# Probing gravity with the DES-CMASS sample and BOSS spectroscopy

S. Lee,1 E. M. Huff,2 A. Choi,3 J. Elvin-Poole,3,4 C. Hirata,3,4 K.
Honscheid,3,4 N. MacCrann,5 A. J. Ross,3 M. A. Troxel,1 T. F. Eifler,6,2 H.
Kong,3,4 A. Ferté,2 J. Blazek,3,7 D. Huterer,8 A. Amara,9 A. Campos,10 A.
Chen,8 S. Dodelson,10 P. Lemos,11,12 C. D. Leonard,13 V. Miranda,6 J. Muir,14
M. Raveri,15 L. F. Secco,16 N. Weaverdyck,8 J. Zuntz,17 S. L. Bridle,18 C.
Davis,14 J. DeRose,19,20 M. Gatti,16 J. Prat,21 M. M. Rau,10 S. Samuroff,10 C.
Sánchez,16 P. Vielzeuf,22 M. Aguena,23,24 S. Allam,25 A. Amon,14 F. Andrade-
Oliveira,26,24 G. M. Bernstein,16 E. Bertin,27,28 D. Brooks,11 D. L.
Burke,14,29 A. Carnero Rosell,30,24,31 M. Carrasco Kind,32,33 J. Carretero,22
F. J. Castander,34,35 R. Cawthon,36 C. Conselice,18,37 M. Costanzi,38,39,40 L.
N. da Costa,24,41 M. E. S. Pereira,8 J. De Vicente,42 S. Desai,43 H. T.
Diehl,25 J. P. Dietrich,44 P. Doel,11 S. Everett,20 A. E. Evrard,45,8 I.
Ferrero,46 P. Fosalba,34,35 J. Frieman,25,15 J. García-Bellido,47 E.
Gaztanaga,34,35 D. W. Gerdes,45,8 T. Giannantonio,48,49 D. Gruen,50,14,29 R.
A. Gruendl,32,33 J. Gschwend,24,41 G. Gutierrez,25 W. G. Hartley,51 S. R.
Hinton,52 D. L. Hollowood,20 B. Hoyle,44,53 D. J. James,54 K. Kuehn,55,56 N.
Kuropatkin,25 O. Lahav,11 M. Lima,23,24 M. A. G. Maia,24,41 M. March,16 J. L.
Marshall,57 F. Menanteau,32,33 R. Miquel,58,22 J. J. Mohr,44,53 R. Morgan,36
A. Palmese,25,15 F. Paz-Chinchón,32,48 A. Pieres,24,41 A. A. Plazas Malagón,59
A. Roodman,14,29 E. Sanchez,42 V. Scarpine,25 M. Schubnell,8 S. Serrano,34,35
I. Sevilla-Noarbe,42 E. Sheldon,60 M. Smith,61 E. Suchyta,62 M. E. C.
Swanson,32 G. Tarle,8 D. Thomas,9 C. To,50,14,29 T. N. Varga,53,63 and J.
Weller53,63 (DES Collaboration) _Affiliations are listed at the end of the
paper_

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The DES-CMASS sample (DMASS) is designed to optimally combine the weak lensing
measurements from the Dark Energy Survey (DES) and redshift-space distortions
(RSD) probed by the CMASS galaxy sample from the Baryonic Oscillation
Spectroscopic Survey (BOSS). In this paper, we demonstrate the feasibility of
adopting DMASS as the equivalent of CMASS for a joint analysis of DES and BOSS
in the framework of modified gravity. We utilize the angular clustering of the
DMASS galaxies, cosmic shear of the DES metacalibration sources, and cross-
correlation of the two as data vectors. By jointly fitting the combination of
the data with the RSD measurements from the CMASS sample and Planck data, we
obtain the constraints on modified gravity parameters
$\mu_{0}=-0.37^{+0.47}_{-0.45}$ and $\Sigma_{0}=0.078^{+0.078}_{-0.082}$. Our
constraints of modified gravity with DMASS are tighter than those with the DES
Year 1 redMaGiC sample with the same external data sets by $29\%$ for
$\mu_{0}$ and $21\%$ for $\Sigma_{0}$, and comparable to the published results
of the DES Year 1 modified gravity analysis despite this work using fewer
external data sets. This improvement is mainly because the galaxy bias
parameter is shared and more tightly constrained by both CMASS and DMASS,
effectively breaking the degeneracy between the galaxy bias and other
cosmological parameters. Such an approach to optimally combine photometric and
spectroscopic surveys using a photometric sample equivalent to a spectroscopic
sample can be applied to combining future surveys having a limited overlap
such as DESI and LSST.

###### keywords:

cosmological parameters – gravitational lensing – large-scale structure of the
Universe

††pubyear: 2021††pagerange: Probing gravity with the DES-CMASS sample and BOSS
spectroscopy–Affiliations

##  1 Introduction

Over the past three decades after the discovery of the accelerating Universe
(Riess et al., 1998; Perlmutter et al., 1999), the $\Lambda$CDM model has been
widely accepted as the simplest and the most successful concordance model in
modern cosmology. This model is based upon a spatially-flat, expanding
Universe governed by Einstein’s General Relativity (GR) and whose components
are dominated by roughly 25% of cold dark matter (CDM) and 70% of dark energy,
which is commonly associated with a cosmological constant. The cosmological
constant, denoted as $\Lambda$, can be cast in the model as a perfect fluid
with the equation-of-state parameter of minus one in order to trigger the
late-time cosmic acceleration. The $\Lambda$CDM model has been thoroughly
validated through a broad range of stringent tests using cosmological data
sets such as the Cosmic Microwave Background (CMB), Type Ia supernovae, baryon
acoustic oscillation (BAO), the large-scale clustering of galaxies, and weak
gravitational lensing. Despite the overall success of the $\Lambda$CDM model
supported by many observations, however, several fundamental puzzles remain.
One notable concern is that the cosmological constant has no explicit physical
theory for its origin. In the context of quantum field theory, one may connect
the cosmological constant with the vacuum energy associated with zero-level
quantum fluctuations. However, this approach can be easily countered by $\sim
120$ orders of magnitude difference in the value of the vacuum energy
predicted by quantum field theory and inferred from the LCDM model (Weinberg,
1989). Such discrepancy leads to the desire for alternative models beyond the
$\Lambda$CDM cosmology.

Modified gravity (MG) has been suggested as one of the strong candidates to
explain the cosmic acceleration without introducing a cosmological constant.
In such a theory, modified GR at cosmological scales naturally produces an
acceleration identical to the one assumed in the $\Lambda$CDM model, without
raising the same issues as a cosmological constant. In a phenomenological
approach, the modification to GR is often parametrized as two MG parameters
added to the gravitational potentials in the Friedmann equations. These MG
parameters modify the growth equations derived from the Friedmann equations,
and thereby any departure from GR appears as a change in the growth of
structure and the deflection of light while keeping the same expansion history
of the Universe as $\Lambda$CDM (see Ishak 2019 for an overview of the theory
and phenomenology of MG models). Hence, it is worth noting that probes
sensitive to the growth of structure play a crucial role in testing deviations
from GR.

Redshift space distortions (RSD) and weak gravitational lensing have been used
together as a popular combination of growth data to test GR (Zhang et al.,
2007; Reyes et al., 2010; Simpson et al., 2013; Blake et al., 2016; de la
Torre et al., 2017; Ferté et al., 2019; Singh et al., 2019; Abbott et al.,
2019; Planck Collaboration, 2020a). A general approach to combine these two
probes is adding an independent measurement of $f(z)\sigma_{8}(z)$ from
redshift-space distortions by a spectroscopic survey to a weak lensing
measurement by a photometric survey. However, several papers (Bernstein & Cai,
2011; Gaztañaga et al., 2012; Cai & Bernstein, 2012) have shown that a
combined analysis of overlapping spectroscopic and weak lensing surveys can
yield much stronger dark energy and growth constraints than a combination of
independent RSD and weak lensing measurements. The motivation for combining
those two overlapping probes comes from the fact that RSD provides the
constraints of the growth parameters only in combination with other
parameters, i.e., the measurements of $f(z)\sigma_{8}(z)$ and $\beta=f(z)/b$
where $f(z)$ is the redshift dependent growth rate, $\sigma_{8}(z)$ is the
amplitude of the matter clustering at redshift $z$, and $b$ is galaxy bias.
Galaxy bias has been a major source of uncertainty in cosmological analyses of
large-scale structure. In the MG framework, its impact is even more
significant because one of the MG parameters modifying the Newtonian potential
enters into $\sigma_{8}(z)$ and $f(z)$ through the growth factor term and is
strongly degenerate with galaxy bias. Without any prior knowledge of galaxy
bias, one cannot constrain $f(z)$ or $\sigma_{8}(z)$ independently, resulting
in the degradation of the measurement of MG parameters111This statement about
degeneracy assumes only linear scales, where MG can be modeled.. Meanwhile,
weak gravitational lensing directly measures the value of $\sigma_{8}$ today.
The cross-correlation of RSD and weak lensing enables us to tighten the
constraint of galaxy bias by breaking the $f$–$b$ degeneracy and allows a more
precise inference of the underlying distribution of matter.

Several studies have taken the aforementioned approach to test GR by combining
the RSD measurements from the Baryonic Oscillation Spectroscopic Survey (BOSS;
Eisenstein et al., 2011) and weak lensing measurements from recent deep
imaging surveys such as the Canada-France-Hawaii Telescope Lensing Survey
(CFHTLenS; Heymans et al., 2012), Dark Energy Survey (DES; Abbott et al.,
2016), Kilo-Degree Survey (KiDS; de Jong et al., 2013) and the Hyper Suprime-
Cam Survey (HSC; Aihara et al., 2018). The BOSS target galaxy samples, LOWZ
and CMASS (Reid et al., 2016), are the largest galaxy spectroscopic samples
yielding the best BAO and RSD measurements in the redshift range of
$0.15<z<0.75$. The large sample size and the availability of spectroscopic
redshifts have turned the galaxies in the samples into a popular candidate for
gravitational lenses in the overlapping regions with the imaging surveys. By
adopting BOSS galaxies as lenses, one can access better deep images while
maintaining the strong constraining power of the galaxy clustering
measurements from BOSS.

Alam et al. (2017a) tested gravity by combining galaxy-galaxy lensing from
CFHTLenS (Miyatake et al., 2015) with redshift space galaxy clustering from
the BOSS CMASS sample (Alam et al., 2015). The galaxy-galaxy lensing signal
was obtained around the CMASS galaxies living in the small overlapping area of
$105{\rm\;deg^{2}}$ between the BOSS and CFHTLenS footprints. Jullo et al.
(2019) performed a similar analysis with the addition of a shape catalogue
from CFHT-Stripe 82 (Moraes et al., 2014) to extend the available area for
weak lensing to $250{\rm\;deg^{2}}$. Unlike the previous cases, the KiDS
survey was intentionally designed to mostly overlap with the BOSS and 2dFLenS
surveys to maximize the number of reliable spectroscopic lenses in their full
footprint (de Jong et al., 2013). Joudaki et al. (2017) and Amon et al. (2018)
fully utilized the KiDS-450 footprint ($450\deg^{2}$) to test gravity in a
phenomenological approach using the BOSS galaxies as lenses on the KiDS
imaging data. Later, Tröster et al. (2020) constrained the $f(R)$ gravity
model with the BOSS galaxies over the KiDS-1000 footprint covering the
increased area of $\sim 1,000{\rm\;deg^{2}}$. As shown in these previous
studies, however, the MG analyses performed with the spectroscopic samples
have to face a limitation due to a fairly small overlapping region with
spectroscopic surveys (mostly within a few hundreds of ${\rm\;deg^{2}}$),
unless imaging surveys are planned in consideration of utilizing the existing
spectroscopic information such as KiDS.

The DES-CMASS galaxy sample from the Dark Energy Survey (hereafter DMASS) has
been developed to optimally combine the measurements of weak lensing from DES
and RSD from BOSS by extending the available area for such analyses beyond the
overlap between BOSS and DES (Lee et al., 2019). The selection algorithm for
DMASS is trained in the overlapping region ($123\deg^{2}$) with galaxy colors
and magnitudes, and then identifies CMASS-like galaxies in the rest of the DES
footprint where the spectroscopic information is not available. The resulting
DMASS sample replicates the statistical properties of the CMASS galaxy sample
and populates the lower region of the DES Year 1 (Y1) wide-area survey
footprint ($1,244\deg^{2}$), excluding the overlap. Using the DMASS galaxies
as lenses, one can obtain the measurements of galaxy clustering and galaxy-
galaxy lensing equivalent to those that would have been measured if CMASS
populated the full footprint of DES. These two clustering measurements, along
with cosmic shear (the correlation of galaxy shapes) from the DES source
galaxies, not only extract the full statistical power of DES, but they can
also be efficiently combined with the measurement of redshift-space distortion
from CMASS without introducing additional systematics parameters such as
galaxy bias.

In this paper, we test the feasibility of this optimal combination method
using the DMASS galaxy sample as lenses in the framework of phenomenological
modified gravity. To isolate the results from any additional complexity
arising from different probes, we adopt a minimal set of data as follows: the
combination of angular galaxy clustering, cosmic shear, and the cross-
correlations of the two (galaxy-galaxy lensing) measured with the DMASS lenses
and DES Y1 sources (hereafter DMASS 3x2pt), the RSD and BAO measurements from
the BOSS CMASS sample (Chuang et al., 2017), and Planck 2018 data (Planck
Collaboration, 2020a). The measurement of angular galaxy clustering used in
this work is described in the original sample paper (Lee et al., 2019). The
measurement of galaxy-galaxy lensing and its calibration procedures are
detailed in (Lee et al., 2021). DES Y1 cosmic shear is adopted from Troxel et
al. (2018). Note that we define ‘DMASS 3x2pt + CMASS RSD/BAO’ as a baseline as
DMASS is designed to harness its maximum power in combination with CMASS. In
Figure 1, we summarize the data sets used in the analysis. We follow the
methodology used in the DES Y1 analysis for modified gravity (hereafter
DESY1MG; Abbott et al., 2019) and compare their MG constraints with ours to
estimate the efficiency of this method.

This paper is organized as follows. In the following section, we introduce the
phenomenological parametrization of MG adopted in this paper. The data sets
and theoretical predictions used to describe the data are detailed in Section
3. In Section 4, we describe our analysis methodology. In Section 5, we
present a series of validation tests for potential systematics that might bias
the results. In Section 6, we briefly describe how we blind the data to avoid
confirmation bias. In Section 7, we present our results and compare them with
the results of DESY1MG. Our conclusions are presented in Section 8.

Figure 1:  Summary of the different data sets in this analysis. The data in
this paper consist of the DMASS 3x2pt measurements, the RSD and BAO
measurements from the BOSS CMASS sample, and the CMB measurement from Planck.
The DMASS 3x2pt measurement in the red box includes angular clustering ($w$),
cosmic shear ($\xi_{\pm}$), the cross-correlation of the two (galaxy-galaxy
lensing, $\gamma_{t}$) measured with the DMASS lenses and the DES Y1
metacalibration sources. The measurement from CMASS provides five constraints
on RSD and BAO in the blue box. See Section 3.2 for a detailed description for
these parameters. The combination of the red and blue boxes is defined as
‘baseline’. The galaxy bias parameter ($b$) is shared in this combination. The
black box represents the CMB and CMB lensing measurements from Planck (see
Section 3.3).

##  2 Parametrization of modified gravity

We parametrize departures from GR in a phenomenological way. This approach has
an advantage in the sense that it does not require exact knowledge of the
specific alternative theory but is still able to capture a generic deviation
of the perturbation evolution from $\Lambda$CDM, by injecting two parameters
into the perturbed Einstein’s equations (see Ishak (2019) for a general
overview of the phenomenological approach and its applications).

The perturbed Friedmann-Lemaître-Robertson-Walker (FLRW) metric describing the
$\Lambda$CDM cosmology is defined in terms of the two gravitational potentials
$\Psi$ and $\Phi$ given as

$\displaystyle\mathrm{d}s^{2}=(1+2\Psi)\mathrm{d}t^{2}-a^{2}(t)(1-2\Phi)\mathrm{d}x^{2}~{}.$
(1)

The evolution of the two gravitational potentials are described by the two
equations as follow:

$\displaystyle k^{2}\Phi$ $\displaystyle=$ $\displaystyle-4\pi
a^{2}G\rho\Delta~{},$ (2) $\displaystyle k^{2}(\Psi-\Phi)$ $\displaystyle=$
$\displaystyle-12\pi Ga^{2}(\rho+P)\sigma~{},$ (3)

where $\Delta$ is the gauge-invariant density contrast, $\rho$ and $\sigma$
are the density and the shear stress, and $P$ is the pressure. For a
negligible shear stress, the combination of the two equations leads to another
set of the Poisson equation as follows:

$\displaystyle k^{2}\Psi$ $\displaystyle=$ $\displaystyle-4\pi
Ga^{2}\rho\Delta~{},$ (4) $\displaystyle k^{2}\Psi_{W}$ $\displaystyle=$
$\displaystyle-4\pi Ga^{2}\rho\Delta~{}.$ (5)

where $\Psi_{W}$ is the Weyl potential defined as $\Psi_{W}=(\Psi+\Phi)/2$
which affects the propagation of light. The deviations from GR can be
encapsulated in two parameters multiplied to these gravitational potentials as
below:

$\displaystyle\Psi(k,a)$ $\displaystyle=$ $\displaystyle[1+\mu(k,a)]\Psi_{\rm
GR}(k,a)~{},$ (6) $\displaystyle\Psi_{W}(k,a)$ $\displaystyle=$
$\displaystyle[1+\Sigma(k,a)]\Psi_{W,{\rm GR}}(k,a)~{}.$ (7)

The gravitational acceleration of non-relativistic particles is determined by
$\Psi$, and the paths of photons depend on $\Psi_{W}$. Therefore, $\mu$ is
sensitive to modifications to the structure growth, whereas $\Sigma$ governs
modifications to the lensing of light. One can break the degeneracy between
$\mu$ and $\Sigma$ by combining the measurements from galaxy clustering
surveys with the measurements from weak lensing.

In this paper, we adopt the time-evolving MG parameters following Ferreira &
Skordis (2010); Simpson et al. (2013) given as

$\displaystyle\mu(a)=\mu_{0}\frac{\Omega_{\Lambda}(a)}{\Omega^{0}_{\Lambda}},~{}~{}~{}\Sigma(a)=\Sigma_{0}\frac{\Omega_{\Lambda}(a)}{\Omega^{0}_{\Lambda}}~{},$
(8)

where $\Omega_{\Lambda}^{0}\equiv\Omega_{\Lambda}(a=1)$ is the dark energy
density today so that $\mu_{0}$ and $\Sigma_{0}$ represent today’s values of
$\mu(1)$ and $\Sigma(1)$, respectively. Note that GR is restored for
$\mu_{0}=\Sigma_{0}=0$.

##  3 Data and measurements

In this section, we explain the data sets and measurements we use for the
analysis. The primary data used in this work are the DMASS galaxy catalog (Lee
et al., 2019) and metacalibration shape catalog (Zuntz & Sheldon et al.,,
2018) from DES. Both catalogs are based on the images taken between Aug. 31,
2013 and Feb. 9, 2014 during the first year observations of DES (Abbott et
al., 2005; Flaugher et al., 2015; Abbott et al., 2018). In Section 3.1, we
briefly describe the two catalogs and the 3x2pt measurements (galaxy
clustering, tangential shear, and cosmic shear) obtained with these catalogs.
We also utilize the RSD and BAO measurements extracted from the galaxy
clustering of the BOSS CMASS galaxies (Chuang et al., 2017). Their
measurements and corresponding covariance matrices are presented in Section
3.2. In Section 3.3, we briefly describe the Planck CMB data we include to
constrain the early universe.

###  3.1 Dark Energy Survey

####  3.1.1 DMASS and metacalibration catalogs

Figure 2:  The spectroscopic redshift distribution of BOSS CMASS (red shaded)
and the photometric redshift distributions of metacalibration (grey shaded).
The maroon color points with error bars show the redshift distribution of
DMASS computed by cross-correlating the sample with the DES Y1 redMaGiC
sample. As the redshift distributions of CMASS and DMASS show good
consistency, we use the redshift distribuion of CMASS as the true redshift
distribution for a theoretical prediction for DMASS. The source sample,
metacalibration is divided into 4 tomographic bins ($0.2<z_{s}<0.43$,
$0.43<z_{s}<0.63$, $0.63<z_{s}<0.90$ and $0.90<z_{s}<1.30$) using the BPZ
photometric redshift code.

Figure 3: The angular galaxy clustering (left) and tangential shear data
(right) from Lee et al. (2019) and Lee et al. (2021). The signals are measured
with the DMASS (lenses) and DES Y1 metacalibration (sources) catalogs. The
dashed lines are the best-fitting MG predictions from the combination of the
baseline case (DMASS 3x2pt + CMASS RSD/BAO) and Planck data. The shaded
regions are discarded in the analysis to exclude the small scales where the
phenomenological parametrization is not valid. The remaining data points are
16 points for $w(\theta)$ and 38 points for $\gamma_{t}(\theta)$. Figure 4:
The cosmic shear data from Troxel et al. (2018) measured with the DES Y1
metacalibration catalog. The bottom triangle contains $\xi_{+}(\theta)$ and
the top triangle contains $\xi_{-}(\theta)$. The dashed lines are the best-
fitting MG predictions from the combination of the baseline case and Planck
data. The labels ($i$, $j$) in the upper-right corner of each panel indicate
the combination of the $i$th and $j$th source tomographic bins used to obtain
each signal. The shaded regions are discarded for the cosmological analysis,
leaving 119 points for $\xi_{+}(\theta)$ and 38 points for $\xi_{-}(\theta)$.

We utilize the DMASS galaxy sample (Lee et al., 2019) as gravitational lenses.
The DMASS sample is a photometric sample designed to replicate the statistical
properties of the BOSS CMASS galaxy sample (Reid et al., 2016). The sample
consists of $117,293$ effective galaxies selected from the DES Y1 GOLD catalog
(Drlica-Wagner et al., 2018), and covers the lower region of the DES Y1 wide-
area survey footprint, excluding the overlapping area with BOSS near the
celestial equator. The full coverage of DMASS is $1,244{\rm\;deg^{2}}$ after
masking.

The feasibility of using the DMASS sample as an equivalent of the CMASS sample
has been studied in Lee et al. (2019) and Lee et al. (2021). In Lee et al.
(2019), the redshift distribution of DMASS was computed by cross-correlating
the sample with the DES Y1 redMaGiC galaxy sample (Rozo & Rykoff et al.,,
2016; Elvin-Poole & Crocce et al.,, 2018). Figure 2 shows the redshift
distributions of CMASS (red shaded region) and DMASS (maroon error bars). The
two distributions show good agreement222The impact of the low- and high-ends
and a bump at $z\sim 0.4$ in the redshift distribution of DMASS has been
tested in the appendices of Lee et al. (2019) and Lee et al. (2021). Both have
a negligible impact on the signals of galaxy clustering and tangential shear..
Therefore, we use the redshift distribution of CMASS as the true redshift
distribution for a theoretical prediction for DMASS. In addition, Lee et al.
(2019) showed the consistency between DMASS and CMASS by comparing various
statistical properties such as the number density, auto-angular clustering,
and cross-angular clustering with the external surveys. The resulting values
of the difference in the mean galaxy bias and mean redshift are $\Delta
b=0.044^{+0.044}_{-0.043}$ and $\Delta z=(3.51^{+4.93}_{-5.91})\times
10^{-3}$, which indicate that the galaxy bias and mean redshift of two samples
are consistent within $1\sigma$. For more description of the galaxy sample and
sample selection algorithm, we refer readers to Lee et al. (2019).

Source galaxies are selected from the DES metacalibration catalog (Huff &
Mandelbaum, 2017; Sheldon & Huff, 2017; Zuntz & Sheldon et al.,, 2018).
Photo-$z$ of individual galaxies are evaluated by the Bayesian Photometric
Redshift (BPZ) algorithm (Coe & Benítez et al.,, 2006; Hoyle & Gruen et al.,,
2018). As in Zuntz & Sheldon et al., (2018); Prat & Sánchez et al., (2018) and
Troxel et al. (2018), sources are split into four redshift bins selected using
BPZ: $0.2<z<0.43$, $0.43<z<0.63$, $0.63<z<0.90$ and $0.90<z<1.30$. The
redshift distributions of the four source bins are plotted in Figure 2 with
the redshift distribution of lenses. The shear multiplicative biases,
photo-$z$ biases, and their uncertainties related to this catalog are
quantified in Zuntz & Sheldon et al., (2018); Hoyle & Gruen et al., (2018) and
employed as priors in our analysis. See Section 4 for a detailed description.

We do not split the lens sample in multiple redshift bins. Instead of using
the five tomographic lens bins as done in DESY1MG, we consider only one lens
bin along with the four source bins. This choice of one tomographic lens bin
is motivated by the two reasons as follows: First, Chuang et al. (2017) split
the LOWZ and CMASS sample into two bins (for a total of four bins) to increase
the sensitivity of redshift evolution, but did not find improvement in terms
of constraining different dark energy model parameters compared to the case of
one bin. Second, splitting the DMASS sample into multiple redshift bins
requires retraining the DMASS algorithm, and a series of validation tests
performed in Lee et al. (2019) should be followed. The combination of the one
lens and four source bins results in one galaxy clustering signal, four
galaxy-galaxy lensing signals, and twenty shear signals. In the following
sections, we will describe the modeling and measurement of these two-point
functions.

####  3.1.2 Angular galaxy clustering

The theoretical prediction for angular galaxy clustering is given as (Kaiser,
1992; LoVerde & Afshordi, 2008)

$\displaystyle
w(\theta)=\frac{1}{2\pi}\int^{\infty}_{0}C_{gg}(\ell)J_{0}(\ell\theta)\ell\mathrm{d}\ell~{},$
(9)

where $J_{n}(x)$ is the $n$th order Bessel function of the first kind. The
galaxy angular power spectrum $C_{gg}$ is the projection along the line of
sight of the 3D power spectrum as given by

$\displaystyle
C_{gg}(\ell)=\int^{\infty}_{0}\mathrm{d}\chi\frac{W_{g}^{2}(k,\chi)}{\chi^{2}}P_{\delta\delta}(k,z(\chi))~{},$
(10)

where $\ell$ denotes the angular multipole, $k=(\ell+1/2)/\chi$, and
$P_{\delta\delta}(k,z(\chi))$ is the matter power spectrum. As the growth
factor term $G^{2}(z)$ is contained in the matter power spectrum at $z$,
modifications to gravity by $\mu$ enters into the matter power spectrum. The
function $W_{g}(k,\chi)$ is the weight function for clustering defined as

$\displaystyle
W_{g}(k,\chi)=b_{g}(k,z(\chi))\frac{n_{g}(z(\chi))}{\bar{n}_{g}}\frac{\mathrm{d}z}{\mathrm{d}\chi}~{},$
(11)

where $b_{g}$ is the galaxy bias of the lens galaxies. In this paper, we adopt
the linear galaxy bias model as we restrict our analysis to sufficiently large
scales.

Lee et al. (2019) computed the angular clustering of DMASS to verify the
consistency between DMASS and CMASS. By following the procedure described in
that paper, we recompute the signal in the same manner, but with the number of
angular bins increased from 10 to 20. The measurement of $w(\theta)$ is
displayed in the first panel of Figure 3 with the best-fitting MG prediction.
As we obtained the same result except for the number of bins, we only briefly
summarize the methodology below and refer readers to Lee et al. (2019) for
details.

The signal is evaluated in 20 logarithmically spaced angular bins over the
range $2.5\arcmin<\theta<250\arcmin$, using the Landy-Szalay estimator (Landy
& Szalay, 1993) as given by

$\displaystyle
w(\theta)=\frac{DD(\theta)-2DR(\theta)+RR(\theta)}{RR(\theta)}~{},$ (12)

where DD, RR, and RR are the number of galaxy pairs, galaxy-random pairs, and
random pairs separated by an angular distance $\theta$. Potential systematics
that can bias the angular clustering were studied and corrected by applying
weights to individual galaxies, as illustrated in Section 4 of Lee et al.
(2019). More details of the measurement procedures are described through
Sections 4 and 5 in the same paper.

####  3.1.3 Galaxy-galaxy lensing

We use the tangential shear as an observable for galaxy-galaxy lensing. The
tangential shear correlating with the $i$th source bin is obtained from the
Fourier transform of the angular power spectrum as follows:

$\displaystyle\gamma^{i}_{t}(\theta)=\frac{1}{2\pi}\int^{\infty}_{0}C^{i}_{g\kappa}(\ell)J_{2}(\ell\theta)\ell\mathrm{d}\ell~{},$
(13)

where $J_{n}(x)$ is the $n$th order Bessel function of the first kind. The
angular power spectrum for galaxy-galaxy lensing correlating with $i$th source
bin takes the form

$\displaystyle
C^{i}_{g\kappa}(\ell)=\int^{\infty}_{0}\mathrm{d}\chi\frac{W_{g}(k,\chi)W^{i}_{\kappa}(\chi)}{\chi^{2}}P_{\delta\delta}(k,z(\chi))~{}.$
(14)

The integral along the line of sight indicates that weak lensing radially
projects the density fluctuations between us and the source galaxies. The
function $W(\chi)$ is the geometric weight function describing the lensing
efficiency defined as

$\displaystyle~{}~{}W^{i}_{\kappa}(\chi)=\frac{3H_{0}^{2}\Omega_{m}}{2c^{2}}\frac{\chi}{a(\chi)}$
(15)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\times\int^{\infty}_{\chi}\mathrm{d}\chi^{\prime}\frac{n^{i}_{\kappa}(z(\chi^{\prime}))\mathrm{d}z/\mathrm{d}\chi^{\prime}}{\bar{n}^{i}_{\kappa}}\frac{\chi^{\prime}-\chi}{\chi}[1+\Sigma(\chi^{\prime})]~{}$
(16)

in terms of the source distribution $n_{\kappa}(\chi)$. Modifications to GR in
lensing appear as $[1+\Sigma(\chi)]$. Modifications by $\mu$ enters in the
matter power spectrum as shown in the case of $w(\theta)$. Thus, galaxy-galaxy
lensing is sensitive to both modifications by matter and relativistic
particles.

We utilize the measurement of tangential shear around the DMASS galaxies
presented in Lee et al. (2021) as a data vector for $\gamma_{t}$ in the
cosmological analysis. With the lens and four source bins, the mean tangential
shear is computed by averaging over lens-source pairs as below:

$\displaystyle\gamma_{t}(\theta)=\langle\gamma_{+}(\theta)\rangle=\frac{1}{\langle
R\rangle}\frac{\sum_{j}w_{ls,j}\gamma_{+,j}}{\sum_{j}w_{ls,j}}~{}.$ (17)

The notation $w_{ls}$ is a combination of weights associated with each lens-
source pair. The value $\langle R\rangle$ in the denominator is the mean shear
response averaged over the sources, which is defined as the sum of the
measured shear response ($R_{\gamma}$) and shear selection bias correction
matrix ($R_{S}$) for metacalibration as follows: $\langle R\rangle=\langle
R_{\gamma}\rangle+\langle R_{S}\rangle$. To ensure the measured signal is free
from various systematic effects, Lee et al. (2021) performed tests for the
mean cross-component of the shear and estimated the impact of observing
conditions. The signals were also corrected for the boost factor. The cross-
correlation coefficient between the amplitude of the galaxy clustering of
CMASS and the tangential shear of DMASS was found to be consistent with unity
down to the scale of $4h^{-1}{\rm\;Mpc}$. The measurement is shown in Figure 3
with the best-fitting MG predictions from the combination of the baseline and
Planck data. In Figure 3, the tangential shear signal from the first source
bin ($0.20<z<0.43$) is higher than predicted in theory. The same tendency is
shown in comparison with the $\Lambda$CDM prediction. Lee et al. (2021)
suggests that the interplay between an unmodeled local peak in galaxy bias of
DMASS at low redshifts and the first source bin partially overlapping with the
low-redshift end of the DMASS tomographic bin causes the discrepancy. We will
discuss more details about this discrepancy and its impact on modified gravity
constraints in Section 7. For more description of the tangential shear
measurement, we refer readers to Lee et al. (2021).

####  3.1.4 Cosmic shear

The angular correlation function for cosmic shear correlating the source
redshift bins $i$ and $j$ is expressed in terms of the cosmic shear power
spectrum $C_{\kappa\kappa}$ given as

$\displaystyle\xi^{i,j}_{\pm}(\theta)=\frac{1}{2\pi}\int^{\infty}_{0}C_{\kappa\kappa}^{i,j}(\ell)J_{0/4}(\ell\theta)\ell\mathrm{d}\ell~{},$
(18)

where $J_{n}(x)$ is the $n$th order Bessel function of the first kind. The
cosmic shear power spectrum is defined with the two lensing kernels as

$\displaystyle
C_{\kappa\kappa}^{i,j}(\ell)=\int^{\infty}_{0}\mathrm{d}\chi\frac{W^{i}_{\kappa}(\chi)W^{j}_{\kappa}(\chi)}{\chi^{2}}P_{\delta\delta}(k,z(\chi))~{}.$
(19)

As shown in the case of tangential shear, both $\mu$ and $\Sigma$ affect the
signal but modifications by $\Sigma$ is dominant due to the double lensing
kernels.

We utilize the shear measurement from Troxel et al. (2018) obtained with the
DES metacalibration galaxies. Troxel et al. (2018) presents the auto- and
cross-correlation functions $\xi_{\pm}^{ij}(\theta)$ of the source galaxies
divided into four redshift bins over scales $2.5\arcmin<\theta<250\arcmin$.
Systematics related to the source catalog are parametrized in the same manner
as described in Troxel et al. (2018). Figure 4 shows the measurements of
$\xi_{\pm}(\theta)$. The bottom triangle contains $\xi_{+}(\theta)$ and the
top triangle contains $\xi_{-}(\theta)$. The dashed lines are the best-fitting
MG predictions from the combination of the DMASS 3x2pt, CMASS RSD/BAO, and
Planck data. The labels ($i$, $j$) in the upper-right corner of each panel
indicate the combination of the $i$th and $j$th source tomographic bins used
to obtain each signal. The lowest index indicates the lowest tomographic bin.
See Troxel et al. (2018) for a detailed description.

####  3.1.5 Angular scale cuts

The phenomenological approach such as the $\mu,\Sigma$ parametrization is
based on the linearly perturbed Einstein equations, so the scales should be
limited to the linear regime only. To deal with the nonlinear scales
accurately, using either semi-analytical methods or N-body simulation based on
a specific MG model is necessary.

Following the Planck 2018 analysis (Planck Collaboration, 2020a), we restrict
ourselves to observables sensitive to the linear scales only. We compute the
difference between the nonlinear and linear-theory predictions in the standard
LCDM model in our fiducial cosmology as $\Delta\chi^{2}=(d_{\rm nl}-d_{\rm
lin})^{T}C^{-1}(d_{\rm nl}-d_{\rm lin})$. The nonlinear predictions are
obtained using halofit (Takahashi et al., 2012) implemented in CosmoSIS. Then,
we identify the single data point that contributes most to this quantity, and
remove that data point, iterating until the quantity reaches
$\Delta\chi^{2}<1$. The shaded regions in Figures 3 and 4 indicate the data
points removed. Through this process, we obtain 119 data points for
$\xi_{+}(\theta)$, 18 points for $\xi_{-}(\theta)$, 38 points for
$\gamma_{t}(\theta)$, and 16 points for $w(\theta)$.

####  3.1.6 Covariance

Our measurements of $w(\theta)$, $\gamma_{t}(\theta)$ and $\xi_{\pm}(\theta)$
are correlated across angular and source redshift bins. The joint covariance
for these measurements is computed by cosmoLike (Krause & Eifler, 2017) using
the halo-based approach (Cooray & Sheth, 2002), assuming a $\Lambda$CDM
cosmology. The covariance is calculated as the sum of Gaussian covariance,
non-Gaussian covariance, and the super-sample covariance as described in
Krause & Eifler et al., (2017).

We assume there is no cross-correlation between surveys as the DES and BOSS
areas used do not overlap333The DMASS sample does not cover the overlapping
area between BOSS and DES since the sources from the area were used to train
the DMASS algorithm. Thereby, the angular clustering and tangential shear
signals used in this work are computed excluding the overlap. The shear
measurement from Troxel et al. (2018) did not use the overlapping area as
well., and the CMB data is most sensitive to higher redshift than the galaxy
surveys are. The galaxy surveys also form a small fraction of the full-sky CMB
measurements.

###  3.2 RSD and BAO measurements from BOSS CMASS

Redshift space distortions (RSDs) are one of the most promising probes for
testing gravity. On large scales, peculiar velocities of galaxies follow
infall of matter towards high-density regions (Kaiser, 1984), and through
that, they are sensitive to the growth rate of structure.

Under the assumption of linear theory, the galaxy power spectrum in the
redshift space ($P^{s}_{g}$) can be related to the real space matter power
spectrum ($P_{m}^{r}$) by

$\displaystyle
P^{s}_{g}(k,\mu,z)=b(z)[1+\beta(z)\mu^{2}]^{2}P_{m}^{r}(k,z)~{},$ (20)

where $b(z)$ is the galaxy bias, $\beta(z)$ denotes the amplitude of the RSD
defined as $\beta\equiv f(z)/b(z)$, $f(z)$ is the structure growth rate
defined as $f\equiv d\ln D/d\ln a$ in terms of the growth factor $D(a)$, $\mu$
is the cosine of the angle with the line of sight. As $\sigma_{8}^{2}$ is the
amplitude of the matter power spectrum, Equation (20) implies that RSD probes
$b(z)\sigma_{8}(z)$ and $f(z)\sigma_{8}(z)$. Modifications to GR emerge in the
term $f(z)\sigma_{8}(z)$ and also in the growth factor $D(z)^{2}$ in the
matter power spectrum.

The measurements of RSD that we use in this study are derived from the Baryon
Oscillation Spectroscopic Survey (BOSS; Eisenstein et al., 2011; Bolton et
al., 2012; Dawson et al., 2013). BOSS targeted two distinct samples known as
LOWZ at $0.15<z<0.43$ and CMASS at $0.43<z<0.75$ (Reid et al., 2016). The
higher redshift sample, CMASS, was designed to select a stellar mass-limited
sample of objects of all intrinsic colors, with a color cut that selects
almost exclusively on redshift (Reid et al., 2016). The DMASS sample we use as
lenses is designed to mimic this sample. Chuang et al. (2017) present
cosmological constraints from galaxy clustering of the LOWZ and CMASS samples.
They provide the growth rate and mean galaxy bias combined with the amplitude
of mass fluctuation ($f(z)\sigma_{8}(z)$ and $b(z)\sigma_{8}(z)$) at different
redshift points444As we do not have a pipeline to analyze the BOSS measurement
of the two-point function directly, we had to rely on the cosmological
inferences instead. Chuang et al. (2017) is the only published measurement of
the BOSS CMASS galaxy clustering that provides the constraint on
$b\sigma_{8}(z)$ and its covariance with other cosmological parameters. Future
analyses using the same strategy will likely need to consider the full
analysis of the two-point function to make use of the BAO reconstruction that
is not included in Chuang et al. (2017).. In this work, we utilize their
results measured at the mean redshift of CMASS ($z=0.59$). The galaxy bias
parameter $b$ here is shared with the galaxy bias incorporated in the angular
galaxy clustering and tangential shear of DMASS.

We also utilize the BAO measurements from the same work to constrain the
geometry of the background universe. The BAO measurements chosen are the
constraints of the Hubble parameter ($H(z)$), the comoving angular diameter
distance ($d_{A}(z)$) at the mean redshift of CMASS ($z=0.59$), and the matter
density fraction ($\Omega_{m}h^{2}$).

We use the full covariance matrix between those RSD and BAO parameters given
in the same work.

###  3.3 CMB & CMB lensing

Cosmic Microwave Background (CMB) is one of the most powerful cosmological
probes. Through its primary anisotropies such as temperature and polarization
power spectra, the CMB tightly constrains the background geometry and the
initial conditions. The secondary anisotropies such as the Integrated Sachs-
Wolfe-Effect (ISW) and the CMB lensing at late times are more relevant to
scalar mode perturbations and the growth of large-scale structure.

The ISW effect is caused by time variations in the gravitational potentials
(Sachs & Wolfe, 1967). When CMB photons travel from the surface of the last
scattering to us, they gain energy while falling down gravitational potential
wells, but they lose it again while climbing out of the potential wells.
However, dark energy components or modifications to gravity can cause
stretching in the potential well, resulting in a net gain in energy for the
photons during their journey through the potential wells.

The resulting temperature perturbation is given by

$\displaystyle\frac{\delta
T}{T}(\hat{n})=-\int^{\eta_{*}}_{\eta_{0}}\mathrm{d}\eta\frac{\partial(\Psi+\Phi)}{\partial\eta}~{},$
(21)

where $T$ is the CMB temperature, $\eta$ is the conformal time defined as
$\mathrm{d}\eta\equiv\mathrm{d}t/a$, $\eta_{*}$ is the conformal time at the
surface of last scattering and $\eta_{0}$ at the observer. Note that
$(\Psi+\Phi)$ in the integral can be expressed in terms of the Weyl potential
as $2\Psi_{W}$. By changing the gravitational potentials and their time
evolution (growth), modifications to GR affect the ISW and modify the
amplitude of the CMB temperature power spectra at the largest angular scales
($\ell<10$).

CMB lensing refers to the deflections of CMB photons by the intervening matter
while traveling from the surface of last scattering to us. Hence, CMB lensing
is sensitive to the distribution and growth rate of large-scale structures and
their associated gravitational potential. The deflections of CMB photons smear
out the CMB temperature power spectra and also induce non-Gaussianities in the
temperature and polarization maps (Bernardeau, 1998; Zaldarriaga & Seljak,
1998; Okamoto & Hu, 2003).

Table 1:  Parameters and priors used to describe the measured galaxy-galaxy
lensing signal. ‘Flat’ is a flat prior in the range given while ‘Gauss’ is a
Gaussian prior with mean $\mu$ and width $\sigma$. Priors for the tomographic
shear and photo-z bias parameters $m^{i}$ and $\Delta z_{\rm src}^{i}$ are
identical to DESY1MG.

| Parameter |  |  |  | Notation |  |  |  | Fiducial |  |  |  | Prior |  |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| Normalized matter density |  |  |  | $\Omega_{m}$ |  |  |  | 0.306 |  |  |  | Flat (0.1, 0.9) |  |
| Normalized baryon density |  |  |  | $\Omega_{b}$ |  |  |  | $4.845\times 10^{-2}$ |  |  |  | Flat (0.03, 0.07) |  |
| Amplitude of primordial power spectrum |  |  |  | $A_{s}$ |  |  |  | $2.196\times 10^{-9}$ |  |  |  | Flat ($5\times 10^{-10},5\times 10^{-9}$) |  |
| Power spectrum tilt |  |  |  | $n_{s}$ |  |  |  | 0.968 |  |  |  | Flat (0.87, 1.07) |  |
| Hubble parameter ($H_{0}=100h$) |  |  |  | $h$ |  |  |  | 0.678 |  |  |  | Flat (0.55, 0.90) |  |
| Normalized neutrino density |  |  |  | $\Omega_{\nu}$ |  |  |  | $6.50\times 10^{-4}$ |  |  |  | Flat ($5\times 10^{-4}$,$10^{-2}$) |  |
| Optical depth |  |  |  | $\tau$ |  |  |  | 0.081 |  |  |  | Flat (0.01, 0.20) |  |
| Modified gravity parameter |  |  |  | $\mu_{0}$ |  |  |  | 0.0 |  |  |  | Flat (-3.0, 3.0) |  |
| Modified gravity parameter |  |  |  | $\Sigma_{0}$ |  |  |  | 0.0 |  |  |  | Flat (-3.0, 3.0) |  |
| Linear galaxy bias (lens) |  |  |  | $b$ |  |  |  | 2.0 |  |  |  | Flat (0.8, 3.0) |  |
| Intrinsic alignment amplitude |  |  |  | $A_{\rm IA}$ |  |  |  | 0.0 |  |  |  | Flat (-5.0, 5.0) |  |
| Intrinsic alignment scaling |  |  |  | $\eta_{\rm IA}$ |  |  |  | 0.0 |  |  |  | Flat (-5.0, 5.0) |  |
| Lens redshift bias |  |  |  | $\Delta z_{\rm lens}$ |  |  |  | 0.0035 |  |  |  | Gauss ( 0.0035, 0.005) |  |
| Source photo-z bias ($i=1$) |  |  |  | $\Delta z^{1}_{\rm src}$ |  |  |  | -0.001 |  |  |  | Gauss (-0.001, 0.016) |  |
| Source photo-z bias ($i=2$) |  |  |  | $\Delta z^{2}_{\rm src}$ |  |  |  | -0.009 |  |  |  | Gauss (-0.009, 0.013) |  |
| Source photo-z bias ($i=3$) |  |  |  | $\Delta z^{3}_{\rm src}$ |  |  |  | 0.009 |  |  |  | Gauss (0.009, 0.011) |  |
| Source photo-z bias ($i=4$) |  |  |  | $\Delta z^{4}_{\rm src}$ |  |  |  | -0.018 |  |  |  | Gauss (-0.018, 0.022) |  |
| Shear calibration bias $(i\in\\{1,2,3,4\\})$ |  |  |  | $m^{i}$ |  |  |  | 0.012 |  |  |  | Gauss (0.012, 0.023) |  |

In this work, we utilize the Planck 2018 likelihood described in Planck
Collaboration (2020a). We use the Planck temperature and polarization auto-
and cross-multipole power spectra denoted as
$C^{TT}_{\ell},C^{TE}_{\ell},C^{EE}_{\ell}$, and $C^{BB}_{\ell}$.
Specifically, we use the full range of
$C^{TT}_{\ell},C^{TE}_{\ell},C^{EE}_{\ell}$ from $29<\ell<2509$ and the
low-$\ell$ polarization data including $C^{BB}_{\ell}$ from $2<\ell<29$. We
also make use of the Planck CMB lensing likelihood from temperature only
(Planck Collaboration, 2020b).

##  4 Analysis

In this section, we describe our methodology of measuring the cosmological
constraints in the framework of modified gravity.

To compute a theoretical prediction, we use
CosmoSIS555https://bitbucket.org/joezuntz/cosmosis (Zuntz et al., 2015) with a
version of MGCamb666https://github.com/sfu-cosmo/MGCAMB (Zhao et al., 2009;
Hojjati et al., 2011; Zucca et al., 2019; Lewis et al., 2000) modified to
include the $\Sigma$ and $\mu$ parametrization. We utilize the halofit
prescription (Takahashi et al., 2012) to compute the nonlinear power spectrum.

We perform Markov Chain Monte-Carlo likelihood analyses using the multinest
algorithm (Feroz & Hobson, 2008; Feroz et al., 2009, 2019) implemented in the
CosmoSIS package. Our analysis spans the parameter set
$\\{\Omega_{m},h_{0},\Omega_{b},n_{s},A_{s},(\tau,\Omega_{\nu})\\}$ where the
parentheses around the optical depth parameter and the neutrino density
indicate that they are used only in the analysis combinations that use the CMB
data. In addition to this set of $\Lambda$CDM parameters, we vary
$\\{\mu_{0},\Sigma_{0}\\}$ to test gravity. Along with the parameter set, we
also vary nuisance parameters describing the photo-z and shear systematics for
different tomographic bins, and model parameters for the intrinsic alignment.
Since we use the identical source sample as DESY1MG, we adopt the same models
for the shear and photo-$z$ systematics (Krause & Eifler et al.,, 2017). The
complete set of varied parameters and priors is summarized in Table 1. The
priors imposed in this analysis are identical to those of DESY1MG except for
the lens galaxy bias ($b$) and lens redshift bias ($\Delta z_{\rm lens}$),
summarized below.

  1. (i)

Lens galaxy bias ($b$): the linear galaxy bias parameter for lenses. Since we
select fairly large scales where the linear assumption is valid, we use a
constant galaxy bias over the redshift range. The redshift evolution of CMASS
and DMASS is negligible, which is illustrated in Salazar-Albornoz et al.
(2017) and Lee et al. (2019). The scale-independence of the DMASS galaxy bias
over the scales of interest is shown in Lee et al. (2021). The galaxy bias
parameter is shared with the galaxy bias in $b\sigma_{8}$ from the BOSS
measurements described in Section 3.2.

  2. (ii)

Lens redshift bias ($\Delta z_{\rm lens}$): we model the redshift distribution
of lenses as $n_{\rm true}(z)=\hat{n}(z-\Delta z_{\rm lens})$, where $\hat{n}$
is the measured redshift distribution, and $\Delta z_{\rm lens}$ is a redshift
bias parameter. Lee et al. (2019) constrained $\Delta z_{\rm lens}$ by jointly
fitting the residuals of the angular correlation and clustering-z
measurements, and obtained $\Delta z=3.5\times 10^{-4}$ with its uncertainty
of $\sigma_{\Delta z}=0.5\times 10^{-3}$. We take those values as the mean and
width of a Gaussian prior, respectively.

The likelihood of the combined probes is evaluated by the sum of individual
log-likelihoods given as

$\displaystyle\ln\mathcal{L}(p)=-\frac{1}{2}\sum_{i}\chi^{2}_{i}(p)~{},$ (22)

where $p$ is the set of varied parameters, the subscript ‘$i$’ denotes the
$i$th data set. The value of $\chi^{2}$ is estimated as below:

$\displaystyle\chi^{2}(p)=\sum_{j,k}({\bf{d}}-{\bf{d}}_{\rm
th}(p))_{j}{\rm{\bf{C}}}^{-1}_{jk}({\bf{d}}-{\bf{d}}_{\rm th}(p))_{k}^{T}~{},$
(23)

where $\bf{d}_{\rm th}$ and $\bf{d}$ are theoretical and measured datavectors,
and $\bf{C}$ is the covariance matrix. We assume there is no cross-correlation
between surveys as stated in Section 3.1.6.

##  5 Robustness tests

Figure 5: Marginalized 1D posterior constraints on
${\Omega_{m},S_{8},b,\Sigma_{0},\mu_{0}}$ obtained from DMASS 3x2pt + CMASS
RSD/BAO (baseline; top) and DMASS 3x2pt + CMASS RSD/BAO + Planck (bottom). The
different rows correspond to the different systematics injected in each of the
simulated data vectors. The systematics tested here are detailed in Section 5.
These data vectors are analyzed by the fiducial pipeline to test the impact of
systematics associated with each data vector on the parameter constraints. For
all cases, the posteriors for these cosmological parameters remain well within
$1\sigma$ of the fiducial constraints.

In this section, we perform a series of tests to show the robustness of our
results to modeling assumptions and approximations. We thoroughly follow the
procedures illustrated in DESY1MG. First, we generate a set of simulated data
vectors of $w(\theta)$, $\gamma_{t}(\theta)$, and $\xi_{\pm}(\theta)$ shifted
with the addition of a systematic effect that is not included in our analysis
pipeline. Then, each of the simulated data vectors is individually analyzed
with our fiducial pipeline by using the methodology described in Section 4.
Finally, we compare the inferred values of cosmological parameters from the
simulated data vectors to the fiducial values. Following DESY1MG, we adopt
$1\sigma$ as the threshold for a bias. If we observe a bias shifted more than
$1\sigma$ from the fiducial quantity, the corresponding effect of systematics
or modeling assumption needs to be corrected or adequately accounted for in
our analysis pipeline.

The modeling assumptions that we consider are summarized below:

  1. (i)

Limber approximation and RSD: the Limber approximation (Limber, 1953, 1954)
enables us to compute the angular two-point statistics efficiently by
simplifying the Bessel function integrals. However, this approximation may not
be sufficient if the constraining power from surveys can no longer tolerate
the errors induced by the approximation. We simulate a data vector using the
exact $w(\theta)$ calculation and include the contribution from redshift space
distortions (Padmanabhan et al., 2007).

  2. (ii)

Magnification: the observed number density of foreground galaxies can be
altered due to the matter between the foreground galaxies and the observer,
leading to a systematic bias to the two-point statistics. For the foreground
redshift $z>0.45$, the effect can bias the inferred dark energy equation of
state by more than $5\%$ (Ziour & Hui, 2008). Accounting for this effect is
critical for the higher redshift samples, such as DMASS. To estimate the
impact of the magnification bias, we simulate a data vector by injecting the
contribution of magnification to $\gamma_{t}(\theta)$ and $w(\theta)$. We
assume that the change in the observed galaxy over-density for lenses produced
by magnification is proportional to the convergence field of lenses. This can
be expressed as $\delta_{g}^{\rm obs}=\delta_{g}^{\rm int}+2(\alpha-1)\kappa$,
where $\delta_{g}^{\rm int}$ is the galaxy over-density that would have been
observed in the absence of magnification, $\alpha$ is the logarithmic slope of
the luminosity function of lenses at its faint end, and $\kappa$ is the
convergence of the lenses (Bartelmann & Schneider, 2001). We adopt the value
of $\alpha=2.62$ computed using the CMASS galaxy mocks in Joachimi et al.
(2020).

  3. (iii)

Intrinsic alignments: our fiducial pipeline accounts for the tidal alignment
of massive elliptical galaxies on large scales. However, the contribution from
“tidal torquing” induced by spiral or less luminous elliptical galaxies is
non-negligible on smaller scales. We simulate a data vector using the Tidal
Alignment and Tidal Torquing model (TATT) described in Blazek et al. (2019).
We set the TATT amplitudes to $A_{1}=0,~{}A_{2}=2$ with no $z$ dependence as
done in Abbott et al. (2018) and DESY1MG. This choice of TATT parameters
corresponds to testing the impact of a pure tidal torquing model when analyzed
with a tidal alignment model.

Since we cut out the majority of the nonlinear scales as described in Section
3.1.5, we do not consider systematics that particularly affects small scales
such as baryonic feedback effects. We do not test for the nonlinear galaxy
bias either as the linearity of the galaxy bias of DMASS on galaxy clustering
and galaxy-galaxy lensing has been verified in Lee et al. (2019) and Lee et
al. (2021) down to the scales of $4h^{-1}{\rm\;Mpc}$. These tests have been
performed by jointly fitting the two-point statistics with other external
probes. DESY1MG tested the aforementioned systematics using their 3x2pt
simulated data vector and external data sets. For the case of the 3x2pt
simulated data vector only, their posteriors of modified gravity were skewed
from true values due to the interplay of weak constraining power with a
relatively flat likelihood profile and prior volume effect. As we use only one
lens bin compared to DES Y1 five bins, we should expect a worse bias in the
posteriors with weaker constraining power. Therefore, we perform the
systematics tests by jointly fitting the two-point statistics with the
simulated BOSS CMASS likelihood. Since the simulated BOSS CMASS likelihood is
not sensitive to any of the systematics correlated with the two-point
functions, bias from the fiducial values is solely originating from the two-
point functions of DMASS. We also perform the systematics tests including the
actual Planck data set as shifts by systematics may appear with higher
constraining power.

The results of the robustness tests are shown in Figure 5, for the two
modified gravity parameters ($\mu_{0}$, $\Sigma_{0}$) and the parameters that
are most sensitive to in this work: ${\Omega_{m},S_{8},b}$. The top panel
shows the constraints obtained from the baseline case, and the bottom panel
shows the baseline+Planck case. The different rows correspond to the different
systematics described above. The topmost error bars in each panel are our
fiducial case. For all cases, our posteriors for these cosmological parameters
remain well within $1\sigma$ of the fiducial constraints.

Throughout the robustness tests, we use an importance sampling pipeline
(Weaverdyck et al., prep) to rapidly compute posterior means for each
systematics. For a given sample from a proposal distribution, the importance
sampling pipeline draws the expected values of parameters under a target
distribution by weighting points with density ratios of the proposal
distribution and target distribution, called importance weights. We adopt the
effective number of samples as a diagnostic tool to evaluate the accuracy of
importance sampling, which is given as $N^{\rm IS}_{\rm
eff}=1/\sum^{N}_{i=1}w_{i}^{2}$, where $N$ is the total sample size and
$w_{i}$ represents normalized importance weights. If the proposal distribution
is equal to the target distribution, the quantity becomes $N^{\rm IS}_{\rm
eff}=N$. We determine $N^{\rm IS}_{\rm eff}/N>0.7$ as the threshold of
reliability and evaluate the quantity for each systematics chain obtained by
the importance sampler. We find that all of the systematics chains pass the
threshold except for the case of intrinsic alignments. Therefore, we use
multinest for the intrinsic alignments and utilize the importance sampler for
the rest of the systematics.

Figure 6:  Constraints in the $\mu_{0}-\Sigma_{0}$ plane (left), and
$\Omega_{m}-S_{8}$ plane (right) obtained from various combinations of the
probes. The blue contours show the constraints obtained from the baseline case
(DMASS 3x2pt + CMASS RSD/BAO). The orange contours are obtained by adding the
Planck data to the baseline case. The black-dashed contours are obtained from
the external data only. The grey contour in the right panel shows the
constraints of $\Omega_{m}$ and $S_{8}$ obtained from CMASS only. Adding the
DMASS 3x2pt data improves the constrains from the external data sets by $28\%$
for $\mu_{0}$, $30\%$ for $\Sigma_{0}$, and $24\%$ for $S_{8}$.

##  6 Blinding

We blinded our results in the parameter-level to avoid confirmation bias. The
measurements of $\gamma_{t}$ from Lee et al. (2021) were blinded in the real
data analysis. No comparison to theoretical predictions at the two-point level
was made in the blinding stage. The cosmological parameter constraints were
scaled and shifted by a random number when plotting. After ensuring that there
were no major systematics that can bias the cosmological constraints through a
series of robustness tests listed in Section 5, we unblinded both the data
vector of $\gamma_{t}$ and cosmological constraints. As described in the next
section, after unblinding we performed an additional test to estimate the
impact of the high signal of $\gamma_{t}$ measured with the first source bin
that was detected after unblinding. No change was made in either the analysis
method or pipeline after unblinding.

##  7 Results

Figure 7:  Constraints obtained with all four tomographic cross-correlations
of $\gamma_{t}$ (solid) and only with the third and fourth cross-correlations
of $\gamma_{t}$ (dashed) from the two highest source bins. All contours are
the case of baseline+Planck. There is no significant bias or degradation found
in the constraints.  Figure 8:  Constraints from this work (baseline+Planck;
orange), DESY1 redMaGiC 3x2pt + CMASS RSD/BAO + Planck (purple), and the
published constraints of the DES Y1 MG analysis (black-dashed) (Abbott et al.,
2019). Compared to the case with redMaGiC (purple), using DMASS (orange)
improves the MG constraints by $29\%$ for $\mu_{0}$ and $21\%$ for
$\Sigma_{0}$ with the same external data sets. The DMASS case is even
comparable to that of DESY1MG (black-dashed) despite the lack of the external
data sets. The shift of $\Omega_{m}$ in DESY1MG is caused by the SNe Ia data
favoring low $\Omega_{m}$. Overall, the constraints in the three cases are
consistent.

###  7.1 Constraints on modified gravity

In this section, we show the resulting MG constraints obtained from the
various combinations of the data sets listed in Section 3. For all cases
reported, we include the CMASS likelihood to DES 3x2pt as default and do not
report the result from DMASS alone because 1) DMASS is designed to harness its
maximum power when its measurements are combined with the measurements of
CMASS, and 2) DMASS only, without the help of external data sets, suffers from
a projection bias in the posteriors of modified gravity as stated in Section
5.

Figure 6 shows the resulting constraints in the $\mu_{0}$–$\Sigma_{0}$ plane
(left), and $\Omega_{m}$–$S_{8}$ plane (right). Table 2 presents a more
detailed summary on our findings, including the constraint of galaxy bias. In
Figure 6, the blue contours show the constraints obtained from the baseline
case (DMASS 3x2pt + CMASS RSD/BAO). The resulting values show a mild deviation
from GR with $\mu_{0}=-1.23^{+0.81}_{-0.83}$ and
$\Sigma_{0}=-0.17^{+0.16}_{-0.15}$. As $\Sigma_{0}$ indicates modifications to
GR by relativistic particles, the value of $\Sigma_{0}$ here is driven by weak
lensing in DES. The orange contours are obtained by adding the CMB data from
Planck to the baseline case. As shown in the figure, adding Planck recovers
GR, yielding the values of $\mu_{0}=-0.37^{+0.47}_{-0.45}$ and
$\Sigma_{0}=0.078^{+0.078}_{-0.082}$. Along with these results, we also show
the results from the external data only (CMASS RSD/BAO + Planck; black-dashed)
and CMASS RSD/BAO only (grey-dashed) to infer contributions from the DMASS
3x2pt data indirectly. We find that the DMASS 3x2pt data improves the
constraints from the external data sets by $28\%$ for $\mu_{0}$, $30\%$ for
$\Sigma_{0}$, and $24\%$ for $S_{8}$. For the case of CMASS RSD/BAO alone,
$\Sigma_{0}$ is not constrained by RSD and $\mu_{0}$ is barely constrained
within the imposed prior range. Therefore, we only look into the constraints
of $S_{8}$ and $\Omega_{m}$. Comparing with the baseline, we find that adding
DMASS 3x2pt to CMASS RSD/BAO effectively reduces errors on constraints even
without Planck, improving $S_{8}$ by $45\%$ and $\Omega_{m}$ by $32\%$. These
notable improvements by adding DMASS 3x2pt are mainly driven by the galaxy
bias parameter shared and constrained by DMASS and CMASS simultaneously. The
galaxy bias from the baseline+Planck case is improved by $\sim 80\%$ compared
to that of the external data alone, which leads to a significant improvement
in other parameters that are degenerate with galaxy bias.

Finally, we test the robustness of our results against unmodeled systematics.
As shown in Figure 3, the tangential shear signal measured with DMASS and the
first source bin of DES Y1 is higher than predicted in theory. Lee et al.
(2021) suggests the local peak in galaxy bias at the low-redshift end of the
original BOSS CMASS sample as a potential reason for this. If the DMASS
algorithm faithfully replicated the properties of CMASS, this peak would have
been replicated in DMASS and even amplified at the low-redshifts edge where
the algorithm poorly works. As the first source bin lies largely in front of
the lens bin, it is possible that the tangential shear signal measured with
the first source bin only captures correlations with the DMASS galaxies at low
redshifts where galaxy bias peaks. Lee et al. (2021) tested the impact of the
unmodeled local peak on the galaxy bias constraint inferred from the
tangential shear measurement. In fixed $\Lambda$CDM cosmology, they found that
the resulting constraint is not biased by the local peak above the scale of
$4h^{-1}{\rm\;Mpc}$. It is less likely that the local peak will affect our
result as we utilize the scales larger than $4h^{-1}{\rm\;Mpc}$, and include
the angular galaxy clustering and shear measurements that are less sensitive
to an irregularity in galaxy bias at the redshift tails. However, it is worth
testing its impact in the frame of modified gravity as well to rule out the
possibility of this unmodeled systematics biasing the result. For this, we
repeat the analysis without the tangential shear signals from the first and
second source bins. The signal from the second source bin is excluded to
completely remove any potential effect related to galaxy bias from the low-
redshift end of DMASS. The results are presented in Figure 7. Each panel shows
the contours of the MG parameters (left) and $\Omega_{m}$–$S_{8}$ (right) with
all tangential shear signals (solid lines) and without the signals from the
first two source bins (dashed lines). Both panels are for the case of
baseline+Planck. As shown in this figure, we do not find significant bias or
degradation in the constraints.

###  7.2 Comparison with the DES Y1 MG analysis

Earlier, the DES Collaboration reported the constraints on $\mu_{0}$ and
$\Sigma_{0}$ obtained from the DES Y1 data in DESY1MG. As this work follows
their analysis methods with the same shear catalog, comparing the results of
this work with DESY1MG enables us to validate the capability of using DMASS as
gravitational lenses to combine the spectroscopic and imaging surveys
efficiently. This section briefly describes the lens sample and external data
sets used in DESY1MG and compares their MG constraints with ours. As we do not
report the result for the DES data alone, we only compare the results measured
with the external data sets.

DESY1MG utilized the 3x2pt data vectors measured with the redMaGiC lenses
(Rozo & Rykoff et al.,, 2016; Elvin-Poole & Crocce et al.,, 2018) and the DES
Y1 metacalibration sources (Sheldon & Huff, 2017; Huff & Mandelbaum, 2017;
Zuntz & Sheldon et al.,, 2018). The redMaGiC sample consists of $660,000$
galaxies over an area of $1321\deg^{2}$. The sample was divided into five
redshift bins, using three different cuts on intrinsic luminosity:
$0.15<z<0.3$, $0.3<z<0.45$, $0.45<z<0.6$, $0.6<z<0.75$, and $0.75<z<0.9$. The
first three bins were selected using a luminosity cut of $L>0.5L_{*}$ with a
spatial density $\bar{n}=10^{-3}(h^{-1}{\rm\;Mpc})^{-3}$, the other two bins
were selected using $L>L_{*}$ with $\bar{n}=4\times
10^{-4}(h^{-1}{\rm\;Mpc})^{-3}$, and $L>1.5L_{*}$ with
$\bar{n}=10^{-4}(h^{-1}{\rm\;Mpc})^{-3}$, respectively. The redshift coverage
of DMASS is comparable to the third and fourth bins of redMaGiC combined, but
the number density of DMASS ($\bar{n}=3\times 10^{-4}$) is much lower than
redMaGiC. The external data sets selected in DESY1MG were the CMB measurements
from Planck 2015 (Planck Collaboration, 2016), the RSD and BAO measurements
from BOSS DR12 (Alam et al., 2017b), and the additional BAO measurements from
Six-degree Field Galaxy Survey (6dF; Beutler et al., 2012), SDSS Main Galaxy
Samples (SDSS MGS; Ross et al., 2015), and lastly, Type Ia supernova
measurements from Pantheon (Scolnic et al., 2018). The resulting MG
constraints from these data sets are $\mu_{0}=-0.11^{+0.42}_{-0.46}$ and
$\Sigma_{0}=0.06^{+0.08}_{-0.07}$.

For a more direct comparison with DESY1MG, we perform an additional analysis
using DES Y1 3x2pt (i.e. redMaGiC lenses) with CMASS RSD/BAO. By doing so, we
can infer the pure improvement by DMASS while excluding contributions from
other external data sets in DESY1MG. The resulting constraints of MG are
$\mu_{0}=-0.29^{+0.63}_{-0.66}$, $\Sigma_{0}=0.098^{+0.102}_{-0.100}$.

Figure 8 shows the constraints of this work (baseline + Planck) in orange,
redMaGiC \+ CMASS RSD/BAO + Planck in purple, and DESY1MG in black-dashed
lines. Compared to the case with redMaGiC (purple), it is noticeable that
using DMASS (orange) improves the MG constraints by $29\%$ for $\mu_{0}$ and
$21\%$ for $\Sigma_{0}$ with the same external data sets despite redMaGiC
consisting of five tomographic bins with a higher number density. Our result
is also comparable to that of DESY1MG (black-dashed) despite the lack of the
external data sets. This can be explained as follows: five tomographic bins of
redMaGiC introduce additional nuisance parameters of galaxy bias for each bin.
When combined with the CMASS likelihood, these five galaxy bias parameters
from redMaGiC and an additional one from CMASS are varied independently, and
hence uncertainties from the total six parameters add to the total statistical
error budget. This especially affects the modified-gravity parameters due to
the strong degeneracy among the galaxy bias and MG parameters. Compared with
the other two cases, DESY1MG (black-dashed) slightly shifts towards a lower
value of $\Omega_{m}$. The shift is possibly caused by the Pantheon supernova
data that favors a relatively lower value of $\Omega_{m}$ than the rest of the
data sets (Scolnic et al., 2018). Overall, the constraints obtained with DMASS
are consistent with those with redMaGiC. We do not find a significant shift in
the central values of $\mu_{0}$ and $\Sigma_{0}$ among the three cases.

Table 2: 1D marginalized posteriors obtained from various combinations of data
sets. ‘baseline’ denotes the combination of the DMASS 3x2pt data and RSD and
BAO measurements from BOSS CMASS.

| Data | $\mu_{0}$ | $\Sigma_{0}$ | $\Omega_{m}$ | $\sigma_{8}$ | $S_{8}$ | $b$ |  |
---|---|---|---|---|---|---|---|---|---
| baseline | $-1.23^{+0.81}_{-0.83}$ | $-0.17^{+0.16}_{-0.15}$ | $0.289^{+0.031}_{-0.028}$ | $0.821^{+0.067}_{-0.068}$ | $0.809^{+0.080}_{-0.079}$ | $1.83^{+0.20}_{-0.17}$ |  |
| baseline + Planck | $-0.37^{+0.47}_{-0.45}$ | $0.078^{+0.078}_{-0.082}$ | $\left(308.2^{+9.0}_{-8.0}\right)\times 10^{-3}$ | $0.762\pm 0.045$ | $0.772^{+0.050}_{-0.046}$ | $2.134^{+0.079}_{-0.078}$ |  |
| DESY1 redMaGiC \+ CMASS + Planck | $-0.29^{+0.63}_{-0.66}$ | $0.098^{+0.102}_{-0.100}$ | $\left(305.0\pm 8.6\right)\times 10^{-3}$ | $0.767^{+0.061}_{-0.065}$ | $0.773^{+0.064}_{-0.067}$ | $-$ |  |
| CMASS + Planck | $0.15^{+0.61}_{-0.67}$ | $0.20^{+0.12}_{-0.11}$ | $0.314^{+0.012}_{-0.011}$ | $0.794^{+0.057}_{-0.062}$ | $0.814^{+0.059}_{-0.068}$ | $2.00^{+0.28}_{-0.26}$ |  |
| CMASS | $-$ | $-$ | $0.319^{+0.052}_{-0.055}$ | $0.792^{+0.087}_{-0.084}$ | $0.819^{+0.114}_{-0.119}$ | $1.980^{+0.348}_{-0.313}$ |  |

##  8 Conclusion

In this paper, we measured the modified gravity parameters $\Sigma_{0}$ and
$\mu_{0}$ using the DMASS sample to demonstrate its feasibility as an
equivalent to BOSS CMASS in the DES Y1 footprint for a joint analysis of BOSS
and DES. We utilized the angular clustering of the DMASS galaxies, cosmic
shear of the DES metacalibration sources, and cross-correlation of the two as
data vectors. Then we derived the constraints on MG by jointly fitting the
data vectors with the RSD and BAO measurements from the BOSS CMASS sample and
CMB measurements from Planck. As detailed in Section 4, we thoroughly followed
the analysis methods of DESY1MG. We employed the same modeling for the two-
point statistics and potential systematics, such as the shear and photo-z
biases. A series of tests were performed as done in DESY1MG to make sure our
results are robust to the modeling assumptions that are not included in the
pipeline.

The resulting constraints are consistent with GR. Adding DMASS 3x2pt
significantly improves the existing constraints from CMASS RSD/BAO + Planck by
$28\%$ for $\mu_{0}$ and $30\%$ for $\Sigma_{0}$. Not only that, we found that
DMASS yields constraints improved by $29\%$ for $\mu_{0}$ and $21\%$ for
$\Sigma_{0}$, compared to the constraints obtained with redMaGiC when the same
external data sets are utilized. The results are also comparable to the
results of DESY1MG performed with additional BAO & RSD and supernova
measurements. This is mainly because of a tightly constrained galaxy bias
parameter shared in the CMASS and DMASS likelihoods, which breaks the
degeneracy between galaxy bias and other parameters. Compared to that, the
cases with redMaGiC treats the galaxy bias parameter from RSD as an additional
one, which varies independently from those of redMaGiC. The uncertainty
induced by the independent use of galaxy bias erases gains from additional BAO
& RSD and supernova measurements.

This approach to optimally combine the DES and BOSS surveys with DMASS can be
easily extended to other types of spectroscopic samples and image-based
surveys. In just a few years, data from the Stage IV surveys will become
available. The survey footprint of LSST777Rubin Observatory’s Legacy Survey of
Space and Time (Ivezić et al., 2019) will occupy the entire southern sky with
a small overlapping area with spectroscopic surveys such as DESI888Dark Energy
Spectroscopic Instrument (DESI Collaboration, 2016) in the northern sky. The
DESI survey produces the Bright Galaxy sample (BGS) at $z<0.4$ and the
Emission Line Galaxy sample (ELG) at $z>0.6$. The equivalent samples to those
DESI targets in the LSST area will be ideal lenses to extract the lensing
signals in LSST that can be efficiently combined with the galaxy clustering of
the DESI galaxy samples as done with DMASS and CMASS. Combining the two
surveys in this manner will embrace almost the entire sky and yield a wealth
of information to test gravity.

## Acknowledgements

We thank D. H. Weinberg for useful conversations in the course of preparing
this work.

AC acknowledges support from NASA grant 15-WFIRST15-0008. During the
preparation of this paper, C.H. was supported by the Simons Foundation, NASA,
and the US Department of Energy.

The figures in this work are produced with plotting routines from matplotlib
(Hunter, 2007) and ChainConsumer (Hinton, 2016). Some of the results in this
paper have been derived using the healpy and HEALPix package (Górski et al.,
2005; Zonca et al., 2019).

Funding for the DES Projects has been provided by the U.S. Department of
Energy, the U.S. National Science Foundation, the Ministry of Science and
Education of Spain, the Science and Technology Facilities Council of the
United Kingdom, the Higher Education Funding Council for England, the National
Center for Supercomputing Applications at the University of Illinois at
Urbana-Champaign, the Kavli Institute of Cosmological Physics at the
University of Chicago, the Center for Cosmology and Astro-Particle Physics at
the Ohio State University, the Mitchell Institute for Fundamental Physics and
Astronomy at Texas A&M University, Financiadora de Estudos e Projetos,
Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro,
Conselho Nacional de Desenvolvimento Científico e Tecnológico and the
Ministério da Ciência, Tecnologia e Inovação, the Deutsche
Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy
Survey.

The Collaborating Institutions are Argonne National Laboratory, the University
of California at Santa Cruz, the University of Cambridge, Centro de
Investigaciones Energéticas, Medioambientales y Tecnológicas-Madrid, the
University of Chicago, University College London, the DES-Brazil Consortium,
the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH)
Zürich, Fermi National Accelerator Laboratory, the University of Illinois at
Urbana-Champaign, the Institut de Ciències de l’Espai (IEEC/CSIC), the
Institut de Física d’Altes Energies, Lawrence Berkeley National Laboratory,
the Ludwig-Maximilians Universität München and the associated Excellence
Cluster Universe, the University of Michigan, NFS’s NOIRLab, the University of
Nottingham, The Ohio State University, the University of Pennsylvania, the
University of Portsmouth, SLAC National Accelerator Laboratory, Stanford
University, the University of Sussex, Texas A&M University, and the OzDES
Membership 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
by the Association of Universities for Research in Astronomy (AURA) under a
cooperative agreement with the National Science Foundation.

The DES data management system is supported by the National Science Foundation
under Grant Numbers AST-1138766 and AST-1536171. The DES participants from
Spanish institutions are partially supported by MICINN under grants
ESP2017-89838, PGC2018-094773, PGC2018-102021, SEV-2016-0588, SEV-2016-0597,
and MDM-2015-0509, some of which include ERDF funds from the European Union.
IFAE is partially funded by the CERCA program of the Generalitat de Catalunya.
Research leading to these results has received funding from the European
Research Council under the European Union’s Seventh Framework Program
(FP7/2007-2013) including ERC grant agreements 240672, 291329, and 306478. We
acknowledge support from the Brazilian Instituto Nacional de Ciência e
Tecnologia (INCT) do e-Universo (CNPq grant 465376/2014-2).

This manuscript has been authored by Fermi Research Alliance, LLC under
Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of
Science, Office of High Energy Physics.

Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the
Participating Institutions, the National Science Foundation, and the U.S.
Department of Energy Office of Science. The SDSS-III web site is
http://www.sdss3.org/.

SDSS-III is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS-III Collaboration including the
University of Arizona, the Brazilian Participation Group, Brookhaven National
Laboratory, Carnegie Mellon University, University of Florida, the French
Participation Group, the German Participation Group, Harvard University, the
Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA
Participation Group, Johns Hopkins University, Lawrence Berkeley National
Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for
Extraterrestrial Physics, New Mexico State University, New York University,
Ohio State University, Pennsylvania State University, University of
Portsmouth, Princeton University, the Spanish Participation Group, University
of Tokyo, University of Utah, Vanderbilt University, University of Virginia,
University of Washington, and Yale University.

This work used resources at the Owens Cluster at the Ohio Supercomputer Center
(OSC, 1987) and the Duke Compute Cluster (DCC) at Duke University.

## Data Availability

The DMASS galaxy catalog used in this work and some of the ancillary data
products shown in the Figures are available through the DES data release page
(https://des.ncsa.illinois.edu/releases/other/y1-dmass). Readers interested in
comparing to these results are encouraged to check out the release page or
contact the corresponding author for additional information.

## References

  * Abbott et al. (2005) Abbott T., et al., 2005, arXiv e-prints, pp astro–ph/0510346
  * Abbott et al. (2016) Abbott T., et al. 2016, MNRAS, 460, 1270
  * Abbott et al. (2018) Abbott T. M. C., et al. 2018, Phys. Rev. D, 98, 043526
  * Abbott et al. (2019) Abbott T. M. C., et al. 2019, Phys. Rev. D, 99, 123505
  * Aihara et al. (2018) Aihara H., et al. 2018, PASJ, 70, S4
  * Alam et al. (2015) Alam S., et al. 2015, The Astrophysical Journal Supplement Series, 219, 12
  * Alam et al. (2017a) Alam S., et al. 2017a, MNRAS, 465, 4853
  * Alam et al. (2017b) Alam S., et al. 2017b, MNRAS, 470, 2617
  * Amon et al. (2018) Amon A., et al. 2018, MNRAS, 479, 3422
  * Bartelmann & Schneider (2001) Bartelmann M., Schneider P., 2001, Phys. Rep., 340, 291
  * Bernardeau (1998) Bernardeau F., 1998, A&A, 338, 767
  * Bernstein & Cai (2011) Bernstein G. M., Cai Y.-C., 2011, MNRAS, 416, 3009
  * Beutler et al. (2012) Beutler F., et al. 2012, MNRAS, 423, 3430
  * Blake et al. (2016) Blake C., et al. 2016, MNRAS, 456, 2806
  * Blazek et al. (2019) Blazek J. A., et al. 2019, Phys. Rev. D, 100, 103506
  * Bolton et al. (2012) Bolton A. S., et al. 2012, AJ, 144, 144
  * Cai & Bernstein (2012) Cai Y.-C., Bernstein G., 2012, MNRAS, 422, 1045
  * Chuang et al. (2017) Chuang C.-H., et al. 2017, MNRAS, 471, 2370
  * Coe et al. (2006) Coe D., et al. 2006, AJ, 132, 926
  * Cooray & Sheth (2002) Cooray A., Sheth R., 2002, Phys. Rep., 372, 1
  * DESI Collaboration (2016) DESI Collaboration 2016, arXiv e-prints, p. arXiv:1611.00036
  * Dawson et al. (2013) Dawson K. S., et al. 2013, AJ, 145, 10
  * Drlica-Wagner et al. (2018) Drlica-Wagner A., et al. 2018, The Astrophysical Journal Supplement Series, 235, 33
  * Eisenstein et al. (2011) Eisenstein D. J., et al. 2011, AJ, 142, 72
  * Elvin-Poole et al. (2018) Elvin-Poole J., et al. 2018, Phys. Rev. D, 98, 042006
  * Feroz & Hobson (2008) Feroz F., Hobson M. P., 2008, MNRAS, 384, 449
  * Feroz et al. (2009) Feroz F., Hobson M. P., Bridges M., 2009, MNRAS, 398, 1601
  * Feroz et al. (2019) Feroz F., et al. 2019, The Open Journal of Astrophysics, 2, 10
  * Ferreira & Skordis (2010) Ferreira P. G., Skordis C., 2010, Phys. Rev. D, 81, 104020
  * Ferté et al. (2019) Ferté A., et al. 2019, Phys. Rev. D, 99, 083512
  * Flaugher et al. (2015) Flaugher B., et al. 2015, AJ, 150, 150
  * Gaztañaga et al. (2012) Gaztañaga E., et al. 2012, MNRAS, 422, 2904
  * Górski et al. (2005) Górski K. M., et al. 2005, ApJ, 622, 759
  * Heymans et al. (2012) Heymans C., et al. 2012, MNRAS, 427, 146
  * Hinton (2016) Hinton S. R., 2016, The Journal of Open Source Software, 1, 00045
  * Hojjati et al. (2011) Hojjati A., Pogosian L., Zhao G.-B., 2011, J. Cosmology Astropart. Phys., 2011, 005
  * Hoyle et al. (2018) Hoyle B., et al. 2018, MNRAS, 478, 592
  * Huff & Mandelbaum (2017) Huff E., Mandelbaum R., 2017, arXiv e-prints, p. arXiv:1702.02600
  * Hunter (2007) Hunter J. D., 2007, Computing in Science and Engineering, 9, 90
  * Ishak (2019) Ishak M., 2019, Living Reviews in Relativity, 22, 1
  * Ivezić et al. (2019) Ivezić Ž., et al. 2019, ApJ, 873, 111
  * Joachimi et al. (2020) Joachimi B., et al. 2020, arXiv e-prints, p. arXiv:2007.01844
  * de Jong et al. (2013) de Jong J. T. A., et al. 2013, Experimental Astronomy, 35, 25
  * Joudaki et al. (2017) Joudaki S., et al. 2017, MNRAS, 471, 1259
  * Jullo et al. (2019) Jullo E., et al. 2019, A&A, 627, A137
  * Kaiser (1984) Kaiser N., 1984, ApJ, 284, L9
  * Kaiser (1992) Kaiser N., 1992, ApJ, 388, 272
  * Krause & Eifler (2017) Krause E., Eifler T., 2017, MNRAS, 470, 2100
  * Krause et al. (2017) Krause E., et al. 2017, arXiv e-prints, p. arXiv:1706.09359
  * Landy & Szalay (1993) Landy S. D., Szalay A. S., 1993, ApJ, 412, 64
  * Lee et al. (2019) Lee S., et al. 2019, MNRAS, 489, 2887
  * Lee et al. (2021) Lee S., et al. 2021, arXiv e-prints, p. arXiv:2104.11319
  * Lewis et al. (2000) Lewis A., Challinor A., Lasenby A., 2000, ApJ, 538, 473
  * Limber (1953) Limber D. N., 1953, ApJ, 117, 134
  * Limber (1954) Limber D. N., 1954, ApJ, 119, 655
  * LoVerde & Afshordi (2008) LoVerde M., Afshordi N., 2008, Phys. Rev. D, 78, 123506
  * Miyatake et al. (2015) Miyatake H., et al. 2015, ApJ, 806, 1
  * Moraes et al. (2014) Moraes B., et al. 2014, in Revista Mexicana de Astronomia y Astrofisica Conference Series. pp 202–203
  * OSC (1987) OSC 1987, Ohio Supercomputer Center, http://osc.edu/ark:/19495/f5s1ph73
  * Okamoto & Hu (2003) Okamoto T., Hu W., 2003, Phys. Rev. D, 67, 083002
  * Padmanabhan et al. (2007) Padmanabhan N., et al. 2007, MNRAS, 378, 852
  * Perlmutter et al. (1999) Perlmutter S., et al. 1999, ApJ, 517, 565
  * Planck Collaboration (2016) Planck Collaboration 2016, A&A, 594, A13
  * Planck Collaboration (2020a) Planck Collaboration 2020a, A&A, 641, A6
  * Planck Collaboration (2020b) Planck Collaboration 2020b, A&A, 641, A8
  * Prat et al. (2018) Prat J., et al. 2018, Phys. Rev. D, 98, 042005
  * Reid et al. (2016) Reid B., et al. 2016, MNRAS, 455, 1553
  * Reyes et al. (2010) Reyes R., et al. 2010, Nature, 464, 256
  * Riess et al. (1998) Riess A. G., et al. 1998, AJ, 116, 1009
  * Ross et al. (2015) Ross A. J., et al. 2015, MNRAS, 449, 835
  * Rozo et al. (2016) Rozo E., et al. 2016, MNRAS, 461, 1431
  * Sachs & Wolfe (1967) Sachs R. K., Wolfe A. M., 1967, ApJ, 147, 73
  * Salazar-Albornoz et al. (2017) Salazar-Albornoz S., et al. 2017, MNRAS, 468, 2938
  * Scolnic et al. (2018) Scolnic D. M., et al. 2018, ApJ, 859, 101
  * Sheldon & Huff (2017) Sheldon E. S., Huff E. M., 2017, ApJ, 841, 24
  * Simpson et al. (2013) Simpson F., et al. 2013, MNRAS, 429, 2249
  * Singh et al. (2019) Singh S., et al. 2019, MNRAS, 482, 785
  * Takahashi et al. (2012) Takahashi R., et al. 2012, ApJ, 761, 152
  * de la Torre et al. (2017) de la Torre S., et al. 2017, A&A, 608, A44
  * Tröster et al. (2020) Tröster T., et al. 2020, arXiv e-prints, p. arXiv:2010.16416
  * Troxel et al. (2018) Troxel M. A., et al. 2018, Phys. Rev. D, 98, 043528
  * Weaverdyck et al. (prep) Weaverdyck N., et al., prep
  * Weinberg (1989) Weinberg S., 1989, Reviews of Modern Physics, 61, 1
  * Zaldarriaga & Seljak (1998) Zaldarriaga M., Seljak U., 1998, Phys. Rev. D, 58, 023003
  * Zhang et al. (2007) Zhang P., et al. 2007, Phys. Rev. Lett., 99, 141302
  * Zhao et al. (2009) Zhao G.-B., et al. 2009, Phys. Rev. D, 79, 083513
  * Ziour & Hui (2008) Ziour R., Hui L., 2008, Phys. Rev. D, 78, 123517
  * Zonca et al. (2019) Zonca A., et al. 2019, Journal of Open Source Software, 4, 1298
  * Zucca et al. (2019) Zucca A., et al. 2019, J. Cosmology Astropart. Phys., 2019, 001
  * Zuntz et al. (2015) Zuntz J., et al. 2015, Astronomy and Computing, 12, 45
  * Zuntz et al. (2018) Zuntz J., et al. 2018, MNRAS, 481, 1149

## Affiliations

1 Department of Physics, Duke University Durham, NC 27708, USA
2 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak
Grove Dr., Pasadena, CA 91109, USA
3 Center for Cosmology and Astro-Particle Physics, The Ohio State University,
Columbus, OH 43210, USA
4 Department of Physics, The Ohio State University, Columbus, OH 43210, USA
5 Department of Applied Mathematics and Theoretical Physics, University of
Cambridge, Cambridge CB3 0WA, UK
6 Department of Astronomy/Steward Observatory, University of Arizona, 933
North Cherry Avenue, Tucson, AZ 85721-0065, USA
7 Institute of Physics, Laboratory of Astrophysics, École Polytechnique
Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix,
Switzerland
8 Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
9 Institute of Cosmology and Gravitation, University of Portsmouth,
Portsmouth, PO1 3FX, UK
10 Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania
15312, USA
11 Department of Physics & Astronomy, University College London, Gower Street,
London, WC1E 6BT, UK
12 Department of Physics and Astronomy, Pevensey Building, University of
Sussex, Brighton, BN1 9QH, UK
13 School of Mathematics, Statistics and Physics, Newcastle University,
Newcastle upon Tyne, NE1 7RU, UK
14 Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450,
Stanford University, Stanford, CA 94305, USA
15 Kavli Institute for Cosmological Physics, University of Chicago, Chicago,
IL 60637, USA
16 Department of Physics and Astronomy, University of Pennsylvania,
Philadelphia, PA 19104, USA
17 Institute for Astronomy, University of Edinburgh, Edinburgh EH9 3HJ, UK
18 Jodrell Bank Center for Astrophysics, School of Physics and Astronomy,
University of Manchester, Oxford Road, Manchester, M13 9PL, UK
19 Department of Astronomy, University of California, Berkeley, 501 Campbell
Hall, Berkeley, CA 94720, USA
20 Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA
21 Department of Astronomy and Astrophysics, University of Chicago, Chicago,
IL 60637, USA
22 Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain
23 Departamento de Física Matemática, Instituto de Física, Universidade de São
Paulo, CP 66318, São Paulo, SP, 05314-970, Brazil
24 Laboratório Interinstitucional de e-Astronomia - LIneA, Rua Gal. José
Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil
25 Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510,
USA
26 Instituto de Física Teórica, Universidade Estadual Paulista, São Paulo,
Brazil
27 CNRS, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France
28 Sorbonne Universités, UPMC Univ Paris 06, UMR 7095, Institut
d’Astrophysique de Paris, F-75014, Paris, France
29 SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA
30 Instituto de Astrofisica de Canarias, E-38205 La Laguna, Tenerife, Spain
31 Universidad de La Laguna, Dpto. Astrofisica, E-38206 La Laguna, Tenerife,
Spain
32 Center for Astrophysical Surveys, National Center for Supercomputing
Applications, 1205 West Clark St., Urbana, IL 61801, USA
33 Department of Astronomy, University of Illinois at Urbana-Champaign, 1002
W. Green Street, Urbana, IL 61801, USA
34 Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain
35 Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans,
s/n, 08193 Barcelona, Spain
36 Physics Department, 2320 Chamberlin Hall, University of Wisconsin-Madison,
1150 University Avenue Madison, WI 53706-1390
37 University of Nottingham, School of Physics and Astronomy, Nottingham NG7
2RD, UK
38 Astronomy Unit, Department of Physics, University of Trieste, via Tiepolo
11, I-34131 Trieste, Italy
39 INAF-Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143
Trieste, Italy
40 Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014
Trieste, Italy
41 Observatório Nacional, Rua Gal. José Cristino 77, Rio de Janeiro, RJ -
20921-400, Brazil
42 Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas
(CIEMAT), Madrid, Spain
43 Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India
44 Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679
Munich, Germany
45 Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA
46 Institute of Theoretical Astrophysics, University of Oslo. P.O. Box 1029
Blindern, NO-0315 Oslo, Norway
47 Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, 28049
Madrid, Spain
48 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge
CB3 0HA, UK
49 Kavli Institute for Cosmology, University of Cambridge, Madingley Road,
Cambridge CB3 0HA, UK
50 Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford,
CA 94305, USA
51 Department of Astronomy, University of Geneva, ch. d’Écogia 16, CH-1290
Versoix, Switzerland
52 School of Mathematics and Physics, University of Queensland, Brisbane, QLD
4072, Australia
53 Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse,
85748 Garching, Germany
54 Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA
55 Australian Astronomical Optics, Macquarie University, North Ryde, NSW 2113,
Australia
56 Lowell Observatory, 1400 Mars Hill Rd, Flagstaff, AZ 86001, USA
57 George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and
Astronomy, and Department of Physics and Astronomy, Texas A&M University,
College Station, TX 77843, USA
58 Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona, Spain
59 Department of Astrophysical Sciences, Princeton University, Peyton Hall,
Princeton, NJ 08544, USA
60 Brookhaven National Laboratory, Bldg 510, Upton, NY 11973, USA
61 School of Physics and Astronomy, University of Southampton, Southampton,
SO17 1BJ, UK
62 Computer Science and Mathematics Division, Oak Ridge National Laboratory,
Oak Ridge, TN 37831
63 Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians
Universität München, Scheinerstr. 1, 81679 München, Germany