscil-lation (BAO) feature in the galaxy distribution [32], andfor galaxy cluster abundance [33]. Additionally, cosmo-logical results have been reported for the first three years(Y3) of supernova data [34], as well as for the combinedanalysis of Y3 SNe with Y1 galaxy clustering, weak lens-ing, and BAO [1]. Analyses of DES Y3 clustering andlensing data are currently underway. The results pre-sented in this paper are based on a multi-probe analysislike that of Ref. [1].

Because weak lensing and large scale structure probeslike those measured by DES mix information from growthand geometry [28, 35–39], rather than purely comparingΛCDM constraints from two datasets, we introduce newparameters to facilitate this comparison. As we explainin more detail in Sect. II, we define separate “growth”and “geometry” versions of a subset of cosmological pa-rameters Θ: Θgrow and Θgeo. By constraining growthand geometry parameters simultaneously, we can answerquestions like

• Are DES constraints driven more by growth or ge-ometric information?

• Are the data consistent with the predictions ofΛCDM— that is, with Θgrow = Θgeo?

• Is the DES preference for low σ8 compared toPlanck driven more by its sensitivity to backgroundexpansion (geometry) or by its measurement of theevolution of inhomogeneities (growth)?

Our analysis thus serves as both a model-independentsearch for new physics affecting structure growth and anapproach to building a deeper understanding of how DESobservables contribute cosmological information.

The closest predecessors to the present work areRefs. [40–42] which introduce similar growth-geometryconsistency tests and apply them to data. These analyseshave the same general idea and approach as the presentanalysis, but differ in several important aspects of howthey implement the theoretical modeling of observables intheir split parameterization. In a similar spirit, Ref. [43]explores growth-geometry consistency without introduc-ing new parameters, using instead dataset comparisons ina search for discordance with ΛCDM. These approachesare complemented by other attempts at model indepen-dent tests of dark energy and modified gravity [44, 45],

4

including analyses involving meta parameters analogousto our split parameterization [46–48], as well as other pa-rameterizations which allow structure growth to deviatefrom expectations set by general relativity. These includeanalyses that have constrained free amplitudes multiply-ing the growth rate fσ8 [49], or the “growth index” pa-rameter γ [27, 50]. The commonly studied Σ − µ modelof modified gravity [31, 51–54] is also in this category. Infact, the analysis presented below can be viewed as anal-ogous to a Σ-µ study like that in Refs. [55, 56], with Σfixed to its GR value, though differences in our physicalinterpretation of the added parameters changes how weapproach analysis choices related to nonlinear scales.

A. Plan of analysis

Our goal is to test the consistency between DES Year 1constraints from expansion and those from measurementsof the growth of large scale structure. We will do thisusing using three different combinations of data:

1. DES data alone (including DES galaxy clusteringand weak lensing, BAO, and supernova measure-ments) — henceforth, “DES-only” or just “DES”;

2. As above, plus external data constraining geome-try only from Planck 2015 and BOSS DR12 BAOmeasurements — henceforth, “DES+Ext-geo”;

3. As above, plus external growth information fromBOSS DR12 RSD measurements — henceforth,“DES+Ext-all”.

Our main results will come from the combination of allof these datasets, but we will use the DES-only andDES+Ext-geo subsets to aid our interpretation of howdifferent probes contribute information.1

The motivation for this growth-geometry split param-eterization is to study the mechanism behind late-timeacceleration, so we focus on splitting parameters associ-ated with dark energy properties. Primarily, we will focuson the case where we split the matter density parameterΩm in flat ΛCDM, that is

Ωm → {Ωgeom ,Ωgrowm } [Split Ωm].

As we discuss in more detail below, with some caveats,this split essentially means that Ωgeom controls quantitieslike comoving and angular distances, while Ωgrowm controlsquantities like the growth factor. Because we impose therelation Ωm + ΩΛ = 1, this means we also split ΩΛ, and

Ωgrowm 6= Ωgeom necessarily implies ΩgrowthΛ 6= ΩgeoΛ .

1 We do not include constraints from Planck 2018 [57], eBOSSDR14 [58–60], or eBOSS DR16 [61] because those likelihoodswere not available when we set up this analysis. At the end ofthis paper, in Sect. VII E, we will briefly discuss how updatingto use those datasets might influence our results.

We will additionally show limited wCDM results wherewe split both Ωm and the dark energy equation of state,w, that is

{Ωm, w} → {Ωgeom ,Ωgrowm wgeo, wgrow} [Split Ωm, w].

Similarly to the split Ωm case, wgeo enters into calcu-

lations of comoving distances, while wgrow is used tocompute, e.g., the growth factor. We wish to calculatethe posteriors for the split parameters given the afore-mentioned data, and in particular test their consistency(whether Θgeo = Θgrow) and identify any tensions.

For the split Ωm model we will additionally exam-ine how fitting in the extended growth-geometry splitparameter space affects constraints on other parame-ters, with an eye toward understanding degeneraciesbetween the split parameters and

∑mν , σ8, h ≡

H0/[100 km s−1 Mpc−1

], and AIA. This will allow us to

build a deeper understanding of how the various datasetswe consider provide growth and geometry information. Itwill also allow us to weigh in on whether non-standardcosmological structure growth could potentially alleviatetensions between late- and early-Universe measurementsof σ8 and h.

Unless otherwise noted, we use the same modeling andanalysis choices as the DES Year 1 cosmology analysesdescribed in Refs. [1, 31, 62]. In order to ensure thatour results are robust against various modeling choicesand priors, we will follow similar blinding and validationprocedures to those used in Ref. [31]’s analyses of DESY1 constraints on beyond-wCDM physics.

The paper is organized as follows. In Sect. II we de-scribe how we model observables in our growth-geometrysplit parameterization, and in Sect. III we introducethe data used to measure those observables. Sect. IVdiscusses our analysis procedure, including the stepstaken to protect our results from confirmation bias inSect. IV A, and our approach to quantifying tensions andmodel comparison in Sect. IV B. We present our main re-sults, which are constraints on the split parameters andtheir consistency with ΛCDM, in Sect. V. Sect. VI con-tains additional results characterizing how our growth-geometry split parameterization impacts constraints onother cosmological parameters, including σ8. We con-clude in Sect. VII. We discuss validation tests in detailin Appendices A-D, and in Appendix E we show plots ofresults supplementing those in the main body of the text.

II. MODELING GROWTH AND GEOMETRY

We consider several cosmological observables in ouranalysis: galaxy clustering and lensing, BAO, RSD, su-pernovae and the CMB power spectra. We model theseobservables in a way that explicitly separates informationfrom geometry (i.e. expansion history) and growth. Theseparation of growth and geometry is immediately clearfor some probes; supernovae, for instance, are purelygeometric because they directly probe the luminosity

5

distance. For other probes, however, this split is notobvious, or even necessarily unique. Throughout, weendeavor to make physically motivated, self-consistentchoices, and will note where past studies of growth andgeometry differ. We emphasize that we are not develop-ing a new physical model, but are rather developing aphenomenological split of ΛCDM.

Since one of our primary interests is in probing thephysics associated with cosmic acceleration, we will use“growth” to describe the evolution of density perturba-tions in the late Universe. Below, we describe our ap-proach to modeling the observables we consider, and sum-marize this information in Table I.

Because structure growth depends primarily on thematter density via ρm ∝ h2Ωm and we would liketo decouple this from expansion-based constraints onh, for both our split parameterizations we addition-ally split the dimensionless Hubble parameter h ≡H0/(100 km s

−1 Mpc−1). In practice we fix hgrow to afiducial value because it has almost no effect on growthobservables: varying h across its full prior range re-sults in fractional changes that are less than a percentfor all observables considered. We demonstrate in Ap-pendix D that altering this choice by either not splittingh or marginalizing over hgrow has little impact on ourresults.

A. Splitting the matter power spectrum

Several of the observables that we consider depend onthe matter power spectrum, namely galaxy clusteringand lensing, RSD, and the CMB power spectrum. Thematter power spectrum P (k, z) contains both growth andgeometric information, so there is not a unique choice forhow to compute it within our split parameterization. Wechoose a simple-to-implement and physically motivatedapproach. Because we use “growth” to describe the evo-lution of perturbations in the late Universe, we assumethat the early-time shape of the power spectrum is de-termined by geometric parameters.

More concretely, we construct the split linear powerspectrum as a function of wavenumber k and redshift

z, P splitlin (k, z), by combining linear matter power spectracomputed separately using geometric or growth parame-ters:

P splitlin (k, z) ≡P geolin (k, zi)

P growlin (k, zi)P growlin (k, z), (1)

where P geolin and Pgrowlin are the linear matter power spec-

tra computed in ΛCDM using the geometric and growthparameters, respectively, and zi is an arbitrary red-shift choice, to be discussed below. This definitionhas several desirable properties. First, if the growthand geometric parameters are the same, then it reducesto the standard ΛCDM linear power spectrum. Sec-ond, ignoring scale-dependent growth from neutrinos,P growlin (k, z)/P

growlin (k, zi) = D

2(z)/D2(zi), where D(z) is

102

104

P(k,z

=0.0

)[(

Mp

c/h

)3]

ΛCDM (fid)

Ωm ↓ 20%Ωm ↑ 20%Ωgeom ↓ 20%

Ωgeom ↑ 20%Ωgrowm ↓ 20%Ωgrowm ↑ 20%

10−4 10−2 100

k [h/Mpc]

0.5

1.0

1.5

P(k

)/P

fid(k

)

FIG. 1. Dependence of the nonlinear matter power spectrumon Ωgrowm and Ω

geom . Gray lines show the impact of changing

Ωm by ±20% in ΛCDM, red lines show changes to Ωgeom , andblue lines show changes to Ωgrowm . The fiducial model usesΩgrowm = Ω

geom = 0.295. Solid lines correspond to an increase

in the relevant Ωm parameter to 0.354, while dotted lines showa decrease to 0.236.

the linear growth factor. Consequently, the growth pa-rameters will effectively control the growth of perturba-tions from zi to z. Third, for z � zi, this ratio of growthfactors approaches one, so the early time matter powerspectrum is controlled by the geometric parameters, asdesired.

We compute nonlinear corrections to the matter powerspectrum using halofit [63–65]. halofit provides arecipe, calibrated on simulations, for converting the linearmatter power spectrum into the nonlinear power spec-trum. As arguments to the halofit fitting function,we use the mixed linear power spectrum from Eq. (1),and use the growth versions of the cosmological param-eters. By using the growth parameters as arguments tohalofit, we ensure that nonlinear evolution is controlledby the growth parameters, and that if Θgrow = Θgeo, theresultant power spectrum agrees with that computed inthe standard DES analyses of e.g., Ref. [4]. Althoughhalofit has not been explicitly validated for our growth-geometry split model, using it is reasonable because weare performing a consistency test against ΛCDM ratherthan implementing a real physical model.

Fig. 1 shows how the full nonlinear power spectrumP (k, z = 0) is affected by 20% changes to Ωgrowm (blue)and Ωgeom (red). For comparison, we also show the effectof changes to Ωm in ΛCDM (gray). The main effects ofchanging Ωgeom are a scaling of the normalization of the

6

TABLE I. Modeling summary.

Observable Modeling Ingredient Described in Geometry GrowthGalaxy clustering and lensing P (k) shape at zi Sect. II A X

P (k) evolution since zi Sect. II A XProjection to 2PCF Sect. II B XIntrinsic alignments Sect. II B X

BAO Distances Sect. II C XRSD f(z)σ8(z)/σ8(0) Sect. II D X

σ8(z = 0) Sect. II D X XSupernovae (SN) Distances Sect. II E XCMB Compressed likelihood Sect. II F X

power spectrum and a change in the wavenumber where itpeaks. This amplitude change occurs because the Poissonequation relates gravitational potential fluctuations Φ tomatter density fluctuations δ via

k2 Φ(k, z) = 4πGρmδ =32H

20 Ωm(1 + z) δ(k, z). (2)

Thus, for fixed primordial potential power spectrum, thematter power spectrum’s early-time amplitude is propor-tional to (Ωgeom )

−2. The peak of the power spectrum oc-curs at the wavenumber corresponding to the horizonscale at matter-radiation equality, keq ∝ Ωmh2, so in-creasing Ωgeom shifts the peak to higher k. Thus, the neteffect of increasing Ωgeom is a decrease in power at low kand an increase in power at high k. Changing Ωgrowm ,on the other hand, impacts the late time growth, lead-ing to a roughly scale-independent change in the powerspectrum. Nonlinear evolution at small scales breaks thisscale indepdence.

We use zi = 3.5 as our fiducial value for the redshift atwhich growth parameters start controlling the evolutionof the matter power spectrum. This choice is motivatedby the fact that z = 3.5 is before the dark energy dom-inated era and is well beyond the redshift range probedby the DES samples. Raising zi will slightly increase thesensitivity to growth because it means that the growthparameters control a greater portion of the history ofstructure growth between recombination and the present.However, as long as zi is high enough, this has only asmall effect on observables. For the values of Ωgrowm andΩgeom shown in Fig. 1, we confirm that increasing zi to 5or 10 results in changes of less than one percent at all

wave numbers of P splitlin (k, z = 0), and also at all angularscales of the DES galaxy clustering and weak lensing 2ptfunctions. Therefore, the combined constraints of DESand external data are weakly sensitive to the choice of zias we show in Appendix A.

B. Weak lensing and galaxy clustering

For a photometric survey like DES, galaxy and weaklensing correlations are typically measured via angulartwo-point correlation functions (2PCF). To make theorypredictions for 2PCF we first compute the angular power

spectra. Assuming flat geometry and using the Limberapproximation [66, 67], the angular power spectrum be-tween the ith redshift bin of tracer A and the jth redshiftbin of tracer B is

CijAB(`) =

∫dz

H(z)

c χ2(z)W iA(z)W

jB(z)P (k, z)

∣∣∣∣k=

(`+

12

)/χ(z)

.

(3)Here χ is the comoving radial distance andH(z)/(c χ2(z)) is a volume element that translates three-dimensional density fluctuations into two-dimensionalprojected number density per redshift. The terms W iAand W jB are window functions relating fluctuationsin tracers A and B to the underlying matter densityfluctuations whose statistics are described by the powerspectrum P (k, z). The window functions for galaxynumber density g and weak lensing convergence κ are,respectively

W ig(z, k) =ni(z) bi(z, k), (4)

W iκ(z) =

(3H20 Ωm

2c2

)(χ(z)

a(z)H(z)

)

×∫ ∞

z

dz′ni(z′)χ(z′)− χ(z)

χ(z′). (5)

In these expressions, ni(z) is the normalized redshift dis-tribution of galaxies in sample i while bi(z, k) is theirgalaxy bias. Following the DES Y1 key-paper analysis[4], we will assume a constant linear bias for each sam-ple, denoted with the parameter bi.

In our growth-geometry split framework, we computethe power spectrum P (k, z) in Eq. (3) via the proceduredescribed in Sect. II A. We treat all projection operationsin Eqs. (3)-(9) as geometric. This choice means that theusual σ8-Ωm weak lensing degeneracy will occur between

between σ8 (computed with Psplitlin ) and Ω

geom , so we define

S8 ≡ σ8√

Ωgeom /0.3.

We include contributions to galaxy shear correlationsfrom intrinsic alignments between galaxy shapes via anon-linear alignment model [68] which is the same in-trinsic alignment model used in previous DES Y1 analy-ses [62]. This model adds a term to the shear convergence

7

window function,

W iκ(z)→W iκ(z)−AIA[(

1 + z

1 + z0

)αIA C1ρm0D(z)

]ni(z).

(6)Here AIA and αIA are free parameters which should bemarginalized over when performing parameter inference.The normalization C1 = 0.0134/ρcrit is a constant cali-brated based on SuperCOSMOS observations [68], ρm0is the present-day physical matter density, and D(z) isthe linear growth factor. Because intrinsic alignments arecaused by cosmic structures, in our split formulation, wecompute these quantities using growth parameters.

To obtain real-space angular correlation functionswhich can be compared to DES measurements, we thentransform the angular power spectra of Eq. (3) using Leg-endre and Hankel transformations. The correlation be-tween galaxy positions in tomographic bins i and j is

wij(θ) =∑

`

2`+ 1

4πP`(cos θ)C

ijgg(`), (7)

where P`(x) is the Legendre polynomial of order `. Shearcorrelations are computed in the flat-sky approximationas

ξij+ (θ) =

∫d` `

2πJ0(`θ)C

ijκκ(`),

ξij− (θ) =

∫d` `

2πJ4(`θ)C

ijκκ(`).

(8)

In these expressions, Jm(x) is a Bessel function of thefirst kind of order m. Finally, the correlation betweengalaxy positions in bin i and tangential shears in bin j —the so-called“galaxy-galaxy lensing” signal — is similarlycomputed via

γijt (θ) =

∫d` `

2πJ2(`θ)C

ijgκ(`). (9)

In our analysis, we perform these Fourier transformationsusing the function tpstat_via_hankel from the nicaeasoftware.2 [69]

Several astrophysical and measurement systematicsimpact observed correlations for galaxy clustering andweak lensing. In addition to intrinsic alignments, whichwe addressed above, these include shear calibration andphotometric redshift uncertainties. We model these ef-fects following the previously published DES Y1 anal-yses [62], introducing several nuisance parameters thatwe marginalize over when performing parameter estima-tion. This includes a shear calibration parameters mi foreach redshift bin i where shear is measured and a photo-metric redshift bias parameter ∆zi for each redshift bini. These systematic effects are not cosmology dependentand so are not impacted by the growth-geometry split.

2 www.cosmostat.org/software/nicaea

C. BAO

Baryon acoustic oscillations (BAO) rely on a charac-teristic scale imprinted on galaxy clustering which is setby the sound horizon scale at the end of the Comptondrag epoch. That characteristic physical sound horizonscale is

rd =

∫ ∞

zd

cs(z)

H(z)dz, (10)

where cs is the speed of sound, zd is the redshift ofdrag epoch, and H(z) is the expansion rate at redshiftz. Measurements of the BAO feature in galaxy cluster-ing in directions transverse to the line of sight constrainDM (z)/rd, where DM (z) = (1+z)DA(z) is the comovingangular diameter distance and DA(z) is the physical an-gular diameter distance. Line-of-sight measurements, onthe other hand, constrain H(z)rd. In practice constraintsfrom BAO analyses are reported in terms of dimension-less ratios,

α⊥ =DM (z)r

fidd

DfidM (z)rd(11)

and

α‖ =Hfid(z)rfiddH(z)rd

, (12)

where the superscript “fid” indicates that the quantity iscomputed at a fiducial cosmology.

The cosmological information here comes fundamen-tally from measures of distances via the comparison be-tween the observed scale of the BAO feature and thephysical distance rd. Given this, in our split parameteri-zation we compute the expressions in Eqs. (11) and (12)using geometric parameters.

D. RSD

Redshift-space distortions (RSD) measure anisotropiesin the apparent clustering of matter in redshift space.These distortions are caused by the infall of matter intooverdensities, so the RSD allow us to measure the rateof growth of cosmic structure. RSD constraints arepresented in terms of constraints on f(z)σ8(z), wheref(z) = d lnD/d ln a for linear density fluctuation am-plitude D and scale factor a = (1 + z)−1. In our splitparameterization, the amplitude σ8 ≡ σ8(z = 0) shouldmatch the value computed using the mixed power spec-

trum P splitlin from Eq. (1), while the time evolution ofσ8(z)/σ8(0) and the growth rate f(z) should be governedby growth parameters.

To achieve this, we proceed as follows. First, follow-ing the method used in Planck analyses [70] (see theirEq. 33), we use our growth parameters to compute

f(z)σgrow8 (z) =

[σ

(δv)8 (z)

]2

σgrow8 (z). (13)

www.cosmostat.org/software/nicaea

8

Here the superscript on σgrow8 denotes that it was com-puted within ΛCDM using the growth parameters. The

quantity σ(δv)8 is the smoothed density-velocity correla-

tion; it is defined similarly to σ8(z), but instead of usingthe matter power spectrum P (k, z) it is computed byintegrating over the linear cross power between the mat-ter density fluctuations δ and the divergence of the darkmatter and baryon (but not neutrino) peculiar velocityfields in Newtonian-gauge, v = −∇vN/H. Ref. [70] mo-tivates this definition by noting that it is close to what isactually being probed by RSD measurements.

In order to make σ8 consistent with our split mat-ter power spectrum definition from Eq. (1), we multiply

Eq. (13) by the z = 0 ratio of σ8, computed from Psplitlin ,

and σgrow8 . The quantity that we use to compare theorywith RSD measurements is therefore:

f(z)σ8(z) = f(z)σgrow8 (z)×

σ8(0)

σgrow8 (0). (14)

This expression will be consistent with our method ofdefining the linear power spectrum in Eq. (1) as long asit is evaluated at z < zi.

E. Supernovae

Cosmological information from supernovae comes frommeasurements of the apparent magnitude of Type Ia su-pernovae as a function of redshift. Because the absoluteluminosity of Type Ia supernovae can be calibrated toserve as standard candles, the observed flux can be usedas a distance measure. Even when the value of that abso-lute luminosity is not calibrated with more local distancemeasurements, the relationship between observed super-nova fluxes and redshifts contains information about howthe expansion rate of the Universe has changed over time.

Measurements and model predictions for supernovaeare compared in terms of the distance modulus µ, whichis related to the luminosity distance dL via

µ = 5 log [dL/10pc] . (15)

The observed distance modulus is nominally given by thesum of the apparent magnitude, mB , and a term account-ing for the combination of the absolute magnitude andthe Hubble constant, M0.

We follow the approach to computing this used inthe DES Y3 supernovae analysis [71], also described inRef. [72], and use the CosmoSIS module associated withthe latter paper to perform the calculations. In practice,computing the distance requires a few additional model-ing components. These include the width x1 and color Cof the light curve, which are used to standardize the lumi-nosity of the Type Ia supernovae, as well as a parameterGhost which introduces a step function to account for cor-relations between supernova luminosity and host galaxystellar mass Mhost (Ghost is +1/2 if Mhost > 10

10M�,

−1/2 if Mhost < 1010M�). The final expression for thedistance modulus in terms of these parameters is

µ = mB + αx1 − βC +M0 + γGhost + ∆µbias. (16)

Here the calibration parameters α, β, and γ are fit todata using the formalism from Ref. [73], and the selec-tion bias ∆µbias is calibrated using simulations [74]. Theparameter M0 is marginalized over during parameter es-timation.

The cosmological information in supernova observa-tions comes from distance measurements, so in our splitparameterization we compute these quantities using geo-metric parameters.

F. CMB

The cosmic microwave background (CMB)anisotropies in temperature and polarization are arich cosmological observable with information aboutboth growth and geometry. The geometric informationprimarily consists of the distance to the last scatteringsurface and the sound horizon size at recombination.Two parameters encapsulate how these distances (andthrough them, the cosmological parameters) impact theobserved CMB power spectra: the shift parameter [75],

Rshift ≡√

Ωm(100h)2DM (z∗)/c, (17)

which describes the location of the first power spectrumpeak, and the angular scale of the sound horizon at lastscattering `A = π/θ∗,

`A ≡ πDM (z∗)/rs(z∗). (18)

Here z∗ is the redshift of recombination, DM is the co-moving angular diameter distance at that redshift, andrs is the comoving sound horizon size. In our split pa-rameterization, we use geometric parameters to computethese quantities.

The CMB is sensitive to late-time structure growth ina few different ways. The ISW effect adds TT powerat low-` in a way that depends on the linear growthrate, and weak lensing from low-z structure smooths thepeaks of the CMB power spectra at high-`. To be self-consistent, the calculation of these effects should use thesplit power spectrum described in Sect. II A. Adaptingthe ISW and CMB lensing predictions to our split pa-rameterization would therefore require a modification ofthe CAMB software3 [76, 77] we use to compute powerspectra. In order to simplify our analysis, we focus on asubset of measurements from the CMB that are closelytied to geometric observables, independent of late-timegrowth.

3 http://camb.info

http://camb.info

9

We do this via a compressed likelihood which describesCMB constraints on Rshift, `A, Ωbh

2, ns, and As aftermarginalizing over all other parameters, including

∑mν

and ALens. This approach is inspired by the fact thatthe CMB mainly probes expansion history, and thus darkenergy, via the geometric information provided by the lo-cations of its acoustic peaks [78], and by the compressedPlanck likelihood provided in Ref. [53]; see Sect. III B 2below for details. In this formulation, we have con-structed our CMB observables to be independent of late-time growth, so we compute the model predictions forthem with geometric parameters.

G. Modeling summary and comparison to previous work

Table I summarizes the sensitivity of the probes dis-cussed above to growth and geometry. Briefly, we deriveconstraints from structure growth from the LSS observ-ables — galaxy clustering, galaxy-galaxy lensing, weaklensing shear, and RSD — while all probes we considerprovide some information about geometry. Constraintsfrom BAO, supernovae, and the scale of the first peaksof the CMB provide purely geometric information. TheLSS observables mix growth and geometry via their de-pendence on the power spectrum: its shape is set bygeometry, while its evolution since zi = 3.5 is governedby growth parameters. All projection translating fromthree-dimensional matter power to two-dimensional ob-served correlations are geometry dependent.

We now compare our choices to previous work.For the CMB, our geometry-growth split choices are

motivated by simple implementation and (since our fo-cus is on DES data) the ease of interpretation. In thiswe roughly follow the approach in Ref. [41], which alsoconsiders a compressed CMB likelihood that is governedpurely by geometry. In contrast, Ref. [40] describesCMB fluctuations (and so the sound horizon scale) us-ing growth parameters, then uses geometry parametersin converting physical to angular scales. Ref. [42] splitsthe growth and geometric information in the CMB bymultipole, using the TT, TE, and EE power spectra at` > 30 to constrain geometric parameters, and the low `(

10

TABLE II. Table summarizing datasets included and abbreviations for plots.

Combination Datasets Described in Geometry GrowthDES DES Y1 3× 2pt (galaxy clustering and WL) Sect. III A 1 X X

DES Y1 BAO Sect. III A 2 XDES Y3 + lowZ SNe Sect. III A 3 X

Ext-geo Compressed 2015 Planck likelihood Sect. III B 2 XBOSS DR12 BAO Sect. III B 1 X

Ext-all Ext-geo XBOSS DR12 RSD Sect. III C X X

0

1

2

3

4

5

6

7

8

Nor

mal

ized

coun

ts

LensesredMaGiC

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

Redshift

0

1

2

3

4

5

Nor

mal

ized

coun

ts

SourcesMETACALIBRATION

FIG. 2. Redshift distribution of source and lens galaxies usedin the DES Y1-3×2pt analysis. The vertical shaded bandsrepresent the nominal range of the redshift bins, while thesolid lines show their estimated true redshift distributions,given their photometric-redshift-based selection.

ity cuts to have relatively small photo-z errors. They aresplit into five redshift bins with nominal edges at z ={0.15, 0.3, 0.45, 0.6, 0.75, 0.9}. Weak lensing shears aremeasured from the source galaxy sample, which includes26 million galaxies. These were selected from the Y1 Goldcatalog using the Metacalibration [86, 87] and NG-MIX4 algorithms, and the BPZ algorithm [88] is used toestimate redshifts. The source galaxies are split into fourredshift bins with approximately equal densities, withnominal edges at z = {0.2, 0.43, 0.63, 0.9, 1.3} [89, 90].For each source bin a multiplicative shear calibrationparameter mi for i ∈ {1, 2, 3, 4} is introduced in orderto prevent shear measurement noise and selection effectsfrom biasing cosmological results. Metacalibrationprovides tight Gaussian priors on these parameters. Theredshift distributions for the lens and source galaxies used

4 https://github.com/esheldon/ngmix

in the DES Y1 galaxy clustering and weak lensing mea-surements are shown in Fig. 2. Uncertainties in pho-tometric redshifts are quantified with nine nuisance pa-rameters ∆zxi which quantify translations of each redshiftbin’s distribution to nxi (z−∆zxi ), where i labels the red-shift bin and x = source or lens.

The 2PCF measurements that comprise the Y1-3×2ptdata are presented in Ref. [91] (galaxy-galaxy), Ref. [92](galaxy-shear), and Ref. [93] (shear-shear). Each 2PCFis measured in 20 logarithmic bins of angular separa-tion from 2.5′ to 250′ using the Treecorr [94] algo-rithm. Angular scale cuts are chosen as described inRef. [62] in order to remove measurements at small angu-lar scales where our model is not expected to accuratelydescribe the impact of nonlinear evolution of the mat-ter power spectrum and baryonic feedback. The result-ing DES Y1-3×2pt data vector contains 457 measured2PCF values. The likelihood for the 3 × 2pt analysis isassumed to be Gaussian in that data vector. The covari-ance is computed using Cosmolike [95], which employs ahalo-model-based calculation of four-point functions [96].Refs. [95, 97] present more information about the calcu-lation and validation of the covariance matrix.

2. DES Y1 BAO

The measurement of the signature of baryon acous-tic oscillations (BAO) in DES Y1 data is presented inRef. [32]. That measurement is summarized as a likeli-hood of the ratio between the angular diameter distanceand the drag scale DA(z = 0.81)/rd. This result was de-rived from the analysis of a sample of 1.3 million galaxiesfrom the DES Y1 Gold catalog known as the DES BAOsample. These galaxies in the sample have photomet-ric redshifts between 0.6 and 1.0 and were selected usingcolor and magnitude cuts in order to optimize the highredshift BAO measurement, as is described in Ref. [? ].An ensemble of 1800 simulations [99] and three differentmethods for measuring galaxy clustering [100–102] wereused to produce the DES BAO likelihood.

The DES BAO sample is measured from the same sur-vey footprint as the samples used in the DES Y1-3×2ptanalysis, so there will be some correlation between thetwo measurements. Following Ref. [1], we neglect thiscorrelation when combining the two likelihoods. This

https://github.com/esheldon/ngmix

11

can be motivated by the fact that the intersection be-tween the 3× 2pt and BAO galaxy samples is estimatedto be about 14% of the total BAO sample, and the factthat no significant BAO signal is measured in the 2PCFmeasured for the 3× 2pt analysis.

3. DES Y3 + lowZ Supernovae

The cosmological analysis of supernova magnitudesfrom the first three years of DES observations is pre-sented in Ref. [34]. The 207 supernovae used in thisanalysis were discovered via repeated deep-field obser-vations of in a 27 deg2 region of the sky taken betweenAugust 2013 and February 2016, and are in the redshiftrange 0.07 < z < 0.85. A series of papers describe thesearch and discovery [84, 103, 104], calibration [105, 106],photometry [107], spectroscopic follow-up [108], simula-tions [109], selection effects [110], and analysis methodol-ogy [71] that went into those results. Following the DESsupernova analysis [1, 34] (but not the fiducial choices ofthe multi-probe analysis of Ref. [1]), we additionally in-clude in the supernova sample the so-called low-z subset:122 supernovae at z < 0.1 that were measured as partof the Harvard-Smithsonian Center for Astrophysics Sur-veys [111, 112] and the Carnegie Supernova Project [113].

The DES supernova likelihood is a multivariate Gaus-sian in the difference between the predicted and measuredvalues of the distance modulus µ. The likelihood is im-plemented in our analysis pipeline using the CosmoSISPantheon [72] module, adapted to use the DES measure-ments instead of the original Pantheon supernova sample.

B. External geometric data

1. BOSS DR12 BAO

We use BAO information from the constraints pre-sented in the BOSS Data Release 12 [49]. The likelihoodprovided by BOSS has a default fiducial rd, and mea-surements on DM (z) and H(z) (described in Sect. II C)at the redshifts z = {0.38, 0.51, 0.61}. These constraintsinclude measurements of the Hubble parameter H(z) andcomoving angular diameter distance dA(z) at redshiftsz = {0.38, 0.51, 0.61}. Specifically, we use the post-reconstruction BAO-only consensus measurements datafile BAO_consensus_results_dM_Hz.txt and covariancefiles BAO_consensus_covtot_dM_Hz.txt provided on theBOSS results page.5 No covariance with other data is as-sumed.

5 https://www.sdss3.org/science/boss_publications.php

2. Compressed Planck likelihood

In order to extract information from Planck data thatis independent of our growth parameters, we make ourown version of the compressed Planck likelihood pre-sented in Ref. [53]. This likelihood is a five-dimensionalGaussian likelihood extracted from a Multinest chain runwith the Planck lite 2015 likelihood using the tempera-ture power spectrum (TT) and low-` temperature andpolarization, with no lensing. We ran this chain usingthe same settings as used for the Planck constraints re-ported in the DES Y1 papers [4], which includes fixingw = −1 and marginalizing over neutrino mass. We alsomarginalize over the lensing amplitude ALens to reducethe possible impact of growth via weak lensing on thetemperature power spectrum. From that chain we ex-tracted a 5D mean and covariance for the parametervector [Rshift, `A,Ωbh

2, ns, 109As]. The compressed like-

lihood is then a five-dimensional multivariate Gaussian inthose parameters. We confirm that this compressed like-lihood is an accurate representation of the Planck con-straints in this five-dimensional parameter space — inother words, that the Planck likelihood is approximatelyGaussian — by checking that the chain samples for thefull Planck likelihood follow a χ2 distribution when eval-uated relative to the mean and covariance used in thecompressed likelihood.

C. External growth data (RSD)

We include an external growth probe using the BOSSDR12 combined results [49]. We use the full-power-spectrum-shape-based consensus measurements data filefinal_consensus_results_dM_Hz_fsig.tx and covari-ance file final_consensus_covtot_dM_Hz_fsig.txtprovided on the BOSS results page.5 This includes con-sensus measurements of DM (z), H(z), and f(z)σ8(z) atthe same three redshifts z = {0.38, 0.51, 0.61} as theBAO-only likelihood. The reported values are the com-bined results from seven different measurements usingdifferent techniques and modeling assumptions, wherethe covariances between those results have been assessedusing mock catalogues [114, 115].

As a slight complication, we note that these BOSSresults use both the post-reconstruction BAO-only fitsdescribed in Sect. III B 1, and those from the full-shapeanalysis of the pre-reconstruction data. The combinationof the post-reconstruction BAO and pre-reconstructionfull-shape fits tightens constraints on DM (z) by around10% and on H(z) by 15-20%. This means that in addi-tion to adding growth information from RSD, our Ext-alldata combination will also have slightly tighter geometricconstraints than Ext-geo.

https://www.sdss3.org/science/boss_publications.php

12

IV. ANALYSIS CHOICES AND PROCEDURE

We use the same parameters and parameter priors asprevious DES Y1 analyses [1, 4, 31]. For our split param-eters, we use the same prior as their unsplit counterparts’priors in those previous analyses:

Ωm, Ωgeom , Ω

growm ∈ [0.1, 0.9] (19)

w, wgeo, wgrow ∈ [−2.0, 0.33] (20)

We use the same angular scale cuts for the DES Y1 weaklensing and LSS measurements, leaving 457 data points inthe weak lensing and galaxy clustering combined 3× 2ptdata vector. The DES BAO likelihood contributes an-other measurement (of DA(z = 0.81)/rd), and the DESSNe likelihood is based on measurements of 329 super-novae (207 from DES, 122 from the low-z sample). Thismeans that our DES-only analysis is based on a total of787 data points. The DES+Ext-geo analysis thereforehas 798 data points (787 from DES, 5 from compressedPlanck, 6 from BOSS BAO), and the DES+Ext-all anal-ysis has 801 (same as DES+Ext-geo plus 3 BOSS RSDmeasurements).

Calculations were done in the CosmoSIS6 softwarepackage [116], using the same pipeline as the Y1KP, mod-ulo changes to implement the growth-geometry split. Forvalidation tests, chains were run with Multinest sam-pler [117–119], with low resolution fast settings of 250 livepoints, efficiency 0.3, and tolerance 0.01. For fits to datawhere we need both posteriors and Bayesian evidence, weuse Polychord [120] with 250 live points, 30 repeats,and tolerance of 0.01. Summary statistics and contourplots from chains are done using the GetDist [121] soft-ware with a smoothing kernel of 0.5.

As noted in Sect. I A, our main results will be productsof parameter estimation and model comparison evaluatedfor

• Split Ωm constrained with DES+Ext-geo, and

• Split Ωm constrained with DES+Ext-all.

This choice was based on simulated analyses performedbefore running parameter estimation on real data. Inthese analyses we computed model predictions for observ-ables at a fiducial cosmology, then analyzed those predic-tions as if they were measurements. By studying the re-lationship between the resulting posteriors and the inputparameter values we identified which model-data combi-nations are constraining enough so that parameter esti-mates are unbiased by parameter-space projection effects.This is described in more detail in Appendix B. For theDES+Ext-geo and DES+Ext-all constraints on split Ωm,we confirm that the input parameter values are containedwithin the 68% confidence intervals of the synthetic-data

6 https://bitbucket.org/joezuntz/cosmosis/

versions of all marginalized posteriors plotted in this pa-per.

We consider two additional sets of constraints:

• Split Ωm constrained by DES only, and

• Split Ωm and w constrained by DES+Ext-all.

Our simulated analyses revealed that the one-dimensional marginalized posteriors are impactedby significant projection effects. Given this, for thesecases we do not report numerical parameter estimatesor error bars, but we will still report model comparisonstatistics (to be discussed in Sect. IV B) and show theirtwo-dimensional confidence regions on plots. We donot consider constraints splitting both Ωm and w forDES-only and DES+Ext-geo because these datasets areless constraining than DES+Ext-all and so are expectedto suffer from even more severe projection effects. Amore detailed discussion of these projection effects andthe parameter degeneracies which cause them can befound in Appendix B.

We follow a procedure similar to that used in Ref. [31]to validate our analysis pipeline. Our goal is to charac-terize the robustness of our results to reasonable changesto analysis choices, as well as to astrophysical or mod-eling systematics. The analysis presented in this paperwas blinded in the sense that all analysis choices werefixed and we ensured that the pipeline passed a numberof predetermined validation tests before we looked at thetrue cosmological results. The blinding procedure andthese tests are described below.

A. Validation

In planning and executing this study, we took severalsteps to protect the results against possible experimenterbias, following a procedure similar to the parameter-level blinding strategy used in previous DES Y1 beyond-ΛCDM analyses. Key to this were extensive simulatedanalyses, in which we analyzed model predictions for ob-servables with known input parameters as if they weredata. All analysis choices are based on these simulatedanalyses, including which datasets we focus on and howwe report results. Before running our analysis pipelineon real data, we wrote the bulk of this paper’s text, in-cluding the plan of how the analysis would proceed, andsubjected that text to a preliminary stage of DES internalreview.

When performing parameter estimation on the realdata, we concealed the cosmology results using the fol-lowing strategies:

• We avoided over-plotting measured data and theorypredictions for observables.

• We post-processed all chains so that the mean ofthe posterior distributions lay on our fiducial cos-mology.

https://bitbucket.org/joezuntz/cosmosis/

13

• We do not look at model comparison measures be-tween our split parameterization and ΛCDM.

We maintained these restrictions until we confirmed thatthe analysis passed several sets of validation tests:

• We confirm that our results cannot be signifi-cantly biased by any one of the sample systematicsadopted in our validation tests. To do this we checkthat the parameter estimates we report change byless than 0.3σ when we contaminate synthetic in-put data with a number of different effects, includ-ing non-linear galaxy bias and a more sophisticatedintrinsic alignment model. This test is discussedAppendix C.

• We confirm that non-offset ΛCDM chains give re-sults consistent with what Ref. [1] reports.7

• We studied whether our main results are robust tochanges in our analysis pipeline. We found that pa-rameter constraints shift by less than 0.3σ when weapply more aggressive cuts to removing non-linearangular scales, and when we use an alternative setof photometric redshifts.

Our results did change when we replaced the intrin-sic alignment model defined Eq. (6) with one wherethe amplitude AIA varies independently in eachsource redshift bin. Upon further investigation, de-tailed in Appendix D, we found that a similar pos-terior shift manifests in the analysis of syntheticdata, so we believe that it is due to a parameter-space projection effect rather than a property ofthe real DES data. We therefore proceed with theplanned analysis despite failing this robustness test,but add an examination of how intrinsic alignmentproperties covary with our split parameters to thediscussion in Sect. VI.

After passing another stage of internal review, we thenfinalized the analysis by updating the plots to show non-offset posteriors, computing tension and model compari-son statistics, and writing descriptions of the results. Af-ter unblinding a few changes were made to the analysis:First, we discovered that our real-data results had ac-cidentally been run using Pantheon [72] supernovae, sowe reran all chains to include correct DES SNe data.While doing this, we additionally made a small changeto our compressed Planck likelihood, centering its Gaus-sian likelihood on the full Planck chain’s mean parametervalues, rather than on maximum-posterior sample. Thischoice was motivated by the fact that sampling error inthe maximum posterior estimate means that compressedlikelihood is more accurate when centered on the mean.

7 The data combinations we use are slightly different than thosein Ref. [1], so we simply require that our ΛCDM results be rea-sonably consistent with theirs, rather than identical.

We estimate that centering on the maximum posteriorsample was causing the compressed likelihood to be bi-ased by ∼ 0.2σ relative to the mean, though we avoidedlooking at the direction of this bias in parameter spacein order to prevent our knowledge of that direction frominfluencing this choice.

B. Evaluating tensions and model comparison

There are two senses in which measuring tension is rel-evant for this analysis. First, we want to check for tensionbetween different datasets in order to determine whetherit is sensible to report their combined constraints. Sec-ond, we want to test whether our split-parameterizationresults are in tension with ΛCDM (or wCDM in the caseof split w). For both of these applications, we evaluatetension using Bayesian suspiciousness [5, 122], which wecompute using anesthetic.8 [123]

Suspiciousness S is a quantity built from the Bayesianevidence ratio R designed to remove dependence of thetension metric on the choice of prior. Let us define Sdat tomeasure the tension between two datasets A and B. TheBayesian evidence ratio between A and B’s constraintsis

Rdat =ZABZAZB

, (21)

where ZX =∫dΘP(Θ|X) is the Bayesian evidence for

dataset X with posterior P(Θ|X). Generally, values ofRdat > 1 indicate agreement between A and B’s con-straints, while Rdat < 1 indicates tension, though thetranslation of R values into tension probability dependson the choice of priors [5, 124].

The Kullback-Leibler (KL) divergence

DX =∫dΘP(Θ|X) log [P(Θ|X)/π(Θ)], (22)

measures the information gain between the prior and theposterior for constraints based on dataset X. The com-parison between KL divergences can be used to quantifythe probability, given the prior, that constraints fromdatasets A and B will agree. This information is encap-sulated in the information ratio,

log Idat = DA +DB −DAB , (23)

where DAB is the KL divergence for the combined anal-ysis of A and B. To get Bayesian suspiciousness we sub-tract the information ratio from the Bayesian evidence:

logSdat = logRdat − log Idat. (24)

8 https://github.com/williamjameshandley/anesthetic

https://github.com/williamjameshandley/anesthetic

14

This subtraction makes S insensitive to changes in thechoice of priors, as long as those changes do not signif-icantly impact the posterior shape. As with R, largervalues of S indicate greater agreement between datasets.

To translate this into a more quantitative measureof consistency, we use the fact that the quantity d −2 logS approximately follows a χ2d probability distribu-tion, where d is the number of parameters constrainedby both datasets. In practice we determine d by com-puting the Bayesian model dimensionality d [125], whichaccounts for the extent to which our posterior is uncon-strained (prior-bounded) in some parameter-space direc-tions. The model dimensionality for a single set of con-straints X is defined as

d̃X2

=

∫dΘP(Θ|X) (log [P(Θ|X)/π(Θ)])2 −D2X . (25)

This measures the variance of the gain in informationprovided by X’s posterior. Though d̃ is generally non-integer, it can be interpreted as the effective number ofconstrained parameters. To get the value of d that weuse for our tension probability calculation, we compute

d ≡ dA∩B = d̃A + d̃B − d̃AB . (26)

Since any parameter constrained by either A or B willalso be constrained by their combination, this subtrac-tion will remove the count for any parameter constrainedby only one dataset. Thus, d is the effective number ofparameters constrained by both datasets. As we notedabove, the quantity d − 2 logS approximately follows aχ2d probability distribution, so we compute the tensionprobability

p(S > Sdat) =

∫ ∞

d−2 logSχ2d(x) dx, (27)

which quantifies the probability that the datasets A andB would be more discordant than measured. If in ouranalysis we find p(S > Sdat) < 5%, we will considerthe two datasets to be in tension and will not reportparameter constraints from their combination.

We will also use the Bayesian Suspiciousness in orderto perform model comparison. One can interpret theBayesian Evidence Ratio and Suspiciousness defined inEqs. (21)- (24) as a test of the hypothesis that datasetsA and B are described by a common set of cosmologicalparameters as opposed to two independent sets. That canbe directly translated into what we would like to deter-mine: are the data in tension with a single set of param-eters describing both growth and geometric observables?We therefore compute

Rmod = ZΛCDM/Zmod (28)Imod = DΛCDM −Dmod (29)

logSmod = logRmod − log Imod (30)

We use the label “mod” to identify these as model com-parison statistics. As before we translate this into a ten-sion probability by computing the Bayesian model di-mensionality,

d = dmod − dΛCDM, (31)and integrating the expected χ2d distribution as inEq. (27). The resulting quantity p(S > Smod) measuresthe probability to exceed the observed tension betweengrowth and geometric observables.

To convert a probability p to an equivalent Nσ scale,we compute N such that p is the probability that |x| > Nfor a standard normal distribution,

p = 1− 2∫ N

0

(2π)−1/2e−x2/2 dx = erfc

(N√

2

), (32)

N =√

2 erfc−1(p). (33)

Unless otherwise noted, this double-tail equivalent prob-ability is what will be used to convert probabilities toNσ. In the specific case when we are testing in Sect. V Cwhether the difference between the corresponding growthand geometry parameter is greater than zero, a single-tailprobability is relevant instead; in that case, we simplymultiply p in Eq. (33) by a factor of two.

V. RESULTS: SPLIT PARAMETERS

Here we present our main results, which are constraintson split parameters and an assessment of whether or notthe data are consistent with Θgrow = Θgeo. Sect. V Areports results for splitting Ωm (with w = −1), whileresults for splitting both Ωm and w are presented inSect. V B. We summarize the results in Sect. V D, report-ing constraints, tension metrics, and model comparisonstatistics in Table III.

All datasets considered fulfill the p(S > Sdat) ≥ 0.05prerequisite set in Sect. IV B for reporting combined con-straints. Note, however, that while this is strictly true,the ΛCDM constraints from DES and Ext-geo, as wellthe split Ωm constraints from DES and Ext-all are foundto have tensions at the 2σ threshold. Thus, while we willreport these combined results, they should be interpretedwith caution.

Note that while one might assume that the 2σ tensionfound between DES and Ext-geo constraints in ΛCDMis related to the familiar Planck-DES σ8 offset, this isnot necessarily the case. This is because the σ8 tensionis generally studied in terms of the constraints from thefull CMB power spectrum, while we are only using lim-ited, geometric information from the CMB. When we doexamine marginalized ΛCDM posteriors (not shown), wefind substantial overlap between the 1σ regions of themarginalized DES and Ext-geo constraints on σ8. Sim-ilarly, we find no obvious incompatibility between DESand Ext-geo constraints on any other individual param-eter. This 2σ tension therefore appears to be related tothe higher-dimensional properties of the two posteriors.

15

A. Splitting Ωm

Fig. 3 shows the 68 and 95% confidence regions forΩgrowm and Ω

geom for various data combinations. We study

three different comparisons: a comparison between ourfiducial DES dataset and a version without the BAOand SNe in the top panel; DES plus external geomet-ric (DES+Ext-geo) data in the middle panel, and DESplus external data including RSD (Ext-all) in the bottompanel. The diagonal gray line corresponds to Ωgrowm =Ωgeom . Marginalized parameter constraints and tensionmetrics for both data combination and model compari-son are reported in Table III.

Looking at DES-only results in the top panel, we findthat, as expected including the (geometric) DES BAOand SNe likelihoods tightens the constraints on Ωgeom butonly weakly affects Ωgrowm . We find that constraints onΩgeom are much stronger than those on Ω

growm for both the

3 × 2pt-only and the fiducial DES constraints. In fact,the DES constraints on Ωgeom are only slightly weaker thanΛCDM constraints on Ωm, implying that most of DES’constraining power is derived from geometric informa-tion. This might be surprising, since one might expect aLSS survey to have more growth sensitivity. However, itis consistent with the findings summarized in Ref. [39],which discusses how distance and growth factor measure-ments can place comparable constraints on the dark en-ergy equation of state when other cosmological parame-ters are held fixed [37, 38], but the growth weakens whenone marginalizes over more parameters [28, 35]. The factthat the confidence regions intersect the Ωgrowm = Ω

geom

line but are asymmetrically distributed around it is re-flected in the Bayesian Suspiciousness measurement of1.5σ tension with ΛCDM.

In the middle panel of Fig. 3 we show the combinationof the DES data with external geometric measurementsfrom the CMB and BAO (Ext-geo). As expected, theexternal geometric data alone put tight constraints onΩgeom but do not constrain Ω

growm at all. The combined

constraints on Ωgeom are straightforwardly dominated bythose from the external data, while the DES+Ext-geoconstraints on Ωgrowm are counterintuitively bounded frombelow but not above. To understand the appearance ofthe lower bound, note that the DES-only measurement ofa given late-time density fluctuation amplitude allows ar-bitrarily small values of Ωgrowm because little or no struc-ture growth over time can be compensated by a largeprimordial amplitude As. Adding the Planck constraintsprovides an early-time anchor for As, and therefore re-quires Ωgrowm to be above some minimal value in order toaccount for the evolution of structure growth between re-combination and the redshifts probed by DES. The rea-son DES’ upper bound on Ωgrowm does not translate tothe DES+Ext-geo constraints can also be understood interms of degeneracies in our model’s larger parameterspace. We will explore this in more detail in Sect. VI.

Finally, the bottom panel of Fig. 3 shows constraintsfrom DES and Ext-all, which adds BOSS RSD con-

0.24 0.30 0.36 0.42

Ωgeom

0.2

0.4

0.6

0.8

Ωgro

wm

ΛCDM

Split ΩmDES Y1 3x2pt

DES Y1 all

0.24 0.30 0.36 0.42

Ωgeom

0.2

0.4

0.6

0.8

Ωgro

wm

ΛCDM

Split ΩmDES Y1 all

Ext geo

DES + ext geo

0.24 0.30 0.36 0.42

Ωgeom

0.2

0.4

0.6

0.8

Ωgro

wm

ΛCDM


Ext all

DES + ext all

FIG. 3. The 68 and 95% confidence regions for Ωgrowm and Ωgeom

for our various data combinations. The diagonal gray linesshow where Ωgrowm = Ω

geom . Note that the three plots have the

same axis ranges, and that vertical axes cover a much largerrange of values than the horizontal axes. The blue outline-only contours in the top plot are the same as the shaded bluecontours in the other plots.

16

straints on growth to the previously considered externalgeometric measurements. We see that compared to themiddle panel’s Ext-geo results, adding RSD allows Ext-all to place a lower bound on Ωgrowm , and when combinedwith DES, Ωgrowm is bounded on both sides. The factthat there is not very much overlap between the DESand Ext-all contours, with Ext-all preferring somewhathigher Ωgrowm than DES, reflects their weak 2σ tension.The shape of the Ext-all constraints here, as well as howDES adds information, is related to a degeneracy be-tween Ωgrowm and

∑mν , which we will discuss further in

Sect. VI.

B. Splitting Ωm and w

Fig. 4 shows the 68% and 95% confidence contourswhen splitting both Ωm and w for DES+Ext-all con-straints, showing the parameters Ωgeom , Ω

growm , w

geo, andwgrow. The most notable feature is the strong degeneracybetween the two growth parameters, Ωgrowm and w

grow.We interpret this to mean that while DES+Ext-all canseparately constrain growth and geometry, the data can-not distinguish between Ωm-like and w-like deviationsfrom the structure growth history expected from wCDM.This behavior is also consistent with Ref. [27]’s findingthat, for a given Ωm(z), w only weakly effects growthrates. This makes it unsuprising that it is difficult torobustly constrain wgrow separately from Ωgrowm .

Because of this degeneracy, even using our most in-formative “DES+Ext-all” data combination wgrow is un-constrained, and the upper and lower bounds placed onΩgrowm are entirely dependent on the choice of prior forwgrow. As discussed in Appendix B, our analyses of simu-lated data showed that projection effects associated withthis degeneracy significantly affect the one-dimensionalmarginalized constraints on both Ωgrowm and w

grow. Be-cause of this we do not report parameter constraints forthis model.

C. Consistency with Θgrow = Θgeo

Ultimately the question we would like to ask is whetherthe results above are consistent with ΛCDM, or withwCDM, in the case where we split both Ωm and w. Thereare several ways we can assess this. We begin simply bylooking at the two-dimensional confidence regions shownin Figs. 3 and 4, noting whether or not they intersectthe lines corresponding to ΛCDM (in Fig. 3) and wCDM(in Fig. 4). We see that when we split Ωm, the 68%confidence intervals for DES and DES+Ext-geo intersectthe Ωgrowm = Ω

geom line, while that of DES+Ext-all just

touches the ΛCDM line, preferring Ωgrowm > Ωgeom . When

we split both Ωm and w, both the Ωgrowm = Ω

geom and

wgrow = wgeo lines goes directly through the DES+Ext-all 68% confidence intervals.

0.25 0.30 0.35

Ωgeom

−2.0

−1.5

−1.0

−0.5

wgro

w

0.3

0.6

Ωgro

wm

−1.25

−1.00

−0.75

wgeo

−1.25 −1.00 −0.75

wgeo0.3 0.6

Ωgrowm

−2.0 −1.5 −1.0 −0.5

wgrow

Split Ωm, w

DES Y1 all

Ext all

DES + ext all

FIG. 4. Marginalized constraints from DES and external datawhen both Ωm and w are split. The diagonal panels show nor-malized one-dimensional marginalized posteriors, while theoff-diagonal panels show 68% and 95% confidence regions.Solid gray lines show the wCDM parameter subspace whereΩgrowm = Ω

geom and w

grow = wgeo.

To assess consistency with Θgrow = Θgeo in our fullparameter space, we use Bayesian Suspiciousness Smod,as described in Eq. (30) of Sect. IV B. As we did whenwe used Suspiciousness to evaluate concordance betweendatasets, we use p(S > Smod) to report the probability toexceed the observed Suspiciousness, and “1-tail equiv. σ”as the number of normal distribution standard deviationswith equivalent probability. Here, larger Smod, smallerp(S > Smod), and larger σ indicate more tension withΘgrow = Θgeo. Numbers for all of these quantities areshown in Table III. According to this metric, when wesplit Ωm we find the DES-only results to have a 1.5σtension with ΛCDM. This becomes 1.9σ for DES+Ext-geo, and 1.0σ for DES+Ext-all. When we split both Ωmand w we find tensions with wCDM to be 1.6σ for DES-only and 1.4σ for DES+Ext-all.

As another way of quantifying compatibility of thesplit-Ωm constraints with ΛCDM, in Fig. 5 we show themarginalized posterior for the difference Ωgrowm − Ωgeom .When we assess the fraction of the posterior volumeabove and below 0, we find that the fraction of theposterior volume with Ωgrowm > Ω

geom is 30% for DES-

only, equivalent to a normal distribution single-tail prob-ability of 0.5σ. These numbers become 91% (1.3σ) forDES+Ext-geo, and 95% (1.6σ) for DES+Ext-all.

We note two points of caution in interpreting theΩgrowm − Ωgeom marginalized posterior. First, because ofthe difference in constraining power on Ωgrowm and Ω

geom

17

−0.2 0.0 0.2 0.4 0.6Ωgrowm − Ωgeom

PSplit ΩmDES Y1 all

DES + ext geo

DES + ext all

FIG. 5. Marginalized posterior of the difference Ωgrowm −Ωgeom ,from fitting the split-Ωm model to the DES, DES+Ext-geo,and DES+Ext-all data combinations.

there is some asymmetry expected in these marginalizedposteriors even if the data are consistent with ΛCDM.This can be seen in Fig. 12 of Appendix C, which showsversions of this plot for synthetic data generated withΩgrowm = Ω

geom . Additionally, the posterior distribution is

impacted by the priors on Ωgrowm and Ωgeom . While the

GetDist software allows us to correct for the impactof hard prior boundaries for the parameters we sampleover, it is unable to do so for derived parameters. Thismeans that in cases where the shape of the posterior isinfluenced by the prior boundary of e.g. Ωgrowm , this willnecessarily affect the shape of the marginalized posteriorfor Ωgrowm −Ωgeom . Accounting for these caveats and com-paring to the simulated results in Appendix C, we seethat the DES+Ext-all probability distribution is shiftedto higher Ωgrowm −Ωgeom than was found in simulated anal-yses. The DES-only and DES+Ext-geo distributions donot appear to be significantly different from what mightbe expected given parameter space projection effects inΛCDM.

D. Summary of main results

The results discussed in this section are summarizedin Table III. In it, for the split Ωm model we show one-dimensional marginalized constraints on Ωgrowm and Ω

geom

from DES+Ext-geo and DES+Ext-all, along with ΛCDMand wCDM constraints for comparison. For each param-eter we show two-sided errors corresponding to the 68%confidence interval one-dimensional marginalized poste-rior. Because we expect the one-dimensional marginal-ized posteriors to be subject to significant projection ef-fects for DES-only constraints on the split Ωm model andfor the DES+Ext-all constraints when splitting both Ωmand w, as discussed in Sect. IV and Appendix B we donot report parameter bounds for those cases.

TABLE III. Summary of results. Parameter errors quotedare 68% confidence intervals, and d̃ is the Bayesian modeldimensionality. The quantity S is Bayesian Suspiciousness,with the superscript “dat” denoting an assessment of tensionbetween two datasets, and“mod”denoting model comparison.

ΛCDM DES DES+Ext-geo DES+Ext-all

Ωm 0.296+0.020−0.022 0.301

+0.009−0.008 0.302

+0.007−0.008

d̃ 14.0± 0.7 18.2± 0.8 15.8± 0.8logSdat - −1.8± 0.3 −0.8± 0.2

p(S > Sdat) - 0.04± 0.03 0.25± 0.05equiv. σ - 2.0± 0.4 1.2± 0.1

Split Ωm DES DES+Ext-geo DES+Ext-all

Ωgrowm − Ωgeom - 0.126+0.228−0.129 0.116+0.100−0.084

Ωgeom - 0.300+0.009−0.008 0.304

+0.009−0.008

Ωgrowm - 0.425+0.232−0.131 0.421

+0.102−0.089

d̃ 15.0± 0.7 18.1± 0.9 19.8± 1.0logSdat - −1.1±−0.2 −2.0± 0.3

p(S > Sdat) - 0.13± 0.06 0.05± 0.03equiv. σ - 1.5± 0.3 2.0± 0.3logSmod −0.6± 0.2 −1.3± 0.2 −0.4± 0.3

p(S > Smod) 0.14± 0.08 0.06± 0.05 0.31± 0.07equiv. σ 1.5± 0.4 1.9± 0.4 1.0± 0.1

p(Ωgrowm > Ωgeom ) 0.30 0.91 0.95

1-tail equiv. σ 0.5 1.3 1.6

wCDM DES DES+Ext-all

Ωm 0.292+0.022−0.022 0.301

+0.009−0.008

w −0.911+0.073−0.076 −0.997+0.048−0.050

d̃ 16.3± 0.8 17.6± 0.8logSdat - −1.5± 0.2

p(S > Sdat) - 0.17± 0.04equiv. σ - 1.4± 0.1

Split Ωm, w DES DES+Ext-all

d̃ 15.2± 0.7 18.2± 0.9logSdat - −2.2± 0.2

p(S > Sdat) - 0.09± 0.03equiv. σ - 1.7± 0.2logSmod −0.5± 0.2 −0.6± 0.2

p(S > Smod) 0.11± 0.07 0.15± 0.09equiv σ 1.6± 0.4 1.4± 0.4

For all model-data combinations considered we useBayesian Suspiciousness as defined in Sect. IV B toreport data tension and model comparison statistics.In Table III, d̃ is the Bayesian model dimensionality(Eq. (25)) quantifying the effective number of parame-ters constrained, Sdat is the Bayesian suspiciousness as-

18

sessing agreement between pairs of datasets (Eq. (24)),and Smod is the model-comparison Bayesian Suspicious-ness (Eq. (30)), quantifying tension or agreement withΘgrow = Θgeo. The quantities p(S > SX), for X ∈[dat,mod] is the probability that a random realizationexceeds the observed Suspiciousness SX , and “equiv. σ”translates that probability into the number standard de-viations with an equivalent double-tail probability for anormal distribution (Eq. (33)). Large S, small p, andlarge equivalent σ indicate tension, while small S, largep, and small equivalent σ indicate concordance. For allquantities the numbers quoted in Table III are the meanand standard deviation from sampling error reported byAnesthetic.

As an alternative model-comparison statistic for thesplit-Ωm model, we additionally report p(Ω

growm > Ω

geom ),

the fraction of the posterior volume with Ωgrowm > Ωgeom .

For this part of the table, the “equiv. σ” is the number ofnormal distribution standard deviations with equivalentsingle-tail probability.

VI. RESULTS: IMPACT OF GROWTH-GEOMETRYSPLIT ON OTHER PARAMETERS

Here we explore how our split parameterization, focus-ing on splitting only Ωm, affects the inference of othercosmological parameters. In this discussion we will pri-marily reference Fig. 6, which shows two-dimensionalmarginalized posteriors of DES+Ext-all constraints onΩgrowm , Ω

geom , the difference Ω

growm − Ωgeom ,

∑mν , S8 ≡

σ8√

Ωgeom /0.3, h, and AIA. For comparison, we also showa DES+Ext-geo version of this plot in Fig. 17 of Ap-pendix E. We use this higher dimensional visalization ofthe posterior to characterize how additional degrees offreedom in the relationship between expansion historyand structure growth change considerations in cosmolog-ical analyses, both in terms of how we model of astro-physical effects (

∑mν , AIA) and in terms of commonly

studied tensions (S8, h).In the off-diagonal panels of Fig. 6, 68% and 95% con-

fidence regions are shown for DES-only as blue shadedcontours, Ext-all as pink shaded contours, and the com-bination DES+Ext-all as dark purple outlines. The diag-onal panels show normalized one-dimensional marginal-ized posteriors for each parameter. Solid gray lines showthe ΛCDM subspace where Ωgrowm =Ω

geom , and grey dashed

lines show the DES+Ext-all posterior for ΛCDM.

A. Effect of split on neutrino mass

Because the combination of Planck, BOSS BAO, andBOSS RSD are able to tightly constrain cosmological pa-rameters in ΛCDM, it may be surprising that DES addsinformation at all when combined with the Ext-all data.Looking at Fig. 6, we see that it does so because theexternal data exhibits a significant degeneracy between

Ωgrowm and the sum of neutrino masses∑mν . The Ext-all

degeneracy occurs because changes in∑mν and Ω

growm

have competing effects on the matter power spectrum:higher neutrino mass suppresses structure formation atsmall scales (k & 10−2h Mpc−1), while raising Ωgrowm re-sults produces more late-time structure. DES data addsconstraining power because it provides an upper boundon Ωgrowm which breaks that degeneracy.

Looking at the marginalized constraints on∑mν , we

see that both the DES+Ext-all (Fig. 6) and DES+Ext-geo (Fig. 17) constraints produce a detection of neutrinomass at

∑mν = 0.4±0.1 eV, which is significantly higher

than the upper bounds obtained from the combined anal-ysis of BOSS DR12 and the full Planck temperature andpolarization power spectra [49, 57]. The DES-only poste-rior gives a weak lower bound on neutrino mass, thoughwe suspect that this may be at least in part causedby parameter-space projection effects. In ΛCDM, theExt-all constraints on

∑mν become an upper bound of∑

mν < 0.45 eV at 95% confidence, which is is consistentwith the BOSS results (though weaker because we do notuse the full Planck likelihood), while the DES preferencefor high

∑mν remains. This causes the DES+Ext-all

ΛCDM posterior, shown as a gray dashed line in Fig. 6,to peak at

∑mν = 0.2± 0.1 eV.

To begin interpreting the preference for high∑mν ,

we can look at the∑mν-Ω

growm panel of Fig. 6 and note

that the Ext-all constraints exhibit a preference for thehigh-

∑mν , high-Ω

growm part of parameter space. That

preference combined with the DES upper bound on Ωgrowmlikely drives the 2σ tension between Ext-all and DES,and it appears to be responsible for pulling the combinedDES+Ext-all constraints away from the ΛCDM Ωgrowm =Ωgeom line.

It is instructive to examine how the constituent Planckand BOSS likelihoods combine to produce the Ext-allcontours. We show this in Fig. 7, with the compressedPlanck posterior in yellow, BOSS BAO+RSD in orange,and their combination, Ext-all, as black outlines. Thecompressed Planck likelihood approximately defines aplane in the σ8-

∑mν-Ω

growm parameter space because

Planck’s measurement of As can be extrapolated forwardto predict σ8, but the effects of Ω

growm and

∑mν on late-

time structure growth loosen that predictive relationship.The BOSS data probe late-time structure more directly,so the combined BAO and RSD results can be thoughtof as roughly providing a measurement of σ8 that is in-sensitive to

∑mν and only weakly dependent on Ω

growm .

Putting all of this together, we see that the shape of theExt-all posterior strongly depends on the relationship be-tween Planck’s measurement of As, BOSS’s measurementof σ8, as well as the extent to which late-time degrees offreedom impact how deterministically Planck’s As con-straint maps to σ8. For example: if the Planck As con-straints were lowered slightly, or the BOSS σ8 constraintswere raised, this would move the Ext-all constraints to-wards lower

∑mν and consequently, lower Ω

growm . The

DES+Ext-geo constraints have a similar property: we

19

0.0 0.6

Ωgrowm − Ωgeom

0

1

2

AIA

0.6

0.8

h

0.4

0.8

∑mν

[eV

]

0.6

0.8

1.0

S8

0.3

0.6

Ωgro

wm

0.25

0.30

0.35

Ωgeom

0.25 0.30 0.35

Ωgeom

0.3 0.6

Ωgrowm

0.6 0.8 1.0

S8

0.4 0.8∑mν [eV]

0.6 0.8

h

0 1 2

AIA


Ext all

DES + ext all

ΛCDM

FIG. 6. Constraints from the DES and the Ext-all external dataset, which includes the compressed Planck likelihood, BOSSDR12 BAO, and BOSS DR12 RSD. The off-diagonal panels show the 68 and 95% confidence intervals for each data combination,while the diagonal panels show normalized one-dimensional marginalized posteriors on parameters. DES-only results are shownin blue, Ext-all results are pink, and their combination is shown using unshaded purple contours. The gray dashed curves showDES+Ext-all constraints in ΛCDM and the gray solid lines show where Ωgrowm = Ω

geom .

can see in the S8-∑mν panel of Fig. 17 that slight rela-

tive changes to the Planck As or DES S8 constraints canhave a significant impact on the

∑mν posterior. In other

words, our results’ preference for high∑mν (and conse-

quently, high Ωgrowm ) can be interpreted as a manifestationof the early-versus-late-Universe σ8 tension discussed inthe Introduction.

Our findings here are in line with several previousstudies which report a preference for

∑mν ∼ 0.3 eV

when modeling degrees of freedom affecting structuregrowth are introduced to combined CMB and LSS analy-ses. These include the growth-geometry split analysis ofRefs. [41, 42], as well as examinations of neutrino massin conjunction with ALens [126, 127] (which describes theamount of lensing-induced smoothing of the CMB power

spectrum), time-dependent dark energy [128], and mod-ified gravity [129]. Notably, however

des y1 results: splitting growth and geometry to test cdm · 2020. 10. 14. · year 1 (y1) galaxy...

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