# The Dark Energy Survey Year 3 and eBOSS: constraining galaxy intrinsic
alignments across luminosity and colour space

S. Samuroff,1,2 R. Mandelbaum,2,3 J. Blazek,1 A. Campos,2,3 N. MacCrann,4 G.
Zacharegkas,5 A. Amon,6,7 J. Prat,5,8 S. Singh,2,3 J. Elvin-Poole,9 A. J.
Ross,10 A. Alarcon,11 E. Baxter,12 K. Bechtol,13 M. R. Becker,11 G. M.
Bernstein,14 A. Carnero Rosell,15,16,17 M. Carrasco Kind,18,19 R. Cawthon,20
C. Chang,5,8 R. Chen,21 A. Choi,22 M. Crocce,23,24 C. Davis,25 J. DeRose,26 S.
Dodelson,2,3 C. Doux,14,27 A. Drlica-Wagner,5,8,28 K. Eckert,14 S. Everett,29
A. Ferté,29 M. Gatti,14 G. Giannini,5,8,30 D. Gruen,31 R. A. Gruendl,18,19 I.
Harrison,32 K. Herner,28 E. M. Huff,29 M. Jarvis,14 N. Kuropatkin,28 P.-F.
Leget,25 P. Lemos,33 J. McCullough,25 J. Myles,25,34 A. Navarro-Alsina,35 S.
Pandey,14 A. Porredon,10,36 M. Raveri,37 M. Rodriguez-Monroy,38 R. P.
Rollins,39 A. Roodman,25,40 G. Rossi,41 E. S. Rykoff,25,40 C. Sánchez,24,14 L.
F. Secco,5,8 I. Sevilla-Noarbe,38 E. Sheldon,42 T. Shin,43 M. A. Troxel,21 I.
Tutusaus,23,24,43 N. Weaverdyck,44,26 B. Yanny,29 B. Yin,2 Y. Zhang,45 J.
Zuntz36, M. Aguena,16 O. Alves,45 J. Annis,28 D. Bacon,47 E. Bertin,48,49 S.
Bocquet,31 D. Brooks,50 D. L. Burke,34,40 J. Carretero,30 M. Costanzi,51,52,53
L. N. da Costa,16 M. E. S. Pereira,54 J. De Vicente,38 S. Desai,55 H. T.
Diehl,28 J. P. Dietrich,31 P. Doel,50 I. Ferrero,56 B. Flaugher,28 J.
Frieman,28,5 J. García-Bellido,57 S. R. Hinton,58 D. L. Hollowood,59 K.
Honscheid,10 D. J. James,60 K. Kuehn,61,62 O. Lahav,50 J. L. Marshall,63 P.
Melchior,64 J. Mena-Fernández,38 F. Menanteau,18,19 R. Miquel,65,30 J.
Newman,66 A. Palmese,67,2 A. Pieres,16,67 A. A. Plazas Malagón,64 E.
Sanchez,38 V. Scarpine,28 M. Smith,69 E. Suchyta,70 M. E. C. Swanson,50,60 G.
Tarle,45 and C. To10 (DES Collaboration)
Author affiliations are listed at the end of the paper
<EMAIL_ADDRESS>
###### Abstract

We present direct constraints on galaxy intrinsic alignments using the Dark
Energy Survey Year 3 (DES Y3), the Extended Baryon Oscillation Spectroscopic
Survey (eBOSS) and its precursor, the Baryon Oscillation Spectroscopic Survey
(BOSS). Our measurements incorporate photometric red sequence (redMaGiC)
galaxies from DES with median redshift $z\sim 0.2-1.0$, luminous red galaxies
(LRGs) from eBOSS at $z\sim 0.8$, and also a SDSS-III BOSS CMASS sample at
$z\sim 0.5$. We measure two point intrinsic alignment correlations, which we
fit using a model that includes lensing, magnification and photometric
redshift error. Fitting on scales $6<r_{\rm p}<70$ Mpc$/h$, we make a
detection of intrinsic alignments in each sample, at $5\sigma-22\sigma$
(assuming a simple one parameter model for IAs). Using these red samples, we
measure the IA-luminosity relation. Our results are statistically consistent
with previous results, but offer a significant improvement in constraining
power, particularly at low luminosity. With this improved precision, we see
detectable dependence on colour between broadly defined red samples. It is
likely that a more sophisticated approach than a binary red/blue split, which
jointly considers colour and luminosity dependence in the IA signal, will be
needed in future. We also compare the various signal components at the best
fitting point in parameter space for each sample, and find that magnification
and lensing contribute $\sim 2-18\%$ of the total signal. As precision
continues to improve, it will certainly be necessary to account for these
effects in future direct IA measurements. Finally, we make equivalent
measurements on a sample of Emission Line Galaxies (ELGs) from eBOSS at $z\sim
0.8$. We report a null detection, constraining the IA amplitude (assuming the
nonlinear alignment model) to be $A_{1}=0.07^{+0.32}_{-0.42}$ ($|A_{1}|<0.78$
at $95\%$ CL).

###### keywords:

cosmological parameters - cosmology: observations - gravitational lensing:
weak - galaxies: statistics

††pubyear: 2022††pagerange: The Dark Energy Survey Year 3 and eBOSS:
constraining galaxy intrinsic alignments across luminosity and colour space–E

##  1 Introduction

The study of cosmic shear as a probe of the large scale structure of the
Universe has developed rapidly over the past decade. Although its potential
was recognised some time ago (see e.g. Jain & Seljak 1997; Hu 1999), only more
recently have high-precision cosmological constraints been possible. In the
past ten years, data sets have grown to the point where weak lensing
measurements alone have roughly comparable power to constrain certain
cosmological parameters as the Cosmic Microwave Background (CMB) temperature
fluctuations. Galaxy weak lensing and the CMB are both sensitive to the
amplitude of the matter power spectrum in the low-redshift Universe, $S_{8}$.
Whereas lensing allows one to probe the late-time matter field directly, the
primary temperature anisotropies of the CMB provide a somewhat more
complicated route, relying on an extrapolation from the surface of last
scattering to the present day. Ever since the results of Heymans et al. (2013)
lensing measurements have given a typically lower $S_{8}$ than the CMB;
interestingly, this finding holds across multiple lensing surveys, whose
members have implemented their own independent, well tested, blind analyses
(Kilbinger et al. 2013; Dark Energy Survey Collaboration 2016; Jee et al.
2016; Troxel et al. 2018; Hildebrandt et al. 2017, 2020; Hikage et al. 2019;
Hamana et al. 2020; Asgari et al. 2021; Secco, Samuroff et al. 2022; Amon et
al. 2022). The current level of (dis)agreement in the full parameter space, as
assessed using various different metrics (Lemos et al., 2021), is at the level
of up to $\sim 2.5\sigma$ (although it differs significantly between surveys
and probe combinations).

Future lensing surveys will have much smaller statistical uncertainties
compared with the current generation, which will greatly increase the
precision of weak lensing measurements. This will, in turn, improve our
constraining power and help us make sense of the apparent tensions in the
literature. It will also, however, require a much tighter control of modelling
errors in order to avoid our analyses becoming dominated by systematic
uncertainties. Although much progress has been made in recent years, and the
methods for mitigating systematics are highly sophisticated, we still have
some way to go, as a field. One outstanding gap in our understanding is the
treatment of intrinsic alignments (IAs; Joachimi et al. 2015; Troxel & Ishak
2015; Kirk et al. 2015; Kiessling et al. 2015).

IAs are shape correlations induced not by cosmological lensing, but by local
interactions, which can mimic cosmic shear. Most obviously, galaxies that are
physically close by to each other experience the same background tidal field,
which couples their intrinsic shapes, inducing what are known as II
correlations. Additionally, GI (shear-intrinsic) correlations are generated by
the fact that the same foreground matter that interacts with foreground
galaxies also lenses background sources. A significant amount of literature
over the past few years has focused on developing analytic models for IAs,
which allow them to be forward modelled and marginalised in cosmological
analyses (Catelan et al., 2001; Mackey et al., 2002; Hirata & Seljak, 2004;
Bridle & King, 2007; Blazek et al., 2015; Blazek et al., 2019; Vlah et al.,
2020; Fortuna et al., 2021a).

Perhaps the most well established approach is an analytic formalism that
assumes the intrinsic shapes of galaxies are linear in the background tidal
field, and frozen at the point of galaxy formation (Catelan et al., 2001;
Hirata & Seljak, 2004). What became known as the Linear Alignment (LA) model
predicts both the GI and II power spectra and is, by convention, normalised
such that the free amplitude $A_{1}$ is very roughly one for a typical lensing
source sample (i.e. a mixed colour sample, dominated by blue galaxies at
$z\lesssim 1$). A few years later, Hirata et al. (2007) and Bridle & King
(2007) introduced a modification, whereby the full nonlinear matter power
spectrum is used in place of the linear version in the LA model equations.
This has been shown to improve the performance of the model on scales $\sim$ a
few $h^{-1}\mathrm{Mpc}$ (Blazek et al., 2011). More recently, Blazek et al.
(2015) and Blazek et al. (2019) take further steps along this route. The
perturbative model developed in those papers (known as the Tidal Alignment and
Tidal Torque model, TATT) extends the LA model to include higher order terms.
Although in principle there are specific physical mechanisms for how
correlations that are, for example, quadratic in the tidal field arise, in
practice the model is agnostic to the underlying physics. An alternative
approach, which is more closely connected with the physics on sub-halo scales,
is to use a version of the halo model. The basic concept was introduced a
decade ago (Schneider & Bridle, 2010), and more recently Fortuna et al.
(2021a) took significant steps towards developing a practical implementation.

Although a useful tool for learning about IAs, pure theory cannot provide a
complete picture. Real data is very much necessary for properly understanding
their behaviour in the real universe. Broadly, measurements can be classified
as _direct_ (i.e. using a statistic that is dominated by IAs, with little or
no contribution from lensing), or _simultaneous_ (i.e. where IAs contribute
only a small part of the total signal, and are inferred alongside cosmological
and other parameters). There have also been studies that have sought to do
something in between, using particular combinations of lensing data to try to
isolate an IA signal (e.g. Zhang 2010; Blazek et al. 2012). By this
definition, almost all cosmic shear studies to date are simultaneous IA
measurements. Although comparison is complicated by non-trivial differences in
the samples and measurement methods, as well as the high-dimensional model
space, such constraints have typically found IA amplitudes for mixed lensing
samples in the range $A_{1}\sim 0.1-1$ (Hildebrandt et al. 2020; Asgari et al.
2021; Troxel et al. 2018; Secco, Samuroff et al. 2022). A smaller number of
works have attempted to understand how IAs enter simultaneous measurements in
more detail. For example Heymans et al. (2013) split the CFHTLenS source
catalogue into early and late types and perform independent cosmic shear
analyses; they report $A_{1}\sim 0$ in the bluer population, and $A_{1}\sim 5$
in early types (albeit with large error bars). Several years later, Samuroff
et al. (2019) implemented a similar colour-split methodology to explore IAs in
DES Y1, this time analysing red and blue galaxies along with their cross-
correlations simultaneously. Assuming the NLA model, that work found
qualitatively similar results, with blue galaxies consistent with zero
alignments and $A_{1}\sim 3$ in the red population. Using the TATT model, it
found the quadratic alignment amplitude $A_{2}$ to be $<0$ at the level of
$\sim 2\sigma$ in both colour samples.

Direct measurements are typically restricted by the need for precise estimates
for the redshifts of individual galaxies, and for this reason have tended to
focus on bright red samples. A number of such studies have been carried out
over the years (Hirata et al., 2007; Joachimi et al., 2011; Singh et al.,
2015; Johnston et al., 2019; Fortuna et al., 2021b), and the alignment
strength as a function of luminosity is relatively well measured in brighter
populations. Since these samples tend to have compact redshift distributions,
any given study only weakly (if at all) constrains the redshift evolution of
IAs. In the case of bluer galaxies, a handful of direct measurements have been
attempted (Mandelbaum et al., 2011; Tonegawa et al., 2018; Johnston et al.,
2019), but the samples here are typically small; though they make null
detections and place upper limits on the IA amplitude, the error bars are wide
enough to allow for significantly non-zero values. Although analogous IA
measurements can (and have) been made on hydrodynamic simulations, these are
limited by finite box size, difficulty in constructing realistic galaxy
samples and the accuracy of the simulations themselves (Codis et al., 2015;
Hilbert et al., 2017; Samuroff et al., 2021).

This paper falls into the category of direct measurements, and represents the
first such exercise using DES. We use a combination of DES redMaGiC (or the
“red sequence Matched filter Galaxy Catalog”; photometric, but with precise
per-galaxy redshift estimates) and the overlapping BOSS and eBOSS surveys
(spectroscopic) to measure IA correlations in physical space, which we then
fit using a range of IA models. This work follows implicit IA constraints from
DES cosmic shear (Secco, Samuroff et al. 2022; Amon et al. 2022) and galaxy-
galaxy lensing (Prat et al. 2022; Sánchez, Prat et al. 2022). We should note
that, while they make use of the same DES catalogues, the samples in these
earlier works are significantly different from ours, and so we do not expect
the IA signal to be the same.

The paper is structured as follows. In Section 2 we describe the various data
sets used in this work. Section 3 then outlines measurements on these data,
including redshifts, calibrated galaxy shapes, and two point correlations. In
Section 4 we set out the model used fit those measurements, and discuss our
analysis choices such as priors and scale cuts; a range of validation tests of
that theory pipeline, using real and simulated data, are outlined in Section
5. Our main results are discussed in Section 6. We conclude in Section 7.

##  2 Data

In this section we briefly describe the data sets used in this work, and how
the various galaxy samples are defined.

Figure 1: The approximate footprint of the Dark Energy Survey (purple) and SDSS eBOSS (pink). The overlap in the Stripe 82 region (across the beak and head of the hummingbird shape) is roughly 600 square degrees.  | No. of Galaxies | $f_{D}$ | Area [sq. deg.] |  $\bar{n}_{\rm c}$ $[10^{4}h^{3}$Mpc${}^{-3}]$ | $\sigma_{e}$ | Mean redshift $\langle z\rangle$ | $R_{S}\times 10^{4}$ | $\langle M_{r}-M_{z}\rangle$ | $\langle M_{r}\rangle$
---|---|---|---|---|---|---|---|---|---
redMaGiC high-$z$ | 754,574 | 1.13 | 4203 | $3.22$ | 0.28 | 0.78 | 4.8 | 0.43 | -21.6
redMaGiC low-$z$ | 1.61 M | 1.14 | 4203 | $7.64$ | 0.26 | 0.46 | 23.1 | 0.43 | -20.9
eBOSS ELGs | 92,954 | 1.10 | 604 | $1.95$ | 0.24 | 0.84 | -4.2 | - | -
eBOSS LRGs | 22,244 | 3.03 | 604 | $0.55$ | 0.26 | 0.75 | -4.7 | 0.37 | -21.9
CMASS | 49,820 | 3.35 | 604 | $4.11$ | 0.23 | 0.52 | -4.0 | 0.36 | -21.8
Table 1: A summary of the properties of the shape samples used in this work.
From left, we show the total number of galaxies after cuts; the area of the
footprint in square degrees and the mean comoving number density (averaged
over redshift). The value $f_{D}$ indicates the number of galaxies in the
density tracer sample relative to the size of the shape sample. Note that this
reflects both the impact of metacalibration cuts and, in the SDSS samples, the
geometric selection. The ellipticity dispersion $\sigma_{e}$ is defined
according to Heymans et al. 2012 (see also Gatti, Sheldon et al. 2021 Eq. 13).
Also shown are the ensemble mean redshift $\langle z\rangle$; the selection
response for each sample (the shear response due to the galaxy shape catalogue
cuts; see Sheldon & Huff 2017) and the mean rest frame colour and $r-$band
magnitude. The means incorporate the weights described in Section 3.2.

###  2.1 The Dark Energy Survey Year 3

The Dark Energy Survey is a six-year programme encompassing $\sim 5000$ square
degrees of the Southern sky using the Víctor Blanco telescope in Chile. The
approximate footprint is shown (purple) in Figure 1. Images were taken in five
photometric bands ($g$, $r$, $i$, $z$, and $Y$) with a nominal depth of
magnitude $r\sim 24.1$ (at full Y6 depth); although all five are used for
redshift estimation, galaxy shape measurements are limited to $riz$ due to
difficulties in accurately estimating the point spread function in the
$g$-band (Jarvis et al., 2021), and shallower imaging in the $Y$-band (45s
exposures as opposed to 90s in $griz$). The work described in this paper is
based on data collected during the first three years of operation (the Y3
data). These data cover the full area at slightly less than the full depth,
with an average of about 5 exposures per galaxy. A description of the image
processing and reduction pipeline, including background subtraction and object
detection can be found in Morganson et al. (2018) and Sevilla-Noarbe, Bechtol
et al. (2021). The photometric data set, before any further cuts, is known as
the gold catalogue (Sevilla-Noarbe, Bechtol et al., 2021). In Y3, this has a
limiting magnitude $i\sim 23$. Per-galaxy photometry measurements are obtained
using multi-object fitting (MOF; Drlica-Wagner et al., 2018), and shapes are
measured using the Metacalibration algorithm (Gatti, Sheldon et al. 2021; see
Section 3.1).

###  2.2 BOSS & eBOSS

We also make use of galaxies from both the Baryon Oscillation Spectroscopic
Survey (BOSS) and the Extended Baryon Oscillation Spectroscopic Survey (eBOSS)
in this paper, and so we discuss both briefly here. The former is a
spectroscopic sample collected as part of SDSS-III (Eisenstein et al., 2011).
Imaging in five photometric bands ($ugriz$) and spectroscopy for BOSS were
performed using the 2.5 metre Sloan Telescope at Apache Point Observatory in
New Mexico (Gunn et al., 2006; Smee et al., 2013). The observing program took
place between autumn 2009 and spring 2014, and covered more than 1.5M galaxies
across 10,000 square degrees of high latitude sky. One can find a description
of the BOSS spectrographs and other details in Dawson et al. (2013).

The eBOSS data are slightly more recent, being taken over the period between
summer 2014 and spring 2019, as part of SDSS-IV (Blanton et al. 2017; see also
Dawson et al. 2016 for a discussion of the differences between BOSS and
eBOSS). Again, spectroscopy relied on the BOSS spectrographs on the Sloan
Telescope. The method for target selection differs slightly depending on the
nature of the galaxy sample. Emission line galaxies were targeted from the
DECam Legacy Survey (DECaLS), which is deeper than SDSS, detections and
photometry. DECaLS was carried out using the DECam on the Blanco telescope,
and covers an area of 6700 square degrees in the region $-20<\delta<+30$
degrees to a $5\sigma$ limiting magnitude of 24.7, 23.9, and 23.0 in the $g$,
$r$ and $z$ bands respectively (compared with 22.8, 22.3 and 20.4 for SDSS;
Delubac et al. 2017). Luminous red galaxy targets, on the other hand, were
selected using SDSS $riz$ imaging and infrared sky maps from the Wide-field
Infrared Survey Explorer (WISE; Wright et al. 2010).

The BOSS and eBOSS footprints are divided in two approximately equal area
regions; the emission line galaxy (ELG) and luminous red galaxy (LRG) samples
used in this work come from the Southern Galactic Cap (SGC), which contains
roughly $600$ square degrees of overlap with the DES footprint.

###  2.3 Primary Galaxy Samples

In this paper we consider five distinct samples (for a summary, see Table 1).
These are:

  * •

eBOSS Emission Line Galaxies: Our eBOSS ELG sample contains $\sim 100,000$
galaxies in the SGC region. The target selection process in described in more
detail in Raichoor et al. (2017), and further discussion can be found in
Tamone et al. (2020); Raichoor et al. (2021). The cuts that define the sample
are relatively complex, and include a $g-$band magnitude limit at $g=22.8$
mag, as well as selection in colour space designed to limit the redshift
range. In total, we have 92,954 galaxies, with a mean redshift of about 0.8.
Although this sample covers a similar range in redshift to the LRGs described
below, they are significantly bluer than any of our other samples (both in
apparent and rest frame magnitudes), and are thus not expected to exhibit
strong intrinsic alignments.

  * •

eBOSS Luminous Red Galaxies: A sample of luminous red galaxies (LRGs) from
eBOSS DR16, drawn from the SGC region. Selection is performed using the
criteria outlined in Prakash et al. (2016) (see also Ross et al. 2020;
Bautista et al. 2021; Rossi et al. 2021 for details on the LRG sample); the
cuts are primarily on extinction corrected magnitudes and colours ($z<19.95$
mag). The redshift coverage is similar to that of the ELG sample, with a mean
of $z\sim 0.8$, and the total number of galaxies is 22,244.

  * •

BOSS CMASS: The CMASS selection algorithm is described in detail by Reid et
al. (2016) (see their Sec 3.3 and references therein). We additionally impose
redshift cuts at $z<0.6$ to ensure there is no overlap with the eBOSS LRGs,
and at $z>0.4$ to remove outliers. This leaves us with 49,820 galaxies.

  * •

redMaGiC high-$z$ (RMH): A sample of red sequence galaxies from the DES Y3
redMaGiC catalogue. These objects are selected using the algorithm outlined in
Rozo et al. (2016). In brief, all detected galaxies are fitted using a red
sequence template, yielding a best fitting redshift, $z_{\rm redmagic}$, and a
derived luminosity $L$, as well as a corresponding $\chi^{2}$. If the galaxy
falls above a minimum $L$ and below a maximum $\chi^{2}$, it is included in
the catalogue. The $\chi^{2}$ threshold is a function of redshift, such that
the comoving density is constant (Rozo et al. 2016 Sec. 3.3). This process
gives a set of bright red galaxies with both well constrained per-object
photometric redshifts ($\sigma_{z}/(1+z)<0.02$) and well understood redshift
error. Our high-$z$ sample consists of the upper two redshift bins of the lens
sample used in Dark Energy Survey Collaboration (2022), cut at $z>0.6$. The
luminosity threshold is $L_{\rm min}=L_{*}$, where $L_{*}$ is a characteristic
luminosity, as defined in Rozo et al. (2016) Sec. 3.1. The sample before shape
cuts comprises $\sim 0.8$ M galaxies over 4203 square degrees. The redshift
distribution is relatively compact and peaks at a similar value to our eBOSS
samples at $z\sim 0.8$.

  * •

redMaGiC low-$z$ (RML): Our low-$z$ redMaGiC sample is defined in a similar
way to redMaGiC high-$z$, with key differences. Primarily, the luminosity
threshold is lower at $L_{\rm min}=0.5L_{*}$ (Dark Energy Survey
Collaboration, 2022). A cut on $z_{\rm redmagic}$ is imposed at $z<0.6$,
equivalent to the three lower lens bins from Dark Energy Survey Collaboration
(2022). Without shape cuts, the catalogue contains $1.84$ M objects, with a
median redshift of $z\sim 0.5$.

In each case we define density and shape tracer selections. The former make
use of all galaxies passing the baseline cuts described above, and also are
not required to be within the DES-SDSS overlap. In each sample, we obtain
galaxy shape estimates by matching galaxies to the DES Y3 Metacalibration
catalogue (Gatti, Sheldon et al., 2021). We construct a KDTree of
Metacalibration galaxy angular positions, which is used to locate the nearest
DES neighbour for each eBOSS or redMaGiC object. A match tolerance of 1
arcsecond is imposed to exclude spurious matches, and objects outside the
overlap region between the two surveys. To obtain a subset of galaxies with
reliable shapes, we then impose the cuts recommended by Gatti, Sheldon et al.
(2021) (their Section 4.2), which includes selections based on size and
signal-to-noise ratio, as well as a cut designed to remove binary star
contamination. We show the estimated redshift distributions $n(z)$ for each
sample in the top panel of Figure 2 (see Section 3.4 for more detail about how
these are estimated), and the comoving number density $n_{c}(z)$ in the
bottom.

###  2.4 Comparison Sample: LOWZ

In addition to the five catalogues discussed above, we also make use of BOSS
LOWZ (Dawson et al., 2013) as a reference sample. The point of including these
data is to test our measurement and inference pipelines by comparing against
the baseline analysis of Singh et al. (2015). LOWZ is a convenient choice for
this, since there are relatively detailed published results using a very
similar methodology to our own.

LOWZ is a sample of LRGs from BOSS DR11. The sample covers a footprint of 9243
square degrees and is approximately volume-limited over the redshift range
$0.16<z<0.36$; a sharp cut off is imposed at these bounds. Unlike the other
samples, we do not match to Y3 Metacalibration to obtain shape estimates, but
simply use the pre-existing catalogues (Reyes et al., 2012). For all other
catalogue-level quantities (redshifts, $k+e$-corrected magnitudes etc), we
likewise use the pre-computed columns (see Singh et al. 2015 for details).
After cuts, the LOWZ shape and density tracer samples contain 159,620 and
173,854 galaxies respectively.

##  3 Measurements

###  3.1 Shapes

The galaxy shapes for all samples except LOWZ are obtained by matching to the
DES Y3 Metacalibration catalogue. Discussion of the shape measurement
algorithm, and catalogue level tests, can be found in Gatti, Sheldon et al.
(2021). The basic measurement is a maximum likelihood fit of an elliptical
Gaussian to each galaxy. This process uses an MCMC, and is performed over
multiple exposures and in bands $riz$ simultaneously. In order to calibrate
biases due to image noise, model bias and other effects, the fit is repeated
four times using artificially PSF-deconvolved and resheared images, a
technique known as Metacalibration. For details of how the Metacalibration
corrections are applied in this particular context see Section 3.6; for the
general case and validation on simulations see Zuntz et al. (2018); Huff &
Mandelbaum (2017); Sheldon & Huff (2017).

###  3.2 Galaxy Weights

For galaxy clustering and galaxy-shape measurements, we use the recommended
weights for each sample respectively. Descriptions of these can be found in
Raichoor et al. (2017) and Ross et al. (2020) (for eBOSS), Reid et al. (2016)
(for CMASS) and Rodríguez-Monroy et al. (2022) (redMaGiC). These are designed
to correct for correlations between the observed galaxy number density and
various survey properties, which can be induced by systematics. For the SDSS
samples, there are additional weights designed to account for fibre collisions
and redshift failures, which are combined as per the references above.

It is worth noting briefly that previous works (see e.g. Ross et al. 2020)
identified a possible systematic due to variations in the redshift
distributions of the eBOSS samples within the SGC and NGC regions, which is
not explicitly corrected by the weights. Although relatively mild for eBOSS
LRGs, it was found to be significant enough to need correcting for in an RSD
analysis using the ELG sample (Tamone et al. 2020; Bautista et al. 2021; de
Mattia et al. 2021). We do not, however, believe this to be a significant
concern for our analysis, given the fact that our IA constraint (from
$w_{g+}+w_{++}$) is constrained to the DES-eBOSS overlap region, which is a
relatively small part of the overall eBOSS SGC footprint. Although we do use
the full area for $w_{gg}$, given that the result from ELGs is essentially a
null detection (see Section 6.1), we do not expect a small systematic
affecting the galaxy bias to be a significant factor.

The shape catalogues for the different samples are all ultimately subsets of
Y3 Metacalibration, and so we adopt the inverse variance weights discussed in
Gatti, Sheldon et al. (2021).

###  3.3 Magnification Coefficients

In addition to imprinting a coherent pattern in their shapes, lensing by large
scale structure also modulates the observed brightness and size of galaxies,
an effect known as magnification. In order to model the impact on our galaxy
number counts, we require an estimate for the slope of the faint end of the
galaxy luminosity function for each of our density tracer samples (see e.g.
Mandelbaum et al. 2005; Elvin-Poole, MacCrann et al. 2022 and Joachimi &
Bridle 2010’s Appendix A). Our fiducial estimates are derived via what we
refer to as the “flux-only” method (Elvin-Poole, MacCrann et al., 2022). In
the cases of eBOSS, CMASS and LOWZ the process is straightforward. For a
particular catalogue containing $N_{0}$ galaxies, with a given pre-existing
selection function, we apply a small achromatic shift $\delta m$ to the
observed magnitudes. We reapply the magnitude cuts using this perturbed
catalogue, and count how many galaxies are lost to the bright-end cut $\delta
N_{-}$. The sign of $\delta m$ is then flipped, and the process repeated to
estimate the number shifted up over the faint-end threshold $\delta N_{+}$.
The total change in number counts is then simply:

$\delta N(\delta\kappa)=\delta N_{+}-\delta N_{-}$ (1)

with $\delta\kappa=0.5(10^{-\delta m/2.5}-1)$. For small perturbations we can
measure the slope of $\delta N(\delta\kappa)/N_{0}$ with $\delta\kappa$
numerically. This gives us a quantity Elvin-Poole, MacCrann et al. (2022)
refer to as $C_{\rm sample}$, which describes the linear response of the
observed galaxy number density to a small change in $\kappa$. We define a
quantity referred to as the magnification coefficient as $\alpha=C_{\rm
sample}/2$ (see Table 3 and Section 4 for how this enters the theory
predictions).

For redMaGiC, the sample selection is more complex. For this reason, we start
with the Y3 GOLD catalogue (Sevilla-Noarbe, Bechtol et al., 2021), perturb the
magnitudes, and re-run the redMaGiC algorithm for each $\delta m$. We then
estimate $\alpha$ in the same way as before. We find $\alpha^{\rm RML}=1.101$
for redMaGiC low-$z$ and $\alpha^{\rm RMH}=1.719$ for redMaGiC high-$z$. For
our LRG, ELG and CMASS samples we find $\alpha^{\rm LRG}=2.020$, $\alpha^{\rm
ELG}=1.177$ and $\alpha^{\rm CMASS}=0.529$ respectively.

In addition to the flux-only estimates, we have alternative values, derived
using an algorithm called balrog (Suchyta et al., 2016; Everett et al., 2022):
$\alpha^{\rm RMH}=2.11\pm 0.32$ and $\alpha^{\rm RML}=0.20\pm 0.29$ for
redMaGiC high-$z$ and low-$z$. balrog works by inserting additional synthetic
galaxies into real photometric images. By running the detection and
measurement processes on the altered balrog images, one can sample the
selection function of the survey and explore effects such as magnification and
blending. Although these, in principle, capture size selection effects that
the flux-only numbers cannot (see Elvin-Poole, MacCrann et al. 2022 for
discussion), they are also relatively noisy. We also have balrog estimates for
the redMaGiC samples only, and not CMASS/eBOSS. We thus use the flux-only
estimates as our fiducial choice; we do, however, confirm that in the two
redMaGiC samples our basic conclusions are unaffected by this choice (see
Section 6.3 and Figure 11 specifically).

###  3.4 Redshift Distributions

Figure 2: Top: The estimated redshift distributions of the various shape
samples used in this work. Each $n(z)$ is normalised to integrate to 1 over
the redshift range shown. The $n(z)$s for the spectroscopic samples are shown
as shaded curves, and are estimated as the histogram of single-galaxy $z$
estimates. The unshaded $n(z)$s are estimated by stacking random samples from
the redMaGiC redshift PDFs. Bottom: The same, but showing comoving number
density as a function of redshift. Note that $n_{c}$ is weakly cosmology-
dependent, and so we assume the fiducial cosmology specified in Section 4.
Note that both the $n(z)$ and $n_{c}(z)$ are qualitatively the same for the
density tracer samples. The shape cuts remove galaxies, but do not change the
shape or mean redshift of these distributions significantly.  Figure 3: The
$r$-band luminosity distributions of the red samples used in this work. We
divide each sample, with the exception of eBOSS LRGs, into roughly equal
number bins in luminosity, as indicated by the shading. See Section 3.5 for
details.

####  3.4.1 Spectroscopic Redshifts for BOSS & eBOSS

For details on the BOSS and eBOSS spectroscopic redshift pipelines see
Comparat et al. (2016), Hutchinson et al. (2016) and Bolton et al. (2012). In
brief, galaxy spectra are collected using the BOSS spectrographs on the Sloan
Telescope (Smee et al., 2013); the instrument has two identical spectrographs,
each of which has a red and a blue camera, collectively covering the
wavelength range $360-1040$ nm, and 1000 optical fibres, 3 arcsec in diameter,
and with a collision scale of 62 arcsec (corresponding to a physical scale of
$\sim$ 0.6 $h^{-1}\mathrm{Mpc}$ at $z=0.8$). Each object is observed in
multiple exposures, which are 15 minutes in duration and can be distributed
across several nights. All good data for a particular galaxy are co-added
together during the spectroscopic data reduction process. Fits are made to
each observed spectrum using a number of templates and combinations of
templates evaluated for all allowed redshifts. A point estimate redshift is
then obtained by maximising the likelihood. The estimated redshift
distributions used in our theory modelling of the CMASS and eBOSS samples (the
shaded curves in the top panel of Figure 2) are, then, histograms of these
point estimates. Note that in making these histograms, we apply the
Metacalibration weights described in Section 3.2.

####  3.4.2 Photometric Redshifts

Unlike with the SDSS samples, we do not have spectroscopic redshifts for our
DES redMaGiC samples. Rather, for each galaxy, we have a redshift PDF, which
is obtained using DES photometry. The redMaGiC algorithm (Rozo et al., 2016)
relies on the fact that red sequence galaxies have a relatively tight
magnitude-colour-redshift relation, which can be calibrated using overlapping
spectroscopic data (Cawthon et al., 2022). Each candidate galaxy above some
fixed luminosity threshold $L_{\rm min}$ is fit to obtain a likelihood
$\mathcal{L}(z)$ and best $\chi^{2}$. The latter acts as a selection
criterion, with $\chi^{2}_{\rm max}$ adjusted as a function of redshift to
ensure approximately constant comoving density. Where it is necessary to have
point redshift estimates (e.g. for the binning in Section 3.6), we use the
value that maximises the likelihood, $z_{\rm redmagic}$. We follow Dark Energy
Survey Collaboration (2022) and estimate the ensemble $n(z)$s by stacking
samples from the full non-Gaussian redshift PDFs (see also Porredon et al.
2021 for discussion). These are shown in Figure 2 (upper panel).

In addition to the $n(z)$ for each sample and point estimates themselves, our
modelling also requires an estimate for the per-galaxy redshift _uncertainty_
as a function of redshift. In the cases of eBOSS and CMASS, the spectral
resolution allows very precise redshift estimates, and so we can assume this
to be negligible. In the case of redMaGiC we obtain error estimates using a
representative subsample of the Y3 redMaGiC catalogues with spectra (see
Pandey et al. 2022). Specifically, we divide the sample into bins of $z_{\rm
spec}$, and within each bin we evaluate the histogram $p([z_{\rm samp}-z_{\rm
spec}]/[1+z_{\rm spec}])$, where $z_{\rm samp}$ are the PDF draws used in
estimating the $n(z)$ above. Since we have four PDF samples per galaxy, we
compute four histograms, and average them, giving us a noisy estimate for the
redshift error in the bin centred on $z_{\rm spec}$. We find that the
histograms are well approximated by a Gaussian distribution, and so we fit
each histogram to obtain a width $\sigma_{z}$. This process leaves us with
$\sigma_{z}(z)$, an estimate for the redshift scatter as a function of
redshift, which we interpolate and incorporate into the modelling described in
Section 4.4.2. Although there is some slight variation with redshift, a
constant $\sigma_{z}/(1+z)\sim 0.01$ is a reasonable approximation, with
$\sigma_{z}/(1+z)<0.02$ over the range $z=0.2-1.1$ (see Porredon et al. 2021,
and in particular their Fig. 1).

###  3.5 Luminosities, Colours & Absolute Magnitudes

To obtain rest frame absolute magnitudes for our galaxy catalogues, we first
convert the best-fit $r$-band fluxes from Metacalibration to apparent
magnitudes, $r=30-2.5\mathrm{log}f_{r}$. The corresponding absolute magnitude
is then given by:

$M_{r}^{i}=r^{i}-5\left(\mathrm{log}D_{l}(z^{i})-1\right)-K(z^{i}),$ (2)

where the index $i$ denotes a galaxy, $z^{i}$ is the best point estimate
redshift for that galaxy, and $D_{l}$ is the corresponding luminosity
distance. Note that $D_{l}$ _is in units of pc $/h$_. We calculate a
$k+e-$correction $K$ for each galaxy based on the redshift using the stellar
synthesis models of Bruzual & Charlot (2003). In brief, we employ two models:
one assuming a passively evolving Spectral Energy Distribution (SED), and the
other passive but with a single instantaneous burst of star formation at
$z=9.84$. These models give us predicted colours and a $k+e-$correction as a
function of $z$. For each galaxy $i$, we then compare the observed $r-i$
colour with the model predictions; if the observed colour is redder than the
predicted one from the passive model, we use that model. If it is bluer than
the one from the passive plus star formation burst model, then we use that
one. Otherwise, we calculate a weighted average of the two $k+e-$corrections.
In all cases, we correct the magnitudes to $z=0$. Note that these star
formation models are designed to describe elliptical galaxies, and we do not
apply them to our ELG sample.

The above procedure is based on the assumption that the overall stellar
population in a given galaxy sample is a mixture of two sub-populations, such
that the observed colours are a linear combination of the colours of those
components; these observed colours are therefore subject to a linear
combination of the associated $k+e-$corrections. Note that in practice the
templates do not differ enormously over the redshift range of our samples.
Indeed, we recompute the $k+e-$corrections using the two models separately,
and find no significant change in the distributions shown in Figure 3.

The luminosity relative to a pivot $L_{0}$ is then given by
$\log{(L_{r}/L_{0})}=-(M_{r}-M_{0})/2.5$, where $M_{0}$ is a fixed reference
magnitude; we adopt a value $M_{0}=-22$ for the sake of comparability with
previous results. For the purposes of constraining trends in alignment
properties, we subdivide our red galaxy samples into luminosity bins. These
are shown in Figure 3, and are defined such that they contain roughly equal
numbers of galaxies (with the exception of the bright end of redMaGiC
high$-z$, where the signal-to-noise was sufficient to allow us to further
split the highest $L$ bin in two.). Between them, our four samples cover a
range of roughly $\log{(L_{r}/L_{0})}=[-0.9,0.4]$, with redMaGiC low-$z$ in
particular providing excellent coverage of the fainter end. We also show the
rest frame colour magnitude diagram for these red samples (as well as LOWZ) in
the top panel of Figure 4. As can be seen here, although we group these
samples together as “red”, there is some significant variation in colour at
fixed luminosity. We will return to this in the context of our main results in
Section 9. The lower panel shows the same colour-magnitude space, but using
apparent magnitudes. Here the distributions are relatively elongated,
primarily due to the colour-redshift degeneracy; that is, a galaxy of given
rest-frame magnitude and colour observed at high redshift will appear both
fainter and redder than the same object observed at low redshift.

Figure 4: Top: The rest frame colour magnitude diagram for the samples used in
this work. The quantities here are $k+e-$corrected magnitudes in the DES
filters. In the main galaxy samples (solid lines), these are estimated using
Metacalibration fluxes. For LOWZ, which is kept as a validation sample, we use
the pre-computed absolute magnitudes described in Singh et al. (2015). The
contours are defined relative to the peak, at $0.5\times$ and $0.25\times$ the
maximum density. Bottom: The equivalent colour-magnitude space, but using
apparent magnitudes. Note that we do not have $k+e-$corrections for eBOSS
ELGs, so they appear in the lower panel only.

###  3.6 Two-Point Correlations

Following a number of previous intrinsic alignment studies, our primary
measurements are constructed using a modified Landy-Szalay estimator (Landy &
Szalay, 1993). For two-point galaxy clustering, this has the form:

$\xi_{gg}(r_{\rm p},\Pi)=\frac{(D-R_{D})(D-R_{D})}{R_{D}R_{D}}.\\\ $ (3)

The measurement is made on a grid of line-of-sight and perpendicular
(comoving) separation, $r_{\rm p}$ and $\Pi$. For a particular sample of
galaxies, we have a density tracer catalogue and a second catalogue of random
points tracing the same footprint and redshift distribution. $DD$, $R_{D}D$
and $R_{D}R_{D}$ are the weighted counts of galaxy-galaxy, galaxy-random and
random-random pairs in a given bin of $r_{\rm p}$ and $\Pi$. To reduce shot
noise, the randoms $R_{D}$ are oversampled relative to the actual data by a
factor of $>10$.

Similarly one can estimate the shape-density cross correlation:

$\xi_{g+}(r_{\rm p},\Pi)=\frac{S_{+}(D-R_{D})}{R_{D}R_{S}},\\\ $ (4)

where again $D$ represents the density sample, and $R_{S}$ and $R_{D}$ are
randoms matching the shape and density samples respectively. The shape-shape
correlation is constructed in a similar way:

$\xi_{++}(r_{\rm p},\Pi)=\frac{S_{+}S_{+}}{R_{S}R_{S}}.$ (5)

We also define:

$S_{+}S_{+}\equiv\sum_{\alpha,\beta;\beta\neq\alpha}e_{+}(\beta|\alpha)e_{+}(\alpha|\beta),$
(6) $S_{+}D\equiv\sum_{\alpha,\beta;\beta\neq\alpha}e_{+}(\beta|\alpha).$ (7)

Here the sum runs over galaxies (or random points) at a given separation drawn
from the two catalogues; $e_{+}(\beta|\alpha)$ is the tangential ellipticity
component of galaxy $i$, defined by the separation vector with galaxy $\beta$.
One can write down a set of analogous equations for $\xi_{g\times}$,
$\xi_{+\times}$ and $\xi_{\times\times}$, which are identical to the above,
but with galaxy ellipticities rotated by 45 degrees. Any astrophysical
contribution to these, however, is expected to be negligible (due to parity
arguments) and for this reason they are commonly used for null testing.

The $\xi(r_{\rm p},\Pi)$ measurements are then projected along the line of
sight as

$w_{ab}(r_{\rm p})=\int_{-\Pi_{\rm max}}^{\Pi_{\rm max}}\xi_{ab}(r_{\rm
p},\Pi)\mathrm{d}\Pi.$ (8)

We use TreeCorr 111http://rmjarvis.github.io/TreeCorr (version 4.1.1)  (Jarvis
et al., 2004) for all two point measurements with $\mathrm{bin\\_slop}=0.0$.
We use 20 logarithmically spaced bins in $r_{\rm p}$, over the range $0.1-200$
$h^{-1}\mathrm{Mpc}$. For the line of sight binning we set $\Pi_{\rm max}=100$
$h^{-1}\mathrm{Mpc}$, with 20 linearly spaced bins between $\pm\Pi_{\rm max}$.
The resulting data vectors are shown in Figure 5. For parts of our analysis,
we also make use of data vectors in bins of luminosity. These are shown in
Figure 6. The shaded regions here indicate physical scales excluded from our
fiducial analysis. Further discussion of the fits to these data can be found
in Section 6. The choice of $\Pi_{\rm max}$ is driven by signal-to-noise
considerations in the photometric samples, and follows Singh et al. (2015). We
also note that since we are including a galaxy-galaxy lensing (Section 4) term
in our model, our choice here is not limited by the need to suppress such
contributions.

Figure 5: Intrinsic alignment correlations measured from DES Y3 and eBOSS. The
rows show (top/bottom) position-shape and shape-shape correlations, as defined
in the text. The shaded bands indicate scales excluded from our NLA (light
grey) and TATT (darker grey) fits. The solid lines are the best fitting NLA
predictions for each data set at $r_{\rm p}>6h^{-1}\mathrm{Mpc}$. For
reference, we also show the various terms arising from IAs, lensing and
magnification separately. For clarity, the total lensing + magnification
contribution, shown as a dashed line, is scaled up by a factor of 10. Note
that the vertical axes vary between panels. Figure 6: The measured galaxy-
shape correlations from redMaGiC and eBOSS LRGs. The columns (left to right)
show bins of $k+e-$corrected $r$-band luminosity (as defined in Table 4). The
rows show (top to bottom, colours the same as in Figure 5) redMaGiC low-$z$
(purple), redMaGiC high-$z$ (red), CMASS (green) and LRGs (pink). Note that
the luminosity bins are not the same in the three cases; we use the $LX$
notation for convenience, but the bin edges and widths are defined for a
particular sample (shown in the upper left of each panel; see also Section 3.5
for discussion of how the luminosity bins are defined). We also show the best
fitting NLA model prediction for each measurement (solid line). As above,
points within the shaded grey regions are excluded from the fits. Note that we
do not fit the TATT model to our luminosity-binned measurements and so, unlike
in Figure 5, there is only one set of grey bands shown here.

The DES shape catalogues make use of a technique called Metacalibration for
accurately inferring an underlying shear signal from galaxy shape estimates.
We apply response corrections in exactly the same way as in the DES Y3
cosmology analyses (Gatti, Sheldon et al., 2021). That is, we have a mean
scale-independent factor $\langle R\rangle$, which is applied at the level of
the two-point measurements as:

$\xi_{g+}\rightarrow\frac{1}{\langle
R\rangle}\xi_{g+},\;\;\;\;\xi_{++}\rightarrow\frac{1}{\langle
R\rangle^{2}}\xi_{++},$ (9)

where the angular brackets indicate an average over galaxies, and $\langle
R\rangle=\langle R_{\gamma}+R_{\rm sel}\rangle$, or the sum of a shear
response and a selection term for the shape sample in question.

Note that $R_{\rm sel}$ corrects only for shape catalogue selection cuts.
Since we do not have either redMaGiC re-runs or the eBOSS selection on sheared
images, any bias induced by the basic sample selection is not included in this
correction. That said, the estimated selection response for an early/late
split of the DES Y1 catalogues was found to be of the order of $10^{-4}$,
which is easily subdominant to our uncertainties (see Samuroff et al. 2019
Section 4.1). We thus judge it safe to ignore this missing correction for the
purposes of our analysis.

##  4 Modelling & Analysis Choices

The following section sets out our theoretical modelling choices. Our aim here
is to connect an observed joint data vector, $w_{gg}+w_{g+}+w_{++}$, with
underlying physical quantities, which can be calculated from theory. Each of
these data vector components is a combination of two observable fields
$\hat{\delta_{g}}$ and $\hat{\gamma}$, or the observed galaxy overdensity in
counts and shapes. If we assume the former is the sum of contributions from
gravitational clustering and magnification,
$\hat{\delta}_{g}=\delta_{g}+\delta_{\mu}$, and the latter is the combination
of intrinsic shape alignments and lensing,
$\hat{\gamma}=\gamma^{I}+\gamma^{\rm G}$, we have a total of four correlations
contributing to each observable. The sections below will set out how we
evaluate these model ingredients

In reality, the observed shear is weighted by the overdensity of shape
galaxies, $\hat{\gamma}\rightarrow(1+\delta_{g,\mathrm{S}})\hat{\gamma}$ (see
e.g. Hirata & Seljak 2004 eq. (6)). This contributes an additional intrinsic
alignment term, which is explicitly included in TATT (although not NLA; Blazek
et al. 2019). For conciseness, we absorb this factor into the definition of
$\gamma^{I}$ when discussing the TATT model. Note that since the overdensity
weighting applies to the total observed shear, not just the intrinsic
component, it also gives rise to terms that scale as $\gamma^{\rm
G}\times\delta_{g,\mathrm{S}}$, an effect known as source clustering. An
analogous effect called source magnification enters in a similar way. These
extra terms, however, are expected to be small at the level of projected
observables, and so we neglect them here (see e.g. Krause et al. 2021 Sec 5B
and Schmidt et al. 2009 for discussion).

We start in Sections 4.1\- 4.3 by describing how we calculate the 3D power
spectra that enter each of our models. Section 4.4 then sets out how these are
combined and projected to give predictions for the observable correlations. We
discuss how the covariance matrix of the data is estimated in Section 4.5.
Finally, Section 4.6 discusses how we choose a set of scale cuts, which
restrict our analysis to the regime where our model is thought to be
sufficient.

When it is necessary to assume a background cosmology, we use a flat
$\Lambda$CDM model $\mathbf{p}_{\rm
cos}=(\Omega_{\mathrm{m}},\Omega_{\mathrm{b}},\sigma_{8},h,n_{\mathrm{s}},\Omega_{\nu}h^{2})=(0.3,0.048,0.82,0.69,0.97,0.00083$222This
corresponds to a total sum of the neutrino masses, $\sum m_{\nu}=0.077$ eV.).
Although our results are not strongly cosmology dependent, we do quantify the
impact of this choice in Section 6. The linear matter power spectrum is
computed using the Boltzmann code CAMB 333http://camb.info/ (Lewis et al.,
2000), with nonlinear corrections using halofit (Takahashi et al., 2012).
Parameter inference is performed within CosmoSIS
444https://bitbucket.org/joezuntz/cosmosis/wiki/Home; v1.6, “des-y3” branch of
cosmosis-standard-library, “develop” branch of cosmosis (Zuntz et al., 2015)
using the MultiNest nested sampling algorithm555v3.6;
$\mathrm{efficiency}=0.3$, $\mathrm{live\\_points}=500$ (Feroz et al., 2013).

Parameter | Description | TATT Prior | NLA Prior
---|---|---|---
$A_{1}$ | Lin. IA amplitude | $\mathrm{U}[-8,8]$ | $\mathrm{U}[-8,8]$
$A_{2}$ | Quadratic IA amplitude | $\mathrm{U}[-8,8]$ | $\delta[0]$
$b_{\rm TA}$ | Density wt. coefficient | $\mathrm{U}[-6,6]$ | $\delta[0]$
$b_{1}$ | Lin. galaxy bias | $\mathrm{U}[0,3]$ | $\mathrm{U}[0,3]$
$b_{2}$ | 2nd order galaxy bias | $\mathrm{U}[-3,3]$ | $\mathrm{U}[-3,3]$
Table 2: The free parameters and priors for the models discussed in this
paper. The upper three rows are IA model parameters, while the lower two
describe galaxy bias. We include two sets of priors here: one for our TATT
model analyses (which extends to $r_{\rm p}>2h^{-1}\mathrm{Mpc}$), and one for
NLA ($r_{\rm p}>6h^{-1}\mathrm{Mpc}$).

###  4.1 Intrinsic Alignment Power Spectra

To model the power spectra of the intrinsic alignment GI and II signals
(respectively the correlation between $\gamma^{I}$ and $\gamma^{\rm G}$, and
$\gamma^{I}$ with itself), we use the Tidal Alignment and Tidal Torque model
(TATT; Blazek et al. 2019). The basic idea is that the intrinsic shape field
$\gamma^{I}$ can be expressed as an expansion in powers of the background
tidal field $s$ and matter overdensity $\delta$:

$\displaystyle\gamma^{I}_{ij}=C_{1}s_{ij}+C_{1\delta}\delta
s_{ij}+C_{2}\sum_{k}s_{ik}s_{kj}+\cdots$ (10)

Note that while $\delta$ is a scalar at any given position $\mathbf{x}$,
$\gamma^{I}$ and $s$ are $3\times 3$ tensors. The above expansion can be
propagated to the two-point level to give expressions for $P_{\rm GI}$ and
$P_{\rm II}$ (see Blazek et al. 2019). Our implementation of the TATT model is
identical to that of Secco, Samuroff et al. (2022) and Krause et al. (2021).
We refer the reader to those papers for specifics, and in particular Sec. D2
and eq. (21) and (22) of Secco, Samuroff et al. (2022) for the full
expressions.

The TATT model has three free parameters, which we refer to as $A_{1}$,
$A_{2}$ and $b_{\rm TA}$. One can also parameterise the redshift dependence of
all the contributions, if desired, as in previous cosmological analyses. Since
our individual samples do not have a particularly wide redshift range,
however, this is not an especially useful thing to do in our case. On the
other hand, one can look at the evolution _between_ samples. Considering
galaxies with similar colour and luminosity properties in Appendix E, we find
no evidence for $z$ evolution over a significantly wider range than the
coverage of any one of our samples alone. The two amplitudes modulate the
strength of IA contributions that are linear and quadratic in $s$:

$C_{1}=-A_{1}\frac{\rho_{\rm
crit}\Omega_{\mathrm{m}}\bar{C}_{1}}{D(z)}\;\;\;\;\;\;\;C_{2}=A_{2}\frac{5\rho_{\rm
crit}\Omega_{\mathrm{m}}\bar{C}_{1}}{D^{2}(z)},$ (11)

where $D(z)$ is the linear growth factor, $\rho_{\rm crit}$ is the critical
density and $\bar{C}_{1}$ is a constant, which by convention is fixed to a
value of $5\times 10^{-14}M_{\odot}\mathrm{Mpc}^{2}/h^{2}$. The other
parameter, $b_{\rm TA}$, is known as the density weighting coefficient, and
controls the size of the $C_{1\delta}$ term above as $C_{1\delta}=b_{\rm
TA}C_{1}$.

We also consider a nested subspace of the full TATT model. The simplest
subspace, known as the nonlinear alignment model (NLA; Hirata et al. 2007;
Bridle & King 2007), has only one free parameter, $A_{1}$. The $\gamma^{I}$
field is assumed to be purely linear in the tidal field (effectively setting
$C_{2}=0$, $C_{1\delta}=0$ in Equation (10)), and so the IA power spectra have
the same shape as the nonlinear matter power spectra, but with a scaling
factor applied.

In all samples considered, we vary the IA parameters with wide flat priors, as
given in Table 2.666Note that these differ slightly from those used in Dark
Energy Survey Collaboration 2022. Although the DES Y3 priors were chosen to be
uninformative for that particular sample, we are considering significantly
different (often much redder) populations of galaxies. We thus opt to allow
for more extreme IA values.

###  4.2 Galaxy Power Spectrum

Galaxy bias, or the mapping between the matter and galaxy overdensity fields,
is an important source of uncertainty in any analysis that relies on galaxy-
shape correlations. Similarly to $\gamma^{I}$ in Section 4.1, one can expand
the galaxy overdensity in terms of $\delta$ (McDonald, 2006; Baldauf et al.,
2010; Saito et al., 2014):

$\displaystyle\delta_{g}=b_{1}\delta+\frac{1}{2}b_{2}\left(\delta^{2}-\langle\delta^{2}\rangle\right)+\frac{1}{2}b_{s^{2}}\left(s^{2}-\langle
s^{2}\rangle\right)+b_{3\mathrm{nl}}\psi+\cdots$ (12)

Here $\psi$ is the sum of several different third-order terms with the same
scaling (see Saito et al. 2014). On large enough scales, it is often
sufficient to assume a simple linear relation $\delta_{g}=b_{1}\delta$; in
this case the galaxy power spectrum is simply:

$P_{\delta_{g}}(k,z)=b_{1}^{2}P_{\delta}(k,z),$ (13)

where the galaxy bias $b_{1}$ depends on the galaxy sample, but is independent
of wave number. $P_{\delta}$ is the nonlinear matter power spectrum.
Unfortunately, we see evidence of the need for a more sophisticated approach
in some of our samples. This is discussed further in Section 4.6, where we see
that redMaGiC high$-z$ and eBOSS LRGs favour a more complicated bias model,
even on relatively large scales.

Using Equation (12) one can write down a slightly more complete expression for
$P_{\delta_{g}}$ (e.g. in Krause et al. 2021 Eq. 38). Our fiducial model for
the galaxy power spectrum includes all terms in the expansion above, for which
we use the implementation in FastPT (McEwen et al., 2016). Assuming co-
evolution, however, we can reduce the number of free parameters to two, with
$b_{s^{2}}=-4/7(1-b_{1})$ and $b_{3\mathrm{nl}}=b_{1}-1$ (see Saito et al.
2014, and also Pandey et al. 2020 and Krause et al. 2021 for further
discussion). For all samples, we marginalise these galaxy bias parameters with
wide flat priors $b_{1}=[0.05,3]$, $b_{2}=[-3,3]$.

For the power spectra entering $w_{g+}$ we assume linear bias (despite using
Equation (12) for $P_{\delta_{g}}$):

$P_{\delta_{g}\mathrm{I}}(k,z)=b_{1}P_{\rm GI}(k,z),$ (14)

and similarly:

$P_{\delta_{g}\delta}(k,z)=b_{1}P_{\delta}(k,z),$ (15)

where $b_{1}$ is the same as in Equation (12) above. In principle, nonlinear
galaxy bias, and also various cross terms between TATT parameters and higher
order bias are expected to contribute to $w_{g+}$. In all cases considered
here, however, $w_{g+}$ has significantly lower signal-to-noise than the
equivalent galaxy-galaxy correlations, and thus the latter dominate the fits
for galaxy bias. To within the level of uncertainty the TATT model is able to
sufficiently describe the potential impact of correlations between nonlinear
galaxy bias and IA through the free $b_{\rm TA}$ parameter (see the similarity
of these nonlinear terms in Blazek et al. 2015). For fits using the NLA model,
we exclude scales where nonlinear bias correlations are significant (see
Section 4.6 for discussion of how the scale cuts are chosen). Although $b_{\rm
TA}$ cannot absorb the galaxy-galaxy lensing signal $\delta_{g}\delta$ so
easily, this term is subdominant on all scales and for all samples ($\sim
5-10\%$ of the total $w_{g+}$ signal; see Section 6.3). We test the impact by
substituting the fully nonlinear $P_{\delta_{g}\delta}$ in place of the
approximation in Equation (15) above. Using the best fit bias parameters for
each of our samples, we find a roughly $10\%$ change in
$w_{\delta_{g}\mathrm{G}}$ on scales $<6h^{-1}\mathrm{Mpc}$; compared to the
full signal, however, the impact is at the sub-percentage level. Implementing
a fully consistent nonlinear model is a work in progress, but we do not expect
this to have a significant impact given the statistical uncertainties in
current data sets.

###  4.3 Magnification & Lensing Power Spectra

As well as contributions due to galaxy clustering and intrinsic shape
correlations, magnification can have an effect on direct IA measurements. Its
impact is to alter the observed galaxy number density in a patch of sky as
$\hat{\delta}_{g}=\delta_{g}+\delta_{\mu}$. Similarly, the observed shear in a
set of galaxies has both an IA contribution, and one from cosmological
lensing: $\hat{\gamma}=\gamma^{I}+\gamma^{\rm G}$. At the two-point level, one
has two additional terms in the $gg$ correlation (galaxy-magnification and
magnification-magnification; $\delta_{g}\delta_{\mu}$ and
$\delta_{\mu}\delta_{\mu}$), and two in the galaxy-shape correlation
(magnification-intrinsic and magnification-lensing; $\delta_{\mu}\gamma^{I}$
and $\delta_{\mu}\gamma^{\rm G}$). Similarly, $w_{++}$ has contributions from
the standard II and GI power spectra, but also a pure cosmic shear term
$\gamma^{\rm G}\gamma^{\rm G}$. On large scales, the additional magnification
power spectra are all related to galaxy and IA power spectra via magnification
coefficients $\alpha$ (see Table 3, and Joachimi & Bridle 2010; von
Wietersheim-Kramsta et al. 2021; Joachimi et al. 2021; Elvin-Poole, MacCrann
et al. 2022 for discussion).

A number of different methods for constraining $\alpha$ have been discussed in
the literature. We describe how we estimate $\alpha$ for each density sample
in Section 3.3. In short, our fiducial estimates are obtained by artificially
perturbing the observed galaxy magnitudes (i.e. a flux-only estimate). For the
two redMaGiC samples, we have estimates from balrog, which we use for
validation (see Section 6.2.3).

###  4.4 Modelling Projected Correlation Functions

####  4.4.1 Modelling Spectroscopic Data

Given power spectra from any model, one can convert into projected correlation
functions of the sort discussed in Section 3.6 via Hankel transforms. In the
case of perfect knowledge of individual galaxy redshifts (i.e. spectroscopic
redshifts) one has:

$w^{ij}_{g+}(r_{\rm
p})=-\int\mathrm{d}z\mathcal{W}^{ij}(z)\int\frac{\mathrm{d}k_{\perp}k_{\perp}}{2\pi}J_{2}(k_{\perp}r_{\rm
p})P_{\delta_{g}\mathrm{I}}(k_{\perp},z),$ (16)

with the Roman indices indicating the two galaxy samples, and $J_{\nu}$ being
a Bessel function of the first kind of order $\nu$. The projection kernel is
given by (see Mandelbaum et al. 2011’s Appendix A)

$\mathcal{W}^{ij}(z)=\frac{n^{i}(z)n^{j}(z)}{\chi^{2}(z)\mathrm{d}\chi/\mathrm{d}z}\times\left[\int\mathrm{d}z\frac{n^{i}(z)n^{j}(z)}{\chi^{2}(z)\mathrm{d}\chi/\mathrm{d}z}\right]^{-1}.$
(17)

In the above, $n^{i}(z)$ is the estimated redshift distribution for sample
$i$, and $\chi(z)$ is the comoving line-of-sight distance corresponding to a
redshift $z$. The other two-point correlations follow by analogy as:

$w^{ij}_{gg}(r_{\rm
p})=b^{i}_{g}b^{j}_{g}\int\mathrm{d}z\mathcal{W}^{ij}(z)\int\frac{\mathrm{d}k_{\perp}k_{\perp}}{2\pi}J_{0}(k_{\perp}r_{\rm
p})P_{\delta_{g}}(k_{\perp},z),$ (18)

and

$w^{ij}_{++}(r_{\rm p})=\int\mathrm{d}z\mathcal{W}^{ij}(z)\\\
\int\frac{\mathrm{d}k_{\perp}k_{\perp}}{2\pi}\left[J_{0}(k_{\perp}r_{\rm
p})+J_{4}(k_{\perp}r_{\rm p})\right]P_{\rm II}(k_{\perp},z).$ (19)

In each case, the theory prediction amounts to a projection of a power
spectrum along the redshift axis, and then a Bessel integral.

####  4.4.2 Modelling IA correlations in the presence of photo-$z$ error

When dealing with spectroscopic galaxy samples, one can in general safely
assume that the associated redshift error is much smaller than the distance
scales of interest. This assumption does not hold for photometric samples such
as redMaGiC, which means the modelling is slightly more complicated. The
impact of redshift error is to scatter galaxies along the line of sight; this
in effect shuffles galaxies between $\Pi$ bins and so redistributes power out
along the line of sight. In principle the effect should wash out when
integrating over a sufficiently large range in $\Pi$. In reality, however, one
must choose finite $\Pi$ limits, and widening the integration range to large
separations is not necessarily desirable, since it can degrade the signal-to-
noise of the measurement. This leads to an overall suppression of the measured
correlations due to photo$-z$ scatter. Another impact of photo-$z$ error is
that it can boost additional (non-IA) signals. That is, galaxy pairs allocated
to small $\Pi$ bins may actually be physically quite distant. Such pairs carry
little local II signal, but they do tend to increase the lensing and
magnification contributions. The consequence of this is that one must account
for the $\Pi$ cut-off in the model. To do so we follow the method set out in
Joachimi et al. (2011), of which we provide an outline below.

To begin, we compute angular spectra from the IA and galaxy-galaxy power
spectra. Incorporating all of the magnification and lensing contributions to
number counts and shear, one has:

$C^{ij}_{\hat{\delta}_{g}\hat{\delta}_{g}}=C^{ij}_{\delta_{g}\delta_{g}}+C^{ij}_{\delta_{\mu}\delta_{\mu}}+C^{ij}_{\delta_{\mu}\delta_{g}}+C^{ij}_{\delta_{g}\delta_{\mu}}$
(20)
$C^{ij}_{\hat{\delta}_{g}\hat{\gamma}}=C^{ij}_{\delta_{g}\gamma^{I}}+C^{ij}_{\delta_{\mu}\gamma^{I}}+C^{ij}_{\delta_{g}\gamma^{\rm
G}}+C^{ij}_{\delta_{\mu}\gamma^{\rm G}}$ (21)
$C^{ij}_{\hat{\gamma}\hat{\gamma}}=C^{ij}_{\gamma^{I}\gamma^{I}}+C^{ij}_{\gamma^{I}\gamma^{\rm
G}}+C^{ij}_{\gamma^{\rm G}\gamma^{I}}+C^{ij}_{\gamma^{\rm G}\gamma^{\rm G}},$
(22)

where the subscripts $\delta_{\mu}$, $\gamma^{I}$, $\delta_{g}$ and
$\gamma^{\rm G}$ indicate magnification, intrinsic shape, gravitationally
induced galaxy overdensity and gravitational shear. Implicitly, the II term
here is the E-mode autocorrelation, $C_{\gamma^{I}\gamma^{I}}^{\mathrm{EE}}$.
In principle, one could also include $C_{\gamma^{I}\gamma^{I}}^{\mathrm{BB}}$,
which can be calculated assuming a particular IA model. We do not include this
in our model because (a) typically any IA induced B-modes are small (Hirata &
Seljak, 2004; Blazek et al., 2019) and (b) they contribute only to $w_{++}$,
where the signal-to-noise of our measurements is low. The Limber integrals
used to compute each of the angular power spectra then have the form:

$C^{ij}_{ab}(\ell|z_{1},z_{2})=\int_{0}^{\chi_{\rm
hor}}\mathrm{d}\chi^{\prime}\\\
\frac{q_{a}^{i}\left(\chi^{\prime}|\chi(z_{1})\right)q_{b}^{j}\left(\chi^{\prime}|\chi(z_{2})\right)}{\chi^{\prime
2}}P_{ab}\left(k=\frac{\ell+0.5}{\chi^{\prime}},z(\chi^{\prime})\right).\\\ $
(23)

Correlation | Kernel | Power Spectrum | Correlation Function
---|---|---|---
$\delta_{g}\delta_{g}$ | $p^{i}p^{j}$ | $P_{\delta_{g}}$ | $w_{gg}$
$\delta_{\mu}\delta_{\mu}$ | $g^{i}g^{j}$ | $4(\alpha^{i}-1)(\alpha^{j}-1)P_{\delta}$ | $w_{gg}$
$\delta_{\mu}\delta_{g}$ | $g^{i}p^{j}$ | $2(\alpha^{i}-1)P_{\delta_{g}\delta}$ | $w_{gg}$
$\delta_{g}\gamma^{I}$ | $p^{i}p^{j}$ | $P_{\delta_{g}\mathrm{I}}$ | $w_{g+}$
$\delta_{\mu}\gamma^{I}$ | $g^{i}p^{j}$ | $2(\alpha^{i}-1)P_{\rm GI}$ | $w_{g+}$
$\delta_{g}\gamma^{\rm G}$ | $p^{i}g^{j}$ | $P_{\delta_{g}\delta}$ | $w_{g+}$
$\delta_{\mu}\gamma^{\rm G}$ | $g^{i}g^{j}$ | $2(\alpha^{i}-1)P_{\delta}$ | $w_{g+}$
$\gamma^{I}\gamma^{I}$ | $p^{i}p^{j}$ | $P_{\rm II}$ | $w_{++}$
$\gamma^{\rm G}\gamma^{I}$ | $g^{i}p^{j}$ | $P_{\rm GI}$ | $w_{++}$
$\gamma^{\rm G}\gamma^{\rm G}$ | $g^{i}g^{j}$ | $P_{\delta}$ | $w_{++}$
Table 3: A summary of the various contributing terms to our observables
$w_{gg}$, $w_{g+}$ and $w_{++}$. The kernel column lists all the possible
combinations for $q_{a}^{i}q_{b}^{j}$ in Equation (23) (where each $q$ is
either the lensing kernel $g$ or the galaxy PDF $p$). For each one we show the
kernel (either lensing efficiency or redshift distribution), and the relevant
power spectrum included in the Limber integral. The prefactors $\alpha$ are
magnification coefficients, which are defined in Section 3.3.

The kernel $q$ is either the lensing efficiency $g$, or the error distribution
$p$, according to Table 3. The power spectrum $P_{ab}$ corresponding to a
given $C(\ell)$ are also shown in Table 3. Here $p_{i}(z^{\prime}|z)$ is the
conditional probability distribution for the true redshift of a galaxy from
sample $i$, which has a best-estimate redshift at $z$. The estimates for $p$
at any given $z$ are obtained using the method described in Section 3.4. Note
that this is different from the more common form of the Limber integral in the
context of cosmological lensing, which uses the ensemble redshift distribution
$n(z)$, not the per-galaxy PDF. One can then transform from harmonic to
angular space as follows:

$\xi^{ij}_{ab}(\theta|z_{1},z_{2})=\frac{1}{2\pi}\int^{\infty}_{0}\mathrm{d}\ell\ell
J_{\nu}(\ell\theta)C_{ab}(\ell|z_{1},z_{2}),$ (24)

where the order of the Bessel function $\nu$ depends on the type of
correlation ($\nu=0$ for $ab=\hat{\delta}_{g}\hat{\delta}_{g}$, $\nu=2$ for
$ab=\hat{\delta}_{g}\hat{\gamma}$ or $\nu=(0,4)$ for
$ab=\hat{\gamma}\hat{\gamma}$), as in Section 4.4.1. As argued in Joachimi et
al. (2011), from here one can obtain the photometric correlation function
$\xi^{ij}_{ab}(r_{\rm p},\Pi,z_{\rm m})$ using a simple coordinate
transformation (see equation A11 in that paper, which also defines $z_{\rm
m}=(z_{1}+z_{2})/2$). Finally, the projected correlation function as a
function of perpendicular physical separation is expressed as,

$w^{ij}_{ab}(r_{\rm p})=\int^{\Pi_{\rm max}}_{-\Pi_{\rm
max}}\mathrm{d}\Pi\int\mathrm{d}z_{\rm m}\mathcal{W}^{ij}(z_{\rm
m})\xi^{ij}_{ab}(r_{\rm p},\Pi,z_{\rm m})$ (25)

With these ingredients, the recipe for generating a theory prediction for the
cross correlation between photometric samples $i$ and $j$ is as follows.

  * •

Choose an initial value of $\Pi$ and $z_{\rm m}$. Use Joachimi et al. (2011)’s
equation A11 to obtain $z_{1}$ and $z_{2}$, and evaluate the per-galaxy error
distributions for the two samples at these redshift values.

  * •

Carry out the Limber integral in Eq. (23) with these error distributions to
obtain $C(\ell|z_{1},z_{2})$.

  * •

Carry out the Hankel transform in Eq. (24) with the appropriate Bessel kernel
to obtain $\xi(\theta|z_{1},z_{2})$.

  * •

Perform the coordinate transform, such that
$\xi(\theta|z_{1},z_{2})\rightarrow\xi(r_{\rm p}|\Pi,z_{\rm m})$.

  * •

Repeat the above steps with varying $\Pi$ and $z_{\rm m}$, to give a three
dimensional grid $\xi(r_{\rm p},\Pi,z_{\rm m})$.

  * •

Integrate over the redshift kernel $\mathcal{W}(z)$ and then over line-of-
sight separation with the appropriate $\Pi_{\rm max}$ to obtain $w_{ab}(r_{\rm
p})$.

We confirm that our implementation of this method returns the same results as
Eqs. (16)-(19) in the limit of narrow photo-$z$ distributions and wide $\Pi$
bounds. We also verify that, with a matching cosmology and set of input
parameters, our modelling code can reproduce Fig. 5 from Joachimi et al.
(2011). Our fiducial modelling setup is to use the steps above to predict
$w_{g+}$ and $w_{++}$.

For $w_{gg}$, however, it is not sufficient to assume redshift space
distortions have negligible impact (see Appendix A and Figure 12). For this
reason, we instead choose to use a sum over Legendre polynomials to obtain the
anisotropic galaxy-galaxy correlation $\xi_{gg}$, which we then integrate over
$\Pi$ (eq. (36)-(41)). We do, however, still need to account for lensing,
magnification and photo-$z$ suppression. Unlike with $w_{g+}$, where the
combined impact of these effects are seen to have some non-trivial scale
dependence, this is much less true for $w_{gg}$; using the recipe set out
above, we generate theory data vectors for each sample with and without
photo-$z$ scatter, lensing, and magnification, finding that correction factor,
$a(r_{\rm p})=w^{\rm full}_{gg}/w_{gg}$, is flat with $r_{\rm p}$ to good
approximation over scales $2<r_{\rm p}<70h^{-1}\mathrm{Mpc}$. Given this, we
derive a single multiplicative factor for each sample, which we apply to the
theory predictions as $w_{gg}\rightarrow aw_{gg}$. We obtain $a_{\rm
RMH}=0.83$ and $a_{\rm RML}=0.87$ for our two redMaGiC samples respectively.

###  4.5 Covariance Matrix

We estimate the covariance of our data using an analytic prescription. This
approach has a number of advantages over data-based estimators such as
jackknife, which have been widely used in the past (Hirata et al., 2007;
Mandelbaum et al., 2011; Joachimi et al., 2011; Singh et al., 2015; Fortuna et
al., 2021b). For example, it can be used on large scales where jackknife
breaks down, and it is unaffected by noise in the data. Note, though, we are
assuming here that the covariance of our data is dominated by the Gaussian
component, and any connected 4pt and super sample covariance contributions are
negligible for our purposes (see e.g. Takada & Jain 2009; Takada & Hu 2013).

The covariance is assumed to be dominated by two components: one from cosmic
variance, and one from shape and shot noise $\mathrm{Cov}=\mathrm{Cov}^{\rm
CV}+\mathrm{Cov}^{\rm SN}$. For any two elements of our data vector in scale
bins centred on $r_{\mathrm{p},m}$ and $r_{\mathrm{p},n}$, the cosmic variance
part is given by:

$\mathrm{Cov}^{\rm
CV}\left[w_{ij}(r_{\mathrm{p},m})w_{kl}(r_{\mathrm{p},n})\right]=\\\
\frac{1}{\mathcal{A}(z_{c})}\int\frac{k\mathrm{d}k}{2\pi}\Theta_{ij}(kr_{\mathrm{p},m})\Theta_{kl}(kr_{\mathrm{p},n})\times\\\
\left[P_{ik}(k)P_{jl}(k)+P_{il}(k)P_{kj}(k)\right],$ (26)

where the lower indices define the tracer type (i.e. $g$ or $+$). The term
$\Theta_{ij}(x)$ is a Bessel function of the first kind (or a sum of two);
specifically $J_{2}$, $J_{0}$ and $J_{0}+J_{4}$ for $ij=g+,gg$ and $++$
respectively. The power spectra are $P_{\delta_{g}\mathrm{I}}$ for $ij=g+$,
$P_{\delta_{g}}$ for $gg$ and $P_{\rm II}$ for $++$. Note that we do not
include secondary contributions from magnification and lensing, but this is
not expected to significantly change our results. The prefactor $\mathcal{A}$
is the projected comoving area of the footprint (including masking), at a
characteristic redshift $z_{c}$.

The noise contribution is simply given by (Schneider et al. 2002; Hu & Jain
2004; Joachimi & Bridle 2010):

$\mathrm{Cov}^{\rm
SN}\left[w_{g+}(r_{\mathrm{p},m})w_{g+}(r_{\mathrm{p},n})\right]=\delta_{mn}\frac{\sigma^{2}_{\mathcal{\epsilon}}}{N^{g+}_{\rm
p}},$ (27)

$\mathrm{Cov}^{\rm
SN}\left[w_{gg}(r_{\mathrm{p},m})w_{gg}(r_{\mathrm{p},n})\right]=\delta_{mn}\frac{1}{N^{gg}_{\rm
p}},$ (28) $\mathrm{Cov}^{\rm
SN}\left[w_{++}(r_{\mathrm{p},m})w_{++}(r_{\mathrm{p},n})\right]=\delta_{mn}\frac{\sigma^{4}_{\mathcal{\epsilon}}}{N^{++}_{\rm
p}},$ (29)

for our three observables respectively. Since our measurements with TreeCorr
give us the number of galaxy pairs in each bin $N_{\rm p}(r_{\mathrm{p},m})$
without extra computational cost, we use these exact numbers here. The shape
dispersion $\sigma_{\mathcal{\epsilon}}$ is measured for each sample, using
the Heymans et al. (2012) definition, and incorporating the correct response
weighting (see Gatti, Sheldon et al. 2021 eq. 13).

We perform initial fits using a preliminary covariance matrix, which we then
replace with an updated version with best fit values of $A_{1}$, $b_{1}$ and
$b_{2}$ entering Equation (26) above. Since $w_{g+}$ is shape noise dominated
for all samples and on all but the largest scales (and $w_{++}$ entirely so,
on all scales), this update makes little difference to the final IA parameter
constraints.

We compare our analytic predictions with jackknife estimates in Appendix C,
and find good agreement on scales $2h^{-1}\mathrm{Mpc}<r_{\rm
p}<70h^{-1}\mathrm{Mpc}$.

###  4.6 Scale Cuts

We impose scale cuts on all three of our measured correlations when fitting,
to mitigate model uncertainty. In brief, our minimum scales are $r_{\rm
p,min}=(2,6,6)h^{-1}\mathrm{Mpc}$ for NLA and $r_{\rm
p,min}=(2,2,2)h^{-1}\mathrm{Mpc}$ for TATT (where the ordering is $w_{gg}$,
$w_{g+}$, $w_{++}$). For the latter two this is driven by the fact that we
know our IA models start to break down on certain scales, and rely on
assumptions that are valid only in specific regimes (NLA on scales above $\sim
5-10h^{-1}\mathrm{Mpc}$, TATT down to $\sim 1-2h^{-1}\mathrm{Mpc}$; Bridle &
King 2007; Blazek et al. 2015). The motivation behind the $w_{gg}$ scale cuts
is discussed in more detail in Section 4.6.1 below.

We also impose an upper cut at $70h^{-1}\mathrm{Mpc}$, a choice motivated by
the null tests in Appendix B. This maximum scale is applied to to all three
correlations for all samples. Large scale systematics, most prominently PSF
modelling error, are known to modulate galaxy number counts at large $r_{\rm
p}$, but are difficult to model analytically. We thus choose to remove the
affected scales.

####  4.6.1 Galaxy clustering

Figure 7: The difference in Bayesian Information Criterion between two simple
models of effective galaxy bias, as a function of the lower scale cut
$r_{p,\mathrm{min}}$ applied to $w_{gg}$. Negative values indicate that the
linear bias assumption is justified by the data; positive values, on the
contrary, indicate that the dark matter only correlation $w_{\delta\delta}$
with a scale-independent multiplicative factor is not a good representation of
$w_{gg}$. The different colours show fits to the galaxy-galaxy correlation
from various DES and eBOSS samples. As in Figure 5, the shaded region shows
scales excluded in our fiducial analysis setup.

Since scale cuts are designed to mitigate modelling uncertainty, the choice of
$r_{\mathrm{p,min}}$ for $w_{gg}$ is unavoidably connected to the choice of
galaxy bias model. We first seek to test whether there are a set of cuts that
will allow us to use a simple, scale independent linear bias model. For each
of our samples, one can estimate an effective bias

$b_{g}^{{}^{\prime}}(r_{\rm p})=\sqrt{\frac{\hat{w}_{gg}(r_{\rm
p})}{w_{\delta\delta}(r_{\rm p})}},$ (30)

where $\hat{w}_{gg}$ is the measured projected galaxy-galaxy correlation. The
matter-matter correlation in the denominator is the theoretical prediction,
and so assumes a particular cosmology; we test the impact of switching between
reasonably different cosmologies (specifically, the best fitting values from
DES Y1 and Planck 2018), and find our results are only very weakly sensitive
to this choice. For each sample, we fit $b_{g}^{{}^{\prime}}$ twice, once
using a scale independent constant $b_{g}^{{}^{\prime}}(r_{\rm p})=b$, and
again using a linear-exponential function $b_{g}^{{}^{\prime}}(r_{\rm
p})=ae^{-r_{\rm p}}+b$. Although this is not a physically motivated bias
model, it has qualitatively the correct behaviour, increasing rapidly on small
scales and converging to a constant on large scales. The exact form was
motivated by Fig. D1 of Samuroff et al. (2021), where the bias in IllustrisTNG
is seen to scale roughly as $e^{-r_{\rm p}}$ plus a constant. In each case, we
compute the Bayesian Information Criterion (Schwarz, 1978),

$\mathrm{BIC}=k\,\mathrm{log}N_{\rm pts}+\chi^{2},$ (31)

where $k$ is the number of model parameters (either 1 or 2, in the
constant/log-linear cases respectively) and $N_{\rm pts}$ is the number of
data points included in the fit. The $\chi^{2}$ for model $M$ is computed
using the full data covariance matrix, as
$\chi^{2}_{M}=[\hat{w}_{gg}-b_{g,M}^{{}^{\prime}}w_{\delta\delta}]\mathbf{C}^{-1}[\hat{w}_{gg}-b_{g,M}^{{}^{\prime}}w_{\delta\delta}]$.
The difference $\Delta\mathrm{BIC}$ then gives us an indicator of which model
is preferred by the data – that is, whether linear bias is sufficient, in a
statistical sense, to describe the measured $w_{gg}$. We repeat this process
using a range of lower scale cuts $r_{\rm p,min}$, resulting in the curves
shown in Figure 7. Although eBOSS ELGs and redMaGiC low-$z$ appear to be
relatively consistent with a linear bias model, even down to small scales,
this is not true in all of our samples. The picture is slightly different in
the case of redMaGiC high-$z$ and eBOSS LRGs, with the latter in particular
preferring the more complicated bias scaling for almost any choice of minimum
scale.

The above test indicates that, at least for some of our samples, even at
relatively large scales (above $6h^{-1}\mathrm{Mpc}$), the linear bias
approximation does not provide a good description of the data. Motivated by
these findings, our fiducial scale cuts are as follows. We fit $w_{gg}$ for
all samples down to $r_{\rm p,min}=2h^{-1}\mathrm{Mpc}$, with a model that
includes nonlinear galaxy bias (as described in Section 4.4). At
$2h^{-1}\mathrm{Mpc}$ we are still well outside the one-halo regime, even for
the largest objects in our samples, and so the perturbative expansion in
Equation 12 may still be sufficiently accurate. To help further validate this
choice, we perform additional fits to $w_{gg}$ alone, using very large scales
($>10h^{-1}\mathrm{Mpc}$) and linear bias. For each sample, we calculate the
shift relative to the $b_{1}$ value obtained using the fiducial setup, and
verify that it is not sufficient to produce an appreciable bias in $w_{g+}$.

####  4.6.2 Intrinsic alignment correlations

Figure 8: The impact of higher order effects on galaxy-shape correlations. We
include both redMaGiC high-$z$ and redMaGiC low-$z$ to show that our
conclusions hold across the redshift range of our samples. As before, the
shaded bands indicate scales removed in our fiducial NLA (light grey) and TATT
(dark grey) analyses. We include here: the impact of baryonic feedback, as
represented by OWLS-AGN (solid line); the impact of one-halo alignment
physics, as represented by the model of Schneider & Bridle (2010) (dotted) and
signal boosting due to anisotropy in the 3D correlation (Singh & Mandelbaum
2016; dashed). In all cases, the unmodelled effects appear at the level of a
few percent or less on the scales we use, which is below the level of our
statistical precision.

For $w_{g+}$ and $w_{++}$, $r_{\rm p,min}=6h^{-1}\mathrm{Mpc}$ for our NLA
fits (see Section 6), primarily driven by IA modelling uncertainty. The TATT
model allows us to push to slightly smaller scales, and so here we adopt
$r_{\rm p,min}=2h^{-1}\mathrm{Mpc}$.

We test the robustness of our chosen $w_{g+}$ cuts to a number of unmodelled
effects in Figure 8. Specifically, we generate theory data vectors containing
(a) a matter power spectrum contaminated with OWLS-like baryonic feedback
(considered as an extreme scenario; see the next paragraph); (b) a one-halo
intrinsic alignment signal and (c) projection effects in the 3D correlation
function. In each case, we choose a reference IA model $A_{1}=3.5$ (roughly
the NLA best fit for redMaGiC in Section 6). We do this using both redMaGiC
high-$z$ and redMaGiC low-$z$ redshift distributions, since these are
representative of the range in $z$ covered by our samples. The fractional
differences in Figure 8 are calculated relative to a reference data vector,
which does not contain the contamination.

For (a), we follow Krause et al. (2021) in taking the OWLS-AGN scenario
(Schaye et al., 2010; van Daalen et al., 2011) as an upper limit on the
extremity of baryon feedback in the late-time matter power spectrum (see also
Secco, Samuroff et al. 2022 Fig. 5 and the accompanying discussion). The
baseline matter power spectrum from CAMB is modified in such a way as to
preserve the original cosmology but introduce high$-k$ distortions which mimic
the impact of baryons in the OWLS hydrodynamic simulations (Dark Energy Survey
Collaboration 2016, eq. 8). As we can see in Figure 8, baryonic feedback is
entirely negligible at $r_{\rm p}>6h^{-1}\mathrm{Mpc}$. Its impact increases
rapidly in the intermediate ($2<r_{\rm p}<6h^{-1}\mathrm{Mpc}$) window, but is
still only $\sim 2-5\%$ at $r_{\rm p}=2h^{-1}\mathrm{Mpc}$, which is well
below the level of statistical error on these scales. If we take the TATT best
fit from each redMaGiC sample (fit on scales $r_{\rm p}>2h^{-1}\mathrm{Mpc}$;
see Section 6.2.2), we obtain $\Delta\chi^{2}_{\rm RMH}=0.05$ and
$\Delta\chi^{2}_{\rm RML}=0.45$.

We carry out a similar exercise with small scale alignments. To get a rough
gauge of the impact of one-halo contributions, we use the fitting formulae
provided by Schneider & Bridle (2010). We choose to update the overall
amplitude of the model to the value found by Singh et al. (2015)
($a_{h}=0.08$); since this matches LOWZ LRGs, which are somewhat brighter and
redder than any of our samples, we expect this to be an upper estimate for the
impact of intra-halo physics. Shown by the dotted line in Figure 8, we again
see the effect to be vanishing on scales $r_{\rm p}>6h^{-1}\mathrm{Mpc}$ and
slightly larger but still subdominant to errors at $r_{\rm
p}>2h^{-1}\mathrm{Mpc}$.

Finally, we also test the impact of a kind of projection effect that induces
anisotropy in $\xi_{g+}(r_{\rm p},\Pi)$ (dashed lines in Figure 8). First
identified by Singh & Mandelbaum 2016, the idea is that galaxy alignments
along the $\Pi$ direction cannot be measured using shapes measured in 2D
projected space; the result is a suppression of the observed alignment signal
that scales as $f=r_{\rm p}/(r_{\rm p}^{2}+\Pi^{2})^{\frac{1}{2}}$. As one
might intuitively expect from the geometry, $f$ diverges from 1 as $\Pi$
increases, at fixed $r_{\rm p}$. Looking at Figure 8 we can see that the
impact is primarily at large $r_{\rm p}$. To understand this, consider the
fact that particular $\Pi$ scales do not contribute equally to the projected
correlation at all $r_{\rm p}$ (for an illustration of this, see Singh &
Mandelbaum 2016 Fig. 10c). That is, at $r_{\rm p}=1h^{-1}\mathrm{Mpc}$, even
considerable suppression at large $\Pi$ matters very little; that regime
contributes almost nothing to the line of sight integral, since $\xi_{g+}$
scales approximately as $1/r^{2}$, which at small $r_{\rm p}$ and large $\Pi$
is essentially $1/\Pi^{2}$. At $r_{\rm p}=30h^{-1}\mathrm{Mpc}$, on the other
hand, large $\Pi$ scales contribute much more. Although the geometric
suppression at a given $\Pi$ is less important for this larger $r_{\rm
p}$value, the background scaling of $\xi_{g+}$ dominates, and so the overall
impact on $w_{g+}$ is larger. This can be modelled in an analogous way to
redshift space distortions in galaxy clustering. Although not included in our
fiducial model, we can assess the impact in the NLA case using the recipe set
out in Singh & Mandelbaum (2016) Sec. 2.3. Fortunately we see the impact is
largely contained at separations above $r_{\rm p}>70h^{-1}\mathrm{Mpc}$, which
are already removed by the upper $r_{\rm p}$ cut. Within the range of scales
used for our fits, the impact is comfortably smaller than our error bars.

##  5 Pipeline Testing

In this section we describe the various tests of the analysis pipeline, and
the measurements themselves. These include tests of the theory predictions by
comparing different code implementations. We seek to validate the pipeline by
reanalysing an existing data set and comparing with published results. Finally
we discuss null tests on the data, designed to be sensitive to residual
systematics.

###  5.1 Reanalysing LOWZ

For the purposes of validating our measurements and demonstrating
comparability with previous results, we partially reanalyse the BOSS LOWZ
catalogues of Singh et al. (2015) (see also Section 2). LOWZ makes a good test
data set for several reasons – not least that it has documented, relatively
high signal-to-noise $w_{g+}$ measurements in the literature, and the redshift
catalogue is publicly available. We repeat all measurement steps downstream
from shape catalogues using our pipeline, and then fit the resulting
correlation functions with our modelling setup. At the level of data vectors,
our pipeline can reproduce the galaxy clustering and galaxy-shape
correlations, $w_{gg}$ and $w_{g+}$, of Singh et al. (2015) to sub-percent
precision on scales $[2,70]$ $h^{-1}\mathrm{Mpc}$.

We also analyse the LOWZ data on large scales, and compare our results to
those of Singh et al. (2015); when matching their analysis choices exactly
(NLA model, linear bias, $r_{\rm p}>6h^{-1}\mathrm{Mpc}$), we recover their
reported best fit in the $A_{1}-b_{1}$ plane to $<<1\sigma$. Our fiducial
analysis configuration differs from the published LOWZ paper in a number of
ways. Most significantly, these include:

  * •

Our assumed cosmology is that set out in Section 4, instead of WMAP9 (Hinshaw
et al., 2013). This results in a slight increase in the amplitude of the
matter power spectrum, which in turn results in a slightly lower alignment
amplitude. Note that our fiducial cosmology includes massive neutrinos, which
modify the nonlinear $P(k)$ slightly. The difference in $h$ also alters the
measurement of the two point functions earlier in the pipeline (via the
redshift to distance conversion), although this difference is minimal.

  * •

Our fiducial data vector includes galaxy-galaxy, galaxy-shape and shape-shape
correlations, whereas Singh et al. (2015) include only the former two.

  * •

We use the Takahashi et al. (2012) version of halofit to compute the nonlinear
matter power spectrum, whereas Singh et al. (2015) use a slightly older
release (Smith et al., 2003).

  * •

We use an analytic calculation to estimate the data vector covariance matrix,
instead of jackknife. While the two agree relatively well, slight differences
in the relative weighting of different scales in both $w_{gg}$ and $w_{++}$
are apparent.

  * •

We include contributions from lensing and magnification in our model. Although
this has little impact on a low redshift spectroscopic data set such as LOWZ,
it has a larger bearing on our eBOSS and redMaGiC samples.

We show a more detailed comparison at the parameter level in Appendix C. In
short, when matching the analysis choices of Singh et al. (2015), we can
reproduce their published IA results almost exactly. Switching to our fiducial
NLA setup produces a very similar result, with a small reduction in the size
of the error bars.

###  5.2 Null Tests

A number of systematics (e.g. PSF modelling errors) can lead to a non-zero
mean shear. Unlike multiplicative biases, we can look for such effects
directly using the data. We find no evidence of such a signal in any of the
samples considered here, with $|\left\langle e_{i}\right\rangle|\sim 10^{-4}$
in all cases. A number of other tests for systematic induced signals are
presented in Gatti, Sheldon et al. (2021); they find no evidence for
correlations between the response-corrected shear and PSF shape and size, or
for a statistically significant B-mode signal.

We also measure one additional null signal. Constructing $w_{g\times}$
involves the same basic quantities as $w_{g+}$, but measuring the shape
component that is rotated 45 degrees with respect to the radial/tangential
direction. Like lensing, astrophysical processes such as intrinsic alignments,
to first order, should induce only tangential/radial correlations777Although
some IA models predict a non-zero B-mode contribution (see, e.g., Catelan et
al. 2001; Hirata & Seljak 2004; Blazek et al. 2019), which translates into
correlations in the cross ellipticity component, such effects appear only in
the II alignment spectra. Given that our constraints are dominated by $g+$
correlations, these terms are thought to be easily subdominant to noise in
current surveys.. Non-zero detection of a cross signal, then, is a red flag
for residual measurement systematics. For all samples considered, we find
these additional measurements to be consistent with zero within the scales
$r_{\rm p}<70h^{-1}\mathrm{Mpc}$. Details of the measurements can be found in
Appendix B.

##  6 Results

This section presents the results of our analyses on the various samples.
Although we will focus on intrinsic alignments, it is worth bearing in mind
that each analysis also includes two free galaxy bias parameters. The
constraints on the bias parameters are strongly dominated by $w_{gg}$, and so
they contribute relatively little to the marginal uncertainties on IA
parameters. The bias does, however, also enter $w_{g+}$, and so it is
important to model it accurately. In every case, the linear bias falls within
the bounds of expectation from previous studies ($b_{1}\sim 1.5-2.0$,
depending on the sample), and $b_{2}$ is small ($b_{2}\sim 0-0.3$). We note
that all samples appear to be fit reasonably well by our model (as quantified
by the best $\chi^{2}$ obtained from fits to the joint $w_{gg}+w_{g+}+w_{++}$
data vector). For more detail on the bias constraints, see Appendix D. It is
also worth bearing in mind that all parameters (bias and IA) are constrained
within the prior bounds. As we note below, although some samples provide only
weak constraints, the priors in Table 2 are sufficiently wide to allow the
contours to close in all cases. In Section 6.1 below we discuss our results on
ELGs, which amount to a null detection. We then move onto the various red
samples in Section 6.2, presenting first large-scale results using NLA in
Section 6.2.1, and then extending to slightly smaller scales with TATT in
Section 6.2.2. Section 6.3 then considers more carefully the level of
contribution from lensing and magnification.

###  6.1 Emission Line Galaxies

Our first, and perhaps easiest to interpret, results are based on eBOSS
Emission Line Galaxies. The data vector is shown in blue in Figure 5. We fit
the NLA model on large scales (the unshaded region in Figure 5), and obtain a
null detection,

$A_{1}^{\rm ELG}=-0.42^{+0.50}_{-0.50}\;\;\;\;\;\;(r_{\rm
p}>6h^{-1}\mathrm{Mpc})$ (32)

with $\chi^{2}/\mathrm{dof}=1.17$ (with a corresponding $p-$value $p=0.32$).
This is expected, given the sample: a non-zero IA signal has never been
detected in ELGs (or in any colour-selected sample of blue galaxies more
generally; Mandelbaum et al. 2011; Samuroff et al. 2019; Johnston et al.
2019). The additional (non $\delta_{g}$I) terms are also seen to be small, for
a number of reasons: first, the magnification coefficient is small
$(\alpha-1)\sim 0.1$, for ELGs, which scales down the $\mu$I and $\mu$G
contributions. Second, the limits of the line of sight integral tend to
suppress the lensing contributions to the signal; integrated out to
$\Pi_{\max}=1000h^{-1}\mathrm{Mpc}$, $\mu$G tends to dominate on larger
scales. In practice, however, with integral limits at $\pm
100h^{-1}\mathrm{Mpc}$, the largest term by some way is $\delta_{g}$I, with
$w_{\delta_{g}\gamma^{I}}/(w_{\delta_{g}\gamma^{\rm
G}}+w_{\delta_{\mu}\gamma^{\rm G}}+w_{\delta_{\mu}\gamma^{I}})\sim 14$ at
$6h^{-1}\mathrm{Mpc}$ (as evaluated at the best fitting parameters).
Similarly, for the shape-shape correlation, the ratio of II to other terms is
$\sim 18$. The end result is a combined best fitting theory prediction that is
below the level of shape noise.

Since the signal-to-noise is relatively low, and there is no visible structure
in $w_{g+}$, we also repeat our NLA fits with slightly less stringent cuts,
$r_{\rm p}>2h^{-1}\mathrm{Mpc}$. This tightens the bounds on the alignment
amplitude to:

$A_{1}^{\rm ELG}=0.07^{+0.32}_{-0.42}\;\;\;\;\;\;(r_{\rm
p}>2h^{-1}\mathrm{Mpc})$ (33)

Indeed, even considering scales down to $0.1h^{-1}\mathrm{Mpc}$ in Figure 5,
we still see no hints of signal in $w_{g+}$ or $w_{++}$. Computing the null
$\chi^{2}$ on all scales $r_{\rm p}<70h^{-1}\mathrm{Mpc}$, we find
$\chi^{2}=17.5$ for 16 data points ($p=0.42)$. This is interesting, since it
suggests that there is not a strongly scale dependent one halo (1h) signal of
the sort seen in the redMaGiC and CMASS samples (or at least, not one that is
detectable above the level of shape noise).

In terms of sample, the closest results in the literature are those of
Mandelbaum et al. (2011) and Tonegawa et al. (2018). These both use blue
emission line galaxies, from WiggleZ and Subaru respectively, and also make
null detections of $A^{\rm M11}_{1}=0.15^{+1.03}_{-1.07}$ and $A_{1}^{\rm
T18}=0.49^{+3.56}_{-3.56}$ respectively. Our results tighten the null
constraint, imposing an upper limit of $|A_{1}|<0.78$ at $95\%$ CL. In terms
of redshift, our eBOSS ELG measurements sits between the earlier two ($z\sim
0.8$, compared with $z\sim 0.5$ for WiggleZ and $z\sim 1.4$ for Subaru). It is
worth exercising some caution here, however, since in both cases it is not
clear that the sample matches ours closely. In particular, the Mandelbaum et
al. (2011) sample is a relatively bright selection of starburst galaxies with
specific colour cuts (see their Sec 3.1). That said, the best-fitting bias
values are relatively similar to our own ($b_{1}\sim 1.4$ for eBOSS ELGs,
$b_{1}\sim 1.5$ for WiggleZ). The Tonegawa et al. (2018) sample on the other
hand, has both a complicated spectroscopic selection function, and additional
shape catalogue cuts that remove $\sim 50\%$ of objects. Although all three
results (including our own) present IA results consistent with zero in blue-
ish samples across a range of redshifts, it is not clear the results are
directly equivalent.

It is also worth stressing here that although very different from (and much
bluer than) an LRG or redMaGiC type sample, eBOSS ELGs are not exactly
representative of a typical weak lensing catalogue either. Indeed, the eBOSS
ELG selection is designed to facilitate a high S/N BAO measurement, and is
based on the presence of particular emission lines, as determined via a
complex set of magnitude and colour cuts (Raichoor et al., 2017). The
completeness of the sample given the cut is difficult to quantify (Guo et al.,
2019). In contrast, lensing samples tend to have much simpler (if any) colour
selection, and cuts designed to minimise lensing measurement biases and
optimise a weak lensing measurement. The two are designed for different
scientific purposes, and so we should not expect them to match. For this
reason, caution is required when trying to extrapolate these results.

###  6.2 Red Galaxies

####  6.2.1 Constraints on Large Scale Intrinsic Alignments

We next consider our other galaxy samples, redMaGiC, eBOSS LRGs, and CMASS,
which we fit on large scales (again, $>6h^{-1}\mathrm{Mpc}$) using NLA. In
each case, we find a clear detection, with our three parameter model of the
joint data vector $(A_{1},b_{1},b_{2})$ providing a good
$\chi^{2}/\mathrm{dof}$. The constraints and the goodness of fit statistics
can be found in the upper four rows of Table 4. Defining the signal-to-noise
according to Eq. (2) of Becker et al. (2016)888The expression is
$S/N=(\hat{\mathbf{w}}\mathbf{C}^{-1}\mathbf{w}^{\rm model})/(\mathbf{w}^{\rm
model}\mathbf{C}^{-1}\mathbf{w}^{\rm model})^{\frac{1}{2}}$, where
$\hat{\mathbf{w}}$ is the observed (noisy) data vector, $\mathbf{w}^{\rm
model}$ is the best fitting theory prediction, and $\mathbf{C}^{-1}$ is the
inverted covariance matrix. This is slightly different from the common
definition using $\hat{\mathbf{w}}$ only, which is known to be biased high if
noise is present., we find $S/N=22$ in redMaGiC low-$z$ and $S/N=18$ in
redMaGiC high-$z$. Given the smaller area, the detections in our SDSS samples
are slightly weaker, at $S/N=6$ for CMASS and $S/N=5$ for eBOSS LRGs. The best
fitting model predictions can be seen in Figure 5. As we saw with the ELGs in
Section 6.1, CMASS and eBOSS LRGs are dominated by the primary IA signal
($\delta_{g}$I for $w_{g+}$, II for $w_{++}$). For redMaGiC, however, the
picture is slightly different; photo-$z$ scatter tends to increase the maximum
distance galaxies can be physically separated (by shifting well-separated
objects below $\Pi_{\rm max}$), and so boosts the lensing and magnification
terms. With redMaGiC low-$z$ this is partly cancelled out by the fact that
$\alpha$ is very close to $1$, and that the mean redshift is relatively low.
These things are less true for redMaGiC high-$z$, however, and so we see a
stronger $\delta_{g}$G contribution. The $\delta_{g}$I signal is also slightly
stronger, however, and the additional terms still account for only order of a
few percent of the total signal.

Figure 9: Intrinsic alignment strength as a function of $k+e$-corrected $r$-band luminosity in red galaxies. By convention, the luminosities are defined relative to a pivot $L_{0}$, which corresponds to an absolute magnitude $M_{r}=-22$. Open points show previous results from the literature (as indicated in the legend). For illustrative purposes we also show two power law fits from the literature, one fit to GAMA+SDSS at low-mid $L$ (dotted blue), and the other to MegaZ at high $L$ (solid purple). The filled points show our measurements on redMaGiC low-$z$ (stars; five points), redMaGiC high-$z$ (red dots, four points), eBOSS LRGs (pink triangle; one point) and CMASS (green upside down triangles; three points). Note that all results shown here assume the one-parameter NLA model.  Sample | $\langle L_{r}\rangle/L_{0}$ | $\langle z\rangle$ | $\langle M_{r}-M_{z}\rangle$ | $A_{1}$ | $p(>\chi^{2})$
---|---|---|---|---|---
RMH (all $L$) | $0.68$ | $0.78$ | 0.43 | $3.54^{+0.18}_{-0.18}$ | $0.20$
RML (all $L$) | $0.40$ | $0.46$ | 0.43 | $3.34^{+0.13}_{-0.13}$ | $0.12$
CMASS (all $L$) | $0.84$ | $0.52$ | 0.36 | $2.72^{+0.47}_{-0.47}$ | $0.10$
LRGs (all $L$) | $1.20$ | $0.75$ | 0.39 | $5.78^{+1.19}_{-1.19}$ | $0.89$
RMH $L0$ | 0.49 | 0.77 | 0.44 | $3.47^{+0.34}_{-0.34}$ | $0.11$
RMH $L1$ | 0.63 | 0.77 | 0.42 | $3.01^{+0.31}_{-0.30}$ | $0.52$
RMH $L2$ | 0.83 | 0.78 | 0.44 | $4.13^{+0.33}_{-0.34}$ | $0.15$
RMH $L3$ | 1.16 | 0.78 | 0.45 | $5.22^{+1.11}_{-1.11}$ | $0.40$
RML $L0$ | 0.19 | 0.32 | 0.50 | $1.95^{+0.23}_{-0.23}$ | $0.28$
RML $L1$ | 0.26 | 0.43 | 0.42 | $2.86^{+0.29}_{-0.29}$ | $0.22$
RML $L2$ | 0.35 | 0.47 | 0.40 | $3.01^{+0.52}_{-0.51}$ | $0.62$
RML $L3$ | 0.49 | 0.50 | 0.40 | $4.39^{+0.20}_{-0.20}$ | $0.79$
RML $L4$ | 0.73 | 0.53 | 0.40 | $5.00^{+0.20}_{-0.21}$ | $0.14$
CMASS $L0$ | 0.52 | 0.48 | 0.42 | $2.23^{+0.92}_{-0.92}$ | $0.47$
CMASS $L1$ | 0.79 | 0.52 | 0.36 | $3.00^{+0.62}_{-0.61}$ | $0.47$
CMASS $L2$ | 1.24 | 0.55 | 0.29 | $3.78^{+0.49}_{-0.49}$ | $0.70$
Table 4: NLA model constraints from the various red galaxy samples discussed
in this work, as a function of $r$-band luminosity. The upper four rows refer
to the full samples, while the lower twelve refer to subsets in luminosity
bins, as defined in Section 3.5 and Figure 6. The first three columns show the
mean luminosity, redshift and rest frame colour of each sample. The final two
are the posterior mean IA amplitude, and the corresponding $p-$value. In all
cases the model is seen to provide a reasonable fit to the data. These fits
were performed using large scales only $(r_{\rm p}>6h^{-1}\mathrm{Mpc})$ in
$w_{g+}$ and $w_{++}$.

Another useful exercise is to divide the samples into luminosity bins, and map
out the dependence of the alignment signal. For each of the bins shown in
Figures 3 and 6, we fit an NLA amplitude (we also fit for galaxy bias, but
since we only split the shape sample by luminosity, that doesn’t change
significantly between $L$ bins). The results are shown in Figure 9, with
numerical parameter constraints in Table 4. Note that we also include a
selection of previous measurements from the literature, denoted by open
points. There are a number of trends worth considering here. First, taken
naively, our results are consistent with the qualitative picture of a broken
power law dependence on luminosity: the trend in $A_{1}$ below
$\mathrm{log}L/L_{0}\sim-0.2$ is much shallower than above it. The
$(L/L_{0})^{\beta}$ power law parameterisation was first introduced as an
empirical scaling by Joachimi et al. (2011). Although there is little physical
motivation, it has been adopted relatively widely both in direct IA
measurements (Singh et al., 2015; Johnston et al., 2019; Fortuna et al.,
2021b) and in forecasts (Krause et al., 2016; Fortuna et al., 2021a), as it
was simple to implement and appeared to fit the available data relatively
well. In recent years, the picture has become more complicated, as evidence
has begun to emerge of a weaker relationship at low $L$ (see e.g. Johnston et
al. 2019). Again, however, this is empirical, and there is no first-principles
reason to expect a double power law in particular (or any other form). Our
results appear to reinforce that evidence. Our redMaGiC low-$z$ sample in
particular provides a significant improvement in the constraints on the
fainter end of the $A_{1}-L$ relation (by a factor of 3 or more in the error
bars). At the brighter end, our CMASS, LRG and redMaGiC high-$z$ samples also
appear qualitatively consistent with previous results, following a
considerably steeper slope. Taken at face value, given the error bars, we
could interpret this as ruling out a single power law with relatively high
significance. We stress, however, that it is worth being cautious here.
Despite all being “red”, there are differences between the composition of the
samples, as we will come to below – it is possible these population
differences may be partly responsible for the apparent trends in $L-A_{1}$
space. For this reason, we do not present best-fitting constraints on $\beta$,
but rather a more qualitative discussion of how to interpret our results.

It is noticeable that CMASS (and to a lesser extent redMaGiC high-$z$) tends
to lie roughly $\sim 1-3\sigma$ below the best fit single power law from the
literature (the purple line in Figure 9). Since the error bars here are mostly
shape noise dominated and the luminosity bins are disjoint, the points should
also be uncorrelated to good approximation, meaning it is unlikely this is
random scatter. We can perhaps understand these trends in terms of colour
differences. In the upper panel of Figure 4 we can see that CMASS is
considerably bluer than LOWZ. Indeed, while most extreme for CMASS, all of our
samples tend to peak lower than LOWZ in $M_{r}-M_{z}$ space. Not only this,
there is also some difference in colour between luminosity bins for a given
sample. For example, the galaxies in the upper CMASS bin ($L2$), on average,
have slightly bluer rest-frame magnitudes than the lower two bins. Although
for the sake of convenience, it has been useful to split galaxies into binary
“red” and “blue” categories, our results suggest that this may be an over-
simplification for modelling purposes, given the precision of current data
sets. They suggest that a more sophisticated modelling may be needed, which
accounts for colour and luminosity (and potentially other properties such as
satellite fraction) simultaneously. This will be the focus of future work.

We can compare these results with those of Singh et al. (2015), who consider
colour bins within the LOWZ LRG sample. Although that work reported no clear
trend across five bins in $g-i$ rest frame colour, it should be noted that
LOWZ covers a fairly narrow range in colour space (see the black contour in
Figure 4). Even the bluest bin in that paper still represents a relatively
bright red sample compared with the galaxies considered here. It seems
plausible that our wider coverage allows us to see a trend that is not
detectable in a relatively homogeneous sample like LOWZ. We thus consider the
two results qualitatively consistent.

One other feature worth mentioning, although we do not seek to quantify it, is
the behaviour of $w_{g+}$ on very small scales. Both redMaGiC samples appear
to exhibit a strongly luminosity-dependent 1h contribution to $w_{g+}$ (Figure
5, purple and red). Noticeably, the RML L3 and RMH L0 bins, though having very
similar mean luminosities, have qualitatively different one halo signals. In
the case of eBOSS LRGs and CMASS we see no such trends, but this is quite
possibly simply the result of low $S/N$, even on smaller scales. In all of
these cases, it is worth bearing in mind that the density tracer samples
differ. Although the differences between the small scale behaviour of $w_{g+}$
in the various bins/samples _could_ be a result of the 1h IA signals, they
could also be partly down to differences in, for example, comoving density and
galaxy bias.

We also briefly test for redshift evolution in our red galaxy samples. Since
we are interested in isolating inherent evolution in the IA signal (as opposed
to changes in sample composition), we compare samples at roughly the same
luminosity. Specifically, we define two narrow bins in Figure 9 (one in the
low luminosity regime, at $\mathrm{log}L/L_{0}\sim-0.3$, and the other at
higher $L$, $\mathrm{log}L/L_{0}\sim-0.05$). Plotting out $A_{1}(z)$ in these
two slices, we find no evidence for redshift evolution over the range
$z=[0.25,0.8]$. Although we see the same trend with CMASS being slightly lower
than other samples at the same $L$, there is no evidence that this is the
result of an underlying redshift trend. Since there is no statistically
significant correlation, we do not include the figure in the main body of the
paper; for completeness, however, it is shown along with redshift power-law
constraints in Appendix E, Figure 16.

####  6.2.2 Model comparison: NLA & TATT

Figure 10: $68\%$ and $95\%$ confidence intervals for the TATT model
constraints from our two DES Y3 redMaGiC samples, on scales $r_{\rm
p}>2h^{-1}\mathrm{Mpc}$. As before, we show redMaGiC high-$z$ in red and
redMaGiC low-$z$ in purple. For comparison, the marginalised NLA constraints
on $A_{1}$ are shown as dashed curves in the upper left panel. Note that the
scales here differ from the fiducial NLA analyses, and so the best fits are
different from those shown in Table 4. The dotted black lines mark the zero
points for the non-NLA parameters.

In Section 6.2.1 we explored the behaviour of IAs on very large scales, where
the NLA model is thought to be sufficient. We next turn to a slightly
different question: on what scales precisely does the simple model break down?
Based on theory, there is thought to be an intermediate regime, outside the
one halo regime, but where higher order correlations (such as those included
in the TATT model) become significant. It is still, however, and open question
as to _how_ significant and on exactly _which_ scales.

To explore this, we repeat our analysis with the minimum scale reduced
slightly to $r_{\rm p}>2h^{-1}\mathrm{Mpc}$ in $w_{g+}$ and $w_{++}$. Note
that the cuts on $w_{gg}$ are fixed, and the modelling there does not change
(i.e., there are always 2 free galaxy bias parameters). For each sample, we
fit both NLA (1 free IA parameter) and TATT (3 IA parameters). Our results are
summarised in Figure 10.

Unsurprisingly given Figure 5, CMASS, eBOSS ELGs and eBOSS LRGs provide very
broad constraints on the TATT parameters, and so are not shown. The signal-to-
noise in these samples is still relatively low. Although one can fit a single
amplitude relatively well, there is little constraining power left for the
shape of the correlation function. The picture is slightly different, however,
in our redMaGiC samples. The cosmological volume here is significantly larger,
and the $S/N$ higher. Starting with the slightly larger sample, redMaGiC
low-$z$, we find:

$A_{2}^{\rm RML}=1.24^{+0.74}_{-0.80}\;\;\;\;\;\;b_{\rm TA}^{\rm
RML}=-0.43^{+0.18}_{-0.13}.$ (34)

In words, our measurements favour (albeit relatively weakly) a combination
$A_{2}>0$ and $b_{\rm TA}<0$. Comparing with the NLA fits on the same scales,
we find $p(>\Delta\chi^{2})=0.01$ ( $\chi_{\rm NLA}^{2}=31.2$ and $\chi_{\rm
TATT}^{2}=21.9$).

In the case of redMaGiC high-$z$, the data appear to prefer a non-zero $b_{\rm
TA}$ at the level of roughly $2.5\sigma$, with:

$A_{2}^{\rm RMH}=-0.36^{+1.63}_{-1.64}\;\;\;\;\;\;b_{\rm TA}^{\rm
RMH}=0.92^{+0.41}_{-0.36}.$ (35)

Again, comparing the TATT and NLA fits, we obtain
$p(>\Delta\chi^{2})=0.00004$, suggesting a statistically significant
preference for TATT on these scales; the respective goodness-of-fit statistics
are $\chi_{\rm NLA}^{2}=33.9$ and $\chi_{\rm TATT}^{2}=13.6$, and
$\Delta\mathrm{dof}=2$.

Interestingly, the deviations from NLA manifest in quite different ways at the
data vector level (see Figure 5). In the case of redMaGiC high-$z$, the
positive $b_{\rm TA}$ increases the power on intermediate scales $(2<r_{\rm
p}<10h^{-1}\mathrm{Mpc})$, resulting in a significantly flatter $w_{g+}$. For
redMaGiC low-$z$, on the other hand, we see the opposite effect; a reduction
in the amplitude of $w_{g+}$ in the $(2<r_{\rm p}<6h^{-1}\mathrm{Mpc})$ range
is accompanied by a slight increase around $20-30h^{-1}\mathrm{Mpc}$. This
gives a slightly steeper theory prediction, which matches the shape of the
measured correlation relatively well. Taken together, this amounts to a $\sim
3.8\sigma$ difference in $b_{\rm TA}$ between the two redMaGiC samples.

Although interesting, it is worth being cautious here. Since TATT is a
relatively flexible model (albeit a physically motivated one), it is possible
that non-zero $A_{2}$ and $b_{\rm TA}$ values could arise due to other
untreated systematics. Baryonic physics, for example, tends to appear on
small-intermediate scales, and modulates the power spectrum in a scale
dependent way. It is also true, however, that baryons tend to _suppress_ power
on smaller scales; this is the case in (almost) all hydrodynamic simulations
at all redshifts. Since redMaGiC high-$z$ prefers a TATT model that does the
opposite (relative to NLA), it seems unlikely that baryons are driving the
non-zero $b_{\rm TA}$ value here. Even considering the redMaGiC low-$z$ case,
it seems very unlikely that we are simply seeing residual baryonic feedback.
In Section 4.6, we saw that the OWLS-AGN scenario (itself an extreme case)
produced at most $\Delta\chi^{2}=0.45$ at the TATT best fits quoted above. The
difference between the NLA and TATT goodness-of-fit are more than an order of
magnitude larger than this; for baryonic feedback alone to explain the non-
zero TATT parameters would require a significantly more extreme scenario than
OWLS-AGN.

Another possible effect here is nonlinear galaxy bias. Our model for it is
incomplete in the sense that while we include nonlinear bias in our $w_{gg}$
model, only $b_{1}$ enters the $w_{g+}$ prediction. Incorporating nonlinear
bias, and all the TATT-bias cross terms, into the $w_{g+}$ model is the focus
of ongoing work. We can, however, make a rough estimate for the impact based
on our $w_{gg}$ fits. For redMaGiC low-$z$, we find the data consistent with
linear bias ($b_{2}=-0.09\pm 0.07$). The equivalent value for redMaGiC
high-$z$ is slightly larger, but still small $b_{2}=0.39\pm 0.08$. Since the
additional terms contributing to $w_{g+}$ will be proportional to $b_{2}$
multiplied by the various IA coefficients, it seems likely that they should be
relatively small compared with the IA-only contributions.

Finally, we also consider the possibility that our results here could be the
result of a non-local lensing contribution from small scales. Such
contributions add to the galaxy-galaxy lensing ($g$G) term, and tend to boost
its power on small to intermediate scales (see e.g. Baldauf et al. 2010;
MacCrann et al. 2020). Fortunately, even considerably different halo mass
profiles produce approximately the same contribution on scales well outside
the virial radius, behaving effectively as an enclosed point mass and scaling
as $1/r_{\rm p}^{2}$. To test this, we generate NLA-only theory data vectors
from the NLA fits on large scales; we add a point mass term (Eq. 7-8 of
MacCrann et al. 2020), and adjust $\delta M$ until the NLA+PM theory
prediction for $w_{g+}$ matches the data on scales $2>r_{\rm
p}>6h^{-1}\mathrm{Mpc}$. Although a $1/r_{\rm p}^{2}$ scaling _can_ match the
data in the redMaGiC high$-z$ case, we find the mass required to do this is
$\sim 3\times 10^{15}M_{\odot}/h$, which is much larger than the typical halo
mass expected for DES redMaGiC (see, for example, Pandey et al. 2022;
Zacharegkas et al. 2022). This would also require the point mass contribution
to dominate $g$G out to scales of $\sim 20h^{-1}\mathrm{Mpc}$, which again is
not thought to be realistic. Moreover, the point mass explanation should lead
to an excess $w_{g+}$ on small scales for both redMaGiC samples, contrary to
the observed behaviour. For all of these reasons, we conclude that a point
mass term cannot explain the deviations from NLA on small scales for redMaGiC
high-$z$.

Overall, these tests seem to suggest a real IA signal (or, at least, a
significant systematic that we have not considered). This is interesting from
a modelling perspective. It implies some dependence in the TATT parameters
with galaxy properties ($b_{\rm TA}$ most obviously, going from negative in
redMaGiC low$-z$ to positive in redMaGiC high$-z$, but also potentially
$A_{2}$). As we discussed in the previous section, the redMaGiC high- and
low$-z$ samples differ in redshift, but also in galaxy properties like colour
and luminosity. Although, given this, differences somewhat expected, this is
not something on which there are previous results to guide us. Disentangling
what exactly is driving the differences is an interesting question, but a
potentially difficult one to answer. We leave this for future work.

####  6.2.3 Robustness to Cosmology and $X_{\rm lens}$

In this section we seek to test the robustness of our analysis to various
sources of systematic error. One such potential contaminant is the effect
known as $X_{\rm lens}$. One can find extensive early discussion in the DES Y3
results papers (Dark Energy Survey Collaboration, 2022; Pandey et al., 2022),
but essentially $X_{\rm lens}$ is a multiplicative factor of unknown origin
between the amplitudes of the galaxy-galaxy lensing and galaxy clustering
measurements. This offset was seen to be scale and redshift independent, and
to impact only Y3 redMaGiC, and not the fiducial magnitude limited lens
sample, MagLim. Subsequent tests have pointed towards a systematic in the
photometry, which affects the redMaGiC selection (see Pandey et al. 2022 Sec.
VG). The magnitude of $X_{\rm lens}$ is constrained relatively well by the
$3\times 2$pt data in Dark Energy Survey Collaboration 2022, to $X_{\rm
lens}=0.877^{+0.026}_{-0.019}$, which is roughly the size of the fractional
error bar, $(w_{g+}-\sigma_{w_{g+}})/w_{g+}$, for our redMaGiC samples in the
range $2<r_{\rm p}<70h^{-1}\mathrm{Mpc}$. Since it is scale independent, we
expect the impact to be completely degenerate with $A_{1}$. Given these
things, we do not expect $X_{\rm lens}$ to have a qualitative impact on our
results. Although it may modulate the best fit $A_{1}$ in our redMaGiC (not
CMASS or eBOSS) at the level of $\sim 10\%$, comparison between samples is
already uncertain to at least this level due to differences in colour space.
Given that the TATT parameters primarily alter the shape of $w_{g+}$, we do
not expect $X_{\rm lens}$ to alter the findings of Section 6.2.2.

We also briefly consider the impact of our choice of cosmology; to fit for
IAs, we need to assume a particular set of cosmological parameters (e.g. for
calculating the matter power spectrum). As discussed in Section 4, we assume a
flat $\Lambda$CDM universe with massive neutrinos and a clustering amplitude
$\sigma_{8}$ similar to that reported by Planck. For each of our samples, in
addition to the best fitting NLA data vector, we generate a second with a
perturbed cosmology; for this we choose the DES Y3 $1\times 2$pt best fit. By
comparing $w_{gg}$ at the two cosmologies, we can compute an effective shift
in large scale bias $\Delta b_{1}$. This in hand, plus the observed impact on
$w_{g+}$, we can estimate the shift in the best fit $A_{1}$. The end result is
a change of at most one or two percent. That is, the difference between
plausible cosmologies is not sufficient to significantly affect our results.

###  6.3 Assessing the Contribution of Magnification & Lensing

In addition to the main IA signal, our measurements have contributions from
lensing and magnification (see the discussion in Section 4). These are always
included in our modelling, but it is interesting to briefly discuss their
effect. The fractional impact of the various terms is shown in Figure 11. Note
that the data vectors here are evaluated at the NLA best fit for each sample,
and so $b_{1}$ and $A_{1}$ differ somewhat between the panels.

Figure 11: The impact of lensing and magnification on our best fitting theory
data vectors. Each contribution to $w_{g+}$ is shown as a fraction of the
total signal ($\delta_{g}$I + $\delta_{g}$G + $\mu$G + $\mu$I). Clockwise from
top left, we have: redMaGiC low-$z$ (purple), CMASS (green), eBOSS LRGs
(pink), eBOSS ELGs (blue) and redMaGiC high-$z$ (red). In the left-hand column
we show our two photometric samples, which are more strongly affected by
lensing and magnification. On the right are our three spectroscopic samples.
Within each column, the samples are arranged vertically by redshift, starting
with low redshift samples at the top. As in previous figures, the shaded bands
represent scales excluded from fits to these data vectors. Different line
styles indicate different signal components, with the sum of the non
$\delta_{g}$I terms shown as a solid line. Note that the $\mu$I term is
negligible on all scales and for all samples, and so we do not show it
separately. In the two redMaGiC cases, we include a second solid line (black).
This demonstrates the impact of using an alternative estimate for the
magnification coefficients $\alpha$, obtained using balrog source injection.
Unlike in the other samples, the CMASS $\mu$G term is positive. This is
because the magnification coefficient for this particular sample is small
($\alpha<1$, see Section 3.3); in physical terms this means the geometric
effect from magnification, which expands the apparent area of a patch of sky,
outweighs the boost in number density due to the brightening of the apparent
galaxy fluxes. Note that the scale on the vertical axis differs between
panels.

Consider first the two photometric samples, redMaGiC low-$z$ and redMaGiC
high-$z$ (upper two panels, purple and red in Figure 11). Here we see a total
non-$\delta_{g}$I contribution of roughly $8\%$ and $4\%$ in the high- and
low-$z$ samples respectively. For context, the $1\sigma$ uncertainty on
$A_{1}$ in these samples is $\sim 4-5\%$ (see Table 4); including these
effects in the modelling of $w_{g+}$ is clearly necessary to avoid bias at the
current precision. The closest comparison in the literature is Fig. 5 in
Joachimi et al. (2011); again, we confirm that our pipeline can reproduce
those results. We see a similar ordering of the terms to their figure, with
$\delta_{g}$G dominant, followed by $\mu$G, and with $\mu$I the smallest, at
sub-percent level (although note there are some important differences between
the comparison in Figure 11 and that of Joachimi et al. 2011, and so one
should not expect the details to match perfectly).

In the two left-hand panels, we also show the results of the same exercise,
but using alternative estimates for the magnification coefficients from balrog
(black lines; see Section 3.3 for the actual values). Note that $\alpha$ is
the only quantity that changes here; we do not re-fit the data vectors, and so
the dominant $\delta_{g}$I term in the denominator is fixed. The new $\alpha$
values are thought to include additional effects omitted by the flux-only
method described in Section 3.3. Unfortunately, however, we do not have balrog
injections covering the whole DES footprint, and so the resulting estimates
are noisy. We also cannot easily produce balrog mocks closely matching our
CMASS and eBOSS samples, which somewhat limits its use for our purposes. The
overall impact, however, is seen to be relatively small. That is, while
ignoring magnification altogether could have a significant impact on our
results, the choice of one $\alpha$ estimate over another is unlikely to.

Although it is common to assume spectroscopic IA measurements are immune to
lensing effects, we see in Figure 11 that this is not entirely the case. The
two main terms are the same as with redMaGiC: $\mu$G and $\delta_{g}$G. It is
interesting that the latter (dashed) still dominates; even in the case of very
narrow per-galaxy redshift distributions, where $p(z)\rightarrow\delta_{z}$,
the ensemble redshift distribution is sufficiently broad to allow a non-
negligible galaxy-galaxy lensing contribution. The $\mu$G term (dotted), we
also note, does not depend on the quality of the redshifts. Galaxies along the
same line of sight are affected by magnification and lensing due to the same
foreground structure, which modulates $w_{g+}$ at the level of a few percent
at $z\sim 0.8$.

It is worth briefly considering what these results mean for future IA
measurements. The Stage IV spectroscopic survey DESI is expected to obtain
spectra for a sample of LRGs over a comparable redshift range to ours, but for
a considerably wider area and greater number density ($n_{c}\sim 6\times
10^{-4}h^{3}\mathrm{Mpc}^{-3}$ over 14,000 square degrees; Zhou et al. 2020).
Similarly, one can expect at least an order of magnitude increase in the
number of ELGs available for the type of measurement we perform here (Raichoor
et al., 2020). Euclid and the Roman Space Telescope will also have
spectroscopic instruments, which will further add to the pool of data
available for IA measurements. Likewise, Stage IV photometric surveys such as
Rubin, Euclid and Roman will probe a similar selection of galaxies to DES, but
over a much wider area and to a greater depth (see, for example, Euclid
Collaboration 2021). These will allow measurements using a redMaGiC-like
sample similar to ours, but with significantly improved S/N and finer binning
in colour/redshift/luminosity. Used together – with photometry providing shape
inference, and accurate redshift information from either spectroscopic data or
high quality photometric measurements – the next generation of surveys will
provide a powerful tool for studying intrinsic alignments. Given what we see
here, it seems very likely that direct IA measurements using these upcoming
data will need to account for lensing and magnification.

##  7 Conclusions

This paper presents direct constraints on intrinsic alignments from the Dark
Energy Survey Year 3 shape catalogues. The Y3 Metacalibration catalogue is
used to provide shape estimates for 2.4M redMaGiC galaxies from across the DES
footprint, as well as $\sim$50,000 CMASS galaxies, $\sim$22,000 eBOSS LRGs and
$\sim$100,000 ELGs. We make a high significance detection of intrinsic
alignments in all of these samples, with the exception of eBOSS ELGs, where we
place upper bounds on the magnitude of the possible alignment amplitude.

The key conclusions of this paper are:

  * •

Fitting for $A_{1}$ in red galaxies, our data support the qualitative picture
of a broken power law in $r$-band luminosity of the form $A_{1}\propto
L^{\beta}$, with $\beta$ differing between high and low $L$. Our redMaGiC
low$-z$ sample provides a significant improvement constraints at the faint end
of the $A_{1}-L$ relation, where the slope is shallower than at the bright end
(by a factor of several; see Figure 9).

  * •

Amongst red galaxy samples, however, we find noticeable colour dependence in
the IA-luminosity trend. This is most obvious in CMASS and redMaGiC high-$z$,
which both lying below the bulk of previous measurements at a similar $L$.
These differences can be qualitatively explained by differences in the colour
space distributions. This raises potential questions about the sufficiency of
a simple red/blue binary split for modelling IAs, and whether joint modelling
of luminosity and colour dependence may be needed.

  * •

We find no statistically significant signal in our ELG sample on any scale.
Using the combination of $w_{g+}$ and $w_{++}$, we impose an upper limit on
the large scale NLA amplitude in ELGs at $|A_{1}|<0.3$ ($68\%$ CL). This is an
improvement on the null constraint from WiggleZ at $|A_{1}|<1.03$ (Mandelbaum
et al., 2011).

  * •

The one-parameter NLA model is seen to fit all of our red galaxy samples
reasonably on scales $r_{\rm p}>6h^{-1}\mathrm{Mpc}$. In our redMaGiC samples,
which give the highest signal-to-noise measurements, we do see deviations from
the NLA prediction in the range $2-6h^{-1}\mathrm{Mpc}$. These deviations are
more pronounced in the higher redMaGiC redshift sample.

  * •

Allowing additional flexibility via the TATT model, we can obtain a good fit
to both redMaGiC samples on intermediate scales, $r_{\rm
p}=2-6h^{-1}\mathrm{Mpc}$. We thus place constraints on the additional
parameters (see Figure 10).

  * •

We show that lensing and magnification can have a potentially significant
impact on direct IA measurements. The extra terms are dominated by a galaxy-
galaxy lensing like contribution $\delta_{g}\delta$, and the magnification-
lensing cross correlation, $\delta_{\mu}\delta$. Together they make up $\sim
2-20\%$ of the total signal, depending on the sample. This is relevant for our
higher S/N samples, and will certainly be significant for future measurements
of a similar kind, even those relying only on spectroscopic samples.

A weak lensing cosmology analysis is underway using the Dark Energy Survey
Year 6 data, and similar efforts are ongoing on the KiDS legacy and HSC Y3
results. Understanding astrophysical systematics such as intrinsic alignments,
on both large and small scales, will clearly be important for the success of
these ongoing cosmology projects, as well as future surveys such as the Vera
C. Rubin Observatory Legacy Survey of Space and Time (LSST), Euclid and the
Nancy Grace Roman Space Telescope. Our ability to accurately model IAs and
mitigate their impact is, however, still somewhat limited; even given detailed
information about the redshift and rest frame colour of a sample (which
typically is not available in a photometric survey), we do not have sufficient
a priori understanding of the physical processes to predict the IA signal for
an arbitrary selection of galaxies. We can, however, make measurements of IAs
in a range of samples, and map out the dependence on galaxy properties. In
this way, we can start to build up a phenomenological understanding of
intrinsic alignments, which will feed into the next generation of analyses.
The longer term goal is to develop more accurate models of intrinsic